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static/img/12.29.png create mode 100644 static/img/12.3.png create mode 100644 static/img/12.30.png create mode 100644 static/img/12.31.png create mode 100644 static/img/12.32.png create mode 100644 static/img/12.4.png create mode 100644 static/img/12.5.png create mode 100644 static/img/12.6.png create mode 100644 static/img/12.7.png create mode 100644 static/img/12.8.png create mode 100644 static/img/12.9.png create mode 100644 static/img/20.1.png create mode 100644 static/img/20.2.png create mode 100644 static/img/20.3.png create mode 100644 static/img/20.4.png diff --git a/archetypes/notes.md b/archetypes/notes.md new file mode 100644 index 00000000..9ec01533 --- /dev/null +++ b/archetypes/notes.md @@ -0,0 +1,9 @@ +--- +title: "{{ .Name | humanize | title }}" +weight: 1 +# bookFlatSection: false +bookToc: false +# bookHidden: false +# bookCollapseSection: false +# bookComments: true +--- diff --git a/assets/_custom.scss b/assets/_custom.scss index 3651bbfd..6653bca9 100644 --- a/assets/_custom.scss +++ b/assets/_custom.scss @@ -2,6 +2,21 @@ display: none; } -body::-webkit-scrollbar { - display: none; +// body::-webkit-scrollbar { +// display: none; +// } + +@import "plugins/scrollbars"; + +img { + width: 80%; } + +@media (prefers-color-scheme: dark) { + :root { + --body-background: #212121 + } + img { + filter: invert(87.05%); + } +} \ No newline at end of file diff --git a/config.yaml b/config.yaml index 14da08a8..fd36be5d 100644 --- a/config.yaml +++ b/config.yaml @@ -66,7 +66,7 @@ params: # (Optional, default docs) Specify root page to render child pages as menu. # Page is resoled by .GetPage function: https://gohugo.io/functions/getpage/ # For backward compatibility you can set '*' to render all sections to menu. Acts same as '/' - BookSection: docs + BookSection: '*' # Set source repository location. # Used for 'Last Modified' and 'Edit this page' links. diff --git a/content/_index.md b/content/_index.md index 6123c282..66c6c2ad 100644 --- a/content/_index.md +++ b/content/_index.md @@ -1,6 +1,6 @@ --- title: Introduction -type: docs +type: notes --- # Acerbo datus maxime diff --git a/content/docs/example/_index.md b/content/docs/example/_index.md deleted file mode 100644 index 4835b7ca..00000000 --- a/content/docs/example/_index.md +++ /dev/null @@ -1,71 +0,0 @@ ---- -weight: 1 -bookFlatSection: true -title: "Example Site" ---- - -# Introduction - -## Ferre hinnitibus erat accipitrem dixi Troiae tollens - -Lorem markdownum, a quoque nutu est *quodcumque mandasset* veluti. Passim -inportuna totidemque nympha fert; repetens pendent, poenarum guttura sed vacet -non, mortali undas. Omnis pharetramque gramen portentificisque membris servatum -novabis fallit de nubibus atque silvas mihi. **Dixit repetitaque Quid**; verrit -longa; sententia [mandat](http://pastor-ad.io/questussilvas) quascumque nescio -solebat [litore](http://lacrimas-ab.net/); noctes. *Hostem haerentem* circuit -[plenaque tamen](http://www.sine.io/in). - -- Pedum ne indigenae finire invergens carpebat -- Velit posses summoque -- De fumos illa foret - -## Est simul fameque tauri qua ad - -Locum nullus nisi vomentes. Ab Persea sermone vela, miratur aratro; eandem -Argolicas gener. - -## Me sol - -Nec dis certa fuit socer, Nonacria **dies** manet tacitaque sibi? Sucis est -iactata Castrumque iudex, et iactato quoque terraeque es tandem et maternos -vittis. Lumina litus bene poenamque animos callem ne tuas in leones illam dea -cadunt genus, et pleno nunc in quod. Anumque crescentesque sanguinis -[progenies](http://www.late.net/alimentavirides) nuribus rustica tinguet. Pater -omnes liquido creditis noctem. - - if (mirrored(icmp_dvd_pim, 3, smbMirroredHard) != lion(clickImportQueue, - viralItunesBalancing, bankruptcy_file_pptp)) { - file += ip_cybercrime_suffix; - } - if (runtimeSmartRom == netMarketingWord) { - virusBalancingWin *= scriptPromptBespoke + raster(post_drive, - windowsSli); - cd = address_hertz_trojan; - soap_ccd.pcbServerGigahertz(asp_hardware_isa, offlinePeopleware, nui); - } else { - megabyte.api = modem_flowchart - web + syntaxHalftoneAddress; - } - if (3 < mebibyteNetworkAnimated) { - pharming_regular_error *= jsp_ribbon + algorithm * recycleMediaKindle( - dvrSyntax, cdma); - adf_sla *= hoverCropDrive; - templateNtfs = -1 - vertical; - } else { - expressionCompressionVariable.bootMulti = white_eup_javascript( - table_suffix); - guidPpiPram.tracerouteLinux += rtfTerabyteQuicktime(1, - managementRosetta(webcamActivex), 740874); - } - var virusTweetSsl = nullGigo; - -## Trepident sitimque - -Sentiet et ferali errorem fessam, coercet superbus, Ascaniumque in pennis -mediis; dolor? Vidit imi **Aeacon** perfida propositos adde, tua Somni Fluctibus -errante lustrat non. - -Tamen inde, vos videt e flammis Scythica parantem rupisque pectora umbras. Haec -ficta canistris repercusso simul ego aris Dixit! Esse Fama trepidare hunc -crescendo vigor ululasse vertice *exspatiantur* celer tepidique petita aversata -oculis iussa est me ferro. diff --git a/content/docs/example/collapsed/3rd-level/4th-level.md b/content/docs/example/collapsed/3rd-level/4th-level.md deleted file mode 100644 index aa451f19..00000000 --- a/content/docs/example/collapsed/3rd-level/4th-level.md +++ /dev/null @@ -1,12 +0,0 @@ -# 4th Level of Menu - -## Caesorum illa tu sentit micat vestes papyriferi - -Inde aderam facti; Theseus vis de tauri illa peream. Oculos **uberaque** non -regisque vobis cursuque, opus venit quam vulnera. Et maiora necemque, lege modo; -gestanda nitidi, vero? Dum ne pectoraque testantur. - -Venasque repulsa Samos qui, exspectatum eram animosque hinc, [aut -manes](http://www.creveratnon.net/apricaaetheriis), Assyrii. Cupiens auctoribus -pariter rubet, profana magni super nocens. Vos ius sibilat inpar turba visae -iusto! Sedes ante dum superest **extrema**. diff --git a/content/docs/example/collapsed/3rd-level/_index.md b/content/docs/example/collapsed/3rd-level/_index.md deleted file mode 100644 index cc0100f3..00000000 --- a/content/docs/example/collapsed/3rd-level/_index.md +++ /dev/null @@ -1,26 +0,0 @@ -# 3rd Level of Menu - -Nefas discordemque domino montes numen tum humili nexilibusque exit, Iove. Quae -miror esse, scelerisque Melaneus viribus. Miseri laurus. Hoc est proposita me -ante aliquid, aura inponere candidioribus quidque accendit bella, sumpta. -Intravit quam erat figentem hunc, motus de fontes parvo tempestate. - - iscsi_virus = pitch(json_in_on(eupViral), - northbridge_services_troubleshooting, personal( - firmware_rw.trash_rw_crm.device(interactive_gopher_personal, - software, -1), megabit, ergonomicsSoftware(cmyk_usb_panel, - mips_whitelist_duplex, cpa))); - if (5) { - managementNetwork += dma - boolean; - kilohertz_token = 2; - honeypot_affiliate_ergonomics = fiber; - } - mouseNorthbridge = byte(nybble_xmp_modem.horse_subnet( - analogThroughputService * graphicPoint, drop(daw_bit, dnsIntranet), - gateway_ospf), repository.domain_key.mouse(serverData(fileNetwork, - trim_duplex_file), cellTapeDirect, token_tooltip_mashup( - ripcordingMashup))); - module_it = honeypot_driver(client_cold_dvr(593902, ripping_frequency) + - coreLog.joystick(componentUdpLink), windows_expansion_touchscreen); - bashGigabit.external.reality(2, server_hardware_codec.flops.ebookSampling( - ciscNavigationBacklink, table + cleanDriver), indexProtocolIsp); diff --git a/content/docs/example/collapsed/_index.md b/content/docs/example/collapsed/_index.md deleted file mode 100644 index e954f087..00000000 --- a/content/docs/example/collapsed/_index.md +++ /dev/null @@ -1,4 +0,0 @@ ---- -bookCollapseSection: true -weight: 20 ---- diff --git a/content/docs/example/hidden.md b/content/docs/example/hidden.md deleted file mode 100644 index df7cb9eb..00000000 --- a/content/docs/example/hidden.md +++ /dev/null @@ -1,52 +0,0 @@ ---- -bookHidden: true ---- - -# This page is hidden in menu - -# Quondam non pater est dignior ille Eurotas - -## Latent te facies - -Lorem markdownum arma ignoscas vocavit quoque ille texit mandata mentis ultimus, -frementes, qui in vel. Hippotades Peleus [pennas -conscia](http://gratia.net/tot-qua.php) cuiquam Caeneus quas. - -- Pater demittere evincitque reddunt -- Maxime adhuc pressit huc Danaas quid freta -- Soror ego -- Luctus linguam saxa ultroque prior Tatiumque inquit -- Saepe liquitur subita superata dederat Anius sudor - -## Cum honorum Latona - -O fallor [in sustinui -iussorum](http://www.spectataharundine.org/aquas-relinquit.html) equidem. -Nymphae operi oris alii fronde parens dumque, in auro ait mox ingenti proxima -iamdudum maius? - - reality(burnDocking(apache_nanometer), - pad.property_data_programming.sectorBrowserPpga(dataMask, 37, - recycleRup)); - intellectualVaporwareUser += -5 * 4; - traceroute_key_upnp /= lag_optical(android.smb(thyristorTftp)); - surge_host_golden = mca_compact_device(dual_dpi_opengl, 33, - commerce_add_ppc); - if (lun_ipv) { - verticalExtranet(1, thumbnail_ttl, 3); - bar_graphics_jpeg(chipset - sector_xmp_beta); - } - -## Fronde cetera dextrae sequens pennis voce muneris - -Acta cretus diem restet utque; move integer, oscula non inspirat, noctisque -scelus! Nantemque in suas vobis quamvis, et labori! - - var runtimeDiskCompiler = home - array_ad_software; - if (internic > disk) { - emoticonLockCron += 37 + bps - 4; - wan_ansi_honeypot.cardGigaflops = artificialStorageCgi; - simplex -= downloadAccess; - } - var volumeHardeningAndroid = pixel + tftp + onProcessorUnmount; - sector(memory(firewire + interlaced, wired)); \ No newline at end of file diff --git a/content/docs/example/table-of-contents/_index.md b/content/docs/example/table-of-contents/_index.md deleted file mode 100644 index c7ee0d87..00000000 --- a/content/docs/example/table-of-contents/_index.md +++ /dev/null @@ -1,85 +0,0 @@ ---- -weight: 10 ---- - -# Ubi loqui - -## Mentem genus facietque salire tempus bracchia - -Lorem markdownum partu paterno Achillem. Habent amne generosi aderant ad pellem -nec erat sustinet merces columque haec et, dixit minus nutrit accipiam subibis -subdidit. Temeraria servatum agros qui sed fulva facta. Primum ultima, dedit, -suo quisque linguae medentes fixo: tum petis. - -## Rapit vocant si hunc siste adspice - -Ora precari Patraeque Neptunia, dixit Danae [Cithaeron -armaque](http://mersis-an.org/litoristum) maxima in **nati Coniugis** templis -fluidove. Effugit usus nec ingreditur agmen *ac manus* conlato. Nullis vagis -nequiquam vultibus aliquos altera *suum venis* teneas fretum. Armos [remotis -hoc](http://tutum.io/me) sine ferrea iuncta quam! - -## Locus fuit caecis - -Nefas discordemque domino montes numen tum humili nexilibusque exit, Iove. Quae -miror esse, scelerisque Melaneus viribus. Miseri laurus. Hoc est proposita me -ante aliquid, aura inponere candidioribus quidque accendit bella, sumpta. -Intravit quam erat figentem hunc, motus de fontes parvo tempestate. - - iscsi_virus = pitch(json_in_on(eupViral), - northbridge_services_troubleshooting, personal( - firmware_rw.trash_rw_crm.device(interactive_gopher_personal, - software, -1), megabit, ergonomicsSoftware(cmyk_usb_panel, - mips_whitelist_duplex, cpa))); - if (5) { - managementNetwork += dma - boolean; - kilohertz_token = 2; - honeypot_affiliate_ergonomics = fiber; - } - mouseNorthbridge = byte(nybble_xmp_modem.horse_subnet( - analogThroughputService * graphicPoint, drop(daw_bit, dnsIntranet), - gateway_ospf), repository.domain_key.mouse(serverData(fileNetwork, - trim_duplex_file), cellTapeDirect, token_tooltip_mashup( - ripcordingMashup))); - module_it = honeypot_driver(client_cold_dvr(593902, ripping_frequency) + - coreLog.joystick(componentUdpLink), windows_expansion_touchscreen); - bashGigabit.external.reality(2, server_hardware_codec.flops.ebookSampling( - ciscNavigationBacklink, table + cleanDriver), indexProtocolIsp); - -## Placabilis coactis nega ingemuit ignoscat nimia non - -Frontis turba. Oculi gravis est Delphice; *inque praedaque* sanguine manu non. - - if (ad_api) { - zif += usb.tiffAvatarRate(subnet, digital_rt) + exploitDrive; - gigaflops(2 - bluetooth, edi_asp_memory.gopher(queryCursor, laptop), - panel_point_firmware); - spyware_bash.statePopApplet = express_netbios_digital( - insertion_troubleshooting.brouter(recordFolderUs), 65); - } - recursionCoreRay = -5; - if (hub == non) { - portBoxVirus = soundWeb(recursive_card(rwTechnologyLeopard), - font_radcab, guidCmsScalable + reciprocalMatrixPim); - left.bug = screenshot; - } else { - tooltipOpacity = raw_process_permalink(webcamFontUser, -1); - executable_router += tape; - } - if (tft) { - bandwidthWeb *= social_page; - } else { - regular += 611883; - thumbnail /= system_lag_keyboard; - } - -## Caesorum illa tu sentit micat vestes papyriferi - -Inde aderam facti; Theseus vis de tauri illa peream. Oculos **uberaque** non -regisque vobis cursuque, opus venit quam vulnera. Et maiora necemque, lege modo; -gestanda nitidi, vero? Dum ne pectoraque testantur. - -Venasque repulsa Samos qui, exspectatum eram animosque hinc, [aut -manes](http://www.creveratnon.net/apricaaetheriis), Assyrii. Cupiens auctoribus -pariter rubet, profana magni super nocens. Vos ius sibilat inpar turba visae -iusto! Sedes ante dum superest **extrema**. diff --git a/content/docs/example/table-of-contents/with-toc.md b/content/docs/example/table-of-contents/with-toc.md deleted file mode 100644 index 5345c668..00000000 --- a/content/docs/example/table-of-contents/with-toc.md +++ /dev/null @@ -1,64 +0,0 @@ ---- -title: With ToC -weight: 1 ---- -# Caput vino delphine in tamen vias - -## Cognita laeva illo fracta - -Lorem markdownum pavent auras, surgit nunc cingentibus libet **Laomedonque que** -est. Pastor [An](http://est.org/ire.aspx) arbor filia foedat, ne [fugit -aliter](http://www.indiciumturbam.org/moramquid.php), per. Helicona illas et -callida neptem est *Oresitrophos* caput, dentibus est venit. Tenet reddite -[famuli](http://www.antro-et.net/) praesentem fortibus, quaeque vis foret si -frondes *gelidos* gravidae circumtulit [inpulit armenta -nativum](http://incurvasustulit.io/illi-virtute.html). - -1. Te at cruciabere vides rubentis manebo -2. Maturuit in praetemptat ruborem ignara postquam habitasse -3. Subitarum supplevit quoque fontesque venabula spretis modo -4. Montis tot est mali quasque gravis -5. Quinquennem domus arsit ipse -6. Pellem turis pugnabant locavit - -## Natus quaerere - -Pectora et sine mulcere, coniuge dum tincta incurvae. Quis iam; est dextra -Peneosque, metuis a verba, primo. Illa sed colloque suis: magno: gramen, aera -excutiunt concipit. - -> Phrygiae petendo suisque extimuit, super, pars quod audet! Turba negarem. -> Fuerat attonitus; et dextra retinet sidera ulnas undas instimulat vacuae -> generis? *Agnus* dabat et ignotis dextera, sic tibi pacis **feriente at mora** -> euhoeque *comites hostem* vestras Phineus. Vultuque sanguine dominoque [metuit -> risi](http://iuvat.org/eundem.php) fama vergit summaque meus clarissimus -> artesque tinguebat successor nominis cervice caelicolae. - -## Limitibus misere sit - -Aurea non fata repertis praerupit feruntur simul, meae hosti lentaque *citius -levibus*, cum sede dixit, Phaethon texta. *Albentibus summos* multifidasque -iungitur loquendi an pectore, mihi ursaque omnia adfata, aeno parvumque in animi -perlucentes. Epytus agis ait vixque clamat ornum adversam spondet, quid sceptra -ipsum **est**. Reseret nec; saeva suo passu debentia linguam terga et aures et -cervix [de](http://www.amnem.io/pervenit.aspx) ubera. Coercet gelidumque manus, -doluit volvitur induta? - -## Enim sua - -Iuvenilior filia inlustre templa quidem herbis permittat trahens huic. In -cruribus proceres sole crescitque *fata*, quos quos; merui maris se non tamen -in, mea. - -## Germana aves pignus tecta - -Mortalia rudibusque caelum cognosceret tantum aquis redito felicior texit, nec, -aris parvo acre. Me parum contulerant multi tenentem, gratissime suis; vultum tu -occupat deficeret corpora, sonum. E Actaea inplevit Phinea concepit nomenque -potest sanguine captam nulla et, in duxisses campis non; mercede. Dicere cur -Leucothoen obitum? - -Postibus mittam est *nubibus principium pluma*, exsecratur facta et. Iunge -Mnemonidas pallamque pars; vere restitit alis flumina quae **quoque**, est -ignara infestus Pyrrha. Di ducis terris maculatum At sede praemia manes -nullaque! diff --git a/content/docs/example/table-of-contents/without-toc.md b/content/docs/example/table-of-contents/without-toc.md deleted file mode 100644 index 9b163188..00000000 --- a/content/docs/example/table-of-contents/without-toc.md +++ /dev/null @@ -1,59 +0,0 @@ ---- -title: Without ToC -weight: 2 -bookToc: false ---- - -# At me ipso nepotibus nunc celebratior genus - -## Tanto oblite - -Lorem markdownum pectora novis patenti igne sua opus aurae feras materiaque -illic demersit imago et aristas questaque posset. Vomit quoque suo inhaesuro -clara. Esse cumque, per referri triste. Ut exponit solisque communis in tendens -vincetis agisque iamque huic bene ante vetat omina Thebae rates. Aeacus servat -admonitu concidit, ad resimas vultus et rugas vultu **dignamque** Siphnon. - -Quam iugulum regia simulacra, plus meruit humo pecorumque haesit, ab discedunt -dixit: ritu pharetramque. Exul Laurenti orantem modo, per densum missisque labor -manibus non colla unum, obiectat. Tu pervia collo, fessus quae Cretenque Myconon -crate! Tegumenque quae invisi sudore per vocari quaque plus ventis fluidos. Nodo -perque, fugisse pectora sorores. - -## Summe promissa supple vadit lenius - -Quibus largis latebris aethera versato est, ait sentiat faciemque. Aequata alis -nec Caeneus exululat inclite corpus est, ire **tibi** ostendens et tibi. Rigent -et vires dique possent lumina; **eadem** dixit poma funeribus paret et felix -reddebant ventis utile lignum. - -1. Remansit notam Stygia feroxque -2. Et dabit materna -3. Vipereas Phrygiaeque umbram sollicito cruore conlucere suus -4. Quarum Elis corniger -5. Nec ieiunia dixit - -Vertitur mos ortu ramosam contudit dumque; placabat ac lumen. Coniunx Amoris -spatium poenamque cavernis Thebae Pleiadasque ponunt, rapiare cum quae parum -nimium rima. - -## Quidem resupinus inducto solebat una facinus quae - -Credulitas iniqua praepetibus paruit prospexit, voce poena, sub rupit sinuatur, -quin suum ventorumque arcadiae priori. Soporiferam erat formamque, fecit, -invergens, nymphae mutat fessas ait finge. - -1. Baculum mandataque ne addere capiti violentior -2. Altera duas quam hoc ille tenues inquit -3. Sicula sidereus latrantis domoque ratae polluit comites -4. Possit oro clausura namque se nunc iuvenisque -5. Faciem posuit -6. Quodque cum ponunt novercae nata vestrae aratra - -Ite extrema Phrygiis, patre dentibus, tonso perculit, enim blanda, manibus fide -quos caput armis, posse! Nocendo fas Alcyonae lacertis structa ferarum manus -fulmen dubius, saxa caelum effuge extremis fixum tumor adfecit **bella**, -potentes? Dum nec insidiosa tempora tegit -[spirarunt](http://mihiferre.net/iuvenes-peto.html). Per lupi pars foliis, -porreximus humum negant sunt subposuere Sidone steterant auro. Memoraverit sine: -ferrum idem Orion caelum heres gerebat fixis? diff --git a/content/docs/notes/crews2018.md b/content/docs/notes/crews2018.md deleted file mode 100644 index e6a3d363..00000000 --- a/content/docs/notes/crews2018.md +++ /dev/null @@ -1,34 +0,0 @@ ---- -title: "Crews (2018)" ---- - -# Development of a Collisionless Plasma Kinetic Solver and an Investigation of One-Dimensional Plasma Waves and Instabilities - -Shielded potential of a test electron: - -{{< katex display >}} -\phi(r) = \frac{-e}{4 \pi \epsilon_0 r} e ^{- r / \lambda_D} -{{< /katex >}} - -where the Debye length is {{< katex >}} \lambda_D = \sqrt{\frac{\epsilon_0 T_e}{ n_e e}} {{< /katex >}}. The mean free path between large-angle collisions is estimated as - -{{< katex display >}} -\lambda_{mfp} \sim \frac{\epsilon_0 T_e ^2}{\phi_e n_e \log ( \Lambda)} -{{< /katex >}} -where {{< katex >}}\phi_e = e^2 / 4 \pi \epsilon_0{{< /katex >}} are the constants from the Coulomb force law. - -Smooth out the discreteness of particles via spatial average over small volumes: - -{{< katex display >}} -\rho \rightarrow \langle \rho_c \rangle \qquad \vec E \rightarrow \langle \vec E \rangle + \delta \vec E -{{< /katex >}} - -The mean field {{< katex >}}\langle \vec E \rangle{{< /katex >}} is responsible for collective modes of plasma motion. Estimate the collisionality of the plasma by comparing the length scales {{< katex >}}\lambda_{mfp} / \lambda_D{{< /katex >}} - -{{< katex display >}} -\frac{\lambda_{mfp}}{\lambda_D} \sim \frac{T_e ^{3/2}}{n_e ^{1/2}} -{{< /katex >}} - -Plasma is seen to become collisionless as the temperature becomes high or the plasma becomes more rarified. - -Maybe this will publish... \ No newline at end of file diff --git a/content/docs/shortcodes/buttons.md b/content/docs/shortcodes/buttons.md deleted file mode 100644 index c2ef1e75..00000000 --- a/content/docs/shortcodes/buttons.md +++ /dev/null @@ -1,13 +0,0 @@ -# Buttons - -Buttons are styled links that can lead to local page or external link. - -## Example - -```tpl -{{}}Get Home{{}} -{{}}Contribute{{}} -``` - -{{< button relref="/" >}}Get Home{{< /button >}} -{{< button href="https://github.com/alex-shpak/hugo-book" >}}Contribute{{< /button >}} diff --git a/content/docs/shortcodes/columns.md b/content/docs/shortcodes/columns.md deleted file mode 100644 index 0b8fde83..00000000 --- a/content/docs/shortcodes/columns.md +++ /dev/null @@ -1,45 +0,0 @@ -# Columns - -Columns help organize shorter pieces of content horizontally for readability. - - -```html -{{}} -# Left Content -Lorem markdownum insigne... - -<---> - -# Mid Content -Lorem markdownum insigne... - -<---> - -# Right Content -Lorem markdownum insigne... -{{}} -``` - -## Example - -{{< columns >}} -## Left Content -Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat -stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa -protulit, sed sed aere valvis inhaesuro Pallas animam: qui _quid_, ignes. -Miseratus fonte Ditis conubia. - -<---> - -## Mid Content -Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat -stringit, frustra Saturnius uteroque inter! - -<---> - -## Right Content -Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat -stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa -protulit, sed sed aere valvis inhaesuro Pallas animam: qui _quid_, ignes. -Miseratus fonte Ditis conubia. -{{< /columns >}} diff --git a/content/docs/shortcodes/details.md b/content/docs/shortcodes/details.md deleted file mode 100644 index 248bafd9..00000000 --- a/content/docs/shortcodes/details.md +++ /dev/null @@ -1,22 +0,0 @@ -# Details - -Details shortcode is a helper for `details` html5 element. It is going to replace `expand` shortcode. - -## Example -```tpl -{{}} -## Markdown content -Lorem markdownum insigne... -{{}} -``` -```tpl -{{}} -## Markdown content -Lorem markdownum insigne... -{{}} -``` - -{{< details "Title" open >}} -## Markdown content -Lorem markdownum insigne... -{{< /details >}} diff --git a/content/docs/shortcodes/expand.md b/content/docs/shortcodes/expand.md deleted file mode 100644 index c62520f3..00000000 --- a/content/docs/shortcodes/expand.md +++ /dev/null @@ -1,35 +0,0 @@ -# Expand - -Expand shortcode can help to decrease clutter on screen by hiding part of text. Expand content by clicking on it. - -## Example -### Default - -```tpl -{{}} -## Markdown content -Lorem markdownum insigne... -{{}} -``` - -{{< expand >}} -## Markdown content -Lorem markdownum insigne... -{{< /expand >}} - -### With Custom Label - -```tpl -{{}} -## Markdown content -Lorem markdownum insigne... -{{}} -``` - -{{< expand "Custom Label" "..." >}} -## Markdown content -Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat -stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa -protulit, sed sed aere valvis inhaesuro Pallas animam: qui _quid_, ignes. -Miseratus fonte Ditis conubia. -{{< /expand >}} diff --git a/content/docs/shortcodes/hints.md b/content/docs/shortcodes/hints.md deleted file mode 100644 index 3477113d..00000000 --- a/content/docs/shortcodes/hints.md +++ /dev/null @@ -1,32 +0,0 @@ -# Hints - -Hint shortcode can be used as hint/alerts/notification block. -There are 3 colors to choose: `info`, `warning` and `danger`. - -```tpl -{{}} -**Markdown content** -Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat -stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa -{{}} -``` - -## Example - -{{< hint info >}} -**Markdown content** -Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat -stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa -{{< /hint >}} - -{{< hint warning >}} -**Markdown content** -Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat -stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa -{{< /hint >}} - -{{< hint danger >}} -**Markdown content** -Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat -stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa -{{< /hint >}} diff --git a/content/docs/shortcodes/katex.md b/content/docs/shortcodes/katex.md deleted file mode 100644 index c7459e51..00000000 --- a/content/docs/shortcodes/katex.md +++ /dev/null @@ -1,28 +0,0 @@ -# KaTeX - -KaTeX shortcode let you render math typesetting in markdown document. See [KaTeX](https://katex.org/) - -## Example -{{< columns >}} - -```latex -{{}} -f(x) = \int_{-\infty}^\infty\hat f(\xi)\,e^{2 \pi i \xi x}\,d\xi -{{}} -``` - -<---> - -{{< katex display >}} -f(x) = \int_{-\infty}^\infty\hat f(\xi)\,e^{2 \pi i \xi x}\,d\xi -{{< /katex >}} - -{{< /columns >}} - -## Display Mode Example - -Here is some inline example: {{< katex >}}\pi(x){{< /katex >}}, rendered in the same line. And below is `display` example, having `display: block` -{{< katex display >}} -f(x) = \int_{-\infty}^\infty\hat f(\xi)\,e^{2 \pi i \xi x}\,d\xi -{{< /katex >}} -Text continues here. \ No newline at end of file diff --git a/content/docs/shortcodes/mermaid.md b/content/docs/shortcodes/mermaid.md deleted file mode 100644 index 3a617bcc..00000000 --- a/content/docs/shortcodes/mermaid.md +++ /dev/null @@ -1,38 +0,0 @@ -# Mermaid Chart - -[Mermaid](https://mermaidjs.github.io/) is library for generating svg charts and diagrams from text. - -## Example - -{{< columns >}} -```tpl -{{}} -sequenceDiagram - Alice->>Bob: Hello Bob, how are you? - alt is sick - Bob->>Alice: Not so good :( - else is well - Bob->>Alice: Feeling fresh like a daisy - end - opt Extra response - Bob->>Alice: Thanks for asking - end -{{}} -``` - -<---> - -{{< mermaid >}} -sequenceDiagram - Alice->>Bob: Hello Bob, how are you? - alt is sick - Bob->>Alice: Not so good :( - else is well - Bob->>Alice: Feeling fresh like a daisy - end - opt Extra response - Bob->>Alice: Thanks for asking - end -{{< /mermaid >}} - -{{< /columns >}} diff --git a/content/docs/shortcodes/section/_index.md b/content/docs/shortcodes/section/_index.md deleted file mode 100644 index bd5db38b..00000000 --- a/content/docs/shortcodes/section/_index.md +++ /dev/null @@ -1,15 +0,0 @@ ---- -bookCollapseSection: true ---- - -# Section - -Section renders pages in section as definition list, using title and description. - -## Example - -```tpl -{{}} -``` - -{{
}} diff --git a/content/docs/shortcodes/section/page1.md b/content/docs/shortcodes/section/page1.md deleted file mode 100644 index 96080014..00000000 --- a/content/docs/shortcodes/section/page1.md +++ /dev/null @@ -1 +0,0 @@ -# Page 1 diff --git a/content/docs/shortcodes/section/page2.md b/content/docs/shortcodes/section/page2.md deleted file mode 100644 index f310be33..00000000 --- a/content/docs/shortcodes/section/page2.md +++ /dev/null @@ -1 +0,0 @@ -# Page 2 diff --git a/content/docs/shortcodes/tabs.md b/content/docs/shortcodes/tabs.md deleted file mode 100644 index 096892c6..00000000 --- a/content/docs/shortcodes/tabs.md +++ /dev/null @@ -1,50 +0,0 @@ -# Tabs - -Tabs let you organize content by context, for example installation instructions for each supported platform. - -```tpl -{{}} -{{}} # MacOS Content {{}} -{{}} # Linux Content {{}} -{{}} # Windows Content {{}} -{{}} -``` - -## Example - -{{< tabs "uniqueid" >}} -{{< tab "MacOS" >}} -# MacOS - -This is tab **MacOS** content. - -Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat -stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa -protulit, sed sed aere valvis inhaesuro Pallas animam: qui _quid_, ignes. -Miseratus fonte Ditis conubia. -{{< /tab >}} - -{{< tab "Linux" >}} - -# Linux - -This is tab **Linux** content. - -Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat -stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa -protulit, sed sed aere valvis inhaesuro Pallas animam: qui _quid_, ignes. -Miseratus fonte Ditis conubia. -{{< /tab >}} - -{{< tab "Windows" >}} - -# Windows - -This is tab **Windows** content. - -Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat -stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa -protulit, sed sed aere valvis inhaesuro Pallas animam: qui _quid_, ignes. -Miseratus fonte Ditis conubia. -{{< /tab >}} -{{< /tabs >}} diff --git a/content/menu/index.md b/content/menu/index.md index 810bcfd1..0c8fbf66 100644 --- a/content/menu/index.md +++ b/content/menu/index.md @@ -2,21 +2,3 @@ headless: true --- -- [**Example Site**]({{< relref "/docs/example" >}}) -- [Table of Contents]({{< relref "/docs/example/table-of-contents" >}}) - - [With ToC]({{< relref "/docs/example/table-of-contents/with-toc" >}}) - - [Without ToC]({{< relref "/docs/example/table-of-contents/without-toc" >}}) -- [Collapsed]({{< relref "/docs/example/collapsed" >}}) - - [3rd]({{< relref "/docs/example/collapsed/3rd-level" >}}) - - [4th]({{< relref "/docs/example/collapsed/3rd-level/4th-level" >}}) -
- -- **Shortcodes** -- [Buttons]({{< relref "/docs/shortcodes/buttons" >}}) -- [Columns]({{< relref "/docs/shortcodes/columns" >}}) -- [Expand]({{< relref "/docs/shortcodes/expand" >}}) -- [Hints]({{< relref "/docs/shortcodes/hints" >}}) -- [Katex]({{< relref "/docs/shortcodes/katex" >}}) -- [Mermaid]({{< relref "/docs/shortcodes/mermaid" >}}) -- [Tabs]({{< relref "/docs/shortcodes/tabs" >}}) -
diff --git a/content/notes/UWAA558/01-syllabus.md b/content/notes/UWAA558/01-syllabus.md new file mode 100644 index 00000000..b618c91d --- /dev/null +++ b/content/notes/UWAA558/01-syllabus.md @@ -0,0 +1,50 @@ +--- +title: Syllabus +weight: 10 +--- + + +## Syllabus + +The course topics planned for this section are (in rough order): + +Particle Model, Boltzmann-Maxwell Model, Magnetohydrodynamic (MHD) Model, Region of Validity, Common Assumptions, Ideal MHD Model, General Properties (Equilibrium, Boundary Conditions, Conservation Laws, "Frozen-In" Flux) + +Ideal MHD Equilibrium, Virial Theorem, Magnetic Flux Surfaces + +One-Dimensional Equilibria, Theta-Pinch, Z-Pinch, Screw-Pinch, Safety Factor q + +Two-Dimensional Equilibria, Toroidal Geometry, Grad-Shafranov Equation, Closed Flux Surfaces, Safety Factor q, Magnetic Shear, Magnetic Well, Shafranov Shift, Spheromak, Reversed Field Pinch (RFP), Tokamaks, Stellarators (Elmo Bumpy Torus) + +MHD Stability, General Concepts, Linearized MHD, Exponential (Linear) Stability, Force Operator and Properties, Variational Formulation, Energy Principle, Intuitive Form of delta W, Classification of Instabilities (internal/external, pressure-drive/current-driven, kink/interchange/ballooning) + +Stability of One-Dimensional Equilibria, Modal Analysis, Rayleigh-Taylor, Theta-Pinch, Z-Pinch (Kadomtsev Condition), Screw-Pinch (Kruskal-Shafranov Condition, Suydam Criterion), RFP, "Straight" Tokamak + +Stability of Two-Dimensional Equilibria, Tokamak, Mercier Criterion, Elmo Bumpy Torus + +Resistive (Tearing) Instabilities, Stability of Non-static Equilibria, Nonlinear Stability Effects + +## Course Motivation + +Plasma phenomena tend to be hard to treat because of the span of relevant scales. You have ions, electrons, and photons interacting through electromagnetic interactions. There is a tremendous variation in mass across species, which leads to a large span of both spatial and temporal scales. The species can interact through both short scale collisions and long range interactions through EM forces. In contrast, in normal gas dynamics you may consider only the short-scale interactions. As a consequence, we can describe dispersive plasma waves. + +For comparison, remember in gas dynamics, the speed of sound is + + +{{< katex display >}} +\frac{\omega}{k} = v_s = \pdv{\omega}{k} \qquad \text{(gas)} +{{< /katex >}} + +Here the phase velocity {{< katex >}} \frac{\omega}{k} {{< /katex >}} is equal to the phase velocity {{< katex >}} \pdv{\omega}{k} {{< /katex >}}. In a plasma, we can have non-linear dispersion relations in which the phase and group velocity are different. + +{{< katex display >}} +\frac{\omega}{k} = v(\omega, k) \neq \pdv{\omega}{k} \qquad \text{(plasma)} +{{< /katex >}} + +The number of particles we typically deal with in a laboratory plasma is roughly on the order of a mole of particles + +{{< katex display >}} +\text{particles} \sim O(10^{23}) +{{< /katex >}} + +With long-range interactions, we have a combinatorial explosion of interacting particles! It is not possible to track individual particles at such a scale, so we need much simpler plasma descriptions. These are the plasma models we will discuss in the next chapter. When simplifying our models, we need to pay careful attention to the simplifications we are making because in general, inaccurate physics lead to incorrect conclusions. \ No newline at end of file diff --git a/content/notes/UWAA558/02-plasma-models.md b/content/notes/UWAA558/02-plasma-models.md new file mode 100644 index 00000000..9186507c --- /dev/null +++ b/content/notes/UWAA558/02-plasma-models.md @@ -0,0 +1,124 @@ +--- +title: Plasma Models +weight: 20 +bookToc: false +--- + +# Plasma Models + +## Working towards MHD + +Let's start from a full-particle description with the goal of reaching a continuum description (kinetic model). Then, we'll look at the forces on the separate species and form a multi-fluid model, finally simplifying to a single-fluid MHD model. + +The most important question to ask ourselves is "when is this model going to be useful?" The MHD model is the mathematical model for magnetized plasmas that are treated as a fluid. This means that we can define a fluid element (some lil' box of plasma) and define the physical properties (mass, density, magnetization, etc.) of the element. We need to make some assumptions about scale in order to do this. In terms of spatial scales, we abstract properties below a discrete scale {{< katex >}} a_0 {{< /katex >}} into the properties of a fluid element + +{{< katex display >}} +\frac{a_0}{L} \rightarrow 0 \qquad a_0 = \text{discrete scale} \qquad L = \text{spatial scale of interest} +{{< /katex >}} + +Length scales smaller than the discrete scale will not be properly captured by the model, so scales like the particle radius will be meaningless in our fluid model. + +## Plasma Definition + +A plasma is a quasi-neutral gas of charged and neutral particles which exhibit _collective behavior_. The particles (electrons, ions, neutrals) interact through EM fields and collisions. + +Mathematically you would think that plasmas could be treated as individual particles. Doing so gives an N-body problem with classical interactions through the Lorentz force (Coulomb interaction removed by assumption of quasi-neutrality) and binary collisions. Each particle {{< katex >}} i {{< /katex >}} has a well-defined mass {{< katex >}} m_i {{< /katex >}} and charge {{< katex >}} q_i {{< /katex >}} which do not change in time. The governing equations are + +{{< katex display >}} +\dv{\vec v_i}{t} = \frac{q_i}{m_i} (\vec E + \vec v_i \cross \vec B) + \sum_{j \neq i} \left[ \left. \dv{\vec v_{ij}}{t} \right|_{coll} (\vec r_i - \vec r_j) \right] +{{< /katex >}} +{{< katex display >}} +\dv{\vec r_i}{t} = \vec v_i +{{< /katex >}} + +The fields E and B are described by the Maxwell equations + +{{< katex display >}} +\pdv{B}{t} = - \curl E +{{< /katex >}} +{{< katex display >}} +\frac{1}{c^2} \pdv{E}{t} = \curl B - \mu_0 \sum_i q_i v_i \delta (r - r_i) +{{< /katex >}} +{{< katex display >}} +\div B = 0 +{{< /katex >}} +{{< katex display >}} +\epsilon_0 \div E = \sum_i q_i \delta (r - r_i) +{{< /katex >}} + +## Klimontovich Equation + +Re-writing the force relations as a statement of conservation in phase space, we get Klimontovich equation for species {{< katex >}} \alpha {{< /katex >}} + +{{< katex display >}} +\dv{N_\alpha}{t} = 0 = \pdv{N_\alpha}{t} + \pdv{}{q} \cdot (\dot{q} N_\alpha) + \pdv{}{p} \cdot (\dot{p}N_\alpha) +{{< /katex >}} + +The particle phase space is defined by +{{< katex display >}} +N_\alpha(p, q) = \sum_i \delta(p - p_i) \delta(q - q_i) +{{< /katex >}} +where {{< katex >}} p {{< /katex >}} and {{< katex >}} q {{< /katex >}} are generalized momentum and position coordinates. The resulting {{< katex >}} N(p) {{< /katex >}} looks very spiky, with nonzero values only at the exact values inhabited by particles. Unfortunately, that means that only the tools of discrete mathematics are applicable to the distribution, and we're forbidden from our favorite tool (calculus). To make the analysis possible, we can smooth over the discreteness by performing an ensemble average of the Klimontovich equation. This gives us a statistical description using smooth distribution functions: + +{{< katex display >}} +f(x, v, t) \qquad \frac{(\text{no. of particles})}{(\text{unit distance})^3(\text{unit velocity})^3} +{{< /katex >}} + +That is, {{< katex >}} f(x, v, t) {{< /katex >}} is the number of particles at position {{< katex >}} x {{< /katex >}} with velocity {{< katex >}} v {{< /katex >}} at time {{< katex >}} t {{< /katex >}}. We also work with normalized distributions which give the probability of finding a particle. + +By our definition, we can integrate to get the total number of particles at time {{< katex >}} t {{< /katex >}}. + +{{< katex display >}} +\int \int \dd x \dd v f(x, v, t) = N(t) +{{< /katex >}} + +The ensemble averaging process works well if the the number of particles is very large + +{{< katex display >}} +N \gg 1 +{{< /katex >}} + +

Figure 12.1

+ +See a kinetic theory text (e.g. Krall and Trivelpiece) for a full description of the ensemble averaging process. + +Now we can write our Klimontovich equation in terms of continuous quantities + +{{< katex display >}} +\pdv{f}{t} + \left[ \pdv{}{q} \cdot (\dot{q} f) + \pdv{}{p} \cdot (\dot{p}f) \right] = \text{(cross terms)} +{{< /katex >}} + +The collision terms now have an infinite number of cross terms. We call this the BBGKY (Bogoliubov–Born–Green–Kirkwood–Yvon) hierarchy. + +Expressing as the Boltzmann equation (generally called the Boltzmann-Maxwell equation, since the solution requires solving for the electromagnetic fields of the Maxwell equations) + +{{< katex display >}} +\pdv{f_\alpha}{t} + \vec v \cdot \pdv{f_\alpha}{\vec x} + \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} = \left. \pdv{f_\alpha}{t} \right|_{coll} +{{< /katex >}} + +We leave the collision term in. A lot of the work of kinetic theory is coming up with an applicable form of the collision operator which is appropriate but still simple enough to solve. We often write the collision operator as the product of binary collisions + +{{< katex display >}} + \left. \pdv{f_\alpha}{t} \right|_{coll} = \sum_\beta C_{\alpha \beta} +{{< /katex >}} + +Notice that the terms {{< katex >}} \vec v \cdot \pdv{f_\alpha}{\vec x} {{< /katex >}} and {{< katex >}} \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} {{< /katex >}} are advection equations, advecting in {{< katex >}} \vec x {{< /katex >}} and {{< katex >}} \vec v {{< /katex >}} respectively. If we ignore the collision term (set the RHS to zero) we have the Vlasov equation. + +Note that the fields E and B at any location are generated from the charges and currents of the entire plasma volume, including externally applied fields. As a result, there's an inherent integrating process taking into account the sources across the whole volume that leads to long-range smoothly varying forces. This is in contrast to the collisional effects, which by their nature lead to very short range abrupt forces. It makes sense to make a distinction between the long-range electromagnetic forces and the short range collisional forces. + +Because the Boltzmann-Maxwell model is inherently 6-dimensional, it is a very challenging model to implement. The B-M model provides a complete description, but it is often too detailed to solve. + +If we solve the Vlasov equation for two parallel opposite beams, for example, we see that the + +As it turns out, the integral, centroid, and variance are all that are required to fully describe a Maxwellian distribution + +{{< katex display >}} +f_{M, \alpha} (\vec v) = n_0 \left( \frac{ m_\alpha }{2 \pi T_\alpha} \right)^{3/2} \text{exp}\left[ - \frac{ \frac{1}{2} m_\alpha (v - v_\alpha)^2}{T_\alpha} \right] +{{< /katex >}} + +In other words, we can write the Maxwellian distribution as {{< katex >}} f_M(n_0, \vec v_\alpha, T_\alpha ; \vec v) {{< /katex >}}. We care about Maxwellian distributions so much in plasma physics because it is the solution to the Boltzmann equation for {{< katex >}} \left. \pdv{f_\alpha}{t} \right|_{coll} = 0 {{< /katex >}} (Vlasov equation). That's not saying that there are no collisions, it is saying that there are so many collisions that the effect is isotropic and the overall force is zero. + +Another feature of the Maxwellian distribution is {{< katex >}} \vec v \cdot \pdv{f_\alpha}{\vec x} = 0 {{< /katex >}}. This is a famous result called the Boltzmann H-theorem, and says that any initial distribution will relax (and very quickly) to a Maxwellian distribution. + +By replacing our velocity distribution with the associated Maxwell distribution, we arrive at a Plasma Fluid Model + diff --git a/content/notes/UWAA558/03-plasma-fluid-model.md b/content/notes/UWAA558/03-plasma-fluid-model.md new file mode 100644 index 00000000..299a17d1 --- /dev/null +++ b/content/notes/UWAA558/03-plasma-fluid-model.md @@ -0,0 +1,528 @@ +--- +title: Plasma Fluid Model +bookToc: false +weight: 30 +--- + + +# Plasma Fluid Model + +We take velocity moments of each of the pieces of the kinetic model: + - distribution function, {{< katex >}} f_\alpha {{< /katex >}} {{< katex >}} \rightarrow {{< /katex >}} fluid variables + - Boltzmann Equation {{< katex >}} \rightarrow {{< /katex >}} governing equations describing the evolution of the fluid variables. + +Starting with the zeroth moment (integral) of the distribution function: + +{{< katex display >}} +\int f_\alpha (\vec x, \vec v, t) \dd \vec v = n_\alpha(\vec x, t) +{{< /katex >}} + +1st Moment (momentum): + +{{< katex display >}} +m_\alpha \int \vec v \cdot f_\alpha(\vec x, \vec v, t) \dd \vec v = \vec p_\alpha (\vec x, t) = m_\alpha n_\alpha \vec v_\alpha +{{< /katex >}} + +which is to say that the velocity is the 1st moment divided by the zeroth moment + +{{< katex display >}} +v_\alpha = \frac{\int \vec v f_\alpha}{\int f_\alpha} +{{< /katex >}} + +2nd Moment: + +{{< katex display >}} +\int \vec v \vec v f_\alpha (\vec x, \vec v, t) \dd \vec v = \vec E_\alpha(\vec x, t) (\text{energy tensor}) +{{< /katex >}} + +We can simplify the 2nd moment by taking a reduced 2nd moment. This means that we're going to insert a dot product + +{{< katex display >}} +\int \vec v \cdot \vec v f_\alpha (\vec x, \vec v, t) \dd \vec v +{{< /katex >}} + +Before moving forward, we want to define a "random" velocity. Note that + +{{< katex display >}} +\int \vec v f_\alpha - \vec v_\alpha \int f_\alpha = 0 \rightarrow \int (\vec v - \vec v_\alpha) f_\alpha \dd \vec v = 0 +{{< /katex >}} + +We can define a random velocity {{< katex >}} \vec w = \vec v - \vec v_\alpha {{< /katex >}}. It is random in the sense that it is a fluctuation about the mean velocity, and when we integrate it we get zero. We can use this to define the energy tensor using the mean velocity to get a meaningful result. The pressure tensor is the second moment, using the random velocity + +{{< katex display >}} +\vec P_\alpha = m_\alpha \int \vec w \vec w f_\alpha \dd \vec v = P_\alpha \overline{I} + \overline{\Pi}_\alpha +{{< /katex >}} + +where we've decomposed the pressure into an isotropic value {{< katex >}} P_\alpha {{< /katex >}} and what's called the Braginskii stress tensor {{< katex >}} \overline{\Pi}_\alpha {{< /katex >}}. The average isotropic pressure is given by the reduced 2nd moment: + +{{< katex display >}} +P_\alpha = n_\alpha T_\alpha = \frac{1}{3} \int m_\alpha \vec w \cdot \vec w f_\alpha +{{< /katex >}} + +where the factor of {{< katex >}} 1/3 {{< /katex >}} comes from the number of degrees of freedom in our system. It is related to the thermodynamic factor {{< katex >}} \gamma {{< /katex >}} where {{< katex >}} \gamma = \frac{DOF + 2}{DOF} {{< /katex >}}. Now we can define the temperature {{< katex >}} T_\alpha {{< /katex >}} as + +{{< katex display >}} +T_\alpha (\vec x, t) = \frac{1}{DOF} \frac{\int m_\alpha \vec w \cdot \vec w f_\alpha \dd \vec v}{\int f_\alpha \dd \vec v} +{{< /katex >}} + +Now that we've got {{< katex >}} n_\alpha {{< /katex >}}, {{< katex >}} v_\alpha {{< /katex >}}, and {{< katex >}} T_\alpha {{< /katex >}} we have what we need to define a Maxwellian distribution. Higher moments would be required to describe non-Maxwellian distribution functions. For example, the 3rd moment is called the skewness of the distribution. The 4th moment is the kurtosis. So on and so forth. These give a measure of degree of departure from a Maxwellian distribution, in which case it is often more useful to talk about the excess kurtosis, where the excess kurtosis of a Maxwellian is defined to be zero. You can continue to calculate the moment expansion, and in general it requires an infinite number of moments to describe an arbitrary distribution function. Because of the Boltzmann H-theorem and the tendency of plasmas to quickly relax to Maxwellian, we can usually get away with using just the first three moments. + +Now, what are the governing equations? We get these by taking moments of the Boltzmann equation. Let's proceed carefully in sections, so we'll integrate each piece of the BE in terms. + +### 0th Moment of Boltzmann Equation (Conservation) + +{{< katex display >}} +\int \pdv{f_\alpha}{t} \dd \vec v + \int \vec v \cdot \pdv{f_\alpha}{x} \dd v + \int \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = \int \left. \pdv{f_\alpha}{t} \right|_{coll} \dd \vec v +{{< /katex >}} + +First, we take {{< katex >}} \int \pdv{f_\alpha}{t} \dd \vec v {{< /katex >}}. Because t and v are both independent variables, with an argument of sufficient smoothness we can reverse the order of integration and differentiation + +{{< katex display >}} +\int \pdv{f_\alpha}{t} \dd \vec v = \pdv{}{t}\int f_\alpha \dd v = \pdv{n_\alpha}{t} +{{< /katex >}} + +For {{< katex >}} \int \vec v \cdot \pdv{f_\alpha}{x} \dd v {{< /katex >}} we can perform an integration by parts + +{{< katex display >}} + \int \vec v \cdot \pdv{f_\alpha}{x} \dd v = \int \pdv{}{\vec x} \cdot ( \vec v f_\alpha) \dd \vec v - \int f_\alpha \pdv{}{\vec x} \cdot \vec v \dd \vec v \\ + = \int \pdv{}{\vec x} \cdot ( \vec v f_\alpha) \dd \vec v \\ + = \pdv{}{\vec x} \cdot \int \vec v f_\alpha \dd \vec v \\ + = \pdv{}{\vec x} \cdot (n_\alpha \vec v_\alpha) = \div (n_\alpha \vec v_\alpha) +{{< /katex >}} + +Once again, we've switched the order of integration of {{< katex >}} x {{< /katex >}} and {{< katex >}} v {{< /katex >}}, which we can only do because we have specified that {{< katex >}} f {{< /katex >}} is a distribution function, and as such meets the criterion of sufficient smoothness. + +For the last part, we can write it as a surface integral + +{{< katex display >}} +\int \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = \oint f_\alpha \frac{q_\alpha}{m_\alpha} ( \vec E + \vec v \cross \vec B) \cdot \dd S_v - \int \frac{q_\alpha}{m_\alpha} f_\alpha \pdv{}{\vec v} \cdot (\vec E + \vec v \cross \vec B) \dd \vec v +{{< /katex >}} + +For {{< katex >}} f_\alpha {{< /katex >}} to be well-defined, we require {{< katex >}} \lim_{v \rightarrow \infty} v^3 f_\alpha = 0 {{< /katex >}} so the surface term vanishes and we're left with + +{{< katex display >}} +\int \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = - \int \frac{q_\alpha}{m_\alpha} f_\alpha \pdv{}{\vec v} \cdot (\vec E + \vec v \cross \vec B) \dd \vec v +{{< /katex >}} + +Distributing the divergence through, + +{{< katex display >}} +\pdv{}{\vec v} \cdot \vec E = 0 +{{< /katex >}} +because {{< katex >}} \vec E {{< /katex >}} is independent of {{< katex >}} \vec v {{< /katex >}}. + +{{< katex display >}} +\pdv{}{\vec v} \cdot ( \vec v \cross \vec B) = 0 +{{< /katex >}} +because {{< katex >}} \vec v \cross \vec B {{< /katex >}} is always orthogonal to {{< katex >}} \vec v {{< /katex >}}. + +We've been working through this in normal vector notation for familiarity, but there is another notation known as Einstein Tensor Notation using the lovely Levi-Civita symbol {{< katex >}} \epsilon_{ijk} {{< /katex >}}. + +{{< katex display >}} +\epsilon_{ijk} = 1 \qquad \text{even permutations of i j k} \\ + = -1 \qquad \text{odd permutations of i j k} \\ += 0 \qquad \text{any repeated indexes} +{{< /katex >}} + + +We can write out vector products as products of indices and operators, and any repeated indices are implicitly summed: + +{{< katex display >}} +\vec v \cross \vec B = \epsilon_{ijk} v_j B_k = \sum_{jk} \epsilon_{ijk} v_j B_k +{{< /katex >}} + +where {{< katex >}} \epsilon_{ijk} {{< /katex >}} is defined to be 1 for even permutations of ijk, -1 for negative permutations if ijk, and 0 for any repeated indices. + +We can now write the derivative with respect to {{< katex >}} \vec v {{< /katex >}} as + +{{< katex display >}} +\pdv{}{\vec v} \cdot \vec v \cross \vec B = \partial _{v_i} \epsilon_{ijk} v_j B_k \\ += \epsilon_{ijk} (\partial_{v_i} v_j) B_k +{{< /katex >}} + +We see that we're taking the derivative of the j-th component of velocity with respect to the i-th component of velocity, and _that's_ how we can most easily point out that the quantity is zero without relying on properties of 3-vector products. + +We'll also want to use the divergence in Einstein tensor notation + +{{< katex display >}} +\div \vec A = \partial_i A_j \delta_{ij} = \partial_i A_i +{{< /katex >}} + +Finally, we come back to the collision term in the zeroth moment of the B-M equation + +{{< katex display >}} +\sum_\beta \int C_{\alpha \beta} \dd \vec v = 0 +{{< /katex >}} + +We can say the collision term is zero by making a physical argument, rather than a mathematical one. We assert that collisions cannot create or destroy particles. This assumption is now baked into our equations going forward, but is not always true! Ionization, recombination, fusion reactions all create/destroy species. + +Finally, we've got + +{{< katex display >}} +\pdv{n_\alpha}{t} + \div ( n_\alpha \vec v_\alpha) = 0 \qquad \text{continuity equation} +{{< /katex >}} + +So by taking the 0th moment of the Boltzmann Equation, we've arrived at the continuity equation for {{< katex >}} n_\alpha {{< /katex >}} by introducing {{< katex >}} \vec v_\alpha {{< /katex >}}. + +### 1st Moment of Boltzmann Equation + +{{< katex display >}} +\int \vec v \pdv{f_\alpha}{t} \dd \vec v + \int \vec v \vec v \pdv{f_\alpha}{\vec x} \dd \vec v + \int \vec v \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = \int \left. \vec v \pdv{f_\alpha}{t} \right|_{coll} +{{< /katex >}} + +Term 1: + +{{< katex display >}} +\int \vec v \pdv{f_\alpha}{t} \dd \vec v = \pdv{}{t} \int \vec v f_\alpha \dd \vec v - \int \pdv{\vec v}{t} f_\alpha \dd \vec v \\ += \pdv{}{t} \left( n_\alpha \vec v_\alpha \right) +{{< /katex >}} + +Before we get to the second term, let's have an aside about dyad math. An outer product {{< katex >}} \vec A \vec B {{< /katex >}} gives a second-rank tensor. In Cartesian coordinates, it looks like + +{{< katex display >}} +\vec A \vec B = \begin{bmatrix} +A_x B_x & A_x B_y & A_x B_z \\ +A_y B_x & A_y B_y & A_y B_z \\ +A_z B_x & A_z B_y & A_z B_z \\ + \end{bmatrix} +{{< /katex >}} + +A useful property to do with dot products: + +{{< katex display >}} +\vec A \vec B \cdot \vec C = \vec A ( \vec B \cdot \vec C ) = (\vec A \vec B) \cdot \vec C +{{< /katex >}} + +{{< katex display >}} +( \vec V \cdot \grad) \vec B = \vec V \cdot ( \grad \vec B) +{{< /katex >}} + +Back to term 2: + +{{< katex display >}} +\int \vec v \vec v \cdot \pdv{f_\alpha}{\vec x} \dd \vec v = \int \vec v \cdot \pdv{}{\vec x} ( \vec v f_\alpha) \dd \vec v - \int \vec v f_\alpha \pdv{}{\vec x} \cdot \vec v \dd \vec v \\ + = \int \pdv{}{\vec x} \cdot ( \vec v \vec v f_\alpha) \dd \vec v - \int \vec v f_\alpha \cdot \pdv{\vec v}{\vec x} \dd \vec v \\ + = \pdv{}{\vec x} \cdot \int \vec v \vec v f_\alpha \dd \vec v +{{< /katex >}} + +Re-expanding in terms of random velocity {{< katex >}} \vec v = \vec v_\alpha + \vec w {{< /katex >}} + +{{< katex display >}} += \pdv{}{\vec x} \cdot \int (\vec v_\alpha + \vec w) (\vec v_\alpha + \vec w) f_\alpha \dd \vec v \\ += \pdv{}{\vec x} \cdot \int (\vec v_\alpha \vec v_\alpha + \vec v_\alpha \vec w + \vec w \vec v_\alpha + \vec w \vec w) f_\alpha \dd \vec v \\ += \pdv{}{\vec x} \cdot \left[ \vec v_\alpha \vec v_\alpha \int f_\alpha \dd \vec v + \vec v_\alpha \int \vec w f_\alpha \dd \vec v + \left( \int \vec w f_\alpha \dd \vec v \right) \vec v_\alpha + \int \vec w \vec w f_\alpha f_\alpha \dd \vec v \right] +{{< /katex >}} + +The middle two terms are zero by the very definition of {{< katex >}} \vec v_\alpha {{< /katex >}}. The random velocity {{< katex >}} \vec w {{< /katex >}} is defined such that the integral of the random velocity over all phase space is zero. + +{{< katex display >}} += \pdv{}{\vec x} \cdot \left( \vec v_\alpha \vec v_\alpha n_\alpha \right) + \pdv{}{\vec x} \cdot \frac{ \vec P_\alpha}{m_\alpha} \\ += \div (n_\alpha \vec v_\alpha \vec v_\alpha) + \frac{1}{m_\alpha} \div \vec P_\alpha +{{< /katex >}} + +Substituting in the pressure tensor, + +{{< katex display >}} += \div ( n_\alpha \vec v_\alpha \vec v_\alpha ) + \frac{1}{m_\alpha} \left( \grad P_\alpha + \div \vec \Pi _\alpha \right) +{{< /katex >}} + +Moving on to term 3: + +{{< katex display >}} +\int \vec v \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = \int \frac{q_\alpha}{m_\alpha} \pdv{}{\vec v} \cdot \left[( \vec E + \vec v \cross \vec B) f_\alpha \right] \vec v \dd \vec v - \int \frac{q_\alpha}{m_\alpha} f_\alpha \pdv{}{\vec v} \cdot \left[ \vec E + \vec v \cross \vec B \right] \vec v \dd \vec v \\ += \int \frac{q_\alpha}{m_\alpha} \pdv{}{\vec v} \cdot \left[ f_\alpha (\vec E + \vec v \cross \vec B) \vec v \right] \dd \vec v - \int \frac{q\alpha}{m\alpha} f ( \vec E + \vec v\alpha \cross \vec B) \cdot \pdv{\vec v}{\vec v} \dd \vec v \\ += \frac{q\alpha}{m\alpha} \oint f\alpha ( \vec E + \vec v\alpha \cross \vec B) \vec v \cdot \dd \vec S_v - \frac{q\alpha}{m\alpha} \left[ \vec E \int f \dd \vec v\alpha - \vec B \cross \int \vec v f\alpha \dd \vec v \right] \\ += - \frac{q\alpha}{m\alpha} n\alpha (\vec E + v\alpha \cross \vec B) +{{< /katex >}} + +Finally, the 4th term gives + +{{< katex display >}} + \int \left. \vec v \pdv{f_\alpha}{t} \right|_{coll} = \sum_\beta \int \vec v C_{\alpha \beta} \dd \vec v \\ += \int \vec v C_{\alpha \alpha} \dd \vec v + \sum_{\beta \neq \alpha} \int \vec v C_{\alpha \beta} \dd \vec v +{{< /katex >}} + +Collisions of like particles {{< katex >}} \alpha {{< /katex >}} do not result in a net change of momentum of species {{< katex >}} \alpha {{< /katex >}}, so all we have left is the change in momentum due to collisions of unlike particles + +{{< katex display >}} += \sum_{\beta \neq \alpha} \left[ \vec v_{\alpha} \int C_{\alpha \beta} \dd \vec v + \int \vec w C_{\alpha \beta} \dd \vec v \right] +{{< /katex >}} + +For the same reason as before, we assert that collisions between particles {{< katex >}} \alpha {{< /katex >}} and {{< katex >}} \beta {{< /katex >}} do not lead to the creation or destruction of any species, so the 0th moment {{< katex >}} \int C_{\alpha \beta} \dd \vec v = 0{{< /katex >}}. This leads to the conclusion that only random motion contributes to momentum transfer, not {{< katex >}} \vec v_\alpha {{< /katex >}}. Viscosity and friction are good examples of similar physical processes where bulk velocity does not transfer momentum, but random motion does. + +The momentum transfer from {{< katex >}} \alpha {{< /katex >}} to {{< katex >}} \beta {{< /katex >}} must equal transfer from {{< katex >}} \beta {{< /katex >}} to {{< katex >}} \alpha {{< /katex >}}. Momentum is globally conserved. + +{{< katex display >}} +\int \vec w C_{\alpha \beta} \dd \vec v = - \int \vec w C_{\beta \alpha} \dd \vec v +{{< /katex >}} + +Now we can finally write out the full momentum equation + +{{< katex display >}} +\pdv{}{t} (n_\alpha \vec v_\alpha ) + \div (n_\alpha \vec v_\alpha \vec v_\alpha) + \frac{1}{m_\alpha} \div \vec P_\alpha - \frac{q_\alpha}{m_\alpha} n_\alpha ( \vec E + \vec v _\alpha \cross \vec B) = \sum_{\beta \neq \alpha} \int \vec w C_{\alpha \beta} \dd \vec v +{{< /katex >}} + +The collision term is often represented as the momentum transfer vector {{< katex >}} \vec R_{\alpha \beta} {{< /katex >}} to {{< katex >}} \alpha {{< /katex >}} from {{< katex >}} \beta {{< /katex >}}. + +{{< katex display >}} +\vec R_{\alpha \beta} \equiv m_\alpha \int \vec w C_{\alpha \beta} \dd \vec v +{{< /katex >}} +{{< katex display >}} +\pdv{}{t} (m_\alpha n_\alpha v_\alpha) + \div (m_\alpha n_\alpha \vec v_\alpha \vec v_\alpha) + \grad \vec P_{\alpha} + \div \vec \Pi_{\alpha} - q_\alpha n_\alpha (\vec E + \vec v _\alpha \cross \vec B) = \sum_{\beta} R_{\alpha \beta} +{{< /katex >}} + +Once again, we've written the above in a conservation law form. The terms that aren't strictly conservation terms are the source term {{< katex >}} q_\alpha n_\alpha (\vec E + \vec v _\alpha \cross \vec B) {{< /katex >}} and sink term {{< katex >}} \sum_{\beta} R_{\alpha \beta} {{< /katex >}}. + +Introducing the mass density + +{{< katex display >}} +\rho_\alpha = m_\alpha n_\alpha \qquad \text{mass density} +{{< /katex >}} + +{{< katex display >}} +\pdv{}{t} (\rho_\alpha \vec v _\alpha) + \div (\rho_\alpha \vec v_\alpha v_\alpha + \ldots \\ += \rho_\alpha \pdv{\vec v_\alpha}{t} + \vec v_\alpha \left( \pdv{\rho_\alpha}{t} + \div \rho_\alpha \vec v_\alpha \right) + \rho_\alpha \vec v_\alpha \cdot \grad \vec v_\alpha + \ldots +{{< /katex >}} +The term in parentheses is just the continuity equation, which is zero, and what's left is a more usual form of the momentum equation + +{{< katex display >}} +\text{Momentum Equation:}\\ +\rho_\alpha \left(\pdv{\vec v_\alpha}{t} + \vec v_\alpha \cdot \grad \vec v_\alpha \right) + \grad \vec P_\alpha + \div \vec \Pi_\alpha - q_\alpha n_\alpha (\vec E + \vec v_\alpha \cross \vec B) = \sum_{\beta \neq \alpha} \vec R_{\alpha \beta} +{{< /katex >}} + +So by taking the 1st moment of the BE, we arrive at the momentum equation for {{< katex >}} \vec v_\alpha {{< /katex >}} and we have introduced {{< katex >}} \vec P_\alpha {{< /katex >}}. + +### Momentum transfer (collisions) + +Looking back at {{< katex >}} \vec R_{\alpha \beta} {{< /katex >}}, it's worth noting that the actual situation is complicated by the potential for multi-body collisions. While it is true that the probability of, e.g. three-body collisions is small, but there are many three-body collisional processes which are extremely important in plasma physics. Three-body recombination is the primary loss term in the ionization balance, for example. For now the simplest model is to only include binary collisions that result in small angel deflections. If we suppose we only have two species with distributions that look like the following: + +

Figure 12.2

+ +then we expect collisions between species to tend to drag the distributions towards each other. We should expect the collision-based momentum transfer to be proportional to the distance between the distributions, i.e. the total current density {{< katex >}} \vec j {{< /katex >}}. + +For an ion-electron plasma, the collision momentum transfer vector (to ions from electrons) is + +{{< katex display >}} +\vec R_{ie} = - n_e e \eta \vec j +{{< /katex >}} + +where + +{{< katex display >}} +\eta = \frac{m_e \nu_{ei}}{n e^2} \qquad \text{resistivity} +{{< /katex >}} + +{{< katex display >}} +\vec j = n e (\vec v_i - \vec v_e) \qquad \text {current density for Z = 1} +{{< /katex >}} + +### 2nd Moment of Boltzmann Equation + +Finally we're on to our last expansion term. In general, we would say that the second moment would be + +{{< katex display >}} +\int \vec v \vec v \pdv{f_\alpha}{t} \dd \vec v = \pdv{}{t} \int \vec v \vec v f_\alpha \dd \vec v = \pdv{}{t} \vec E_\alpha / m_\alpha \rightarrow \pdv{}{t} \vec P_\alpha +{{< /katex >}} + +Usually we think of the second moment as giving us energy, a scalar, rather than a tensor equation that we have here. Since the pressure tensor is symmetric, the energy equation will contain 6 independent quantities. Together with the continuity and velocity equations, we have 10 quantities in total, so we call this expansion a 10-moment expansion. + +Instead of the full tensor equation, we're actually going to compute the _reduced 2nd moment_ by contracting {{< katex >}} \vec v \cdot \vec v {{< /katex >}} + +{{< katex display >}} +\int \vec v \cdot \vec v \pdv{f_\alpha}{t} \dd v +{{< /katex >}} + +So, let's get after it: + +{{< katex display >}} +\int v^2 \pdv{f_\alpha}{t} \dd \vec v + \int v^2 \vec v \cdot \pdv{f_\alpha}{\vec x} \dd \vec v + \int v^2 \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = \left. \int v^2 \pdv{f_\alpha}{t} \right|_{coll} \dd \vec v +{{< /katex >}} + +Simplifying... Term 1: + +{{< katex display >}} +\int v^2 \pdv{f_\alpha}{t} \dd \vec v = \int \pdv{}{t} (v^2 f_\alpha) \dd \vec v - \int \pdv{v^2}{t} f_\alpha \dd \vec v \\ += \pdv{}{t} \int (\vec v_\alpha + \vec w) \cdot (\vec v_\alpha + \vec w) f_\alpha \dd \vec v \\ += \pdv{}{t} \left[ \int v_\alpha ^2 f_\alpha \dd \vec v + 2 \int \vec v_\alpha \cdot \vec w f_\alpha \dd \vec v + \int w^2 f_\alpha \dd \vec v \right] \\ += \pdv{}{t} \left[ v_\alpha ^2 \int f_\alpha \dd \vec v + 2 \vec v_\alpha \cdot \int \vec w f_\alpha \dd \vec v + 3 \frac{n_\alpha T_\alpha}{m_\alpha} \right] \\ += \pdv{}{t} \left[ n_\alpha v_\alpha ^2 + 3 \frac{n_\alpha T_\alpha}{m_\alpha} \right] +{{< /katex >}} + +Term 2: + +{{< katex display >}} +\int v^2 \vec v \cdot \pdv{f_\alpha}{\vec x} \dd \vec v = \int \pdv{}{\vec x} \cdot (v^2 \vec v f_\alpha) \dd \vec v - \int f_\alpha \pdv{}{\vec x} \cdot (v^2 \vec v) \dd \vec v \\ += \pdv{}{\vec x} \cdot \int ( \vec v_\alpha + \vec w) \cdot (\vec v_\alpha + \vec w) \vec w f_\alpha \dd \vec v \\ += \pdv{}{\vec x} \cdot \int (v_\alpha ^2 + 2 \vec v_\alpha \cdot \vec w + w^2) \vec v f_\alpha \dd \vec v \\ += \pdv{}{\vec x} \cdot \left[ \int v_\alpha ^2 (\vec v_\alpha + \vec w) f_\alpha \dd \vec v + \int 2 \vec v_\alpha \cdot \vec w (\vec v_\alpha + \vec w) f_\alpha \dd \vec v + \int w^2 (\vec v_\alpha + \vec w) f_\alpha \dd \vec v \right] \\ += \pdv{}{\vec x} \cdot \left[ v_\alpha ^2 \vec v_\alpha \int f_\alpha \dd \vec v + v_\alpha ^2 \int \vec w f_\alpha \dd \vec v + 2 \vec v_\alpha \cdot \left(\int \vec w f_\alpha \dd \vec v \right) \vec v_\alpha \\ +\qquad + 2 \vec v_\alpha \cdot \int \vec w \vec w f_\alpha \dd \vec v + \vec v_\alpha \int w^2 f_\alpha \dd \vec v + \int \vec w w^2 f_\alpha \dd \vec v \right] \\ += \pdv{}{\vec x} \cdot (n_\alpha v_\alpha ^2 \vec v_\alpha) + \pdv{}{\vec x} \cdot 2 \vec v_\alpha \cdot \frac{\vec P_\alpha}{m_\alpha} + \pdv{}{\vec x} \cdot \left(\frac{3 n_\alpha T_\alpha \vec v_\alpha}{m_\alpha} \right) + \pdv{}{\vec x} \cdot \left( \frac{2 \vec h_\alpha}{m_\alpha} \right) +{{< /katex >}} + +In keeping with the progression so far, the very last term would be something like a full 3rd moment {{< katex >}} \int \vec w \vec w \vec w f_\alpha \dd \vec v {{< /katex >}}. Instead, we have a "contracted 3rd moment" of the distribution {{< katex >}} \vec h_\alpha {{< /katex >}} which is actually the _heat flux_, or the random energy flux of random energy. + +Term 3: + +{{< katex display >}} +\int v^2 \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v \\ + = \oint f_\alpha \frac{q_\alpha}{m_\alpha} v^2 ( \vec E + \vec v \cross \vec B) \cdot \dd \vec S_v - \int f_\alpha \frac{q_\alpha}{m_\alpha} \pdv{}{\vec v} \cdot \left[ v^2 ( \vec E + \vec v \cross \vec B) \right] \dd \vec v \\ += - \frac{q_\alpha}{m_\alpha} \int f_\alpha \left[ v^2 \left(\pdv{}{\vec v} \cdot \vec E \right) + \left(\vec E \cdot \pdv{}{\vec v} v^2 \right) \right] \dd \vec v \\ +\qquad - \frac{q_\alpha}{m_\alpha}\int f_\alpha \left[v^2 \pdv{}{\vec v} \cdot (\vec v \cross \vec B) + \vec v \cross \vec B \cdot \pdv{}{\vec v} v^2 \right] \dd \vec v \\ += - \frac{q_\alpha}{m_\alpha} \int f_\alpha (2 \vec E \cdot \vec v) \dd \vec v \\ += - \frac{2 q_\alpha}{m_\alpha} n_\alpha \vec v_\alpha \cdot \vec E +{{< /katex >}} + +We've ended up with the work done on a fluid by the electric field. As expected, there is no work term associated with the bulk magnetic field. + +Term 4: + +{{< katex display >}} +\left. \int v^2 \pdv{f_\alpha}{t} \right|_{coll} \dd \vec v = \int v^2 C_{\alpha \alpha} \dd \vec v + \sum _{\beta \neq \alpha} \int v^2 C_{\alpha \beta} \dd \vec v +{{< /katex >}} + +We can make an energy conservation argument to get rid of the first term. The first term goes to 0 since collisions of like particles results in no net energy exchange. + +{{< katex display >}} += \sum_{\beta \neq \alpha} \int ( \vec v_\alpha + \vec w) \cdot(\vec v_\alpha + \vec w) C_{\alpha \beta} \dd \vec v \\ += \sum_{\beta \neq \alpha}\left( v_\alpha ^2 \int C_{\alpha \beta} \dd \vec v + 2 \vec v_\alpha \cdot \int \vec w C_{\alpha \beta} \dd \vec v + \int w^2 C_{\alpha \beta} \dd \vec v \right) +{{< /katex >}} + +Since collisions (we assume) don't create or destroy particles, the first term {{< katex >}} v_\alpha ^2 \int C_{\alpha \beta} \dd \vec v {{< /katex >}} goes away. The second term {{< katex >}} \vec v_\alpha \cdot \int \vec w C_{\alpha \beta} \dd \vec v {{< /katex >}} we interpret as a frictional force due to the relative motion between {{< katex >}} \alpha {{< /katex >}} and {{< katex >}} \beta {{< /katex >}} + +{{< katex display >}} += \sum_{\beta \neq \alpha} 2 \frac{\vec v_\alpha \cdot \vec R_{\alpha \beta}}{m_\alpha} + 2 \frac{Q_{\alpha \beta}}{m_\alpha} +{{< /katex >}} + +where {{< katex >}} Q_{\alpha \beta} {{< /katex >}} is the heat exchange term (heat generation term in Braginskii). It describes the heat exchange by random collisions of unlike particles, analogous to viscous heating. + +{{< katex display >}} +Q_{\alpha \beta} = \frac{1}{2} \int m_\alpha w^2 C_{\alpha \beta} \dd \vec v +{{< /katex >}} + +If we were to draw distribution functions for species {{< katex >}} \alpha {{< /katex >}} and {{< katex >}} \beta {{< /katex >}} (as shown), even though they have the same centroid they will interact by viscous heating. Species {{< katex >}} \beta {{< /katex >}} will heat up and {{< katex >}} \alpha {{< /katex >}} will cool + +

Figure 12.3

+ +If we multiply by {{< katex >}} \frac{1}{2} m_\alpha {{< /katex >}} then the energy equation becomes + +{{< katex display >}} +\text{Energy Equation: } \\ \qquad \pdv{}{t} \left( \frac{3}{2} n_\alpha T_\alpha + \frac{1}{2} \rho_\alpha v_\alpha ^2 \right) + \\ + \div \left(\frac{3}{2} n_\alpha T_\alpha + \frac{1}{2} \rho_\alpha v_\alpha ^2 \right) \vec v_\alpha + \div (\vec v_\alpha \cdot \vec P_\alpha) + \div \vec h_\alpha - q_\alpha n_\alpha \vec v_\alpha \cdot \vec E \\ = \sum_{\beta \neq \alpha} (v_\alpha \cdot \vec R_{\alpha \beta} + Q_{\alpha \beta}) +{{< /katex >}} + +Notice that the total energy appears + +{{< katex display >}} +E_\alpha = \frac{3}{2} n_\alpha T_\alpha + \frac{1}{2} \rho_\alpha v_\alpha ^2 +{{< /katex >}} + +The factor of {{< katex >}} \frac{3}{2} {{< /katex >}} as usual comes from {{< katex >}} \gamma {{< /katex >}} + +{{< katex display >}} +\frac{3}{2} = \frac{1}{\gamma - 1} = \frac{1}{5/3 - 1} +{{< /katex >}} + +where + +{{< katex display >}} +\gamma = \frac{DOF + 2}{DOF} = \frac{5}{3} +{{< /katex >}} + +We can remove the kinetic portion of the energy equation by subtracting the product of the momentum equation with {{< katex >}} \vec v_\alpha {{< /katex >}}. + +{{< katex display >}} +\rho_\alpha \vec v_\alpha \cdot \pdv{v_\alpha}{t} = \frac{1}{2} \pdv{}{t} (\rho_\alpha v_\alpha ^2) +{{< /katex >}} + +{{< katex display >}} +\pdv{}{t} \left( \frac{3}{2} n_\alpha T_\alpha \right) + \div \left(\frac{3}{2} n_\alpha T_\alpha \vec v_\alpha \right) + \div \left( \vec v_\alpha \cdot \vec P_\alpha \right) - \vec v_\alpha \cdot (\div \vec P_\alpha) + \div \vec h_\alpha = \sum_{\beta \neq \alpha} Q_{\alpha \beta} +{{< /katex >}} +{{< katex display >}} +\frac{3}{2} n_\alpha \left( \pdv{T_\alpha}{t} + \vec v_\alpha \cdot \div T_\alpha \right) + \frac{3}{2} T_\alpha \left[ \pdv{n_\alpha}{t} + \div (n_\alpha \vec v_\alpha) \right] + \vec P_\alpha \cdot \cdot \grad \vec v_\alpha + \div \vec h_\alpha = \sum_{\beta \neq \alpha} Q_{\alpha \beta} +{{< /katex >}} +{{< katex display >}} +\frac{3}{2} n_\alpha \left( \pdv{T_\alpha}{t} + \vec v_\alpha \cdot \div T_\alpha \right) + \vec P_\alpha \cdot \cdot \grad \vec v_\alpha + \div \vec h_\alpha = \sum_{\beta \neq \alpha} Q_{\alpha \beta} +{{< /katex >}} + +Where {{< katex >}} \vec P_\alpha \cdot \cdot \grad \vec v_\alpha {{< /katex >}} is a tensor contraction in two indices. It is a generalization of the dot product, i.e. +{{< katex display >}} +\vec P_\alpha \cdot \cdot \grad \vec v_\alpha = \delta_{ik} \delta_{jl} P_{ij} \partial_k v_l \\ += P_{xx} \partial_x v_x + P_{xy} \partial_y v_x + \ldots +{{< /katex >}} + +{{< katex display >}} +\frac{3}{2} n_\alpha \left( \pdv{T_\alpha}{t} + \vec v_\alpha \cdot \grad T_\alpha \right) + P_\alpha \div \vec v_\alpha + \vec \Pi_\alpha \cdot \cdot \grad \vec v_\alpha + \div \vec h_\alpha = \sum_{\beta \neq \alpha} Q_{\alpha \beta} +{{< /katex >}} + +Now the continuity, momentum, and energy equations describe the evolution of each fluid {{< katex >}} \alpha {{< /katex >}}. The only velocity information we have retained in taking moments is the centroid {{< katex >}} \vec v_\alpha {{< /katex >}} and the width {{< katex >}} T_\alpha {{< /katex >}}. This is the only information we will include in the 5-moment model (5N-moment plasma fluid model). The higher moments (describing skewness, kurtosis) are not evolved with the 5-moment model. There are fluid models that do evolve those moments, but we won't touch on those here. + +### Closure Problem + +We still need to "close" the fluid model (i.e. solving the **Closure Problem**). As we have been finding out, each additional moment of the Boltzmann Equation introduces the next-higher moment. In calculating the 0th moment, we introduce the 1st moment, etc. We need to address the closure problem by relating higher moment variables to lower moment variables. Applying a closure scheme (relating variables that are not evolved directly) is usually equivalent to making a statement on heat flow. + +In our model, we are not evolving the heat flux {{< katex >}} \vec h_\alpha {{< /katex >}}, so we need to relate it to one of the other variables that we do evolve directly. We usually do that by writing down a conductivity relation + +{{< katex display >}} +\vec h_\alpha = - \kappa \grad T_\alpha +{{< /katex >}} + +for some conductivity {{< katex >}} \kappa {{< /katex >}}. + +We're also only evolving the scalar pressure, so we need to relate the stress tensor {{< katex >}} \vec \Pi_\alpha {{< /katex >}} to the other variables. We can do that with a viscosity relation + +{{< katex display >}} +\vec \Pi_\alpha = \nu \grad \vec v_\alpha +{{< /katex >}} + +for some viscosity {{< katex >}} \nu {{< /katex >}}. We call the introduced quantities {{< katex >}} \kappa {{< /katex >}} and {{< katex >}} \nu {{< /katex >}} the **transport coefficients**. They are usually derived by performing an expansion of the equations we've already discussed. They are what lead to "nondimensional" numbers that characterize the flow in our system. + +We can also achieve closure by constraining portions of the system. In an isothermal system + +{{< katex display >}} +\text{isothermal: } \quad T = \text{const.}, p \propto n +{{< /katex >}} +{{< katex display >}} +\text{adiabatic: } \quad p \propto n^\gamma +{{< /katex >}} +{{< katex display >}} +\text{force-free, cold: } \quad p = 0 +{{< /katex >}} + +The expressions that lead us to a closure scheme are the **equations of state**. They are not time-dependent, they simply relate fluid parameters to other properties of the system. + +Maxwell's equations to couple the electromagnetic terms to the fluid variables. +{{< katex display >}} +\pdv{\vec B}{t} = - \curl \vec E +{{< /katex >}} +{{< katex display >}} +\epsilon_0 \pdv{\vec E}{t} = \frac{1}{\mu_0} \curl \vec B - \sum_{alpha} q_\alpha n_\alpha \vec v_\alpha +{{< /katex >}} +{{< katex display >}} +\epsilon_0 \div \vec E = \sum_\alpha q_\alpha n_\alpha +{{< /katex >}} +{{< katex display >}} +\div \vec B = 0 +{{< /katex >}} + +The fluid variables provide the source terms for Ampere's law and Gauss' law, and we've already seen how the electromagnetic fields appear in the momentum equations as source terms for the fluid forces. + +### Stopping after the 2nd moment (5N-Moment Plasma Fluid Model) + +What assumptions have we made by stopping here? It's important to know what they are and to recognize what they mean for the fluid model. + + - Each species {{< katex >}} \alpha {{< /katex >}} is well-represented by a Maxwellian with a small perturbation. The pressure tensor is not strictly diagonal and the heat flux is not zero, so the small perturbations are what lead to the transport coefficients. This is due to the process by which we obtain the transport coefficients, called the Chapman-Enskog expansion. + - Kinetic effects (stream instabilities, counter-flow instabilities, etc.) are not captured. + +The variables of the 5-moment model that we _do_ evolve are +{{< katex display >}} +n, v_x, v_y, v_z, T +{{< /katex >}} + +We can take additional moments. The 10N-moment equation evolves, in addition to the quantities in the 5N-moment model, we evolve all of the independent terms of the pressure tensor +{{< katex display >}} +n, v_x, v_y, v_z, P_{xx}, P_{xy}, P_{xz} , P_{yy} , P_{yz} , P_{zz} +{{< /katex >}} + +The 13-N moment model has everything from the 10N-moment model plus the heat flux tensor {{< katex >}} h_x, h_y, h_z {{< /katex >}} + diff --git a/content/notes/UWAA558/04-two-fluid-plasma-model.md b/content/notes/UWAA558/04-two-fluid-plasma-model.md new file mode 100644 index 00000000..bfb8b368 --- /dev/null +++ b/content/notes/UWAA558/04-two-fluid-plasma-model.md @@ -0,0 +1,195 @@ +--- +title: Two-Fluid Plasma Model +bookToc: false +weight: 40 +--- + + +# Two-Fluid Plasma Model (ions-electrons) + +Restricting our multi-species fluid model to ions and electrons, what can we say about wave behavior in a magnetized 2-fluid plasma? Let's start with a cold plasma approximation ({{< katex >}} p = 0 {{< /katex >}}) and neglect collisions. The momentum equation reduces to + +{{< katex display >}} +m_\alpha \left( \pdv{\vec v_\alpha}{t} + \vec v_\alpha \cdot \grad \vec v_\alpha \right) - q_\alpha (\vec E + \vec v_\alpha \cross \vec B) = 0 +{{< /katex >}} + +From here on out we can avoid some clutter (and wrist strain) by dropping the {{< katex >}} \alpha {{< /katex >}} subscripts and acknowledging that we have sets of equations for ions and electrons. Apply a perturbation to an equilibrium {{< katex >}} g = g_0 + g_1 {{< /katex >}} + +{{< katex display >}} +m \left(\pdv{\vec v_0}{t} + \vec v_0 \cdot \grad \vec v_0 + \pdv{\vec v_0}{t} + \vec v_1 \cdot \grad \vec v_0 + \vec v_0 \cdot \grad \vec v_1 + \vec v_1 \cdot \grad \vec v_1 \right) \\ +\qquad - q (E_0 + E_1) - q (\vec v_0 \cross \vec B_0 + \vec v_1 \cross \vec B_0 + \vec v_0 \cross \vec B_1 + \vec v_1 \cross \vec B_1) = 0 +{{< /katex >}} + +We can drop some terms because equilibrium has to satisfy the original equation. We can balance all of the subscript-0 terms and sum them to get zero. + +{{< katex display >}} +m \left( \pdv{\vec v_1}{t} + \vec v_1 \cdot \grad \vec v_0 + \vec v_0 \cdot \grad \vec v_1 + \vec v_1 \cdot \grad \vec v_1 \right) + q (- E_1 - \vec v_1 \cross \vec B_0 - \vec v_0 \cross \vec B_1 - \vec v_1 \cross \vec B_1) +{{< /katex >}} + +Let's now make the assumption that the perturbation is small, that is {{< katex >}} g_1 \ll g_0 {{< /katex >}}. That means that nonlinear products of perturbation terms are negligible (linearization process). + +{{< katex display >}} +m \left( \pdv{\vec v_1}{t} + \vec v_0 \cdot \grad \vec v_1 + \vec v_1 \cdot \grad \vec v_0 \right) - q \vec E_1 - q (\vec v_1 \cross \vec B_0 + \vec v_0 \cross \vec B_1) = 0 +{{< /katex >}} + +Now, assume that the equilibrium is a **static equilibrium**, that is {{< katex >}} \vec v_0 = 0 {{< /katex >}}. If we decompose into components that are parallel and perpendicular to the equilibrium magnetic field {{< katex >}} \vec B_0 {{< /katex >}}, then + +{{< katex display >}} +\pdv{v_{1, \parallel}}{t} - \frac{q}{m} E_{1, \parallel} = 0 +{{< /katex >}} +{{< katex display >}} +\pdv{\vec v_{1, \perp}}{t} - \frac{q}{m} \left(\vec E_{1, \perp} + B_0 \vec v_{1, \perp} \cross \vu z \right) = 0 +{{< /katex >}} + +The parallel component {{< katex >}} E_{1, \parallel} {{< /katex >}} will lead us to the ordinary wave (**O-wave**). Consideration of the more general case with perpendicular components will lead to the **X-wave**. + +The plasma velocity is related to the fields through the current density (Maxwell equations). Faraday's law gives + +{{< katex display >}} +\pdv{\vec B}{t} = - \curl \vec E +{{< /katex >}} +{{< katex display >}} +\rightarrow \curl \pdv{\vec B}{t} = - \curl \curl \vec E = - \grad (\div \vec E) + \nabla ^2 \vec E +{{< /katex >}} + +Ampere's law gives + +{{< katex display >}} +\epsilon_0 \pdv{\vec E}{t} = \frac{1}{\mu_0} \curl \vec B - \sum_{\alpha} q_\alpha n_\alpha \vec v_\alpha +{{< /katex >}} +{{< katex display >}} +\epsilon_0 \pdv{^2 \vec E }{t ^2} = \frac{1}{\mu_0} \curl \pdv{\vec B}{t} \\ += \frac{1}{\mu_0} \left[ \nabla ^2 \vec E - \grad (\div \vec E) \right] - \sum_\alpha q_\alpha \pdv{}{t} (n_\alpha \vec v_\alpha) +{{< /katex >}} + +Since this is a linear system, assume that the perturbed quantities have a wave-like structure. That is, the perturbed quantities {{< katex >}} g_1 {{< /katex >}} are proportional to {{< katex >}} e^{i(\omega t + \vec k \cdot \vec r)} {{< /katex >}}. This lets us transform the spatial and temporal derivatives into factors of {{< katex >}} \omega {{< /katex >}} and {{< katex >}} \vec k {{< /katex >}} + +{{< katex display >}} +- \epsilon_0 \omega ^2 \vec E_1 = - \frac{1}{\mu_0} \left[k^2 \vec E_1 - \vec k (\vec k \cdot \vec E_1) \right] + i \omega e n_0 \vec v_1 +{{< /katex >}} + +Let's now consider only high frequency oscillations, assuming that only the electrons respond and the ions remain stationary. There's nothing particularly complicated about including the ion response, this just lets us drop the {{< katex >}} \alpha {{< /katex >}} subscripts and focus on a single set of equations. + +{{< katex display >}} +- i \omega \frac{e n_0}{\epsilon_0} \vec v_1 = (\omega ^2 - c^2 k^2) \vec E_1 + c^2 \vec k (\vec k \cdot \vec E_1) +{{< /katex >}} + +Now let's apply the perturbed form to the linearized momentum equation + +{{< katex display >}} +\pdv{v_{1, \parallel}}{t} - \frac{q}{m} E_{1, \parallel} = 0 \\ +\rightarrow i \omega v_{1, \parallel} = - \frac{e}{m} E_{1, \parallel} \\ +\rightarrow i \omega \frac{e n_0}{\epsilon_0} v_{1, \parallel} = \frac{e^2 n_0}{\epsilon_0 m} E_{1, \parallel} +{{< /katex >}} + +Combine the momentum equation and the Maxwell equations to eliminate {{< katex >}} \vec E_1 {{< /katex >}} and {{< katex >}} \vec v_1 {{< /katex >}} + +{{< katex display >}} +\frac{e^2 n_0}{\epsilon_0 m} E_{1, \parallel} = ( \omega ^2 - c^2 k^2 ) E_{1, \parallel} + c^2 k_{\parallel} \vec (k \cdot \vec E_1) +{{< /katex >}} + +Consider different possibilities for the {{< katex >}} \vec k {{< /katex >}} vector. If it is along the magnetic field {{< katex >}} \vec k = k_{\parallel} \vu{e}_\parallel {{< /katex >}} (longitudinal wave) then + +{{< katex display >}} +\frac{e^2 n_0}{\epsilon_0 m} = \omega ^2 - c^2 k^2 + c^2 k^2 = \omega_{pe}^2 +{{< /katex >}} + +For {{< katex >}} \vec k = k_{\perp} \vu e_\perp {{< /katex >}} (transverse wave) then we get the dispersion relation for the O-wave + +{{< katex display >}} +\text{dispersion relation for O-waves:} \qquad \omega^2 - c^2 k^2 = \omega_p ^2 +{{< /katex >}} + +The electric field is in the same direction as the magnetic field {{< katex >}} (\vec E_1 = \vec E_{1, \parallel}) {{< /katex >}}, which means the O-wave is linearly polarized. At large {{< katex >}} k {{< /katex >}} we just have regular light waves, but as we turn the frequency downwards we see a cut-off at the plasma frequency: + +

Figure 12.4

+ +It turns out that the dispersion relation for the X-wave has the same cut-off, but also has another branch with a resonance + +

Figure 12.5

+ + + +The two-fluid plasma model is highly reduced from the full kinetic model, but it is still too complete to be useful when studying gross plasma behavior. Further reductions of the model are possible by making asymptotic assumptions: + +## Low-frequency Asymptotic Assumption + + - Eliminate high frequency, short wavelength phenomena by using pre-Maxwell field equations. Formally, this is {{< katex >}} \epsilon_0 \rightarrow 0 {{< /katex >}}. + +The direct consequences of the low-frequency approximation are + +{{< katex display >}} +c^2 = \frac{1}{\epsilon_0 \mu_0} \qquad c \rightarrow \infty +{{< /katex >}} +{{< katex display >}} +\omega_p ^2 = \frac{n e^2}{\epsilon_0 m} \qquad \omega_p \rightarrow \infty +{{< /katex >}} +{{< katex display >}} +\lambda_D = \frac{v_T}{\omega_p} \rightarrow 0 +{{< /katex >}} + +This means that all phenomena will have {{< katex >}} \omega \ll \omega_p {{< /katex >}}, limiting the frequencies we can resolve to the ion plasma frequency. The characteristic speeds will be limited by the speed of light + +{{< katex display >}} +\frac{\omega}{k} \ll c +{{< /katex >}} + +and all characteristic lengths will be much greater than the Debye length + +{{< katex display >}} +x_0 \gg \lambda_D +{{< /katex >}} + +Looking at Gauss' law, +{{< katex display >}} +\epsilon_0 \div \vec E = \sum_\alpha q_\alpha n_\alpha \rightarrow \sum_\alpha q_\alpha n_\alpha = 0 +{{< /katex >}} + +so we now have charge neutrality everywhere in the domain. For H plasma, locally we have {{< katex >}} n_e = n_i {{< /katex >}} everywhere. + +Looking at Ampere's law, + +{{< katex display >}} +\epsilon_0 \pdv{\vec E}{t} = \frac{1}{\mu_0} \curl \vec B - \sum_\alpha q_\alpha n_\alpha \vec v_\alpha = 0 \\ +\rightarrow \vec j = \frac{1}{\mu_0} \curl \vec B +{{< /katex >}} + +Things we do **not** get from this approximation are {{< katex >}} \vec E = 0 {{< /katex >}} or {{< katex >}} \pdv{\vec E}{t} = 0 {{< /katex >}}. It does mean that plasma dynamics occur on a sufficiently large spatial scale that charge separation is small, and they occur on a sufficiently long temporal scale that electrons respond quickly. + +## Tiny electron asymptotic assumption + +2nd approximation: neglect electron inertia in the momentum equation. Formally, we let the electron mass {{< katex >}} m_e \rightarrow 0 {{< /katex >}} + +{{< katex display >}} +\omega_{pe}^2 = \frac{n e^2}{\epsilon_0 m_e} \rightarrow \infty +{{< /katex >}} +{{< katex display >}} +\omega_{c, e} = \frac{e B}{m_e} \rightarrow \infty +{{< /katex >}} +{{< katex display >}} +v_{T, e} \rightarrow \infty +{{< /katex >}} +The Larmor radius goes to zero +{{< katex display >}} +r_{l, e} = \frac{v_{T, e}}{\omega_{c, e}} \rightarrow 0 +{{< /katex >}} +Importantly, as the gyroradius {{< katex >}} r_{l, e} {{< /katex >}} goes to 0 (because the thermal velocity goes as {{< katex >}} \sqrt{m_e} {{< /katex >}} and the cyclotron frequency goes as {{< katex >}} m_e {{< /katex >}}), this means that the electrons are tied to the magnetic field. +The skin depth is also small. +{{< katex display >}} +\delta_e = \frac{c}{\omega_{p, e}} \rightarrow 0 +{{< /katex >}} + +So all phenomena that we capture must have {{< katex >}} \omega \ll \omega_{p, e} {{< /katex >}}, {{< katex >}} \omega \ll \omega_{c, e} {{< /katex >}}, and {{< katex >}} x_0 \gg r_{L, e} {{< /katex >}}, {{< katex >}} x_0 \gg \delta_e {{< /katex >}} . + +The electron momentum equation becomes + +{{< katex display >}} +\grad P_e + \div \vec \Pi _e + e n_e (\vec E + \vec v \cross \vec B) = \sum_{\beta \neq \alpha} \vec R_{\alpha \beta} +{{< /katex >}} + +The momentum equation is now a state equation, not an evolution equation. It now simply relates the dynamical variables to each other at any point in time. + +Now, note that along magnetic field lines electrons can travel long distances at very fast (finite) speeds which can produce low frequency, long wavelength phenomena. Neglecting electron inertia implies that electrons respond instantaneously, meaning we cannot capture these modes. An example of such a phenomena is drift waves. + +The characteristic speeds {{< katex >}} c {{< /katex >}} and {{< katex >}} v_{T, e} {{< /katex >}} have disappeared from the model. Remaining is {{< katex >}} v_{T, i} {{< /katex >}}. This means that the ion dynamics dictate the plasma evolution. + diff --git a/content/notes/UWAA558/05-mhd-model.md b/content/notes/UWAA558/05-mhd-model.md new file mode 100644 index 00000000..a3d4f13c --- /dev/null +++ b/content/notes/UWAA558/05-mhd-model.md @@ -0,0 +1,666 @@ +--- +title: Magnetohydrodynamic (MHD) Model +bookToc: false +weight: 50 +--- + + +# Magnetohydrodynamic (MHD) Model + +Applying approximations to the two-fluid plasma model will allow us to arrive at a single-fluid (center-of-mass) description. The result is the ideal magnetohydrodynamic model (MHD). + +First, define the MHD variables: + +{{< katex display >}} +\text{mass density:} \qquad \rho = n_i m_i + n_i m_e +{{< /katex >}} + +{{< katex display >}} +\text{fluid velocity:} \qquad \vec v = \frac{n_i m_i \vec v_i + n_e m_e \vec v_e}{n_i m_i + n_e m_e} +{{< /katex >}} + +{{< katex display >}} +\text{current density:} \qquad \vec j = q_i n_i \vec v_i - e n_e \vec v_e \rightarrow \vec v_e = \frac{q_i n_i \vec v_i}{e n_e} - \vec j / e n_e +{{< /katex >}} + +{{< katex display >}} +\text{total pressure:} \qquad p = p_i + p_e = n_e T_e + n_i T_i +{{< /katex >}} + +{{< katex display >}} +\text{total temperature:} \qquad T = \frac{n_i T_i + n_e T_e}{(n_i + n_e)/2} +{{< /katex >}} + +Now let's begin applying asymptotic approximations. For the mass density, applying the first approx (charge neutrality) we have + +{{< katex display >}} +\rho \approx n (m_i + m_e) +{{< /katex >}} + +Using approx 2 (vanishing electron mass) + +{{< katex display >}} +\rho \approx n m_i +{{< /katex >}} +where +{{< katex display >}} +n = n_i \qquad n_e = Z n +{{< /katex >}} + +The center-of-mass velocity (charge neutrality) gives + +{{< katex display >}} +\vec v \approx \frac{m_i \vec v_i + m_e Z \vec v_e}{m_i + Z m_e} +{{< /katex >}} + +with small electron mass approximation: + +{{< katex display >}} +\vec v \approx \vec v_i +{{< /katex >}} + +The current density is (charge neutrality approx) + +{{< katex display >}} +\vec j \approx Z e n (\vec v_i - \vec v_e) \qquad \vec v_e = \vec v_i - \vec j / Z e n +{{< /katex >}} + +The pressure and total temperature are (with charge neutrality) + +{{< katex display >}} +P \approx n (T_i + Z T_e) +{{< /katex >}} + +{{< katex display >}} +T \approx \frac{T_i + Z T_e}{(1 + Z)/2} +{{< /katex >}} + +### MHD Momentum Equation + +Now we combine the two-fluid equations with these asymptotic approximations to obtain the governing equations for the MHD variables. Multiplying the ion continuity equation by the ion mass gives + +{{< katex display >}} +\pdv{}{t} (m_i n_i) + \div (m_i n_i \vec v_i) = 0 \\ +\rightarrow \pdv{\rho}{t} + \div (\rho \vec v) = 0 \quad \text{(MHD continuity eq.)} +{{< /katex >}} + +If we multiply the two-fluid continuity equations by the charge and sum them, we get + +{{< katex display >}} +\pdv{}{t}(q_i n_i - e n_e) + \div (q_i n_i \vec v_i - e n_e \vec v_e) = 0 \\ +\rightarrow \div \vec j = 0 \quad \text{(no accumulation of charge)} +{{< /katex >}} + +If we add the electron and ion momentum equations and apply the small electron mass approximation, and recognizing that {{< katex >}} \vec R_{ei} = - \vec R_{ie} = n e \eta \vec j {{< /katex >}} + +{{< katex display >}} +\rho_i \left(\pdv{\vec i}{t} + \vec v_i \cdot \grad \vec v_i \right) + \grad (P_i + P_e) + \div (\vec \Pi _i + \vec \Pi_e) \\ +\qquad - (q_i n_i - e n_e) \vec E - (q_i n_i \vec v_i - e n_e \vec v_e) \cross \vec B = 0 +{{< /katex >}} +{{< katex display >}} +\text{MHD Momentum Equation} \\ +\rightarrow \rho \left(\pdv{\vec v}{t} + \vec v \cdot \grad \vec v \right) + \grad p - \vec j \cross \vec B = - \div (\vec \Pi _i + \vec \Pi _e) +{{< /katex >}} + +!!! info "MHD Momentum Equation" + + {{< katex display >}} + \rho \left(\pdv{\vec v}{t} + \vec v \cdot \grad \vec v \right) + \grad p - \vec j \cross \vec B = - \div (\vec \Pi _i + \vec \Pi _e) + {{< /katex >}} + +If we take the electron momentum equation, apply the small electron mass asymptotic approximation and introduce the current density, then we have + +### MHD Ohm's Law + +{{< katex display >}} +e n_e (\vec E + \vec v_i \cross \vec B - \vec j \cross \vec B / Z e n) = - \grad P_e - \div \vec \Pi_e + \vec R_{ei} +{{< /katex >}} + +Substituting {{< katex >}} \vec R_{ei} = e n_e \eta \vec j {{< /katex >}} +{{< katex display >}} +\text{MHD Generalized Ohms Law}\\ +\vec E + \vec v \cross \vec B = \eta \vec j + \frac{1}{Z e n} \left(\vec j \cross \vec B - \grad p_e - \div \vec \Pi_e \right) +{{< /katex >}} + +!!! info "MHD Generalized Ohms Law" + + {{< katex display >}} + \vec E + \vec v \cross \vec B = \eta \vec j + \frac{1}{Z e n} \left(\vec j \cross \vec B - \grad p_e - \div \vec \Pi_e \right) + {{< /katex >}} + + +### MHD Energy Equation + +The energy equation is found by adding the ion and electron energies, but first we need to manipulate them into a common form. The ion energy equation is + +{{< katex display >}} +\frac{1}{\gamma - 1} \left[ \pdv{}{t} (n_i T_i) + \div (n_i T_i \vec v_i) \right] + n_i T_i \div \vec v_i = RHS_i +{{< /katex >}} + +where {{< katex >}} RHS_i \equiv - \vec \Pi_i \cdot \cdot \grad \vec v_i - \div \vec h_i + Q_{ie} {{< /katex >}} + +{{< katex display >}} +\frac{1}{\gamma - 1} \left( \pdv{p_i}{t} + \vec v_i \cdot \grad p_i \right) + \frac{\gamma}{\gamma - 1} p_i (\div \vec v_i) = RHS_i +{{< /katex >}} + +Multiply by {{< katex >}} \frac{\gamma - 1}{n_i ^\gamma} {{< /katex >}} and define the total derivative with respect to the ion velocity as + +{{< katex display >}} +\dv{}{t} \equiv \pdv{}{t} + \vec v_i \cdot \grad +{{< /katex >}} + +{{< katex display >}} +\rightarrow \dv{}{t} \left( \frac{p_i}{n_i ^\gamma} \right) - p_i \dv{}{t} \left( \frac{1}{n_i ^\gamma} \right) + \frac{\gamma p_i}{n_i ^\gamma} \div \vec v_i = \frac{\gamma - 1}{n_i ^\gamma} RHS_i +{{< /katex >}} + +Simplify further by recognizing that carrying out the total derivative gives + +{{< katex display >}} +\dv{}{t} \left( \frac{p_i}{n_i ^\gamma} \right) - \frac{\gamma p_i}{n_i ^{\gamma + 1}} \dv{}{t} n_i + \frac{\gamma p_i}{n_i ^\gamma} \div \vec v_i = \frac{\gamma - 1}{n_i ^\gamma} RHS_i +{{< /katex >}} + +The continuity equation says +{{< katex display >}} +\dv{}{t} n_i = - n_i \div \vec v_i +{{< /katex >}} +Multiply by {{< katex >}} m_i ^{- \gamma} {{< /katex >}} and use {{< katex >}} \rho = n m_i {{< /katex >}} and the resulting ion energy equation is + +{{< katex display >}} +\dv{}{t} \left( \frac{p_i}{\rho ^\gamma} \right) = \frac{\gamma - 1}{\rho ^\gamma} RHS_i +{{< /katex >}} + +We want to get the electron energy equation in the same form. The steps are very similar: + +{{< katex display >}} +\dv{}{t} \left( \frac{p_e}{n_e ^\gamma} \right) + \vec v_e \cdot \grad \left( \frac{p_e}{n_e ^\gamma} \right) = \frac{\gamma - 1}{n_e ^\gamma} RHS_e +{{< /katex >}} + +Since we've defined our center-of-mass reference frame to be that of the ions, we can not use the same total derivative + +{{< katex display >}} +\pdv{}{t} + \vec v_e \cdot \grad \neq \dv{}{t} +{{< /katex >}} + +We now apply the charge neutrality approximation + +{{< katex display >}} +n_e = Z n \qquad \vec v_e = \vec v_i - \frac{1}{Z e n} \vec j +{{< /katex >}} + +{{< katex display >}} +\rightarrow \dv{}{t} \left( \frac{p_e}{Z^\gamma n^\gamma} \right) + \vec v_i \cdot \grad \left( \frac{p_e}{Z^\gamma n^\gamma} \right) = \frac{1}{Z e n} \vec j \cdot \grad \frac{p_e}{Z^\gamma n^\gamma} + \frac{\gamma - 1}{Z^\gamma n^\gamma} RHS_e +{{< /katex >}} + +Multiply by {{< katex >}} (Z / m_i)^\gamma {{< /katex >}} and the result is + +{{< katex display >}} +\dv{}{t} \frac{p_e}{\rho ^\gamma} = \frac{1}{Z e n} \vec j \cdot \grad \frac{p_e}{\rho^\gamma} + \frac{\gamma - 1}{\rho^\gamma} RHS_e +{{< /katex >}} + +Finally we can add the ion energy equation to the electron energy equation to get + +{{< katex display >}} +\dv{}{t} \left( \frac{p}{\rho^\gamma} \right) = \\ +\frac{\gamma - 1}{\rho^\gamma} \left[ Q_{ie} + Q_{ei} - \div (\vec h_i + \vec h_e) - \vec \pi_i \cdot \cdot \grad \vec v_i - \vec \Pi_e \cdot \cdot \grad \vec v_e \right] + \frac{\vec j}{Z e n} \cdot \grad \frac{p_e}{\rho^\gamma} +{{< /katex >}} + +!!! info "MHD Energy Equation" + + {{< katex display >}} + \dv{}{t} \left( \frac{p}{\rho^\gamma} \right) = \frac{\gamma - 1}{\rho^\gamma} \left[ Q_{ie} + Q_{ei} - \div (\vec h_i + \vec h_e) - \vec \pi_i \cdot \cdot \grad \vec v_i - \vec \Pi_e \cdot \cdot \grad \vec v_e \right] + \frac{\vec j}{Z e n} \cdot \grad \frac{p_e}{\rho^\gamma} + {{< /katex >}} + +Obviously, we've retained a number of terms that are specific to the behavior of the electrons. It is possible to incorporate the electron behavior by using a single-fluid MHD model with two temperatures {{< katex >}} T_i \neq T_e {{< /katex >}}. One can imagine a hierarchy of models, in which the most simplified is the single-fluid MHD model in which you evolve {{< katex >}} \rho {{< /katex >}}, {{< katex >}} \vec v {{< /katex >}}, and {{< katex >}} T {{< /katex >}}. Moving up a level, you have a MHD model with two temperatures in which you evolve {{< katex >}} \rho {{< /katex >}}, {{< katex >}} \vec v {{< /katex >}}, {{< katex >}} T_i {{< /katex >}}, and {{< katex >}} T_e {{< /katex >}}. Upwards from there you move back into the realm of multi-fluid models. + +Now to relate the fields back to source terms. The low-frequency Maxwell's equations are + +{{< katex display >}} +\pdv{\vec B}{t} = - \curl \vec E +{{< /katex >}} +{{< katex display >}} +\vec j = \frac{1}{\mu_0} \curl \vec B +{{< /katex >}} +{{< katex display >}} +\div \vec B = \div \vec E = 0 +{{< /katex >}} + +Like in the multi-fluid plasma model, we still need to close the system by expressing some of our variables using equations of state ({{< katex >}} \vec h {{< /katex >}}, {{< katex >}} \vec \Pi {{< /katex >}}). + +To simplify further, we can make some assumptions about heat flow + +{{< katex display >}} +\text{isothermal} \qquad \rightarrow p \propto n \qquad T = \text{const.} \qquad \gamma = 1 +{{< /katex >}} +{{< katex display >}} +\text{adiabatic} \qquad \rightarrow p \propto n^\gamma +{{< /katex >}} +{{< katex display >}} +\text{cols plasma / force-free} \qquad \rightarrow p = \text{const.} \qquad \gamma = 0 +{{< /katex >}} + +## Ideal MHD Model + +The extended MHD equations are simpler than the two-fluid model, but they can still be quite complicated. We can often still get useful analysis from further reductions. The ideal MHD model is such a reduction that we can get by dropping (with justification) several terms from the extended model. We justify the simplifications by comparing the magnitude of the neglected terms to the terms that are retained. + +Recall the characteristic speed is {{< katex >}} v_{T, i} {{< /katex >}}. If we say that the characteristic length plasma length is {{< katex >}} L {{< /katex >}}, then we can define characteristic time {{< katex >}} \tau = L / v_{T, i} {{< /katex >}}. + +The derivation of the two-fluid plasma model assumed a Maxwellian velocity distribution. We need the velocity distribution to thermalize, reach local thermodynamic equilibrium, and become Maxwellian. This means that we need many collisions, in fact so many collisions occurring frequently enough that we can ignore collisional effects. There must then be many collisions during the characteristic time {{< katex >}} \tau {{< /katex >}}. + +For ions to be thermalized, + +{{< katex display >}} +\frac{\tau_{ii}}{\tau} \ll 1 +{{< /katex >}} + +And similarly for electrons + +{{< katex display >}} +\frac{\tau_{e}}{\tau} \ll 1 +{{< /katex >}} + +The continuity equation remains unchanged from the extended MHD model + +!!! info "Ideal MHD Continuity Equation" + + {{< katex display >}} + \pdv{\rho}{t} + \div (\rho \vec v) = 0 + {{< /katex >}} + +The momentum equation is + +{{< katex display >}} +\rho \left( \pdv{\vec v}{t} + \vec v \cdot \grad \vec v \right) + \grad p - \vec j \cross \vec B = - \div (\vec \Pi_e + \vec \Pi_e) +{{< /katex >}} +Drop the anisotropic pressure + +!!! info "Ideal MHD Momentum Equation" + + {{< katex display >}} + \rho \left( \pdv{\vec v}{t} + \vec v \cdot \grad \vec v \right) + \grad p - \vec j \cross \vec B = 0 + {{< /katex >}} + + +The generalized Ohm's law is + +{{< katex display >}} +\vec E + \vec v \cross \vec B = \eta \vec j + \frac{1}{Z e n} \left( \vec j \cross B - \grad p_e - \div \vec \Pi_e \right) +{{< /katex >}} + +We're going to drop the entire right hand side + +!!! info "Ideal MHD Generalized Ohm's Law" + + {{< katex display >}} + \vec E + \vec v \cross \vec B = 0 + {{< /katex >}} + +For the energy equation we have + +{{< katex display >}} +\dv{}{t} \left( \frac{p}{\rho^\gamma} \right) = \frac{\gamma - 1}{\rho^\gamma} [Q_{ie} + Q_{ei} - \div (\vec h_i + \vec h_e) - \vec \Pi_i \cdot \cdot \grad \vec v_i - \vec \Pi_e \cdot \cdot \grad \vec v_e] + \frac{\vec j}{Z e n} \cdot \grad \frac{p_e}{\rho^\gamma} +{{< /katex >}} + +We neglect the entire right-hand side + +!!! info "Ideal MHD Energy Equation" + + {{< katex display >}} + \pdv{\rho}{t} + \div (p \vec v) = (1 - \gamma) p \div \vec v + {{< /katex >}} + +### Collision/Pressure terms + +If we assume that the ions and electrons are in thermal equilibrium{{< katex >}} T_i = T_e {{< /katex >}}, we can relate the collision times + +{{< katex display >}} +\tau_{ee} : \tau_{ii} : \tau_{ei} = 1 : \left( \frac{m_i}{m_e} \right) ^{1/2} : \frac{m_i}{m_e} +{{< /katex >}} + +The collision times are specifically collisional relaxation times of the Boltzmann equation + +{{< katex display >}} +\left. \pdv{f}{t} \right|_{coll} = \frac{f - f_{\text{Maxwellian}}}{\tau_{\alpha \beta}} +{{< /katex >}} + +For electrons, the thermalization condition is much stricter for the ions + +{{< katex display >}} +\frac{\tau_{ee}}{\tau} = \left( \frac{m_e}{m_i} \right)^{1/2} \frac{\tau_{ii}}{\tau} \ll 1 +{{< /katex >}} + +Neglect the anisotropic pressure tensor in the momentum and generalized Ohm's law, {{< katex >}} \div \vec \Pi {{< /katex >}}. {{< katex >}} \vec \Pi {{< /katex >}} is primarily the shear stress tensor. The ion thermal speed gives us a characteristic velocity for the plasma, so we use it to characterize the shear stress + +{{< katex display >}} +\vec \Pi_{i, max} \sim 2 \mu \left( \pdv{u}{x} - \frac{1}{3} \div \vec v \right) \sim \mu \frac{v_{T, i}}{L} +{{< /katex >}} + +Standard treatments of the viscosity (Braginskii, etc.) show that viscosity scales with the number density, temperature, and collision time + +{{< katex display >}} +\mu \sim n T_i \tau_{ii} \sim p_i \tau_{ii} +{{< /katex >}} + +{{< katex display >}} +\Pi_{i, max} \sim p_i \frac{\tau_{ii} v_{T, i}}{L} +{{< /katex >}} + +The specific term we want to get rid of is {{< katex >}} \div \vec \Pi {{< /katex >}}, so let's compare it to a term we want to keep {{< katex >}} \grad P {{< /katex >}} + +{{< katex display >}} +\frac{\div \vec \Pi}{\grad p} \sim \frac{p_i \tau_{ii} v_{T, i} / L^2}{p_i / L} \sim \frac{\tau_{ii}}{\tau} +{{< /katex >}} + +So, to neglect the anisotropic pressure term in the momentum equation, once again we require + +{{< katex display >}} +\frac{\tau_{ii}}{\tau} \ll 1 +{{< /katex >}} + +In other words, as long as the plasma is collision-dominated, we can drop the ion anisotropic pressure term. What about associated the electron term? If you can assume {{< katex >}} T_i \approx T_e {{< /katex >}}. Then {{< katex >}} p_i \approx p_e {{< /katex >}} for a neutral plasma, and + +{{< katex display >}} +\frac{\div \vec \Pi_e}{\grad p} \sim \frac{p_i \tau_{ee} v_{T, i} / L^2}{p_i / L} \sim \frac{\tau_{ee}}{\tau} \ll 1 +{{< /katex >}} + +### Magnetic terms + +In the generalized Ohm's law, the diamagnetic drift term is + +{{< katex display >}} +\frac{\grad p_e}{Z e n} \sim \frac{n T_e / L}{Z en} \sim \frac{T_i / L}{Ze} \sim \frac{m_i v_{T, i}^2}{L Z e} +{{< /katex >}} + +Compare {{< katex >}} \grad p_e / Z e n {{< /katex >}} to a term that we're going to keep, which is the dynamo term {{< katex >}} \vec v \cross \vec B {{< /katex >}} + +{{< katex display >}} +\frac{ \grad p_e / Z en}{|\vec v \cross \vec B|} \sim \frac{m_i v_{T, i}^2}{LZe}{v_{T, i} B} \sim \frac{m_i}{ZeB}\frac{v_{T, i}}{L} = \frac{v_{T, i}}{\omega_{c, i}} \frac{1}{L} \sim \frac{r_{L, i}}{L} \ll 1 +{{< /katex >}} + +So to neglect the diamagnetic drift term, we need the plasma to be well-magnetized. This means the Larmor radius must be much less than the plasma characteristic length {{< katex >}} r_{L, i} \ll L {{< /katex >}} + +Now what can we do with the Hall term {{< katex >}} \frac{\vec j \cross \vec B}{Zen} {{< /katex >}}. For a static plasma (or one with subsonic flows): + +{{< katex display >}} +\rho (\pdv{\vec v}{t} + \vec v \cdot \grad \vec v) - \grad p - \vec j \cross \vec B = 0 +{{< /katex >}} + +so by "subsonic" we mean that the static pressure is much larger than the dynamic pressure and we can discard the {{< katex >}} \vec v {{< /katex >}} terms. + +{{< katex display >}} +\vec j \cross \vec B \approx \grad p +{{< /katex >}} + +Comparing the Hall term to the dynamo term {{< katex >}} \vec v \cross \vec B {{< /katex >}} also gives the same requirement + +{{< katex display >}} +\frac{r_{L, i}}{L} \ll 1 +{{< /katex >}} + +How well do some real plasmas hold up to the requirements of ideal MHD? Consider field nulls in a Z-pinch or an FRC, and weakly magnetized plasmas (such as those in a Hall thruster). Clearly, the magnetization requirement does not hold up at all points in space, and ideal MHD does not necessarily apply across the whole domain. + +### Resistivity + +We neglect resistivity and the resistive electric field {{< katex >}} \eta \vec j {{< /katex >}} in the generalized Ohm's law + +{{< katex display >}} +\eta \vec j \sim \eta \frac{\grad p}{B} \sim \frac{ \eta n T}{LB} +{{< /katex >}} + +{{< katex display >}} +\eta = \frac{m_e \nu_{ei}}{n e^2} = \frac{m_e}{n e^2 \tau_{ei}} = \frac{m_e ^2/ m_i}{n e^2 \tau_{ee}} +{{< /katex >}} + +{{< katex display >}} +\rightarrow \eta \vec j \sim \frac{m_e ^2}{e^2 L B} \frac{v_{T, i} ^2}{\tau_{ee}} +{{< /katex >}} + +How does it compare to {{< katex >}} \vec v \cross \vec B {{< /katex >}} + +{{< katex display >}} +\frac{\eta \vec j}{|\vec v \cross \vec B|} \sim \left(\frac{m_e}{m_i} \right)^2 \frac{m_i}{e ^2 B^2} \frac{v_{T, i} ^2}{L^2} \frac{L}{v_{T, i} \tau_{ee}} \sim \left( \frac{m_e}{m_i}\right)^2 \frac{v_{T, i} ^2}{\omega_{c, i} ^2} \frac{1}{L^2} \sim \left( \frac{m_e}{m_i} \right) ^{3/2} \left( \frac{r_{L, i}}{L} \right) ^{2} \frac{\tau}{\tau_{ii}} \ll 1 +{{< /katex >}} + +This now places a lower limit on {{< katex >}} \tau_{ii} {{< /katex >}}; to neglect resistivity {{< katex >}} \tau_{ii} {{< /katex >}} cannot be too low + +{{< katex display >}} +\left( \frac{m_e}{m_i} \right) ^{3/2} \left( \frac{r_{L, i}}{L} \right) ^{2} \ll \frac{\tau_{ii}}{\tau} \ll 1 +{{< /katex >}} + +### Heating sources + +Neglect the collisional heating sources {{< katex >}} Q_{ei} {{< /katex >}}, {{< katex >}} Q_{ie} {{< /katex >}} in the energy equation. We do that by assuming that they are equal and opposite, which only happens when the temperatures are equal. In other words, we are again assuming local thermodynamic equilibrium _between_ electrons and ions. + +{{< katex display >}} +1 \gg \frac{\tau_{ei}}{\tau} = \left( \frac{m_i}{m_e} \right)^{1/2} \frac{\tau_{ii}}{\tau} \rightarrow \frac{\tau_{ii}}{\tau} \ll \left( \frac{m_e}{m_i} \right) ^{1/2} +{{< /katex >}} + +This is a much more restrictive condition than ion collisionality. Alternatively, we could track two temperature independently. + +We also neglect the heat flux terms {{< katex >}} \div (\vec h_i + \vec h_e) {{< /katex >}} in the energy equation. Consider parallel (to the magnetic field) heat conduction which dominates since {{< katex >}} \kappa_\perp \ll \kappa_\parallel {{< /katex >}}, so + +{{< katex display >}} +\div \vec h \approx \grad_\parallel \cdot (\kappa_\parallel \grad_\parallel T) +{{< /katex >}} + +{{< katex display >}} +\grad_\parallel \cdot \left[ ( \kappa_{\parallel, i} + \kappa_{\parallel, e}) \grad_\parallel T \right] +{{< /katex >}} + +{{< katex display >}} +\kappa_{\parallel, e} \sim \frac{n T_e}{m_e} \tau_{ee} \qquad \text{ and } \qquad \kappa_{\parallel, i} \sim \frac{n T_i}{m_i} \tau_{ii} +{{< /katex >}} + +{{< katex display >}} +\frac{\kappa_{\parallel, e}}{\kappa_{\parallel, i}} \sim \frac{\tau_{ee}}{\tau_{ii}} \frac{m_i}{m_e} \sim \left( \frac{m_i}{m_e} \right) ^{1/2} +{{< /katex >}} + +Compare the thermal conductivity to the rate of pressure change {{< katex >}} \pdv{p}{t} {{< /katex >}} + +{{< katex display >}} +\frac{\grad_\parallel \cdot (\kappa_{\parallel, e} \grad_\parallel T)}{\pdv{p}{t}} \sim \frac{n T^2 \tau_{ee}/m_e L^2}{nT/\tau} \sim \left( \frac{m_i}{m_e} \right)^{1/2} \frac{\tau_{ii}}{\tau} \ll 1 +{{< /katex >}} + +We're back to the same requirement that we have ion-electron local thermodynamic equilibrium. + +### Anisotropic pressure terms in energy equation + +Neglect the anisotropic pressure terms in the energy equation. That is, {{< katex >}} \vec \Pi_i \cdot \cdot \grad \vec v_i {{< /katex >}} and {{< katex >}} \vec \Pi_e \cdot \cdot \grad \vec v_e {{< /katex >}}. Skipping ahead, the result is + +{{< katex display >}} +\frac{\tau_{ii}}{\tau} \ll 1 +{{< /katex >}} +{{< katex display >}} +\left( \frac{m_e}{m_i} \right)^{1/2} \frac{\tau_{ii}}{\tau} \frac{r_{L, i}}{L} \ll 1 +{{< /katex >}} + +### Electron convection term + +Lastly, we neglect the electron convection term in the energy equation {{< katex >}} \frac{\vec j}{Zen} \cdot \grad \frac{P_e}{\rho^\gamma} {{< /katex >}}. The result is + +{{< katex display >}} +\frac{r_{L, i}}{L} \ll 1 +{{< /katex >}} + + +## Conservation Law Form of MHD + +{{< katex display >}} +\pdv{}{t} q + \div \vec f = 0 +{{< /katex >}} + +We can express momentum in conservation law form: + +{{< katex display >}} +\pdv{}{t} (\rho \vec v) + \div \left[ \rho \vec v \vec v - \frac{ \vec B \vec B}{\mu_0} + (p + \frac{B^2}{2 \mu_0})\vec 1 \right] = 0 +{{< /katex >}} + +The conservation law form for the magnetic field looks like + +{{< katex display >}} +\pdv{ \vec B}{t} + \div \left[ \vec v \vec B - \vec B \vec v \right ] = 0 +{{< /katex >}} + +And of course the energy equation for + +{{< katex display >}} +E = \frac{1}{\gamma - 1} p + \frac{1}{2} \rho v^2 + \frac{ B^2}{2 \mu_0} +{{< /katex >}} + +{{< katex display >}} +\pdv{E}{t} + \div \left[ \left( E + p + \frac{B^2}{2 \mu_0} \right) - \left( \frac{ \vec B \cdot \vec v}{\mu_0} \right) \vec B \right] = 0 +{{< /katex >}} + +Conservation law forms are particularly useful when considering equilibrium steady-state force balance. This means that in steady-state equilibrium we have + +{{< katex display >}} +\div \left[ \rho \vec v \vec v - \frac{ \vec B \vec B}{\mu_0} + \left( p + \frac{B^2}{2 \mu_0} \right) \vec 1 \right] = 0 +{{< /katex >}} + +We can use this relationship and integrate over various volumes to determine the relationship between the various force balance terms + +### Examples of equilibrium plasma confinement + +For a plasma in static equilibrium {{< katex >}} \vec v = 0 \rightarrow \vec j \cross \vec B = \grad p {{< /katex >}} + +{{< katex display >}} + \div \vec T = 0 = \div \left( \right) + {{< /katex >}} + +{{< katex display >}} +\vec j \cross \vec B = \frac{1}{\mu_0} (\curl \vec B) \cross \vec B = \frac{1}{\mu_0} (\vec B \cdot \grad \vec B - \frac{1}{2} \grad B^2) \\ += \frac{1}{\mu_0} ( B^2 \vu e_B \cdot \grad \vu e_B + \frac{1}{2} \vu e_B \vu e_B \cdot \grad B^2 - \frac{\grad B^2}{2} ) +{{< /katex >}} + +where {{< katex >}} \vu e_B = \vec B / B {{< /katex >}} + +{{< katex >}} \vu e_B \cdot \grad \vu e_B {{< /katex >}} is the curvature of {{< katex >}} \vec B {{< /katex >}}. Write it like a curvature + +{{< katex display >}} +\vec K = - \vu r / R_c +{{< /katex >}} + +{{< katex display >}} +\vu e_B \vu e_B \cdot \grad B^2 +{{< /katex >}} +is gradient of {{< katex >}} B^2 {{< /katex >}} that is parallel to B. Multiply by {{< katex >}} e_B {{< /katex >}} gives the component of gradient along {{< katex >}} e_B {{< /katex >}}. The difference between that and {{< katex >}} \grad B^2 / 2 {{< /katex >}} gives you the perpendicular gradient + +{{< katex display >}} +\vec j \cross \vec B = \frac{1}{\mu_0} (B^2 \vec \kappa - \frac{1}{2} \grad _\perp B^2) = \grad p = \grad_\perp p +{{< /katex >}} + +identify {{< katex >}} B^2 \vec \kappa {{< /katex >}} is magnetic tension resulting from having a bent magnetic field line. {{< katex >}} \frac{1}{2} \grad_\perp B^2 {{< /katex >}} is magnetic pressure. They have to balance the plasma pressure at equilibrium. + +For example, consider a cylindrical plasma that's in equilibrium with a helical magnetic field +{{< katex display >}} +\vec B = B_\theta (r) \vu \theta + B_z (r) \vu z +{{< /katex >}} + +How is plasma pressure profile determined by the different components of the magnetic field? If we want to maximize the amount of pressure we confine, what should be maximized/minimized? + +{{< katex display >}} +\frac{B^2}{\mu_0} \vu \kappa = \grad _\perp ( p + \frac{B^2}{2 \mu_0} ) +{{< /katex >}} + +{{< katex >}} B_z {{< /katex >}} is straight and has no curvature, so the only magnetic tension comes from {{< katex >}} B_\theta {{< /katex >}}, so the magnetic tension from {{< katex >}} B_\theta {{< /katex >}} must balance the total pressure. + +

Figure 12.6

+ +The role of {{< katex >}} B_z {{< /katex >}} is displacing plasma pressure. The utility in defining {{< katex >}} \beta {{< /katex >}} as + +{{< katex display >}} +\beta = \frac{\text{plasma pressure}}{\text{magnetic pressure}} +{{< /katex >}} + +### Conditions of Ideal MHD Validity + +The conditions for ideal MHD to be valid are + +1. High Ion Collisionality: {{< katex >}} \frac{\tau_{ii}}{\tau} \ll 1 {{< /katex >}} +2. Small ion Larmor radius: {{< katex >}} \frac{r_{L, i}}{L} \ll 1 {{< /katex >}} +3. Low resistivity: {{< katex >}} \left(\frac{m_e}{m_i} \right)^{3/2} \left( \frac{r_{L, i}}{L} \right)^2 \ll 1 {{< /katex >}} + +For a given plasma in force balance, we can relate the plasma pressure to the magnetic pressure + +{{< katex display >}} +\beta = \frac{n T}{B^2 / 2 \mu_0} = 4 \times 10^{-16} \frac{n_{cm^{-3}} T_{keV}}{B^2 _{T}} +{{< /katex >}} + +Ion collision time is (Spitzer collisionality) + +{{< katex display >}} +\tau_{ii} = 2.09 \times 10^{7} \frac{T_{eV} ^{3/2} \mu ^{1/2}}{\ln \Lambda n_{cm ^{-3}}} \left[\text{s}\right] +{{< /katex >}} +where {{< katex >}} \mu \equiv m_i / m_p {{< /katex >}}. + +Putting the conditions for ideal MHD in terms of {{< katex >}} \beta {{< /katex >}} and {{< katex >}} \tau_{ii} {{< /katex >}}, + +1. High collisionality: +{{< katex display >}} +\frac{\tau_{ii}}{\tau} = 2.14 \times 10^{12} \frac{T^2}{n L} \ll 1 +{{< /katex >}} +{{< katex display >}} +\rightarrow T_{eV} \ll 6.8 \times 10^{-7} L_{cm} ^{1/2} n_{cm^{-3}} ^{1/2} +{{< /katex >}} +2. Small gyroradius: +{{< katex display >}} +\frac{r_{L, i}}{L} = 2.3 \times 10^7 \frac{1}{Z L} \sqrt{ \frac{\mu B}{n}} \ll 1 +{{< /katex >}} +{{< katex display >}} +\rightarrow n_{cm^{-3}} \gg 5.3 \times 10^{14} \frac{\mu B_{T}}{Z^2 L_{cm} ^2} +{{< /katex >}} +3. Low resistivity: +{{< katex display >}} +\left( \frac{m_e}{m_i} \right) ^{3/2} \left( \frac{r_{L, i}}{L} \right) ^2 \frac{\tau}{\tau_{ii}} = 5.65 \frac{\mu B}{Z^2 L T^2} +{{< /katex >}} +{{< katex display >}} +\rightarrow T_{eV} \gg 2.4 \frac{\mu ^{3/2} \beta^{1/2}}{Z L _{cm} ^{1.2}} +{{< /katex >}} + +

Figure 12.7

+ + +## Perpendicular MHD + +For most configurations for magnetic fusion confinement, we are able to satisfy 2. and 3. but often have densities much too low to meet the high collisionality constraint. However, in practice ideal MHD does accurately model macroscopic behavior of many plasmas. At the same time, magnetized / fusion plasmas are often largely collisionless. We can understand why by re-writing the collisionality requirement as + +{{< katex display >}} +1 \gg \frac{\tau_{ii}}{\tau} = \frac{\tau_{ii} v_{T, i}}{\tau v_{T, i}} \sim \frac{\lambda}{L} +{{< /katex >}} + +where {{< katex >}} \lambda {{< /katex >}} is the mean free path. The ratio {{< katex >}} \lambda / L {{< /katex >}} is also known as the **Knudsen number**. In magnetized plasmas, the mean free path is often very long, but the path between collisions can cover a great distance only by following magnetic field lines. Motion {{< katex >}} \perp {{< /katex >}} to the magnetic field is constrained by {{< katex >}} r_{L, i} {{< /katex >}}. This suggests an approach wherein we divide the plasma model into a 1-D kinetic model to be solved along the magnetic field and a 2-D MHD model {{< katex >}} \perp {{< /katex >}} to the magnetic field. + +We consider diffusivity (terms like viscosity, conductivity) parallel and perpendicular to {{< katex >}} \vec B {{< /katex >}} +{{< katex display >}} +k_\parallel \sim \frac{\lambda ^2}{\tau_{ii}} +{{< /katex >}} +{{< katex display >}} +k_\perp \sim \frac{r_{L, i} ^2}{\tau_{ii}} +{{< /katex >}} +{{< katex display >}} +\frac{k_\parallel}{k_\perp} \sim \left( \frac{\lambda}{r_{L, i}} \right)^2 \sim (\omega_{c, i} \tau_{ii} )^2 +{{< /katex >}} + +This changes one of our conditions for validity. Specifically, we can write + +{{< katex display >}} +\frac{r_{L, i}}{L} \frac{1}{ \omega_{c, i} \tau_{i i}} = 254 \frac{\mu \beta}{Z^2 L_{cm} T_{eV} ^2} +{{< /katex >}} + +or + +{{< katex display >}} +T_{eV} \gg 16 \frac{\mu ^{1/2} \beta^{1/2}}{Z L^{1/2}} +{{< /katex >}} + +This is now only slightly more restrictive than the low-resistivity condition. In fact, most of the plasmas that we looked at before (large tokamaks & toruses, propulsion systems) which had temperatures above the validity range now fall comfortably within the region of validity. The perpendicular MHD model is equivalent to a collisionless model, giving a much wider applicability than the collisional MHD model. + +What about the parallel component? Our collisionality condition still isn't valid parallel to the field. Basically, we ignore the parallel component, which is the same as assuming that {{< katex >}} \rho {{< /katex >}}, {{< katex >}} T {{< /katex >}}, {{< katex >}} p {{< /katex >}} are constant along magnetic field lines, with {{< katex >}} \div \vec v = 0 {{< /katex >}} . Let's write out the expressions for collision-less MHD and for ideal MHD and compare: + +| | Collisionless MHD | Ideal MHD | +| --- | --- | --- | +| Continuity | {{< katex >}} \pdv{\rho}{t} = 0 {{< /katex >}} | {{< katex >}} \pdv{\rho}{t} = - \rho \div \vec v {{< /katex >}} | +| Momentum | {{< katex >}} \rho \dv{\vec v_\perp}{t} = \vec j \cross \vec B - \grad_\perp p {{< /katex >}} | {{< katex >}} \rho \dv{\vec v}{t} = \vec j \cross \vec B - \grad p {{< /katex >}} | +| Parallel constraint | {{< katex >}} \vec B \cdot \grad \frac{v_\parallel}{B} = - \div \vec v _\perp {{< /katex >}} | | +| Energy | {{< katex >}} \dv{p}{t} = 0 {{< /katex >}} | {{< katex >}} \dv{p}{t} = - \gamma p \div \vec v {{< /katex >}} | + +Collisionless MHD reproduces many of the effects of ideal MHD but has a wider region of validity. Corollary: ideal MHD is accurate beyond its region of validity, unless results lead to parallel gradients. For example, we know that MHD is not valid when representing confinement of a plasma confined in a magnetic mirror, which is an inherently kinetic phenomenon. But ideal MHD _can_ generate parallel gradients within its region of validity, and we need to be careful. Ideal MHD does not require different models {{< katex >}} \parallel {{< /katex >}} and {{< katex >}} \perp {{< /katex >}} to the magnetic field, and is therefore preferred. We will continue to use ideal MHD outside of its region of validity. diff --git a/content/notes/UWAA558/06-boundary-conditions.md b/content/notes/UWAA558/06-boundary-conditions.md new file mode 100644 index 00000000..695d86d6 --- /dev/null +++ b/content/notes/UWAA558/06-boundary-conditions.md @@ -0,0 +1,153 @@ +--- +title: Boundary Conditions +bookToc: false +weight: 60 +--- + + +# Boundary Conditions + +Mathematically, a well-posed problem requires both governing equations and a complete set of boundary conditions (the Cauchy data for the problem). The most common boundary conditions we use are perfectly conducting walls (flux surfaces) or a vacuum region. + +### Perfectly Conducting Wall + +For the case where the plasma extends out to a perfectly conducting (impermeable) wall. Perfectly conducting walls do not support tangential electric field: + +{{< katex display >}} +\left. \vec E_t \right|_{wall} = 0 \quad \rightarrow \quad \left. \vu n \cross \vec E \right| _{wall} = 0 +{{< /katex >}} + +Applying Faraday's law at the wall, + +{{< katex display >}} +\left. \vu n \cdot \pdv{\vec B}{t} \right|_{wall} = \left. - \vu n \cdot \curl \vec E \right|_{wall} = \left. \div (\vu n \cross \vec E) \right| _{wall} = 0 +{{< /katex >}} +{{< katex display >}} +\pdv{}{t} \vu n \cdot \vec B |_{wall} = 0 +{{< /katex >}} +If initially there is no normal magnetic field, then +{{< katex display >}} +\vu n \cdot \vec B|_{wall} = 0 \quad \text{if initially true} +{{< /katex >}} + +And of course, for an impermeable wall, + +{{< katex display >}} +\vu n \cdot \vec v |_{wall} = 0 +{{< /katex >}} + +Is this a sufficient set of boundary conditions? Think back to the governing equations in conservation form +{{< katex display >}} +\pdv{}{t} \vec Q + \div \vec F = 0 +{{< /katex >}} +The boundary conditions come into play when defining {{< katex >}} \vec F {{< /katex >}} at the boundary. In particular, we need to know what {{< katex >}} \dd \vec S \cdot \vec F |_{wall} {{< /katex >}} is. In our governing equations, this will involve {{< katex >}} \vec E {{< /katex >}}, {{< katex >}} \vec B {{< /katex >}}, and {{< katex >}} \vec v {{< /katex >}}. + +### Insulating Boundary + +As a slight modification, an insulating boundary can have a tangential electric field. Consider a simple geometry of parallel electrodes with an insulating wall between them. + +

Figure 12.8

+ +From Ohm's law +{{< katex display >}} +\vec E + \vec v \cross \vec B = 0 +{{< /katex >}} +so the only way an electric field tangential to the wall can exist is if {{< katex >}} \vu n \cdot \vec v \neq 0 {{< /katex >}}. + +For either a perfectly conducting or an insulating boundary, the other variables are arbitrary: {{< katex >}} \rho {{< /katex >}} , {{< katex >}} p {{< /katex >}}, {{< katex >}} \vec v_t {{< /katex >}}, {{< katex >}} \vec B_t {{< /katex >}}. + +### Vacuum Region + +The plasma (radius {{< katex >}} R_p {{< /katex >}}) is supported by a region of vacuum out to a perfectly conducting wall at some radius {{< katex >}} R_w {{< /katex >}}. We assume that there is no plasma in the vacuum region. The governing equations in vacuum are just Maxwell's equations + +{{< katex display >}} +\curl \vec B_{vac} = 0 \qquad \text{and} \qquad \div \vec B_{vac} = 0 +{{< /katex >}} + +At the wall, +{{< katex display >}} +\vu n \cross \vec E |_{wall} = 0 +{{< /katex >}} +{{< katex display >}} +\left. \vu n \cdot \pdv{\vec B}{t} \right|_{wall} = 0 +{{< /katex >}} + +What happens at the plasma-vacuum interface? We need to specify jump conditions and continuity conditions. Let's use square brackets to signify a jump: + +{{< katex display >}} +\left[ X \right] = \left. X \right|_{R_p + dr} - \left. X \right|_{R_p - dr} +{{< /katex >}} + +The normal magnetic field has to be continuous. + +{{< katex display >}} +[\vu n \cdot \vec B]_{R_p} = 0 +{{< /katex >}} + +The tangential magnetic field jump is given by the surface current density at the jump + +{{< katex display >}} +\left[ \vu n \cross \vec B \right] _{R_p} = \mu_0 \vec K +{{< /katex >}} + +Integrating {{< katex >}} \grad_\perp (p + \frac{B^2}{2 \mu_0}) = \frac{B^2}{\mu_0} \vec \kappa {{< /katex >}} over a differential volume across the surface gives + +{{< katex display >}} +\left[ p + \frac{B^2}{2 \mu_0} \right] _{R_p} = 0 +{{< /katex >}} + +The plasma shape is determined self-consistently by the wall shape and surface current. This is a free-boundary problem. Another option is to specify the plasma shape, and then determine the required wall shape. This is a fixed-boundary problem. + +The most realistic case includes externally applied magnetic fields coming from source coils, perhaps computed by Biot-Savart law. The vacuum magnetic field is then {{< katex >}} \vec B_{vac} = \vec B_{ext} + \vec B_{plasma} {{< /katex >}}. The crazy coil shapes in the stellarator design come from the 3D geometry computations solving this problem. + +### Conservation of Magnetic Flux ("Frozen-In" Flux) + +Locally, {{< katex >}} \vec E + \vec v \cross \vec B = 0 {{< /katex >}} with Faraday's law +{{< katex display >}} +\pdv{B}{t} = - \curl \vec E = - B \div \vec v + \vec B \cdot \grad \vec v - \vec v \cdot \grad B +{{< /katex >}} +From the continuity equation, +{{< katex display >}} +\pdv{\rho}{t} + \vec v \cdot \grad \rho = - \rho \div \vec v +{{< /katex >}} +Combining we find that +{{< katex display >}} +\dv{\vec B}{t} = \frac{\vec B}{\rho} \dv{\rho}{t} + \vec B \cdot \grad \vec v +{{< /katex >}} +{{< katex display >}} +\rightarrow \dv{}{t} \left( \frac{\vec B}{\rho} \right) = \frac{\vec B}{\rho} \cdot \grad \vec v +{{< /katex >}} + +This says that the field and plasma density move together. Locally, if the magnetic field increases then mass density increases, such that the ratio {{< katex >}} \vec B / \rho {{< /katex >}} remains constant. In the direction parallel to the magnetic field we have a term that involves field line twisting, which is a bit more complicated, but in the perpendicular direction +{{< katex display >}} +\dv{}{t} \left( \frac{\vec B}{\rho} \right) _\perp = 0 +{{< /katex >}} + +If we consider globally the magnetic flux through a moving surface S at velocity {{< katex >}} \vec u {{< /katex >}}. The magnetic flux penetrating the surface is +{{< katex display >}} +\Psi = \int \vec B \cdot \dd \vec S +{{< /katex >}} +or +{{< katex display >}} +\dv{\Psi}{t} = \int \dv{\vec B}{t} \cdot \vu n \dd S +{{< /katex >}} +{{< katex display >}} += \int \pdv{\vec B}{t}\cdot \vu n \dd S + \oint \vec B \cross \vec u \dd \vec l +{{< /katex >}} +Using Faraday's law + +{{< katex display >}} +\dv{\Psi}{t} = \int - \curl \vec E \cdot \vu n \dd S + \oint \vec B \cross \vec u \cdot \dd \vec l +{{< /katex >}} +{{< katex display >}} += \oint (- \vec E + \vec B \cross \vec u) \cdot \dd \vec l +{{< /katex >}} +Using the electric field from Ohm's law +{{< katex display >}} +\dv{\Psi}{t} = \oint(\vec v - \vec u) \cross \vec B \cdot \dd \vec l +{{< /katex >}} +This tells us that if the surface moves with the plasma {{< katex >}} \vec u = \vec v {{< /katex >}} then +{{< katex display >}} +\dv{\Psi}{t} = 0 +{{< /katex >}} +the flux through the surface is constant, and the flux is a constant of the topology. This is a direct consequence of ideal MHD. If we add even a small amount of resistivity, we dramatically alter the results in a process called "tearing" where the magnetic field "tears" and reconnects with itself. \ No newline at end of file diff --git a/content/notes/UWAA558/07-equilibrium-for-fusion.md b/content/notes/UWAA558/07-equilibrium-for-fusion.md new file mode 100644 index 00000000..50e5900d --- /dev/null +++ b/content/notes/UWAA558/07-equilibrium-for-fusion.md @@ -0,0 +1,242 @@ +--- +title: Equilibrium for Fusion +bookToc: false +weight: 70 +--- + + +# Equilibrium for Fusion ({{< katex >}} \beta {{< /katex >}}) + +For a fusion device we would like to determine a magnetic configuration that confines plasma while it fuses. At fusion temperatures, the power required to maintain the equilibrium will be substantial. For a device to be useful, the power required to sustain the equilibrium must be less than the power released from fusion. Important loss terms for a confined plasma are transport (thermal conduction primarily) and radiation terms. The scaling factors are {{< katex >}} P_{Brem} \sim n^2 T^{1/2} {{< /katex >}} and {{< katex >}} P_{cycl} \sim n^2 T^2 {{< /katex >}} for radiation, and {{< katex >}} P_L \sim \frac{3nT}{\tau_E} {{< /katex >}} for thermal losses. + +We know that the fusion source term will primarily come from the DT fusion reaction + +{{< katex display >}} +\text{D} + \text{T} \rightarrow \text{He}^4 (3.5\, MeV) + \text{n} (14.1\, MeV) +{{< /katex >}} + +The primary fusion reaction releases an {{< katex >}} \alpha {{< /katex >}}-particle and a high-energy neutron. The concept of ignition is that the neutron leaves the plasma, and the {{< katex >}} \alpha {{< /katex >}} (with energy {{< katex >}} E_\alpha = 3.5 MeV) {{< /katex >}} remains to heat the plasma. + +{{< katex display >}} +P_\alpha = \frac{1}{4} n^2 \langle \sigma v \rangle E_\alpha \qquad \text{(assuming} \quad n_D = n_T = n/2 \text{)} +{{< /katex >}} + + +{{< katex display >}} +P_\alpha > P_L \quad \rightarrow \quad n \tau_E > \frac{12 T}{E_\alpha \langle \sigma v \rangle} +{{< /katex >}} + +To sustain fusion, we set the fusion heating term above the thermal loss term. The reaction cross-section {{< katex >}} \sigma {{< /katex >}} can be maximized to give the Lawson criterion + +{{< katex display >}} +n \tau_E > 10^{14} s / cm^3 +{{< /katex >}} + +The Lawson criterion only applies at fusion temperatures, but it is a useful parameter even outside of ignition since it gives a ratio of fusion power to lost power + +| {{< katex >}} T_i (keV) {{< /katex >}} | {{< katex >}} \langle \sigma v \rangle (cm^3 / s) {{< /katex >}} | Required {{< katex >}} n \tau_E (s / cm^3) {{< /katex >}} | +| --- | --- | ---| +| 1 | {{< katex >}} 7 \cdot 10^{-21} {{< /katex >}} | {{< katex >}} 5 \cdot 10^{17} {{< /katex >}} | +| 5 | {{< katex >}} 1.4 \cdot 10^{-17} {{< /katex >}} | {{< katex >}} 1.2 \cdot 10^{15} {{< /katex >}} | +| 20 | {{< katex >}} 4.3 \cdot 10^{-16} {{< /katex >}} | {{< katex >}} 1.6 \cdot 10^{14} {{< /katex >}} | +| 60 | {{< katex >}} 8.7 \cdot 10^{-16} {{< /katex >}} | {{< katex >}} 2.4 \cdot 10^{14} {{< /katex >}} | + +We can see that the required {{< katex >}} n \tau_E {{< /katex >}} actually has a minimum around {{< katex >}} 20 keV {{< /katex >}} (at least, as far as the data in the table goes). Even though the maximum cross-section is at a much higher temperature, what we're really concerned with is the ratio of the fusion source term to the thermal loss term, which is linear in temperature. + +MHD equilibrium does not place a limit on the density {{< katex >}} n {{< /katex >}}. Instead, it places a limit on {{< katex >}} \beta {{< /katex >}} in order to achieve equilibrium force-balance {{< katex >}} (\beta = 1) {{< /katex >}} + +{{< katex display >}} +\beta = \frac{n (T_e + T_i)}{B^2 / 2 \mu_0} \rightarrow n = \frac{ \beta B^2}{4 \mu_0 T} +{{< /katex >}} + +In this form, we can more clearly see what our options are to achieve MHD equilibrium. Some devices (large-scale tokamaks) are able to achieve the requisite confinement time at a low {{< katex >}} \beta {{< /katex >}} by making use of very strong magnetic fields. Other devices are able to make use of more modest magnetic fields by working at a higher {{< katex >}} \beta {{< /katex >}}. + +Therefore, + +{{< katex display >}} +\tau_E > \frac{1}{\beta B^2} \frac{48 \mu_0}{E_\alpha} \frac{T^2}{\langle \sigma v \rangle} +{{< /katex >}} + +The term {{< katex >}} \frac{T^2}{\langle \sigma v \rangle} {{< /katex >}} has a minimum at {{< katex >}} 10-20 keV {{< /katex >}}. At 15 keV and a magnetic field of {{< katex >}} 5T {{< /katex >}} (many actual components cannot reasonably exceed such magnetic fields) then + +{{< katex display >}} +\tau_E > \frac{0.1}{\beta} \text{s} +{{< /katex >}} + +For a large-scale toroidal device with {{< katex >}} \beta = 1\% {{< /katex >}}, the confinement time {{< katex >}} \tau_E > 10s {{< /katex >}}. If we consider a common diffusivity (how fast energy will leave due to thermal conductivity) {{< katex >}} D_E \approx 1 m^2 / s{{< /katex >}}, so for a characteristic radius {{< katex >}} a {{< /katex >}} + +{{< katex display >}} +\tau_E \approx \frac{a^2}{4 D_E} \rightarrow a > 6.3 \text{m} +{{< /katex >}} + +This gives you a sense of why low-{{< katex >}} \beta {{< /katex >}} devices need to be so large. Instead, if we consider {{< katex >}} \beta \sim 50\% {{< /katex >}}, {{< katex >}} \tau_E > 0.2 \text{s} {{< /katex >}} and + +{{< katex display >}} +\beta \sim 50\% \rightarrow a > 0.9 \text{m} +{{< /katex >}} + +When you consider that the cost of a device (to first order) scales with the volume of the device, achieving a high {{< katex >}} \beta {{< /katex >}} is very important for fusion equilibrium. However, when we consider MHD stability we are generally forced into lower {{< katex >}} \beta {{< /katex >}} to avoid destructive instabilities. Configuration optimization is the process of balancing this trade-off. + +## Virial Theorem + +Application of the virial theorem to energy balance for the stress tensor {{< katex >}} \vec T {{< /katex >}} tells us that MHD equilibria must be supported by externally supplied currents. Many times you'll hear of theoretical designs for compact toroid devices which can maintain stability under their own currents, but they are the MHD stability equivalent of a perpetual motion machine. A compact toroid cannot exist unsupported. + +Writing static equilibrium: + +{{< katex display >}} +\div \left[ - \frac{ \vec B \vec B}{\mu_0} + \left( p + \frac{B^2}{2 \mu_0}\right) \vec I \right] = \div \vec T = 0 +{{< /katex >}} + +If we define the direction of the magnetic field to be {{< katex >}} \vu e _B = \vu z {{< /katex >}} then + +{{< katex display >}} +\vec T = p_\perp ( \vec I - \vu e_B \vu e_B ) + p_\parallel \vu e_B \vu e_B \\ += \begin{bmatrix} p_\perp & 0 & 0 \\ 0 & p_\perp & 0 \\ 0 & 0 & p_\parallel \end{bmatrix} +{{< /katex >}} +where + +{{< katex display >}} +p_\perp = p + \frac{B^2}{2 \mu_0} +{{< /katex >}} + +and +{{< katex display >}} +p_\parallel = p - \frac{B^2}{2 \mu_0} +{{< /katex >}} + +A gradient vector identity gives + +{{< katex display >}} +\div (\vec r \cdot \vec T) = \vec r \cdot ( \div \vec T) + \vec T \cdot \cdot \grad \vec r +{{< /katex >}} + +Integrating this expression over a volume and assuming that the volume contains a confined MHD equilibrium that is self-contained and self-supported: + +

Figure 12.10

+ +{{< katex display >}} +\int_V \div ( \vec r \cdot \vec T) \dd V = \int_V (\vec r \cdot \overbrace{\cancel{(\div \vec T)}}^{\text{MHD equil.}} + \vec T \cdot \cdot \grad \vec r) \dd V +{{< /katex >}} + +{{< katex display >}} +\grad \vec r = \vec I +{{< /katex >}} + +so +{{< katex display >}} +\vec T \cdot \cdot \grad \vec r = p_\perp + p_\perp + p_\parallel \\ += 3p + \frac{B^2}{2 \mu_0} +{{< /katex >}} + +{{< katex display >}} +\int_V (3p + \frac{B^2}{2 \mu_0} ) \dd V = \int _V \div ( \vec r \cdot \vec T) \dd V = \oint _S (\vec r \cdot \vec T) \cdot \vu n \dd S \\ += \oint _S \left[ \vec r \cdot \vec I p_\perp + \vec r \cdot \vu e_B \vu e_B (p_\parallel - p_\perp) \right]\cdot \vu n \dd S \\ +=\oint \left[ \left( \cancel{p} + \frac{B^2}{2 \mu_0} \right) \vu r \cdot \vu n - \frac{B^2}{\mu_0} (\vec r \cdot \vu e_B)(\vu e_B \cdot \vu n) \right] \dd S +{{< /katex >}} + +Beyond where the plasma is contained, the pressure does not contribute {{< katex >}} p = 0 {{< /katex >}}. If all current sources are contained in the configuration, the magnetic field {{< katex >}} \sim 1/r^3 {{< /katex >}} for a dipole, {{< katex >}} \sim 1/r^4 {{< /katex >}} for a quadrupole, etc. Therefore the right-hand side will fall off like + +{{< katex display >}} +RHS \propto \oint_S B^2 r \dd S \propto \left( \frac{1}{r^3} \right) ^2 r r^2 \propto \frac{1}{r^3} \text{(dipole)} +{{< /katex >}} +so {{< katex >}} RHS \rightarrow 0 {{< /katex >}} as {{< katex >}} r \rightarrow \infty {{< /katex >}}. But what about the left-hand side? Both of the terms in the volume integral are positive definite, so the LHS must be positive finite and the equality can't possibly hold. The assumption that the plasma is self-contained must be invalid. This tells us that we must have external currents. + +## Magnetic Flux Surfaces + +The vast difference in thermal conductivity parallel and perpendicular to the magnetic field in a plasma confinement configuration leads to an avoidance of any open field lines. Magnetic equilibria are generally toroidal to eliminate end losses from open configurations. In general fusion confinement devices, magnetic field lines lie on a set of closed nested toroidal surfaces. This means that we can no longer describe any equilibria in a solely 1D geometry. The minor radius is no longer the only important scale length. + +From {{< katex >}} \vec j \cross \vec B = \grad p {{< /katex >}}, we know that the pressure gradient is perpendicular to {{< katex >}} \vec j {{< /katex >}} and {{< katex >}} \vec B {{< /katex >}}, and therefore both {{< katex >}} \vec B {{< /katex >}} and {{< katex >}} \vec j {{< /katex >}} lie on surfaces of uniform pressure. We call these toroidal surfaces either magnetic surfaces or **flux surfaces**. We can use these surfaces to build a 1-dimensional description. + +

Figure 12.11

+ +As a brief aside, some geometrical vocabulary will be useful when describing toroidal geometry. A toroid is any surface of revolution with a hole in the middle. A torus is the particular case of a toroid in which the revolved figure is a circle. + +We will define our global toroidal coordinate system to consist of the major axis {{< katex >}} (z) {{< /katex >}}, the distance from the major axis {{< katex >}} (R) {{< /katex >}}, and the azimuthal angle around the major axis {{< katex >}} (\phi) {{< /katex >}}. + +

Figure 12.13

+ +We will also make use of a poloidal coordinate system measured by minor radius (distance from the minor axis) {{< katex >}} r {{< /katex >}} and the poloidal angle from the minor axis {{< katex >}} \theta {{< /katex >}}. We will generally refer to a point on the torus relative to the major axis {{< katex >}} (R, z, \phi) {{< /katex >}}, or relative to the minor axis {{< katex >}} (r, \theta, \phi) {{< /katex >}}, or in spherical coordinates {{< katex >}} (R, \theta, \phi) {{< /katex >}}. The major radius {{< katex >}} R_0 {{< /katex >}} is the distance from the major axis to the minor axis. The minor radius {{< katex >}} a {{< /katex >}} is the characteristic distance from the minor axis to the exterior of the revolved figure. Usually we will find symmetry under {{< katex >}} \phi {{< /katex >}}. + +The aspect ratio of a torus is the ratio of the major radius to the minor radius. + +{{< katex display >}} +A = \frac{R_0}{a} +{{< /katex >}} + +When we talk about a "toroidal surface," we mean a cross-section of the toroidal rotation. When we talk about a "poloidal surface" we mean a surface which is coplanar with the minor axis: + +

Figure 12.14

+ +The poloidal flux is determined by the size of the poloidal surface and the poloidal magnetic field: + +{{< katex display >}} +\Psi _p = \int_{S_p} \vec B \cdot \dd \vec S +{{< /katex >}} + +and the toroidal flux is determined by the size of a toroidal surface and the toroidal magnetic field: + +{{< katex display >}} +\Psi_t = \int_{S_t} \vec B \cdot \dd \vec S +{{< /katex >}} + +Considering the poloidal flux, we can see that if we expand the size of the surface towards the minor radius, the flux will increase until eventually we come to a point where the flux begins to decrease. The position of this maximum is called the **magnetic axis**, which does not necessarily correspond to the minor axis. In fact, it is generally displaced from the minor axis. + +

Figure 12.15

+ +To refer back to something more familiar, we'll define the same terms for a cylindrical geometry {{< katex >}}(r, \theta, z) {{< /katex >}}. An axial surface corresponds with a toroidal surface, and an azimuthal surface corresponds with a poloidal surface: + +

Figure 12.16

+ +If we consider the trajectory of a single field line, what sorts of surfaces will it trace out? What surface will contain the field line? As it turns out, there are three options: + +1. Rational surface - the field line closes on itself, and it does so after a finite number of revolutions. One way to quickly visualize such a surface is to draw a Poincaré puncture plot. Choose a toroidal plane and plot a point wherever the field line punctures the surface. A Poincaré puncture plot of a rational surface contains a finite number of points and no continuous curves. +2. Ergodic surface - the field line completely covers an entire surface, which is to say the field line punctures any toroidal surface an infinite number of times. In other words, it never closes on itself and defines an irrational curve. +3. Stochastic region - In this case, there is no definite surface and the field line fills a volume. + +

Figure 12.17

+ +Generally rational surfaces and ergodic surfaces are largely equivalent, but by introducing a small amount of resistivity a rational surface can lead to magnetic islands. One can imagine the addition of resistivity equivalent to allowing a small degree of motion of the magnetic field lines. In an ergodic surface, a flux surface is defined by a single (irrational) field line. If it moves toward itself in one location it will necessarily move away from itself in another location. But in a rational surface, different field lines can lie on the same constant pressure surface and will tend to move towards each other. By concentrating into magnetic islands, the flux surfaces are now more closely spaced, and the pressure gradient increases (a bad thing!) + +

Figure 12.18

+ +**Surface quantities**: Since pressure, current (not current density!), and _flux_ (not field!) are constant along a flux surface, it is convenient to use flux {{< katex >}} \Psi_p {{< /katex >}} as a coordinate. A particular poloidal flux itself uniquely determines a poloidal surface with constant pressure and current. The flux surface quantities are {{< katex >}} p, \, \Psi_p, \, \Psi_t, \, I_p = \int_{S_p} \vec j \cdot \dd \vec S, \, I_t = \int_{S_t} \vec j \cdot \dd \vec S{{< /katex >}}. + +Surface quantities are not independent. The poloidal current {{< katex >}} I_p {{< /katex >}} affects the toroidal field {{< katex >}} B_t {{< /katex >}} and toroidal flux {{< katex >}} \Psi_t {{< /katex >}}. The toroidal current affects the poloidal field {{< katex >}} B_p {{< /katex >}} and poloidal flux {{< katex >}} \Psi_p {{< /katex >}}. + +{{< katex display >}} +B_t = \vec B \cdot \vu \phi +{{< /katex >}} + +{{< katex display >}} +\vec B_p = B_\theta \vu \theta + B_z \vu z \\ += B_r \vu R + B_z \vu z +{{< /katex >}} + +## Toroidal Force Balance + + +Toroidal equilibria solves the end losses of linear equilibrium, but generates a new force which must be balanced. This is a result of the virial theorem. + +### Poloidal Fields (Tire Tube Pressure Force) + +If you think of a flexible bike tire being inflated, as the pressure within the inner tube increases, the major radius will increase! Why is this? The pressure within the tire is isotropic. As we pump up the tire, the pressure increase causes a force imbalance with the atmospheric pressure that causes the tube to expand. To simplify, we can consider a square tube with inner "radius" {{< katex >}} a {{< /katex >}}. The radial force scales with the pressure over the outer and inner surfaces + +{{< katex display >}} +F_r = p S_{outer} - p S_{inner} \\ +\sim p (R_0 + a) a - p (R_0 - a) a \sim 2 a^2 p +{{< /katex >}} + +!!! error "Missing content here on how to balance radial forces for purely poloidal fields (axisymmetric) with a conducting wall" + + +Alternatively, we can use vertical external field coils to increase the field strength near the outer wall. They will also tend to center the plasma between the coils, so they have the advantage of preventing the plasma from drifting upwards or downwards + +

Figure 12.19

+ +### Toroidal Fields + +Now let's consider driving a toroidal field in the plasma using a central coil on the major axis. Driving a toroidal field {{< katex >}} B_\phi {{< /katex >}} will also create a poloidal current {{< katex >}} j_\theta {{< /katex >}}. We know that the driven field will decay as {{< katex >}} 1/R {{< /katex >}} through the plasma. In actuality, the relationship is not perfectly {{< katex >}} 1/R {{< /katex >}} because of the generated poloidal current, which will modify the field within the plasma. Depending on the orientation of {{< katex >}} j_\theta {{< /katex >}}, it could either increase or decrease the poloidal field within the plasma. These are two different operations by which the plasma interacts with the magnetic field. When the induced plasma current tends to increase the field, we have a paramagnetic effect. In the opposite case (decrease) there is a diamagnetic effect. Generally, plasmas are diamagnetic, but there are certain situations where they become paramagnetic. + +

Figure 12.20

+ +Here again, we have a larger field on the in-bore side than on the out-bore side, so there will be a force imbalance tending to push the plasma towards larger radii. We might consider surrounding the plasma with a conducting wall, as we did previously, but we run into a difficulty determining a current distribution in the wall which would balance the radial effect. Any current distribution must be circular; because of the geometry of a torus such a current distribution would have the same force contribution on both the inside and outside edges. There is no way to stabilize the distribution with purely toroidal fields. diff --git a/content/notes/UWAA558/08-1d-equilibria.md b/content/notes/UWAA558/08-1d-equilibria.md new file mode 100644 index 00000000..cbd68a2b --- /dev/null +++ b/content/notes/UWAA558/08-1d-equilibria.md @@ -0,0 +1,397 @@ +--- +title: 1-D Equilibria +bookToc: false +weight: 80 +--- + + +# 1-Dimensional Equilibria + +## The {{< katex >}} \theta {{< /katex >}}-pinch + +In a {{< katex >}} \theta {{< /katex >}} pinch, we have an applied axial field generated by a driven azimuthal current distribution. The way these usually work is that you begin with a plasma generated by some pre-ionization process and zero field. Then you crank up the current to drive an azimuthal current in the plasma (in the opposite direction as the external current). + +{{< katex display >}} +\vec j \cross \vec B = \grad p \quad \rightarrow \quad j_\theta B_z = \dv{p}{r} +{{< /katex >}} + +{{< katex display >}} +j_\theta = - \frac{1}{\mu_0} \dv{B_z}{r} +{{< /katex >}} + +{{< katex display >}} +\dv{p}{r} = - \frac{1}{\mu_0} B_z \dv{B_z}{r} = - \dv{}{r} \left( \frac{B_z ^2}{2 \mu_0} \right) +{{< /katex >}} +{{< katex display >}} +p + \frac{B_z ^2}{2 \mu_0} = \text{constant} = \frac{B_0 ^2}{2 \mu_0} +{{< /katex >}} + +

Figure 12.21

+ +At equilibrium, the magnetic pressure balances the plasma pressure. If we say that the pressure is + +{{< katex display >}} +p = p_0 e^{- r^2 / a^2} +{{< /katex >}} + +with {{< katex >}} p_0 {{< /katex >}} the pressure on-axis, then we can solve for the axial field + +{{< katex display >}} +B_z = B_0 (1 - B_0 e^{-r^2/a^2})^{1/2} +{{< /katex >}} + +We can define the peak {{< katex >}} \beta {{< /katex >}} to be the ratio of the on-axis pressure to the maximum magnetic field + +{{< katex display >}} +\beta_0 = \frac{p_0}{B_0 ^2 / 2 \mu_0} +{{< /katex >}} + +By definition, the peak {{< katex >}} \beta {{< /katex >}} will always be {{< katex >}} \leq 1 {{< /katex >}}. We can define {{< katex >}} \langle \beta \rangle {{< /katex >}} + +{{< katex display >}} +\langle \beta \rangle = \frac{ \langle p \rangle }{B_a ^2 / 2 \mu_0} +{{< /katex >}} + +where {{< katex >}} B_a {{< /katex >}} is a characteristic field value, typically taken to be at the plasma edge. + +{{< katex display >}} +\langle \beta \rangle = \frac{2 \mu_0}{B_0 ^2 \pi a^2} \int _0 ^a 2 \pi r p \, \dd r +{{< /katex >}} +{{< katex display >}} += \frac{2}{a^2} \int_0 ^a \frac{ rp}{B_0 ^2/2 \mu_0} \dd r = \frac{2}{a^2} \int_0 ^a \left( 1 - \frac{B_z ^2}{B_0 ^2} \right) r \dd r +{{< /katex >}} + +In this form, we can see that because {{< katex >}} B_z {{< /katex >}} will everywhere be less than {{< katex >}} B_0 {{< /katex >}}, we can increase {{< katex >}} \langle \beta \rangle {{< /katex >}} by driving {{< katex >}} B_z {{< /katex >}} as low as possible. In this particular example, {{< katex >}} \langle \beta \rangle / \beta_0 = 63\% {{< /katex >}}. + +## Z-pinch + +In the case of a Z-pinch, we only have an applied axial current. + +{{< katex display >}} +\vec j = j_z (r) \vu z +{{< /katex >}} + +For force balance + +{{< katex display >}} +\vec j \cross \vec B = \grad p \quad \rightarrow \quad - j_z B_\theta = \dv{p}{r} +{{< /katex >}} +{{< katex display >}} +j_z = \frac{1}{\mu_0} \frac{1}{r} \dv{}{r} ( r B_\theta) +{{< /katex >}} + +{{< katex display >}} +- \dv{p}{r} = \frac{B_\theta}{\mu_0 r} \dv{}{r} ( r B_\theta) +{{< /katex >}} + +If we find it convenient we can separate this into a magnetic pressure term + +{{< katex display >}} +- \dv{}{r} \left( p + \frac{B_\theta ^2}{2 \mu_0} \right) = \frac{B_\theta ^2}{\mu_0 r} +{{< /katex >}} + +### Bennett Profile + +An example of an achievable distribution is the Bennett profile, which has a diffuse form + +{{< katex display >}} +B_\theta = \frac{\mu_0 I}{2 \pi} \frac{r}{r^2 + a^2} +{{< /katex >}} +{{< katex display >}} +j_z = \frac{I}{\pi} \frac{a^2}{(r^2 + a^2) ^2} +{{< /katex >}} + +{{< katex display >}} +p = \frac{\mu_0 I^2}{8 \pi ^2} \frac{a^2}{(r^2 + a^2)^2} +{{< /katex >}} + +Interestingly, {{< katex >}} j \propto p {{< /katex >}}. For a uniform temperature, {{< katex >}} j \propto n {{< /katex >}}. Since current density is the product of {{< katex >}} \vec v {{< /katex >}} and {{< katex >}} n {{< /katex >}}, this says that we have a uniform drift velocity and all particles are drifting with the same velocity at all points along the profile. If we consider what the equilibrium profile looks like for a Bennett profile: + +

Figure 12.22

+ +So for {{< katex >}} r < a {{< /katex >}} we have magnetic tension and pressure which balance the plasma pressure. For {{< katex >}} r \geq a {{< /katex >}} we have magnetic tension which balances both plasma pressure and magnetic pressure. + +The Z-pinch {{< katex >}} \langle \beta \rangle {{< /katex >}} + +{{< katex display >}} +\langle \beta \rangle \equiv \frac{ \langle p \rangle}{B_a ^2 / 2 \mu_0} \\ += \frac{2 \mu_0}{B_a ^2 \pi a^2} \int_0 ^a 2 \pi r p \dd r +{{< /katex >}} + +If we multiply the force balance by {{< katex >}} r^2 {{< /katex >}} and integrate + +{{< katex display >}} +\int_0 ^a r^2 \dv{p}{r} \dd r + \frac{1}{\mu_0} \int_0 ^a r B_\theta \dv{}{r} (r B_\theta) \dd r = 0 \\ +0 = \left[ r^2 p \right] _0 ^a - \int_0 ^a p \dd (r^2) + \left[ \frac{(r B_\theta)^2}{2 \mu_0} \right] _0 ^a +{{< /katex >}} + +If we have a discrete pinch such that {{< katex >}} p(a) = 0 {{< /katex >}} then the first term vanishes. + +{{< katex display >}} +\int_0 ^a 2 r p \dd r = \frac{(a B_a)^2}{2 \mu_0} +{{< /katex >}} + +If we substitute our definition of {{< katex >}} \langle \beta \rangle {{< /katex >}}, we find {{< katex >}} \langle \beta \rangle = 1 {{< /katex >}}. For a diffuse pinch in which {{< katex >}} p(a) \neq 0 {{< /katex >}} we end up with {{< katex >}} \langle \beta \rangle \leq 1 {{< /katex >}} and we have a wall-supported plasma. Ideal confinement ({{< katex >}} \langle \beta \rangle = 1 {{< /katex >}}) is a very nice property and is what makes the Z-pinch configuration so interesting. + + + +## Stability Considerations + +Instability results if there exists a plasma displacement that leads to a lower energy state. There are several ways to provide stability in the context of MHD. The two most common are **magnetic shear** and **magnetic well**. + +### Magnetic Shear + +In ideal MHD, magnetic field lines can not break or tear. Let's consider some flux surface containing field lines {{< katex >}} \vec B_3 {{< /katex >}}. Behind it, we have another flux surface containing field lines {{< katex >}} B_2 {{< /katex >}} which are not parallel to {{< katex >}} \vec B_3 {{< /katex >}}, and the same for {{< katex >}} \vec B_1 {{< /katex >}}. + +

Figure 12.23

+ +Because the field lines are a different angles to each other, these flux surfaces can _not_ interpenetrate. In other words, if the flux surface pressures are {{< katex >}} P_1 > P_2 > P_3 {{< /katex >}}, we can maintain the pressure gradient and prevent the flux surfaces from moving each other. What prevents the surfaces from achieving a lower energy state is the magnetic shear between flux surfaces. + +Without shear, the surfaces can interpenetrate and exchange positions. In the case of a toroidal geometry, magnetic shear is defined by the rotational transform {{< katex >}} \iota {{< /katex >}}, or by the safety factor + +{{< katex display >}} +q = \frac{2 \pi}{\iota} +{{< /katex >}} + +Generally speaking, {{< katex >}} q {{< /katex >}} is generally referenced for tokamaks and {{< katex >}} \iota {{< /katex >}} is referenced for stellarators. Another way of picturing the safety factor in a toroidal geometry is + +{{< katex display >}} +q \equiv \frac{\text{no. of windings long way}}{\text{no. of windings short way}} \\ += \frac{ \dv{\psi_t}{V}}{\dv{\phi_p}{V}} = \dv{\phi_t}{\phi_p} \\ += \frac{n}{m} = \frac{\text{toroidal transits}}{\text{poloidal transits}} +{{< /katex >}} + +In a cylindrical (1D) geometry it is just + +{{< katex display >}} +q = \frac{\text{longitudinal transits}}{\text{azimuthal transits}} +{{< /katex >}} + +Let's calculate the safety factor for a toroidal geometry: + +

Figure 12.24

+ +{{< katex display >}} +\dv{\phi_p}{V} = \frac{ B_\theta 2 \pi R \dd r}{2 \pi R_0 2 \pi r \dd r} \\ += \frac{ B_\theta}{2 \pi r} \frac{R}{R_0} +{{< /katex >}} + +{{< katex display >}} +\dv{\phi_t}{V} = \frac{B_\phi 2 \pi r \dd r}{2 \pi R_0 2 \pi r \dd r} \\ += \frac{B_\phi}{2 \pi R_0} +{{< /katex >}} + +{{< katex display >}} +q = \frac{r B_\phi}{R B_\theta} +{{< /katex >}} + +In a cylindrical geometry the analysis is even simpler + +{{< katex display >}} +q = 2 \pi \frac{r B_z}{L B_\theta} +{{< /katex >}} + +As a note, it would appear that {{< katex >}} q \rightarrow 0 {{< /katex >}} at the magnetic axis as {{< katex >}} r \rightarrow 0 {{< /katex >}}, but in general {{< katex >}} B_\theta \rightarrow 0 {{< /katex >}} as well, and the safety factor is generally bounded at {{< katex >}} r \rightarrow 0 {{< /katex >}} + +{{< katex >}} q {{< /katex >}} is a flux surface quantity. + +We care about magnetic shear. How does that relate to the safety factor? Magnetic shear is defined as + +{{< katex display >}} +s \equiv 2 \frac{\dd q / q}{\dd V /V} = 2 \dv{\ln (q)}{\ln(V)} +{{< /katex >}} + +Even a uniform {{< katex >}} B_z {{< /katex >}} or {{< katex >}} B_\theta {{< /katex >}} produces a finite magnetic shear because of the way that {{< katex >}} r {{< /katex >}} and {{< katex >}} B_\theta {{< /katex >}} change. The safety factor is often considered synonymous with magnetic shear, and often we don't even compute {{< katex >}} s {{< /katex >}}. + +Shear is generally a stabilizing effect. Interchange between flux surfaces can be prevented/inhibited by shear, or by making it energetically unfavorable. Shear tends to stabilize current-driven instabilities. + +### Magnetic Well + +As before, we can consider two adjacent flux surfaces {{< katex >}} B_1, P_1 {{< /katex >}} and {{< katex >}} B_2, P_2 {{< /katex >}}. If {{< katex >}} B_2 > B_1 {{< /katex >}} and {{< katex >}} P_2 > P_1 {{< /katex >}}, the interchange is energetically favorable. But if {{< katex >}} B_2 > B_1 {{< /katex >}} and {{< katex >}} P_2 < P_1 {{< /katex >}} then the interchange may be unfavorable without any magnetic shear. + +Consider a plasma confined by an externally applied magnetic field generated by a coil {{< katex >}} I {{< /katex >}} + +

Figure 12.25

+ +On the left side, the magnetic field gradient is in the same direction as the plasma pressure gradient, which is a destabilizing configuration. Flux surfaces are able to interchange easily, and the magnetic field is described as having bad curvature. On the right side, the gradients are in the same direction and the magnetic field has a good curvature. + +We can define the "wellness" {{< katex >}} W {{< /katex >}} as + +{{< katex display >}} +W \equiv \frac{ \text{total pressure change relative to mag. pressure}}{\text{relative volume change}} \\ += \frac{\dd \langle p + B^2/2 \mu_0 \rangle / \langle B^2/2 \mu_0 \rangle}{\dd V / V} +{{< /katex >}} +where the angle brackets indicate a quantity integrated along a field line +{{< katex display >}} +\langle Q \rangle \equiv \frac{\int_0 ^L \frac{ Q \dd l}{|B|}}{\int_0 ^l \frac{\dd l}{|B|}} +{{< /katex >}} + +For a stabilizing effect, the wellness must be greater than 0. This means that the magnetic pressure must increase faster than the pressure decreases to prevent pressure-driven instabilities. + +Since {{< katex >}} W {{< /katex >}} is evaluated along a field line, it is also a surface quantity. + +### Application to 1D Equilibria + +{{< katex >}} \theta {{< /katex >}}-pinch: Since {{< katex >}} B_\theta = 0 {{< /katex >}}, {{< katex >}} q \rightarrow \infty {{< /katex >}}, which really just means {{< katex >}} q {{< /katex >}} is not well defined for a {{< katex >}} \theta {{< /katex >}}-pinch. If we consider some small {{< katex >}} \delta B_\theta {{< /katex >}}, we get a very large {{< katex >}} q {{< /katex >}}. From a magnetic shear perspective, a {{< katex >}} \theta {{< /katex >}}-pinch has very large values of shear and very good stability properties. + +The wellness is + +{{< katex display >}} +W = \frac{V}{\langle B^2 \rangle} \dv{}{V} \langle 2 \mu_0 p + B^2 \rangle \\ + = \frac{\pi r^2 L}{B_z ^2} \frac{1}{2 \pi r L} \dv{}{r} (2 \mu_0 p + B_z ^2) \\ + = \frac{\mu_0 r}{B_z ^2} \dv{}{r} \left( p + \frac{B_z ^2}{2 \mu_0} \right) = 0 +{{< /katex >}} +so a {{< katex >}} \theta {{< /katex >}}-pinch has neutral magnetic well. + +Vacuum case: + +{{< katex display >}} +W = \frac{\mu_0 r}{ B_z ^2} \dv{}{r} \left( \frac{ B_z ^2}{2 \mu_0} \right) = 0 +{{< /katex >}} +So vacuum magnetic fields also have neutral wellness. This leads to a general result sometimes referred to as "a plasma cannot dig its own well." In other words, by introducing plasma to a magnetic configuration, it cannot make the configuration more stable than it was. Plasmas make stability more challenging, not less. + +Z-pinch: + +Since {{< katex >}} B_z = 0 {{< /katex >}}, {{< katex >}} q = 0 {{< /katex >}} and there is no magnetic shear. Even for a small value of {{< katex >}} \delta B_z {{< /katex >}} you still get a small {{< katex >}} q {{< /katex >}}. The magnetic well properties of a Z-pinch are + +{{< katex display >}} +W = \frac{V}{\langle B^2 \rangle} \dv{}{V} \langle 2 \mu_0 p + B^2 \rangle \\ += \frac{\mu_0 r}{B_\theta ^2} \dv{}{r} \left( p + \frac{B_\theta ^2}{2 \mu_0} \right) \\ += \frac{\mu_0 r}{B_\theta ^2} \left( - \frac{ B_\theta ^2}{ \mu_0 r} \right) = -1 +{{< /katex >}} + +Recall that {{< katex >}} W > 0 {{< /katex >}} for stability, so the Z-pinch has negative magnetic well and provides no pressure stability. + +In summary, + + - Both {{< katex >}} \theta {{< /katex >}}- and Z-pinch have high {{< katex >}} \beta {{< /katex >}} + - {{< katex >}} \theta {{< /katex >}}-pinch is stable + - Z-pinch is unstable + - End losses in a {{< katex >}} \theta {{< /katex >}} pinch enormous since {{< katex >}} k_\parallel \gg k_\perp {{< /katex >}} + +## Screw Pinch + +A natural extension is to combine a moderate toroidal field and a moderate poloidal field to produce a screw pinch configuration. + +

Figure 12.26

+ +{{< katex display >}} +\curl \vec B = \mu_0 \vec j +{{< /katex >}} +{{< katex display >}} +\rightarrow j_\theta = - \frac{1}{\mu_0 } \dv{B_z}{r} +{{< /katex >}} +{{< katex display >}} +j_z = \frac{1}{\mu_0 r} \dv{}{r} (r B_\theta) +{{< /katex >}} + +For static MHD equilibrium + +{{< katex display >}} +\vec j \cross \vec B = \grad p +{{< /katex >}} +{{< katex display >}} +\dv{}{r} \left( p + \frac{ B_\theta ^2 + B_z ^2}{2 \mu_0} \right) = - \frac{B_\theta ^2}{\mu_0 r} +{{< /katex >}} + +We can define a toroidal {{< katex >}} \beta {{< /katex >}} where {{< katex >}} B_0 = B_z (a) {{< /katex >}} +{{< katex display >}} +\beta_t = \frac{\langle p \rangle }{B_0 ^2 / 2 \mu_0} \\ += \frac{2 \mu_0}{ B_0 ^2} \frac{1}{\pi a^2} \int _0 ^a 2 \pi r p \dd r +{{< /katex >}} + +and in the poloidal direction with {{< katex >}} B_{\theta, a} = B_\theta (a) = \frac{\mu_0 I}{2 \pi a} {{< /katex >}} + +{{< katex display >}} +\beta_p = \frac{\langle p \rangle}{B_{\theta, a} ^2 / 2 \mu_0} \\ += \frac{8 \pi ^2 a^2}{ \mu_0 I_0 ^2} \left( \frac{1}{\pi a^2} \int _0 ^a 2 \pi r p \dd r \right) \\ += \frac{16 \pi ^2}{\mu_0 I_0 ^2} \int_0 ^a r p \dd r +{{< /katex >}} + +To proceed we can multiply the force balance by {{< katex >}} r^2 {{< /katex >}} and integrate + +{{< katex display >}} +\underbrace{\int_0 ^a r^2 \pdv{p}{r} \dd r}_{(1)} + \underbrace{\int _0 ^a \pdv{}{r} \left( \frac{ B_\theta ^2 + B_z ^2 }{2 \mu_0} \right) \dd r}_{(2)} + \underbrace{\int_0 ^a r^2 \frac{B_\theta ^2}{\mu_0 r} \dd r}_{(3)} = 0 +{{< /katex >}} + +{{< katex display >}} +(1) = \int_0 ^a r^2 \dd p = \left. r^2 p \right|_0 ^a - \int_0 ^a p \dd (r^2) = - \int_0 ^a 2 r p \dd r +{{< /katex >}} + +{{< katex display >}} +(2) = \int_0 ^a r^2 \dd \left( \frac{B_\theta ^2 + B_z ^2}{2 \mu_0 } \right) \\ += \left. r^2 \left( \frac{ B_\theta ^2 + B_z ^2}{2 \mu_0 } \right) \right|_0 ^a - \int_0 ^a r \left( \frac{ B_\theta ^2 + B_z ^2}{\mu_0} \right) \dd r +{{< /katex >}} + +{{< katex display >}} +(3) = \int_0 ^a r \frac{B_\theta ^2}{\mu_0} \dd r +{{< /katex >}} + +Combining we have + +{{< katex display >}} +- \int_0 ^a 2 r p \dd r + \frac{a^2 B_{\theta, a}^2}{2 \mu_0} + \overbrace{\frac{a^2 B_0 ^2}{2 \mu_0}}^{B_0 = B_z(r = a)} - \int_0 ^a r \frac{B_z ^2}{\mu_0} \dd r = 0 \\ +- \int_0 ^a 2 r p \dd r + \frac{ \mu_0 I_0 ^2}{8 \pi ^2} + \int_0 ^a r \left( \frac{ B_\theta ^2 - B_z ^2}{\mu_0} \right) \dd r = 0 +{{< /katex >}} + +Dividing {{< katex >}} \int_0 ^a 2 r p \dd r {{< /katex >}} gives + +{{< katex display >}} +\left[ \frac{16 \pi^2}{\mu_0 I^2} \int_0 ^a r p \dd r \right] ^{-1} + \left[\frac{4 \mu_0}{B_0 ^2 a^2} \int_0 ^a r p \dd r \right] ^{-1} \frac{2}{a^2} \int_0 ^a \left(1 - \frac{B_z ^2}{B_0 ^2} \right) r \dd r = 1 +{{< /katex >}} +or + +{{< katex display >}} +\frac{1}{\beta_p} + \frac{\alpha_t}{\beta_t} = 1 +{{< /katex >}} +where +{{< katex display >}} +\alpha_t = \frac{2}{a^2} \int_0 ^a \left( 1 - \frac{B_z^2}{B_0 ^2} \right) r \dd r +{{< /katex >}} +is the diamagnetism. + +{{< katex display >}} +\beta _p = \left( 1 - \frac{\alpha_t}{\beta_t} \right) ^{-1} +{{< /katex >}} + +If we have a diamagnetic current, then {{< katex >}} \alpha > 0 {{< /katex >}}. This maximizes confinement, since we have confinement in the azimuthal field, as well as the axial field. The limit where you have a skin current such that {{< katex >}} B_z = 0 {{< /katex >}} inside the plasma results in the best confinement and {{< katex >}} \alpha _t = 1 {{< /katex >}}. + +Looking at the safety factor, + +{{< katex display >}} +q = \frac{2 \pi r B_z}{L B_\theta} +{{< /katex >}} + +If we look at the edge {{< katex >}} r = a {{< /katex >}}, +{{< katex display >}} +q_a = \frac{2 \pi a B_0}{L B_{\theta, a}} +{{< /katex >}} + +As it turns out, this value of the edge safety factor is critically important, and for stability we require that {{< katex >}} q_a > 1 {{< /katex >}}. + +The magnetic shear of a screw pinch is + +{{< katex display >}} +s = 2 \frac{V}{q} \dv{q}{V} \\ +V = \pi r^2 L \\ +\dd V = 2 \pi L r \dd r = \frac{2 V}{r} \dd r \\ +s = \frac{r}{q} \dv{q}{r} +{{< /katex >}} + +The shear can be adjusted by changing the applied axial field. + +The magnetic well is + +{{< katex display >}} +W = \frac{V}{B^2} \dv{(2 \mu_0 p + B^2)}{V} \\ += \frac{\mu_0 r}{B^2} \dv{}{r} \left( p + \frac{B^2}{2 \mu_0} \right) \\ += \frac{\mu_0 r}{B^2} \left( - \frac{B_\theta ^2}{\mu_0 r} \right) \\ += - \frac{B_\theta ^2}{B^2} \\ += - \frac{B_\theta ^2}{B_\theta ^2 + B_z ^2} \\ += - \left( 1 + \frac{B_z ^2}{B_\theta ^2} \right) ^{-1} +{{< /katex >}} + +So the well is always less than zero, but adding {{< katex >}} B_z {{< /katex >}} improves the well. + +By combining the properties of {{< katex >}} \theta {{< /katex >}}-pinch and Z-pinch, we are able to sacrifice some {{< katex >}} \beta {{< /katex >}} to achieve better stability properties. Of course, we have not addressed the end losses in any way; to do that, we need to connect the ends. + diff --git a/content/notes/UWAA558/09-2d-equilibria.md b/content/notes/UWAA558/09-2d-equilibria.md new file mode 100644 index 00000000..8bdb184c --- /dev/null +++ b/content/notes/UWAA558/09-2d-equilibria.md @@ -0,0 +1,185 @@ +--- +title: 2D Equilibria +bookToc: false +weight: 90 +--- + + +# 2D Equilibria + +Let's connect the ends of our 1D equilibria. Doing so is what gives us inherently toroidal configurations. From the 1-dimensional picture: + +

Figure 12.27

+ +{{< katex display >}} +\vec j \cross \vec B = j_\theta B_z - j_z B_\theta \\ += \grad p = \dv{p}{r} +{{< /katex >}} + +we move to an axisymmetric 2-dimensional torus, replacing our cylindrical coordinate system with a toroidal one + +

Figure 12.28

+ +{{< katex display >}} +\vec j \cross \vec B = \vec j_\theta \cross \vec B_\phi + \vec j_\phi \cross \vec B_\theta = \grad p +{{< /katex >}} + +Eventually, the toroidal force balance will lead to the Grad-Shafranov Equation, which tells us how we can solve for a general equilibrium that solves {{< katex >}} \vec j \cross \vec B = \grad p {{< /katex >}}. + +Let's consider how we might achieve such a configuration. A toroidal magnetic field can be achieved by driving current through a poloidal coil. A more complicated problem is how to drive toroidal current. In general this is done by means of a transformer, where the plasma itself is the secondary circuit. Driving a time-varying current through the primary induces a toroidal current through the plasma. This is called a transformer drive for current. + +## Grad-Shafranov equation + +Computing {{< katex >}} j_\theta {{< /katex >}} and {{< katex >}} B_\phi {{< /katex >}} can be computationally difficult in a toroidal geometry, so let's do some work towards simplifying our force balance expression. The **toroidal magnetic vector potential** is defined as + +{{< katex display >}} +\vec B_\theta = \curl \vec A_\phi +{{< /katex >}} + +If we integrate {{< katex >}} B_\theta {{< /katex >}} over a poloidal surface, Stokes' theorem gives + +{{< katex display >}} +\int _{S_p} \curl \vec A_\phi \cdot \dd \vec S = \oint \vec A_\phi \cdot \dd \vec l \\ += \int _{S_p} B_\theta \cdot \dd \vec S = \Psi _p +{{< /katex >}} + +If the equilibrium is axisymmetric, {{< katex >}} A_\phi {{< /katex >}} must be uniform along {{< katex >}} \dd l {{< /katex >}}, so + +{{< katex display >}} +A_\phi \vu \phi \cdot \oint \dd \vec l = A_\phi 2 \pi R = \Psi _p \\ +\rightarrow A_\phi = \frac{ \Psi_p}{R} \vu \phi +{{< /katex >}} + +where we absorb the factor of {{< katex >}} 2 \pi {{< /katex >}} into the poloidal flux {{< katex >}} \Psi _p {{< /katex >}}. After some manipulation, we can relate {{< katex >}} B_\theta {{< /katex >}} to the poloidal flux + +{{< katex display >}} +\vec B_\theta = \curl \vec A_\phi = - \frac{ \vu R}{R} \pdv{\Psi}{z} + \frac{\vu z}{R} \pdv{\Psi}{R} +{{< /katex >}} + +{{< katex display >}} +\mu_0 j_\phi \cross B_\theta = (\curl \vec B_\theta) \cross \vec B_\theta \\ += \curl \vec B_\theta \cross \left( \grad \Psi \cross \frac{\vu \phi}{R} \right) \\ += - \left[ \pdv{}{R} \left( \frac{1}{R} \pdv{\Psi}{R} \right) + \frac{1}{R} \pdv{\Psi ^2}{z^2} \right] \cdot \left[\frac{ \grad \Psi}{R} ( \vu \phi \cdot \vu \phi) - \frac{ \phi}{R} \cancel{(\grad \Psi \cdot \vu \phi)} \right] +{{< /katex >}} + +That gives the first component of {{< katex >}} \grad p {{< /katex >}}, now let's do the other one + +{{< katex display >}} +\mu_0 \vec j_\theta \cross \vec B_\phi = ( \curl \vec B_\phi) \cross \vec B_\phi \\ + = \left[ - \vu R \pdv{B_\phi}{z} + \vu z \frac{1}{R} \pdv{}{R} ( R B_\phi) \right] \cross \vec B_\phi \\ + = - \frac{B_\phi}{R} \left[ \vu R \pdv{}{R} (R B_\phi) + \vu z \pdv{}{z} (R B_\phi) \right] \\ + = - \frac{B_\phi}{R} \grad (R B_\phi) +{{< /katex >}} + +Finally, since pressure is a flux surface quantity we can write + +{{< katex display >}} +\grad p = \dv{p}{\Psi} \grad \Psi = p' \grad \Psi +{{< /katex >}} + +The toroidal force balance now looks like + +{{< katex display >}} +\mu_0 p' \grad \Psi = - \frac{1}{R} \left( \pdv{}{R} \frac{1}{R} \pdv{\Psi}{R} + \frac{1}{R} \pdv{^2 \Psi}{z^2} \right) \grad \Psi - \frac{B_\phi}{R} \grad(R B_\phi) +{{< /katex >}} + +We notice that the only vector quantities here are {{< katex >}} \grad \Psi {{< /katex >}} and {{< katex >}} \grad (R B_\phi) {{< /katex >}}, so {{< katex >}} \grad (R B_\phi) {{< /katex >}} must be parallel to {{< katex >}} \grad \Psi {{< /katex >}} and is a flux surface quantity. We can define our new flux surface quantity as + +{{< katex display >}} +F(\Phi) \equiv R B_\phi = \frac{\mu_0 I_\theta}{2 \pi} = \frac{\mu_0}{2 \pi} \int_{S_p} \vec j_\theta \cdot \dd \vec S +{{< /katex >}} + +{{< katex display >}} +\grad F = \dv{F}{\Psi} \grad \Psi = F' \grad \Psi +{{< /katex >}} + +Now each term in the toroidal force balance has a factor of {{< katex >}} \grad \Psi {{< /katex >}} attached. Let's multiply through by {{< katex >}} R^2 {{< /katex >}} and factor out the gradient to arrive at the **Grad-Shafranov equation**: + +{{< katex display >}} +R^2 \mu_0 p' = - \Delta ^\star \Psi - F F' +{{< /katex >}} + +where +{{< katex display >}} +\Delta ^\star \equiv R \pdv{}{R} \frac{1}{R} \pdv{}{R} + \pdv{^2}{z^2} +{{< /katex >}} + +To solve the Grad-Shafranov equation, you solve for {{< katex >}} \Psi(R, z) {{< /katex >}}, which determines {{< katex >}} p(\Psi) {{< /katex >}} and {{< katex >}} F(\Psi) {{< /katex >}}, which directly gives you {{< katex >}} p(R, z) {{< /katex >}} and {{< katex >}} F(R, z) {{< /katex >}} and completely defines the equilibrium. + +You can solve for the other terms as well. Since {{< katex >}} \vec B_\theta = \frac{\grad \Psi}{R} \cross \vu \phi {{< /katex >}} + +{{< katex display >}} +\vec j_\phi = - \frac{1}{\mu_0 R} \Delta ^\star \Psi \vu \phi +{{< /katex >}} + +and since {{< katex >}} \vec B_\phi = \frac{F}{R} \vu \phi {{< /katex >}} + +{{< katex display >}} +\vec j_\theta = - \frac{1}{\mu_0 R} \grad (R B_\phi) \cross \vu \phi +{{< /katex >}} + +For the G-S equation to be solvable, you need to specify the equilibrium by specifying {{< katex >}} p(\Phi) {{< /katex >}} and {{< katex >}} F(\Phi) {{< /katex >}}. In practice, this is usually done by making experimental measurements to determine {{< katex >}} p {{< /katex >}} and {{< katex >}} F {{< /katex >}}. A common code that does this is called EFIT, which takes the boundary conditions of the magnetic field and measurements of temperature, density to perform a least-squares fit to solve the G-S equation. + +In general, the Grad-Shafranov equation leads to a matrix equation + +{{< katex display >}} +\overline \vec A \vec \Psi + \vec f(\Psi) = \vec g +{{< /katex >}} + +Depending on the conditions we place on {{< katex >}} \Psi {{< /katex >}}, {{< katex >}} \vec f(\Psi) {{< /katex >}} can be a nonlinear function. + +## Solutions to the Grad-Shafranov equation + +In the limit that {{< katex >}} \vec j \parallel \vec B {{< /katex >}}, then {{< katex >}} \vec j \cross \vec B = 0 = \grad p \rightarrow p' = 0 {{< /katex >}}. These are called force-free states. In the G-S equation, the pressure term vanishes and we're left with + +{{< katex display >}} +\Delta ^\star \Psi + F F' = 0 +{{< /katex >}} + +Spheromaks and RFPs are examples of nearly force-free states in which the current is nearly parallel to the magnetic field. Notice that in completely force-free states, {{< katex >}} \langle \beta \rangle = 0 {{< /katex >}}. + +Another interesting limit is the case where {{< katex >}} F F' \gg \Delta ^\star \Psi {{< /katex >}}. Now we have +{{< katex display >}} +\grad p \approx \vec j_\theta \cross \vec B_\phi +{{< /katex >}} +which looks like a {{< katex >}} \theta {{< /katex >}}-pinch which has been connected at the ends. Remember from the previous section that we can not maintain radial force balance with purely toroidal fields, so the toroidal current is not zero (hence the {{< katex >}} \approx {{< /katex >}}) but is just high enough to maintain radial force balance. This sort of configuration is called a high-{{< katex >}} \beta {{< /katex >}} tokamak. + +The other limit is {{< katex >}} F F' \ll \Delta ^\star \Psi {{< /katex >}} +{{< katex display >}} +\grad p \approx \vec j_\phi \cross \vec B_\theta +{{< /katex >}} +which looks like an end-connected z-pinch. This configuration is usually called an Ohmically heated Tokamak, and the majority of currently operating tokamaks operate this way. As we know, a purely poloidal field has very bad stability properties, so {{< katex >}} \vec B_\phi {{< /katex >}} needs to be added to provide stability. The toroidal {{< katex >}} \beta {{< /katex >}} is very small +{{< katex display >}} +\beta _t \ll 1 \qquad \beta _p \approx 1 +{{< /katex >}} + +## Stability Considerations + +The same stability factors exist in 2D equilibria that we found for 1D equilibria: + +Magnetic shear - the safety factor {{< katex >}} q(\Psi) = \frac{\Delta \phi}{\Delta \theta} {{< /katex >}} for {{< katex >}} \Delta \theta = 2 \pi {{< /katex >}}. We can calculate {{< katex >}} q {{< /katex >}} more easily by integrating along a flux surface in the poloidal plane: + +{{< katex display >}} +q(\Psi) = \frac{1}{2 \pi} \int_0 ^{\Delta \phi} \dd \phi \\ + = \frac{1}{2 \pi} \int _0 ^{2 \pi} \dv{\phi}{\theta} \dd \theta \\ + = \frac{1}{2 \pi} \int_0 ^{2 \pi} \dd \theta \left. \frac{r B_\phi}{R B_\theta} \right|_{\text{along flux surf.}} \\ + = \frac{F(\Psi)}{2 \pi} \oint \frac{ \dd l_p}{R^2 B_\theta} \qquad \dd l_p = r \dd \theta +{{< /katex >}} + +Magnetic well: similarly we can get the magnetic well factor by integrating around a flux surface in the poloidal plane + +{{< katex display >}} +\langle Q \rangle = \frac{\oint \frac{Q \dd l_p}{B_\theta}}{\oint \frac{ \dd l_p}{B_\theta}} +{{< /katex >}} + +## Shafranov Shift + +Remember that when we had an equilibrium which had a toroidal current and a corresponding poloidal magnetic field, and a poloidal magnetic field, then radial force balance will tend to shift the configuration outwards away from the major axis and a conducting wall or external coil will be required to maintain the equilibrium. The radial force balance is really achieved by {{< katex >}} \vec j_\phi \cross B_p {{< /katex >}} + +As we move towards the magnetic axis, {{< katex >}} B_p \rightarrow 0 {{< /katex >}} by definition. With less poloidal field to balance the radial force imbalance, there is more radial expansion. This means that inner portion of the plasma (inner flux surfaces) must shift radially further to achieve radial force balance. + +

Figure 12.29

+ +The shift increases with plasma pressure. This effect is further enhanced if we have low poloidal fields, for example in the high-{{< katex >}} \beta {{< /katex >}} tokamak configurations. + +Low aspect ratios also enhance the effect. Recall that the radial force imbalance increases with smaller aspect ratio, leading to a larger shift. \ No newline at end of file diff --git a/content/notes/UWAA558/10-equilibrium-of-3d-configurations.md b/content/notes/UWAA558/10-equilibrium-of-3d-configurations.md new file mode 100644 index 00000000..14ddc454 --- /dev/null +++ b/content/notes/UWAA558/10-equilibrium-of-3d-configurations.md @@ -0,0 +1,45 @@ +--- +title: Equilibrium of 3D Configurations +bookToc: false +weight: 100 +--- + + +# Equilibrium of 3D Configurations + +In 3 dimensions, we lose the axisymmetry that allowed us to reach the Grad-Shefranov equation and we need to solve the full momentum equation in three dimensions. This is not something that we can actually do in this class, and the existing codes that do this are quite sophisticated. + +Some general features of 3D equilibria are: + + 1. No net toroidal current. This means that they tend to be steady-state configurations. + 2. Radial confinement is accomplished by toroidal fields, as in the end-connected {{< katex >}} \theta {{< /katex >}}-pinch. As we saw, toroidal fields cannot provide radial confinement in a purely axisymmetric configuration, but radial variation with {{< katex >}} \phi {{< /katex >}} _can_ provide confinement. + 3. Toroidal effect (radial force balance) is generated by helical magnetic fields. You can do an expansion of the magnetic field into a toroidal component, and a helical component that traces out a twisted shape as you move around the torus. These twisted shapes are what lead to radial confinement. + + +## ELMO Bumpy Torus (EBT) + +In contrast to most other 3D configurations, even though the EBT is a 3D equilibrium, it has no helical windings. + +

Figure 12.30

+ +Since there are no helical windings, we have to provide radial stability in another way. In the EBT configuration, you also drive hot poloidal electron rings (driven by electron cyclotron resonance) to provide both stability and heating. + +## Stellarator + +The stellarator configuration is composed of a number of helical current lines (generated by helical coils with _alternating_ currents), and a net toroidal field driven by poloidal coils. The direction of the currents alternate, for a total of {{< katex >}} 2l {{< /katex >}} current lines. + +

Figure 12.31

+ +The result is a net magnetic field with a ratio such that {{< katex >}} B_{\text{helical}} \gg B_\phi {{< /katex >}} + +Stellarators raise some very complicated engineering challenges both in the design and construction of the complicated geometry. It is also very difficult to maintain no net current within the plasma, especially during start-up. As you add plasma, you raise from zero {{< katex >}} \beta {{< /katex >}} to a finite {{< katex >}} \beta {{< /katex >}}, introducing things like bootstrap currents that need to be balanced. + +## Torsatron + +Similar to a stellarator, the torsatron does not have alternating currents. All of the helical current lines are in the same direction. There are also no toroidal field coils. + +The engineering is slightly simpler, but it is slightly less efficient at generating the helical magnetic field. + +The flux surfaces in stellarators and torsatrons have geometrical cross-sections depending on the number {{< katex >}} l {{< /katex >}} of helical current lines. About the current lines, the flux surfaces are nearly circular. The flux surfaces within the plasma volume are determined by the separatrix of the helical coil fields. + +

Figure 12.32

diff --git a/content/notes/UWAA558/11-mhd-stability.md b/content/notes/UWAA558/11-mhd-stability.md new file mode 100644 index 00000000..66528af6 --- /dev/null +++ b/content/notes/UWAA558/11-mhd-stability.md @@ -0,0 +1,185 @@ +--- +title: MHD Stability +bookToc: false +weight: 110 +--- + + +# MHD Stability + +Equilibrium is simply a balance of forces that results in a steady state. Beyond equilibrium, stability is the tendency of a perturbation to return to equilibrium, rather than increasing. We are very interested in analyzing the stability of MHD equilibria, including the plasma dynamics, so we need to use the complete ideal MHD model. The MHD equations are non-linear, which means that any evolution/dynamics are also going to be non-linear. We can define the initial deviation from equilibrium to be a linear phenomenon. As usual, we perform this linearization by letting {{< katex >}} Q(r, t) = Q_0 + Q_1(r, t) {{< /katex >}} with {{< katex >}} Q_1 {{< /katex >}} being a small first-order perturbation. Since the equilibrium is both time and space independent, the general form of the perturbation is + +{{< katex display >}} +Q_1(r, t) = \vu Q_1 e^{-i (\omega t - \vec k \cdot \vec r)} +{{< /katex >}} + +{{< katex display >}} +\grad p_0 = \vec j_0 \cross \vec B_0 +{{< /katex >}} +{{< katex display >}} +p = p_0 + p_1 \qquad \rho = \rho_0 + \rho_1 +{{< /katex >}} +{{< katex display >}} +\vec j = \vec j_0 + \vec j_1 \qquad \vec B = \vec B_0 + \vec B_1 +{{< /katex >}} + +and for a static equilibrium +{{< katex display >}} +\vec v = \vec v_1 +{{< /katex >}} + +In our momentum equations of the perturbed quantities, we assume that the static equilibrium holds, so most of the equilibrium terms drop out. We can define a velocity displacement {{< katex >}} \vec \xi = \int_0 ^t \vec v_1 \dd t {{< /katex >}}. As we integrate the field and pressure in time, + +{{< katex display >}} +\pdv{B_1}{t} = \curl (\vec v_1 \cross \vec B_0) +{{< /katex >}} +{{< katex display >}} +\int \pdv{B_1}{t} = \vec B_1 = \curl \int \vec v_1 \cross \vec B_0 \dd t \\ + = \curl (\vec \xi \cross \vec B_0) +{{< /katex >}} + +If we do the same for the pressure equation, we get + +{{< katex display >}} +p_1 = - \vec \xi \cdot \grad p_0 - \Gamma p_0 \div \vec \xi +{{< /katex >}} +where {{< katex >}} \Gamma {{< /katex >}} is the ratio of specific heats, to avoid confusion with typical perturbation growth rate {{< katex >}} \gamma {{< /katex >}}. + +If we combine all of these together, substituting into the momentum equation, we can express the perturbation entirely in terms of {{< katex >}} \vec \xi {{< /katex >}} and the equilibrium properties: + +{{< katex display >}} +\rho_0 \pdv{ ^2 \vec \xi }{t^2} = \grad (\vec \xi \cdot \grad p_0 + \Gamma p_0 \div \vec \xi)\\ + + \frac{1}{\mu_0} \left[(\curl \vec B_0) \cross \curl (\vec \xi \cross \vec B_0) \right] \\ + + \frac{1}{\mu_0} \left[ \curl \curl ( \vec \xi \cross \vec B_0) \cross \vec B_0 \right] +{{< /katex >}} + +We define the right-hand-side as the linearized forcing function of our equilibrium +{{< katex display >}} +\rho_0 \pdv{ ^2 \vec \xi }{t^2} = \vec F(\vec \xi _i , p_0, \vec B_0) +{{< /katex >}} + +For a linear force function, we can also write it in terms of a spring constant tensor + +{{< katex display >}} +\rho_0 \pdv{ ^2 \vec \xi }{t^2} = \vec F(\vec \xi) = - \overline \vec K \cdot \vec \xi +{{< /katex >}} + +We can determine the stability behavior of a configuration by specifying an initial condition + +{{< katex display >}} +\vec \xi (t = 0) = 0 \qquad \text{and} \qquad \left. \pdv{\xi}{t} \right| _{t = 0} = f(\vec r) +{{< /katex >}} +and boundary conditions. A boundary condition may be a rigid wall +{{< katex display >}} +\vec \xi \cdot \vu n |_{wall} = 0 +{{< /katex >}} + +One way we can tell whether a given solution is unstable is to assume a variation of the form +{{< katex display >}} +\vec \xi \propto e^{-i \omega t} +{{< /katex >}} +If {{< katex >}} \omega^2 > 0 {{< /katex >}}, the displacement will oscillate in time without growth, and if {{< katex >}} \omega^2 < 0 {{< /katex >}} then the displacement will grow. In other words, if {{< katex >}} \omega {{< /katex >}} is real, then the mode is stable, and if {{< katex >}} \omega {{< /katex >}} is imaginary then the mode is unstable. The eigenvalue equation to be solved is + +{{< katex display >}} +- \omega ^2 \rho_0 \vec \xi = \vec F(\vec \xi) +{{< /katex >}} +which we can write as a matrix equation +{{< katex display >}} +\overline \vec A\, \overline X = \lambda \overline X +{{< /katex >}} +{{< katex display >}} +\frac{1}{\rho_0} \vec F (\vec \xi) = - \omega ^2 \vec \xi +{{< /katex >}} + +For any arbitrary linear forcing function, we might get an infinite number of eigenvalues. How do we know which ones to look at? It turns out that the linearized force function {{< katex >}} \vec F(\vec \xi) {{< /katex >}} has the property of being self-adjoint, so +{{< katex display >}} +\int \vec \eta \cdot \vec F(\vec \xi) \dd V = \int \xi \cdot \vec F( \vec \eta) \dd V +{{< /katex >}} +where {{< katex >}} \vec \eta {{< /katex >}} and {{< katex >}} \vec \xi {{< /katex >}} are arbitrary displacements that satisfy the same boundary conditions. If {{< katex >}} \vec F {{< /katex >}} is self-adjoint, then the system is Hermitian, which guarantees that we get real eigenvalues ({{< katex >}} \omega^2 {{< /katex >}}) , orthogonal eigenfunctions, and most importantly we are guaranteed to have an ordered spectrum of eigenvalues. That is to say {{< katex >}} \omega_0 ^2 < \omega _1 ^2 < \omega _2 ^2 < \ldots {{< /katex >}}. This means that the eigenvalue of the lowest mode is guaranteed to be the most negative, and therefore dictates the stability of the system. If the lowest eigenvalue is negative, then the system is necessarily unstable, and if the lowest eigenvalue is positive, then we are guaranteed that all modes are stable. + +Because {{< katex >}} \vec F {{< /katex >}} is self-adjoint, we can make use of the energy principle to write the variation in the sum of the kinetic and potential energy as: + +{{< katex display >}} +0 = \dv{}{t} \left[ \frac{1}{2} \int \rho_0 \left( \pdv{\vec \xi}{t} \right) ^2 \dd V - \frac{1}{2} \int \vec \xi \cdot \vec F ( \vec \xi) \dd V \right] +{{< /katex >}} +The kinetic energy term will always be positive, so we can formulate the stability based on the potential energy, often called a {{< katex >}} \delta W {{< /katex >}} approach +{{< katex display >}} +\delta W = - \frac{1}{2} \int \vec \xi \cdot \vec F ( \vec \xi) \dd V +{{< /katex >}} +is the change in potential energy due to a displacement {{< katex >}} \xi {{< /katex >}}. If the potential energy decreases due to a displacement {{< katex >}} \xi {{< /katex >}}, then the kinetic energy must necessarily increase, so {{< katex >}} \delta W < 0 {{< /katex >}} indicates instability. + +We can write the change in kinetic energy for our normal mode decomposition as +{{< katex display >}} +\delta T = \frac{1}{2} \int \rho _0 \left( \pdv{\xi}{t} \right) ^2 \dd V = - \frac{1}{2} \omega ^2 \int \rho _0 \vec \xi ^ \star \cdot \vec \xi \dd V \\ += - \delta W = \frac{1}{2} \int \vec \xi ^\star \cdot \vec F(\vec \xi) \dd V +{{< /katex >}} +{{< katex display >}} +\omega^2 = \frac{- \int \xi ^\star \cdot \vec F \dd V}{\int \rho_0 \xi ^\star \cdot \xi \dd V} = \frac{\delta W}{\frac{1}{2} \int \rho_0 \xi ^\star \cdot \xi \dd V} +{{< /katex >}} + +The denominator is strictly positive, so the sign of {{< katex >}} \omega^2 {{< /katex >}} is determined by the sign of {{< katex >}} \delta W {{< /katex >}} + +{{< katex display >}} +\delta W < 0 \rightarrow \omega^2 < 0 \rightarrow \text{unstable} \\ +\delta W > 0 \rightarrow \omega^2 > 0 \rightarrow \text{stable} +{{< /katex >}} + +Analyzing the form of {{< katex >}} \delta W {{< /katex >}} (within the plasma volume) + +{{< katex display >}} +\delta W = \frac{1}{2} \int_{plasma} \dd V \Gamma p_0(\div \vec \xi) ^2 + \vec \xi \cdot \grad p_0 (\div \vec \xi) \qquad \qquad \\ +\qquad \qquad + \frac{1}{\mu_0} \left[ \curl ( \vec \xi \cross \vec B_0) \right]^2 \\ +\qquad \qquad - \frac{1}{\mu_0} \left[\vec \xi \cross ( \curl \vec B_0) \right] \cdot \left[ \curl ( \vec \xi \cross \vec B_0) \right] +{{< /katex >}} + +Generally speaking, the plasma volume does not extend to infinity, and we care very much about the boundary. The total {{< katex >}} \delta W {{< /katex >}} is the sum of that in the plasma volume {{< katex >}} \delta W_F {{< /katex >}}, the surface {{< katex >}} \delta W_S {{< /katex >}}, and the vacuum region {{< katex >}} \delta W_V {{< /katex >}}. The vacuum term looks like + +{{< katex display >}} +\delta W_V = \frac{1}{2} \int _{vac} \dd V \frac{ (\curl ( \vec \xi \cross \vec B_0))^2}{\mu_0} = \int_{vac} \dd V \frac{\vec B_1 ^2}{\mu_0} > 0 +{{< /katex >}} + +so the vacuum term is always positive, and has a stabilizing influence. The surface contribution offsets this + +{{< katex display >}} +\delta W_S = \frac{1}{2} \oint \dd S ( \vu n \cdot \vec \xi) ^2 \left[ \left[ \grad \left( p_0 + \frac{B_0^2}{2 \mu_0} \right) \right] \right] \cdot \vu n +{{< /katex >}} + +Instabilities can be characterized as: + +- Internal/fixed boundary {{< katex >}} \delta W = \delta W_F {{< /katex >}} +- External/free boundary {{< katex >}} \delta W = \delta W_F + \delta W_S + \delta W_V {{< /katex >}} + +The plasma portion can be re-written slightly as + +{{< katex display >}} +\delta W_F = \frac{1}{2} \int \dd V \frac{ |B_{1, \perp}|^2}{\mu_0} \quad \leftarrow \text{Shear Alfven} \\ + + \mu_0 \left| \frac{B_{1, \parallel}}{\mu_0} - \frac{B_0 \xi \cdot \grad p_0}{B_0} ^2 \right|^2 \quad \leftarrow \text{Fast magnetosonic} \\ + + \Gamma p_0 |\div \xi|^2 \quad \leftarrow \text{Acoustic}\\ + + \frac{\vec j_0 \cdot \vec B_0}{B_0 ^2} (\vec B_0 \cross \vec \xi) \cdot \vec B_1 \quad \leftarrow \text{Current-driven (kink)} \\ +- 2 ( \vec \xi \cdot \grad p_0)(\vec \xi \cdot \vec \kappa) \quad \leftarrow \text{pressure-driven (interchange/balooning)} +{{< /katex >}} + +where {{< katex >}} \vec \kappa {{< /katex >}} is the curvature vector {{< katex >}} \vu e_B \cdot \grad \vu e_B {{< /katex >}}. If we look at each of these terms, the first three terms are all going to be stabilizing effects, which means that all instability is going to come from the last two terms, the current-driven instability term and the pressure-driven instability term. + +Going back to the screw pinch, +{{< katex display >}} +\dv{p}{r} = j_\theta B_z - j_z B_\theta +{{< /katex >}} +we have current in the same direction as magnetic field ({{< katex >}} j_\theta {{< /katex >}} with {{< katex >}} B_\theta {{< /katex >}} and {{< katex >}} j_z {{< /katex >}} with {{< katex >}} B_z {{< /katex >}}), so kink instabilities are possible. We also have a pressure gradient, so interchange instabilities are also possible. + +As a concrete example, look at the pressure driven instability term in a Z-pinch. +{{< katex display >}} +\kappa = - \frac{ \vu r}{r} +{{< /katex >}} +{{< katex display >}} +\vec \xi = \xi _r \vu r +{{< /katex >}} +{{< katex display >}} +\delta W_{F, pressure} = \int \dd V \xi _r \dv{p_0}{r} \frac{\xi_r}{r} \\ + = \int \dd V \frac{2 \xi _r ^2}{r} \dv{p_0}{r} +{{< /katex >}} + +In a Z-pinch, it is always the case that {{< katex >}} \dv{p_0}{r} < 0 {{< /katex >}}. As shown by Kadomtsev (1965) it turns out that these modes can be stabilized by adding {{< katex >}} B_z {{< /katex >}}, but this also introduces kink modes. + +Going back to our stabilizing quantities of wellness and shear, current-driven instabilities are generally managed through shear, and pressure-driven instabilities are stabilized by well. + diff --git a/content/notes/UWAA558/_index.md b/content/notes/UWAA558/_index.md new file mode 100644 index 00000000..31dae7d1 --- /dev/null +++ b/content/notes/UWAA558/_index.md @@ -0,0 +1,5 @@ +--- +bookFlatSection: false +bookCollapseSection: true +title: MHD Theory +--- diff --git a/content/notes/UWAA558/formulary.md b/content/notes/UWAA558/formulary.md new file mode 100644 index 00000000..b296c76e --- /dev/null +++ b/content/notes/UWAA558/formulary.md @@ -0,0 +1,464 @@ +--- +title: Formulary +weight: 999 +bookToc: false +--- + +{{< katex >}} {{< /katex >}} + +# Formulary + +{{< hint info >}} +**"Kinetic Description"** + + +{{< katex display >}} +\dv{\vec v}{t} = \frac{q_i}{m_i} (\vec E + \vec v_i \cross \vec B) + \sum_{j \neq i} \left[ \left. \dv{\vec v_{ij}}{t} \right|_{coll} \delta(\vec r_i - \vec r_j) \right] +{{< /katex >}} + +{{< katex display >}} +\pdv{\vec B}{t} = - \curl \vec E +{{< /katex >}} + +{{< katex display >}} +\frac{1}{c^2} \pdv{\vec E}{t} = \curl \vec B - \mu_0 \sum_i q_i \vec v_i \delta (\vec r - \vec r_i) +{{< /katex >}} + +{{< katex display >}} +\div \vec{B} = 0 +{{< /katex >}} + +{{< katex display >}} +\div \vec E = \frac{1}{\epsilon_0} \sum_i q_i \delta (\vec r - \vec r_i) +{{< /katex >}} + +Klimontovich equation: + +{{< katex display >}} +\dv{N}{t} = 0 = \pdv{N}{t} + \pdv{}{q_i} \cdot (\dot{q_i} N) \\ +N \equiv \sum_i \delta (p - p_i) \delta(q - q_i) +{{< /katex >}} + +{{< /hint >}} + +{{< hint info >}} +**"Plasma Fluid Description"** + +Boltzmann Equation + +{{< katex display >}} +\pdv{f_\alpha}{t} + \vec v \cdot \pdv{f_\alpha}{t} + \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} = \left. \pdv{f_\alpha}{t} \right|_{coll} = \sum_{\beta \neq \alpha} C_{\alpha \beta} +{{< /katex >}} + +Maxwellian distribution: + +{{< katex display >}} +f_\alpha (\vec v) = n_\alpha \left( \frac{m_\alpha}{2 \pi T} \right)^{3/2} e^{- \frac{m_\alpha(\vec v - \vec v_\alpha)^2}{2T}} +{{< /katex >}} +Moments of fluid model (moments of distribution \\( \rightarrow \\) moments of Boltzmann equation: + +{{< katex display >}} +\text{Continuity:} \qquad n_\alpha = \int f_\alpha \dd \vec v \\ +\quad \rightarrow \pdv{n_\alpha}{t} + \div (n_\alpha \vec v_\alpha) = 0 +{{< /katex >}} + +{{< katex display >}} +\text{Momentum:} \qquad n_\alpha \vec v_\alpha = \int \vec v f_\alpha \dd v \\ +\quad \rightarrow \quad \pdv{}{t} (n_\alpha \vec v_\alpha ) + \div (n_\alpha \vec v_\alpha \vec v_\alpha) + \frac{1}{m_\alpha} \div \vec P_\alpha - \frac{q_\alpha}{m_\alpha} n_\alpha ( \vec E + \vec v _\alpha \cross \vec B) = \sum_{\beta \neq \alpha} \int \vec w C_{\alpha \beta} \dd \vec v +{{< /katex >}} + +{{< katex display >}} +\rightarrow \rho_\alpha \left(\pdv{\vec v_\alpha}{t} + \vec v_\alpha \cdot \grad \vec v_\alpha \right) + \grad \vec P_\alpha + \div \vec \Pi_\alpha - q_\alpha n_\alpha (\vec E + \vec v_\alpha \cross \vec B) = \sum_{\beta \neq \alpha} \vec R_{\alpha \beta} +{{< /katex >}} + +{{< katex display >}} +\text{Energy:} \qquad \int \vec v \vec v \pdv{f_\alpha}{t} \dd \vec v = \pdv{}{t} \int \vec v \vec v f_\alpha \dd \vec v = \pdv{}{t} \vec E_\alpha / m_\alpha \rightarrow \pdv{}{t} \vec P_\alpha +{{< /katex >}} + +{{< katex display >}} +\rightarrow \quad \frac{3}{2} n_\alpha \left( \pdv{T_\alpha}{t} + \vec v_\alpha \cdot \grad T_\alpha \right) + P_\alpha \div \vec v_\alpha + \vec \Pi_\alpha \cdot \cdot \grad \vec v_\alpha + \div \vec h_\alpha = \sum_{\beta \neq \alpha} Q_{\alpha \beta} +{{< /katex >}} +Closure relations + +{{< katex display >}} +\vec h_\alpha = - \kappa \grad T_\alpha +{{< /katex >}} + +{{< katex display >}} +\overline \Pi_ \alpha = \nu \grad \vec v_\alpha +{{< /katex >}} + +{{< /hint >}} + +## Ideal MHD + +{{< hint info >}} +**"Ideal MHD"** + +Continuity: + +{{< katex display >}} +\pdv{\rho}{t} + \div (\rho \vec v) = 0 +{{< /katex >}} +Momentum: + +{{< katex display >}} +\rho \left( \pdv{\vec v}{t} + \vec v \cdot \grad \vec v \right) + \grad p - \vec j \cross \vec B = 0 +{{< /katex >}} +Generalized Ohm's Law + +{{< katex display >}} +\vec E + \vec v \cross \vec B = \frac{1}{Zen}\cancel{(\vec j \cross \vec B - \grad p_e)} = 0 +{{< /katex >}} +Energy + +{{< katex display >}} +\dv{}{t} \left( \frac{p}{\rho^\gamma} \right) = 0 +{{< /katex >}} + +{{< /hint >}} + +{{< hint info >}} +"Lawson Criterion" + + +{{< katex display >}} +n \tau_E > 10^4 s/m^3 +{{< /katex >}} + +{{< /hint >}} + +{{< hint info >}} +**"Conservation Law Form of Ideal MHD"** + +Continuity: + +{{< katex display >}} +\pdv{\rho}{t} + \div (\rho \vec v) = 0 +{{< /katex >}} +Momentum: + +{{< katex display >}} +\pdv{(\rho \vec v)}{t} + \div \left[ \rho \vec v \vec v - \frac{\vec B \vec B}{\mu_0} + \left( p + \frac{B^2}{2 \mu_0} \right) \overline{I} \right] = 0 +{{< /katex >}} +Energy: + +{{< katex display >}} +\pdv{\epsilon}{t} + \div \left[ \left( \epsilon + p + \frac{B^2}{2 \mu_0} \right) \vec v - (\vec B \cdot \vec v) \frac{\vec B}{\mu_0} \right] = 0 +{{< /katex >}} + +{{< katex display >}} +\pdv{\vec B}{t} + \div ( \vec v \vec B - \vec B \vec v) = 0 +{{< /katex >}} +where + +{{< katex display >}} +\epsilon = \frac{1}{\gamma - 1} p + \frac{1}{2} \rho v^2 + \frac{B^2}{2\mu_0} +{{< /katex >}} + +Static Equilibrium: + +{{< katex display >}} +\vec j \cross \vec B = \grad p +{{< /katex >}} + +{{< katex display >}} +\frac{B^2}{\mu_0} \vec K = \grad_\perp (p + \frac{B^2}{2 \mu_0}) +{{< /katex >}} + +{{< katex display >}} +\vec K \equiv \frac{\vec B}{|B|} \cdot \grad \frac{ \vec B}{|B|} +{{< /katex >}} + +Conservation of flux: + +{{< katex display >}} +\vec E + \vec v \cross \vec B = 0 +{{< /katex >}} + +{{< katex display >}} +\pdv{\vec B}{t} = - \curl \vec E +{{< /katex >}} + +{{< katex display >}} +\rightarrow \dv{}{t} \left( \frac{\vec B}{\rho} \right) = \frac{\vec B}{\rho} \cdot \grad \vec v +{{< /katex >}} + +{{< /hint >}} + +## 1D Equilibria + +{{< hint info >}} +**"\\( \theta \\)-pinch"** + + +{{< katex display >}} +B_\theta = 0 +{{< /katex >}} + +{{< katex display >}} +j_\theta B_z = \dv{p}{r} +{{< /katex >}} + +{{< katex display >}} +j_\theta = - \frac{1}{\mu_0} \dv{B_z}{r} +{{< /katex >}} + +{{< katex display >}} +\rightarrow p + \frac{B_z ^2}{2 \mu_0} = \frac{B_0 ^2}{2 \mu_0} +{{< /katex >}} + +{{< katex display >}} +\langle \beta \rangle = \frac{2}{a^2} \int_0 ^a \frac{r p}{B_0 ^2 / 2 \mu_0} \dd r +{{< /katex >}} + +{{< katex display >}} +q = \infty +{{< /katex >}} + +{{< katex display >}} +W = \frac{\mu_0 r}{B_z ^2} \dv{}{r} \left( p + \frac{B_z ^2}{2 \mu_0} \right) = 0 +{{< /katex >}} + +{{< /hint >}} + +{{< hint info >}} +**"Z-pinch"** + + +{{< katex display >}} +B_z =0 +{{< /katex >}} + +{{< katex display >}} +\grad p = \dv{p}{r} = - j_z B_\theta +{{< /katex >}} + +{{< katex display >}} +- \dv{}{r} \left( p + \frac{B_\theta ^2}{2 \mu_0} \right) = \frac{B_\theta ^2}{\mu_0 r} +{{< /katex >}} + +{{< katex display >}} +\langle \beta \rangle = \frac{2 \mu_0}{B_0 ^2 \pi a^2} \int _0 ^a 2 \pi r p \dd r = 1 \quad \text{ if } \quad p(a) = 0 +{{< /katex >}} + +{{< katex display >}} +q = S = 0 +{{< /katex >}} + +{{< katex display >}} +W = 1 +{{< /katex >}} + +{{< /hint >}} + +{{< hint info >}} +**"Screw pinch"** + + +{{< katex display >}} +\dv{}{r} \left( p + \frac{B^2}{2 \mu_0} \right) = - \frac{B_\theta ^2}{\mu_0 r} +{{< /katex >}} + +{{< katex display >}} +\beta_t = \frac{2 \mu_0}{B_z (a) ^2} \left( \frac{1}{\pi a^2} \int_0 ^a 2 \pi r p \dd r \right) +{{< /katex >}} + +{{< katex display >}} +\beta_p = \left( 1 - \frac{\alpha_t}{\beta _t} \right)^{-1} \qquad \alpha_t \equiv \frac{2}{a^2} \int_0 ^a \left(1 - \frac{B_z ^2}{B_0 ^2} \right) r \dd r +{{< /katex >}} + +{{< katex display >}} +q = \frac{2 \pi r B_z}{L B_\theta} +{{< /katex >}} + +{{< katex display >}} +q_a = \frac{4 \pi ^2 a^2 B_0}{\mu_0 I_a} +{{< /katex >}} + +{{< katex display >}} +S = \frac{r}{q} \dv{q}{r} +{{< /katex >}} + +{{< katex display >}} +W = - \frac{B_\theta ^2}{B_\theta ^2 + B_z ^2} +{{< /katex >}} + +{{< /hint >}} + +{{< hint info >}} +**"Stability"** + +Shear: + +{{< katex display >}} +S = 2 \frac{ dq / q}{dV / V} = 2 \frac{d \ln q}{d \ln V} +{{< /katex >}} + +{{< katex display >}} +q = \frac{\text{\# long windings}}{\text{\# short windings}} = \dv{\psi_t}{\psi_p} +{{< /katex >}} +Shear for toroid + +{{< katex display >}} +q = \frac{r B_\phi}{R B_\theta} +{{< /katex >}} +Shear for cylinder + +{{< katex display >}} +q = 2 \pi \frac{r B_z}{L B_\theta} +{{< /katex >}} +Well + +{{< katex display >}} +W = \frac{ d \langle p + B^2 / 2 \mu_0 \rangle / \langle B^2 / 2 \mu_0 \rangle}{dV / V} +{{< /katex >}} +For stabilization, \\( B^2 / 2 \mu_0 \\) should increase faster than \\( p \\) decreases +{{< /hint >}} + +## 2D Equilibria + +{{< hint info >}} +**"Grad-Shafranov Equation: Static toroidal equilibrium"** + + +{{< katex display >}} +\grad p = \vec j_\theta \cross \vec B_\phi + \vec j_\phi \cross \vec B_\theta +{{< /katex >}} + +{{< katex display >}} +A_\phi = \frac{\phi}{R} \vu \phi +{{< /katex >}} + +{{< katex display >}} +\phi = \frac{\phi_p}{2 \pi} +{{< /katex >}} + +{{< katex display >}} +\vec B_\theta = - \frac{\vu R}{R} \pdv{\psi}{z} + \frac{ \vu z}{R} \pdv{\psi}{R} = \frac{ \grad \psi}{R} \cross \vu \phi +{{< /katex >}} + +{{< katex display >}} +F \equiv R B_\phi +{{< /katex >}} + +{{< katex display >}} +\Delta ^\star \equiv R \pdv{}{R} \frac{1}{R} \pdv{}{R} + \pdv{^2}{z^2} +{{< /katex >}} + +{{< katex display >}} +\Delta ^\star \psi = \pdv{^2 \psi}{z^2} + \pdv{^2 \psi}{R^2} - \frac{1}{R} \pdv{\psi}{R} +{{< /katex >}} + +{{< katex display >}} +\vec j_\phi = - \frac{1}{\mu_0 R} \Delta ^\star \psi \vu \phi +{{< /katex >}} + +{{< katex display >}} +\vec j_\theta = \frac{1}{\mu_0 R} \grad (F) \cross \vu \phi +{{< /katex >}} + +{{< katex display >}} +R^2 \mu_0 \pdv{p}{\psi} = - \Delta ^\star \psi - F \pdv{F}{\psi} +{{< /katex >}} + +{{< katex display >}} +q(\psi) = \frac{F(\psi)}{2 \pi} \oint_{p} \frac{r \dd \theta}{R^2 B_\theta} +{{< /katex >}} +Limits: + +{{< katex display >}} +\text{Force-free:} \qquad \vec j \parallel \vec B +{{< /katex >}} + +{{< katex display >}} +\rightarrow \Delta ^\star \psi + F F' = 0 +{{< /katex >}} + +{{< katex display >}} +\text{Connected $\theta$ pinch:} \qquad FF' \gg \Delta ^\star \psi +{{< /katex >}} + +{{< katex display >}} +\rightarrow \grad p \approx \vec j_\theta \cross \vec B_\phi +{{< /katex >}} + +{{< katex display >}} +\text{Connected Z-pinch:} \qquad FF' \ll \Delta ^\star \phi +{{< /katex >}} + +{{< katex display >}} +\rightarrow \grad p \approx j_\phi \cross B_\theta +{{< /katex >}} + +{{< /hint >}} + +## MHD Stability + +{{< hint info >}} +**"Linear stability"** + + +{{< katex display >}} +\pdv{\rho_1}{t} = - \vec v_1 \grad \rho_0 - \rho_0 \div \vec v_1 +{{< /katex >}} + +{{< katex display >}} +\pdv{\vec B_1}{t} = \curl ( \vec v_1 \cross \vec B_0) +{{< /katex >}} + +{{< katex display >}} +\rho_0 \pdv{\vec v_1}{t} = - \grad p_1 + \vec j_0 \cross \vec B_1 - \vec j_1 \cross \vec B_0 +{{< /katex >}} + +{{< katex display >}} +\pdv{p_1}{t} = - \vec v_1 \cdot \grad p_0 - \gamma p_0 \div \vec v_1 +{{< /katex >}} +For linear perturbation \\( \vec \xi = \int_0 ^t \vec v_1 \dd t \\) the momentum equation becomes + +{{< katex display >}} +\rho_0 \pdv{^2 \xi}{t^2} = \vec F(\xi) +{{< /katex >}} +where + +{{< katex display >}} +F(\xi) = \grad (\xi \cdot \grad p_0 + \gamma p_0 \div \xi) + \frac{1}{\mu_0} \left[ ( \curl \vec B_0) \cross \curl (\xi \cross \vec B_0) + \curl \curl (\xi \cross \vec B_0) \cross \vec B_0 \right] +{{< /katex >}} + +Eigenvalues of \\( \frac{1}{\rho_0} \vec F (\xi) = \omega^2 \xi \\) are real and ordered. Only need to check \\( n=0 \\) to determine stability/instability of configuration. + +{{< /hint >}} + +{{< hint info >}} +**"\\( \delta W \\) Approach"** + +\\( \delta W = \\) change in potential energy due to a displacement \\( \xi \\) + +{{< katex display >}} +\delta W < 0 \rightarrow \text{instability} +{{< /katex >}} + +{{< katex display >}} +\delta W = - \frac{1}{2} \int \xi \cdot F(\xi) \dd V = \delta W_F + \delta W_S + \delta W_V +{{< /katex >}} +Surface term: + +{{< katex display >}} +\delta W_s = \frac{1}{2} \oint \dd S (\vu n \cdot \xi) ^2 \left( \vu n \cdot \grad p_0 + \left[ \vu n \cdot \grad \frac{B_0 ^2}{2 \mu_0} \right]_{jump} \right) +{{< /katex >}} +Vacuum term: + +{{< katex display >}} +\delta W_V = \int_{vac} \dd V \frac{B_1 ^2}{\mu_0} +{{< /katex >}} +Plasma (free) term: + +{{< katex display >}} +\delta W_F = \frac{1}{2} \int \dd V \frac{ |B_{1, \perp}|^2}{\mu_0} \quad \leftarrow \text{Shear Alfven} \\ ++ \mu_0 \left| \frac{B_{1, \parallel}}{\mu_0} - \frac{B_0 \xi \cdot \grad p_0}{B_0} ^2 \right|^2 \quad \leftarrow \text{Fast magnetosonic} \\ ++ \Gamma p_0 |\div \xi|^2 \quad \leftarrow \text{Acoustic}\\ ++ \frac{\vec j_0 \cdot \vec B_0}{B_0 ^2} (\vec B_0 \cross \vec \xi) \cdot \vec B_1 \quad \leftarrow \text{Current-driven (kink)} \\ +- 2 ( \vec \xi \cdot \grad p_0)(\vec \xi \cdot \vec \kappa) \quad \leftarrow \text{pressure-driven (interchange/balooning)} +{{< /katex >}} +Shear Alfven, fast magnetosonic, and acoustic modes are stabilizing. Current-driven and pressure-driven modes can lead to instability. +{{< /hint >}} diff --git a/content/docs/notes/_index.md b/content/notes/_index.md similarity index 100% rename from content/docs/notes/_index.md rename to content/notes/_index.md diff --git a/content/docs/shortcodes/_index.md b/content/notes/working/_index.md similarity index 100% rename from content/docs/shortcodes/_index.md rename to content/notes/working/_index.md diff --git a/content/notes/working/crews2018.md b/content/notes/working/crews2018.md new file mode 100644 index 00000000..50bc4391 --- /dev/null +++ b/content/notes/working/crews2018.md @@ -0,0 +1,59 @@ +--- +title: "Crews (2018)" +bookToc: false +--- + +# Development of a Collisionless Plasma Kinetic Solver and an Investigation of One-Dimensional Plasma Waves and Instabilities + +Shielded potential of a test electron: + +{{< katex display >}} +\phi(r) = \frac{-e}{4 \pi \epsilon_0 r} e ^{- r / \lambda_D} +{{< /katex >}} + +where the Debye length is {{< katex >}} \lambda_D = \sqrt{\frac{\epsilon_0 T_e}{ n_e e}} {{< /katex >}}. The mean free path between large-angle collisions is estimated as + +{{< katex display >}} +\lambda_{mfp} \sim \frac{\epsilon_0 T_e ^2}{\phi_e n_e \log ( \Lambda)} +{{< /katex >}} +where {{< katex >}}\phi_e = e^2 / 4 \pi \epsilon_0{{< /katex >}} are the constants from the Coulomb force law. + +Smooth out the discreteness of particles via spatial average over small volumes: + +{{< katex display >}} +\rho \rightarrow \langle \rho_c \rangle \qquad \vec E \rightarrow \langle \vec E \rangle + \delta \vec E +{{< /katex >}} + +The mean field {{< katex >}}\langle \vec E \rangle{{< /katex >}} is responsible for collective modes of plasma motion. Estimate the collisionality of the plasma by comparing the length scales {{< katex >}}\lambda_{mfp} / \lambda_D{{< /katex >}} + +{{< katex display >}} +\frac{\lambda_{mfp}}{\lambda_D} \sim \frac{T_e ^{3/2}}{n_e ^{1/2}} +{{< /katex >}} + +Plasma is seen to become collisionless as the temperature becomes high or the plasma becomes more rarified. + +## Phase space mechanics + +To arrive at a kinetic equation governing the collisionless mechanics, consider the one-dimensional motion of a single particle + +{{< katex display >}} +\dot{x} = v \qquad \dot v = F(x) +{{< /katex >}} + +and define the phase space coordinates as {{< katex >}}\vec{\dot r} = \vec F \equiv [ v, F(x) ]{{< /katex >}}. The flux vector is similar to the velocity field of a fluid flow. The streamlines of {{< katex >}}\vec F{{< /katex >}} are the streamlines which a particle will follow if the flux is constant in time. Phase flow is always analogous to that of an incompressible fluid because the flux divergence is zero: + +{{< katex display >}} +\div [v, F(x)] = \pdv{v}{x} + \pdv{F(x)}{v} +{{< /katex >}} + +If the phase fluid density is given by a function {{< katex >}}f(x, v, t){{< /katex >}} where {{< katex >}}t{{< /katex >}} is the time parameter, because any instantiation of a particle can not leave the phase plane, the probability density will be conserved. We can write a conservation law: +{{< katex display >}} +\pdv{f(x, v, t)}{t} = - \div ( f (x, v, t) \vec F) +{{< /katex >}} +and due to the flow's incompressibility +{{< katex display >}} +\div (f \vec F) = f ( \div \vec F) + \vec F \cdot \grad f = \vec F \cdot \grad f \\ +\rightarrow \pdv{f}{t} = - \left[ v, F(x) \right] \cdot \left[ \pdv{f}{x}, \pdv{f}{v} \right] \\ +\rightarrow \pdv{f}{t} + v \pdv{f}{x} + F(x) \pdv{f}{v} = 0 +{{< /katex >}} + diff --git a/docs/404.html b/docs/404.html index 2b463d1b..db59d422 100644 --- a/docs/404.html +++ b/docs/404.html @@ -13,8 +13,8 @@ 404 Page not found | My Notes - - + + I2l#OK}SxySwwhE+`%!jpkNX$rM~E|KQaz?*)UGrh z$?0+b{$rQpBLG}qI3 z|Kc`|0T0$q8V}(p#mhEdxbcp`mk;v^!E%9Z7TlmI{oq)JGxxKW9KM*!p=wC z&TruqMz_b{v%@jsw$!$Z5j<)@XcH!vd=0s7X_85DQdZALlE5$odW?&Mls+~C7<8C& zQI^HvfpO*(%dWVCRu_*GH1PP&Sqz}uH$%{AKrEOc%M7i2isf0GSe~_sU9_W+g8+kAYXKv8p zZIQ+r8E)7nIhcC=bsuqYFkW4HQZYcQc|jpDBM(z>8GsA6EWbL0F$bPFS6FT}`1m2b z`Op6dum0cv2Z4}}Vf48OEdA9RyC$1ec>hSsj$O^n1SFjV-8ay65*rlok z??($1Zr(S%I6#BH9^%YZmdQ}~{u6KMD^96nwO6H+`W53dUv`L}+QdiY>Xdgz9e!ub z%dD@KtOzF>lhagALeD}+TxP`;o>I}P>aF0a+J#>?CA}(I;#Ed#Qk%n9Q6;~4tf_aG zf*8z+w%3+%_d3BZsdh~6chYVWY6bB#l2}A&*@#< z&(`r{4wrSW!+720bKzDNPj5}j_34o-9hJ$>y0Z4NIx??w5Vwq1j2rWf>zNntILG>F zGzpX9cwPMI($%9C<9YrjczM6Z={BvYmLt6rIOXYMLW(z$upVx;-0|>Er!^g-sYY{n zmr;@~l2gueF}a$C9d&gvLM)H!JclaQiSlLi)y7`N`8vHOd`z<`zVpxnZ0@nc2iu%K zmtb}9GC98(J2~z9laoTE`P^z`AE_vWU(q;wzQ*Bclf=v(4$?{t0V3yb)XkNEOI z&;Q7ue`;xCox_S5UAQ$myntY;@8UrMGXW+Fns@MW1oP2a*g}*Y#IC^SoDqYJN3;L z7fCKnD%Gn?-nRm&u|h_hFLv-?^9LvDPdX$5VR+(NF^x&LOuenmEk^A{5MQY<*0nmk zu8>Cn*Xwfxa>W>0{2Ul{XwW(T@IIV=L@Xs3?LNKwBOL$r7lOUwWfT-w5mIEXw#b#cv8<#`C6i@tl!7N;BHLKUft% zvJ-W%exitz-lw?U1_(kwKmg@y8g~G>ikF`?8g*dYp<&11M>+7WkG zB>ehqaX;&0RYNXX@>rAD5p-M-sk;a`4tL*bnku$V{{5B&z^

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- - - - - -
  • - - - - - - Notes - - - - - -
    • - - - -
  • - - - - - - Shortcodes - - - -
      -
    • @@ -233,7 +163,7 @@

      - Buttons + Plasma Fluid Model

      • @@ -246,7 +176,7 @@

      - Columns + Two-Fluid Plasma Model @@ -259,7 +189,7 @@

      - Details + Magnetohydrodynamic (MHD) Model @@ -272,7 +202,7 @@

      - Expand + Boundary Conditions @@ -285,7 +215,7 @@

      - Hints + Equilibrium for Fusion @@ -298,7 +228,7 @@

      - Katex + 1-D Equilibria @@ -311,7 +241,7 @@

      - Mermaid + 2D Equilibria @@ -324,15 +254,11 @@

      - - + Equilibrium of 3D Configurations - - @@ -516,6 +429,7 @@

      window.scrollTo(0,document.body.scrollHeight); }; +
      diff --git a/docs/index.xml b/docs/index.xml index b84148e3..b63d034d 100644 --- a/docs/index.xml +++ b/docs/index.xml @@ -4,161 +4,136 @@ Introduction on My Notes https://peppyhare.github.io/r/ Recent content in Introduction on My Notes - Hugo -- gohugo.io - - - - - - With ToC - https://peppyhare.github.io/r/docs/example/table-of-contents/with-toc/ - Mon, 01 Jan 0001 00:00:00 +0000 - - https://peppyhare.github.io/r/docs/example/table-of-contents/with-toc/ - Caput vino delphine in tamen vias # Cognita laeva illo fracta # Lorem markdownum pavent auras, surgit nunc cingentibus libet Laomedonque que est. Pastor An arbor filia foedat, ne fugit aliter, per. Helicona illas et callida neptem est Oresitrophos caput, dentibus est venit. Tenet reddite famuli praesentem fortibus, quaeque vis foret si frondes gelidos gravidae circumtulit inpulit armenta nativum. - Te at cruciabere vides rubentis manebo Maturuit in praetemptat ruborem ignara postquam habitasse Subitarum supplevit quoque fontesque venabula spretis modo Montis tot est mali quasque gravis Quinquennem domus arsit ipse Pellem turis pugnabant locavit Natus quaerere # Pectora et sine mulcere, coniuge dum tincta incurvae. - - - - Without ToC - https://peppyhare.github.io/r/docs/example/table-of-contents/without-toc/ - Mon, 01 Jan 0001 00:00:00 +0000 - - https://peppyhare.github.io/r/docs/example/table-of-contents/without-toc/ - At me ipso nepotibus nunc celebratior genus # Tanto oblite # Lorem markdownum pectora novis patenti igne sua opus aurae feras materiaque illic demersit imago et aristas questaque posset. Vomit quoque suo inhaesuro clara. Esse cumque, per referri triste. Ut exponit solisque communis in tendens vincetis agisque iamque huic bene ante vetat omina Thebae rates. Aeacus servat admonitu concidit, ad resimas vultus et rugas vultu dignamque Siphnon. -Quam iugulum regia simulacra, plus meruit humo pecorumque haesit, ab discedunt dixit: ritu pharetramque. - - + Hugo -- gohugo.io - - https://peppyhare.github.io/r/docs/example/collapsed/3rd-level/4th-level/ + Syllabus + https://peppyhare.github.io/r/notes/UWAA558/01-syllabus/ Mon, 01 Jan 0001 00:00:00 +0000 - https://peppyhare.github.io/r/docs/example/collapsed/3rd-level/4th-level/ - 4th Level of Menu # Caesorum illa tu sentit micat vestes papyriferi # Inde aderam facti; Theseus vis de tauri illa peream. Oculos uberaque non regisque vobis cursuque, opus venit quam vulnera. Et maiora necemque, lege modo; gestanda nitidi, vero? Dum ne pectoraque testantur. -Venasque repulsa Samos qui, exspectatum eram animosque hinc, aut manes, Assyrii. Cupiens auctoribus pariter rubet, profana magni super nocens. Vos ius sibilat inpar turba visae iusto! + https://peppyhare.github.io/r/notes/UWAA558/01-syllabus/ + Syllabus # The course topics planned for this section are (in rough order): +Particle Model, Boltzmann-Maxwell Model, Magnetohydrodynamic (MHD) Model, Region of Validity, Common Assumptions, Ideal MHD Model, General Properties (Equilibrium, Boundary Conditions, Conservation Laws, &ldquo;Frozen-In&rdquo; Flux) +Ideal MHD Equilibrium, Virial Theorem, Magnetic Flux Surfaces +One-Dimensional Equilibria, Theta-Pinch, Z-Pinch, Screw-Pinch, Safety Factor q +Two-Dimensional Equilibria, Toroidal Geometry, Grad-Shafranov Equation, Closed Flux Surfaces, Safety Factor q, Magnetic Shear, Magnetic Well, Shafranov Shift, Spheromak, Reversed Field Pinch (RFP), Tokamaks, Stellarators (Elmo Bumpy Torus) - - https://peppyhare.github.io/r/docs/example/hidden/ + Plasma Models + https://peppyhare.github.io/r/notes/UWAA558/02-plasma-models/ Mon, 01 Jan 0001 00:00:00 +0000 - https://peppyhare.github.io/r/docs/example/hidden/ - This page is hidden in menu # Quondam non pater est dignior ille Eurotas # Latent te facies # Lorem markdownum arma ignoscas vocavit quoque ille texit mandata mentis ultimus, frementes, qui in vel. Hippotades Peleus pennas conscia cuiquam Caeneus quas. - Pater demittere evincitque reddunt Maxime adhuc pressit huc Danaas quid freta Soror ego Luctus linguam saxa ultroque prior Tatiumque inquit Saepe liquitur subita superata dederat Anius sudor Cum honorum Latona # O fallor in sustinui iussorum equidem. + https://peppyhare.github.io/r/notes/UWAA558/02-plasma-models/ + Plasma Models # Working towards MHD # Let&rsquo;s start from a full-particle description with the goal of reaching a continuum description (kinetic model). Then, we&rsquo;ll look at the forces on the separate species and form a multi-fluid model, finally simplifying to a single-fluid MHD model. +The most important question to ask ourselves is &ldquo;when is this model going to be useful?&rdquo; The MHD model is the mathematical model for magnetized plasmas that are treated as a fluid. - - https://peppyhare.github.io/r/docs/shortcodes/buttons/ + Plasma Fluid Model + https://peppyhare.github.io/r/notes/UWAA558/03-plasma-fluid-model/ Mon, 01 Jan 0001 00:00:00 +0000 - https://peppyhare.github.io/r/docs/shortcodes/buttons/ - Buttons # Buttons are styled links that can lead to local page or external link. -Example # {{&lt; button relref=&#34;/&#34; [class=&#34;...&#34;] &gt;}}Get Home{{&lt; /button &gt;}} {{&lt; button href=&#34;https://github.com/alex-shpak/hugo-book&#34; &gt;}}Contribute{{&lt; /button &gt;}} Get Home Contribute + https://peppyhare.github.io/r/notes/UWAA558/03-plasma-fluid-model/ + Plasma Fluid Model # We take velocity moments of each of the pieces of the kinetic model: + distribution function, \( f_\alpha \) function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: "$$", right: "$$", display: true }, { left: "\\[", right: "\\]", display: true }, { left: "$", right: "$", display: false }, { left: "\\(", right: "\\)", display: false }], macros: { "\\lcm": " - - https://peppyhare.github.io/r/docs/shortcodes/columns/ + Two-Fluid Plasma Model + https://peppyhare.github.io/r/notes/UWAA558/04-two-fluid-plasma-model/ Mon, 01 Jan 0001 00:00:00 +0000 - https://peppyhare.github.io/r/docs/shortcodes/columns/ - Columns # Columns help organize shorter pieces of content horizontally for readability. -{{&lt; columns &gt;}} &lt;!-- begin columns block --&gt; # Left Content Lorem markdownum insigne... &lt;---&gt; &lt;!-- magic separator, between columns --&gt; # Mid Content Lorem markdownum insigne... &lt;---&gt; &lt;!-- magic separator, between columns --&gt; # Right Content Lorem markdownum insigne... {{&lt; /columns &gt;}} Example # Left Content Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat stringit, frustra Saturnius uteroque inter! + https://peppyhare.github.io/r/notes/UWAA558/04-two-fluid-plasma-model/ + Two-Fluid Plasma Model (ions-electrons) # Restricting our multi-species fluid model to ions and electrons, what can we say about wave behavior in a magnetized 2-fluid plasma? Let&rsquo;s start with a cold plasma approximation ( \( p = 0 \) function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: "$$", right: "$$", display: true }, { left: "\\[", right: "\\]", display: true }, { left: "$", right: " - - https://peppyhare.github.io/r/docs/shortcodes/details/ + Magnetohydrodynamic (MHD) Model + https://peppyhare.github.io/r/notes/UWAA558/05-mhd-model/ Mon, 01 Jan 0001 00:00:00 +0000 - https://peppyhare.github.io/r/docs/shortcodes/details/ - Details # Details shortcode is a helper for details html5 element. It is going to replace expand shortcode. -Example # {{&lt; details &#34;Title&#34; [open] &gt;}} ## Markdown content Lorem markdownum insigne... {{&lt; /details &gt;}} {{&lt; details title=&#34;Title&#34; open=true &gt;}} ## Markdown content Lorem markdownum insigne... {{&lt; /details &gt;}} Title Markdown content Lorem markdownum insigne&hellip; + https://peppyhare.github.io/r/notes/UWAA558/05-mhd-model/ + Magnetohydrodynamic (MHD) Model # Applying approximations to the two-fluid plasma model will allow us to arrive at a single-fluid (center-of-mass) description. The result is the ideal magnetohydrodynamic model (MHD). +First, define the MHD variables: + \[\text{mass density:} \qquad \rho = n_i m_i &#43; n_i m_e\] function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: "$$", right: "$$", display: true }, { left: "\\[", right: "\\]", display: true }, { left: " - - https://peppyhare.github.io/r/docs/shortcodes/expand/ + Boundary Conditions + https://peppyhare.github.io/r/notes/UWAA558/06-boundary-conditions/ Mon, 01 Jan 0001 00:00:00 +0000 - https://peppyhare.github.io/r/docs/shortcodes/expand/ - Expand # Expand shortcode can help to decrease clutter on screen by hiding part of text. Expand content by clicking on it. -Example # Default # {{&lt; expand &gt;}} ## Markdown content Lorem markdownum insigne... {{&lt; /expand &gt;}} Expand ↕ Markdown content Lorem markdownum insigne&hellip; With Custom Label # {{&lt; expand &#34;Custom Label&#34; &#34;...&#34; &gt;}} ## Markdown content Lorem markdownum insigne. + https://peppyhare.github.io/r/notes/UWAA558/06-boundary-conditions/ + Boundary Conditions # Mathematically, a well-posed problem requires both governing equations and a complete set of boundary conditions (the Cauchy data for the problem). The most common boundary conditions we use are perfectly conducting walls (flux surfaces) or a vacuum region. +Perfectly Conducting Wall # For the case where the plasma extends out to a perfectly conducting (impermeable) wall. Perfectly conducting walls do not support tangential electric field: - - https://peppyhare.github.io/r/docs/shortcodes/hints/ + Equilibrium for Fusion + https://peppyhare.github.io/r/notes/UWAA558/07-equilibrium-for-fusion/ Mon, 01 Jan 0001 00:00:00 +0000 - https://peppyhare.github.io/r/docs/shortcodes/hints/ - Hints # Hint shortcode can be used as hint/alerts/notification block. -There are 3 colors to choose: info, warning and danger. -{{&lt; hint [info|warning|danger] &gt;}} **Markdown content** Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa {{&lt; /hint &gt;}} Example # Markdown content -Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa Markdown content + https://peppyhare.github.io/r/notes/UWAA558/07-equilibrium-for-fusion/ + Equilibrium for Fusion ( \( \beta \) function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: "$$", right: "$$", display: true }, { left: "\\[", right: "\\]", display: true }, { left: "$", right: "$", display: false }, { left: "\\(", right: "\\)", display: false }], macros: { "\\lcm": "\\mathop{\\mathrm{lcm}}", "\\sen": '\\text{sen}\\,', "\\dd": "\\mathop{\\mathrm{d} #1}", "\\abs": "\\lvert #1 \\rvert", "\\dd": "\\text{d}", "\\cross": "\\times", "\\pdv": "\\frac{\\partial #1}{\\partial #2}" - - https://peppyhare.github.io/r/docs/shortcodes/katex/ + 1-D Equilibria + https://peppyhare.github.io/r/notes/UWAA558/08-1d-equilibria/ Mon, 01 Jan 0001 00:00:00 +0000 - https://peppyhare.github.io/r/docs/shortcodes/katex/ - KaTeX # KaTeX shortcode let you render math typesetting in markdown document. See KaTeX -Example # {{&lt; katex [display] [class=&#34;text-center&#34;] &gt;}} f(x) = \int_{-\infty}^\infty\hat f(\xi)\,e^{2 \pi i \xi x}\,d\xi {{&lt; /katex &gt;}} document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: [ {left: "$$", right: "$$", display: true}, {left: "\\[", right: "\\]", display: true}, {left: "$", right: "$", display: false}, {left: "\\(", right: "\\)", display: false} ], macros: { &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;, &quot;\\sen&quot;: '\\text{sen}\\,', &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;, &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;, &quot;\\dd&quot;: &quot;\\text{d}&quot;, &quot;\\cross&quot; : &quot;\\times&quot;, &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;, &quot;\curl&quot;: &quot;\\nabla \\cross #1&quot;, &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;, &quot;\\dot&quot;: &quot;\\boldsymbol\\cdot&quot;, &quot;\\vdot&quot; : &quot;\\dot&quot;, &quot;\\div&quot;: &quot;\\nabla \\dot #1&quot;, &quot;\\grad&quot; : &quot;\\nabla&quot;, &quot;\\dv&quot; : &quot;\\frac{d\\,#1}{d\\,#2}&quot;, &quot;\\gr&quot; : &quot;\\gamma&quot;, &quot;\\vec&quot; : &quot;\\mathbf{#1}&quot;, &quot;\\tagl&quot; : &quot;\\tag{#1}&quot;, &quot;\\eqref&quot; : &quot;(#1)&quot;, &quot;\\laplacian&quot; : &quot;\\mathbf{\\nabla ^2}&quot;, &quot;\\label&quot; : &quot;{}&quot;, &quot;\\negadelta&quot; : &quot;\\nabla&quot;, } }); }); [f(x) = \int_{-\infty}^\infty\hat f(\xi),e^{2 \pi i \xi x},d\xi] + https://peppyhare.github.io/r/notes/UWAA558/08-1d-equilibria/ + 1-Dimensional Equilibria # The \( \theta \) function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: "$$", right: "$$", display: true }, { left: "\\[", right: "\\]", display: true }, { left: "$", right: "$", display: false }, { left: "\\(", right: "\\)", display: false }], macros: { "\\lcm": "\\mathop{\\mathrm{lcm}}", "\\sen": '\\text{sen}\\,', "\\dd": "\\mathop{\\mathrm{d} #1}", "\\abs": "\\lvert #1 \\rvert", "\\dd": "\\text{d}", "\\cross": "\\times", "\\pdv": "\\frac{\\partial #1}{\\partial #2}" - - https://peppyhare.github.io/r/docs/shortcodes/mermaid/ + 2D Equilibria + https://peppyhare.github.io/r/notes/UWAA558/09-2d-equilibria/ Mon, 01 Jan 0001 00:00:00 +0000 - https://peppyhare.github.io/r/docs/shortcodes/mermaid/ - Mermaid Chart # Mermaid is library for generating svg charts and diagrams from text. -Example # {{&lt; mermaid [class=&#34;text-center&#34;]&gt;}} sequenceDiagram Alice-&gt;&gt;Bob: Hello Bob, how are you? alt is sick Bob-&gt;&gt;Alice: Not so good :( else is well Bob-&gt;&gt;Alice: Feeling fresh like a daisy end opt Extra response Bob-&gt;&gt;Alice: Thanks for asking end {{&lt; /mermaid &gt;}} mermaid.initialize({ "flowchart": { "useMaxWidth":true }, "theme": "default" } ) sequenceDiagram Alice-Bob: Hello Bob, how are you? + https://peppyhare.github.io/r/notes/UWAA558/09-2d-equilibria/ + 2D Equilibria # Let&rsquo;s connect the ends of our 1D equilibria. Doing so is what gives us inherently toroidal configurations. From the 1-dimensional picture: + \[\vec j \cross \vec B = j_\theta B_z - j_z B_\theta \\ = \grad p = \dv{p}{r}\] function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: "$$", right: "$$", display: true }, { left: "\\[", right: "\\]", display: true }, { left: " - - https://peppyhare.github.io/r/docs/shortcodes/section/page1/ + Equilibrium of 3D Configurations + https://peppyhare.github.io/r/notes/UWAA558/10-equilibrium-of-3d-configurations/ Mon, 01 Jan 0001 00:00:00 +0000 - https://peppyhare.github.io/r/docs/shortcodes/section/page1/ - Page 1 # + https://peppyhare.github.io/r/notes/UWAA558/10-equilibrium-of-3d-configurations/ + Equilibrium of 3D Configurations # In 3 dimensions, we lose the axisymmetry that allowed us to reach the Grad-Shefranov equation and we need to solve the full momentum equation in three dimensions. This is not something that we can actually do in this class, and the existing codes that do this are quite sophisticated. +Some general features of 3D equilibria are: + No net toroidal current. This means that they tend to be steady-state configurations. - - https://peppyhare.github.io/r/docs/shortcodes/section/page2/ + MHD Stability + https://peppyhare.github.io/r/notes/UWAA558/11-mhd-stability/ Mon, 01 Jan 0001 00:00:00 +0000 - https://peppyhare.github.io/r/docs/shortcodes/section/page2/ - Page 2 # + https://peppyhare.github.io/r/notes/UWAA558/11-mhd-stability/ + MHD Stability # Equilibrium is simply a balance of forces that results in a steady state. Beyond equilibrium, stability is the tendency of a perturbation to return to equilibrium, rather than increasing. We are very interested in analyzing the stability of MHD equilibria, including the plasma dynamics, so we need to use the complete ideal MHD model. The MHD equations are non-linear, which means that any evolution/dynamics are also going to be non-linear. - - https://peppyhare.github.io/r/docs/shortcodes/tabs/ + Formulary + https://peppyhare.github.io/r/notes/UWAA558/formulary/ Mon, 01 Jan 0001 00:00:00 +0000 - https://peppyhare.github.io/r/docs/shortcodes/tabs/ - Tabs # Tabs let you organize content by context, for example installation instructions for each supported platform. -{{&lt; tabs &#34;uniqueid&#34; &gt;}} {{&lt; tab &#34;MacOS&#34; &gt;}} # MacOS Content {{&lt; /tab &gt;}} {{&lt; tab &#34;Linux&#34; &gt;}} # Linux Content {{&lt; /tab &gt;}} {{&lt; tab &#34;Windows&#34; &gt;}} # Windows Content {{&lt; /tab &gt;}} {{&lt; /tabs &gt;}} Example # MacOS MacOS This is tab MacOS content. -Lorem markdownum insigne. Olympo signis Delphis! + https://peppyhare.github.io/r/notes/UWAA558/formulary/ + \( \) function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: "$$", right: "$$", display: true }, { left: "\\[", right: "\\]", display: true }, { left: "$", right: "$", display: false }, { left: "\\(", right: "\\)", display: false }], macros: { "\\lcm": "\\mathop{\\mathrm{lcm}}", "\\sen": '\\text{sen}\\,', "\\dd": "\\mathop{\\mathrm{d} #1}", "\\abs": "\\lvert #1 \\rvert", "\\dd": "\\text{d}", "\\cross": "\\times", "\\pdv": "\\frac{\\partial #1}{\\partial #2}", "\\curl": "\\nabla \\cross #1" Crews (2018) - https://peppyhare.github.io/r/docs/notes/crews2018/ + https://peppyhare.github.io/r/notes/working/crews2018/ Mon, 01 Jan 0001 00:00:00 +0000 - https://peppyhare.github.io/r/docs/notes/crews2018/ + https://peppyhare.github.io/r/notes/working/crews2018/ Development of a Collisionless Plasma Kinetic Solver and an Investigation of One-Dimensional Plasma Waves and Instabilities # Shielded potential of a test electron: - document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: [ {left: "$$", right: "$$", display: true}, {left: "\\[", right: "\\]", display: true}, {left: "$", right: "$", display: false}, {left: "\\(", right: "\\)", display: false} ], macros: { "\\lcm": "\\mathop{\\mathrm{lcm}}", "\\sen": '\\text{sen}\\,', "\\dd": "\\mathop{\\mathrm{d} #1}", "\\abs": "\\lvert #1 \\rvert" + \[\phi(r) = \frac{-e}{4 \pi \epsilon_0 r} e ^{- r / \lambda_D}\] function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: "$$", right: "$$", display: true }, { left: "\\[", right: "\\]", display: true }, { left: "$", right: "$", display: false }, { left: " - \ No newline at end of file + diff --git a/docs/notes/UWAA558/01-syllabus/index.html b/docs/notes/UWAA558/01-syllabus/index.html new file mode 100644 index 00000000..be82253f --- /dev/null +++ b/docs/notes/UWAA558/01-syllabus/index.html @@ -0,0 +1,716 @@ + + + + + + + + + + + + +Syllabus | My Notes + + + + + + + + + + + + + + +
      + + +
      +
      + +
      + + + Syllabus + + +
      + + + + + + +
      + + + +

      + Syllabus + # +

      +

      The course topics planned for this section are (in rough order):

      +

      Particle Model, Boltzmann-Maxwell Model, Magnetohydrodynamic (MHD) Model, Region of Validity, Common Assumptions, Ideal MHD Model, General Properties (Equilibrium, Boundary Conditions, Conservation Laws, “Frozen-In” Flux)

      +

      Ideal MHD Equilibrium, Virial Theorem, Magnetic Flux Surfaces

      +

      One-Dimensional Equilibria, Theta-Pinch, Z-Pinch, Screw-Pinch, Safety Factor q

      +

      Two-Dimensional Equilibria, Toroidal Geometry, Grad-Shafranov Equation, Closed Flux Surfaces, Safety Factor q, Magnetic Shear, Magnetic Well, Shafranov Shift, Spheromak, Reversed Field Pinch (RFP), Tokamaks, Stellarators (Elmo Bumpy Torus)

      +

      MHD Stability, General Concepts, Linearized MHD, Exponential (Linear) Stability, Force Operator and Properties, Variational Formulation, Energy Principle, Intuitive Form of delta W, Classification of Instabilities (internal/external, pressure-drive/current-driven, kink/interchange/ballooning)

      +

      Stability of One-Dimensional Equilibria, Modal Analysis, Rayleigh-Taylor, Theta-Pinch, Z-Pinch (Kadomtsev Condition), Screw-Pinch (Kruskal-Shafranov Condition, Suydam Criterion), RFP, “Straight” Tokamak

      +

      Stability of Two-Dimensional Equilibria, Tokamak, Mercier Criterion, Elmo Bumpy Torus

      +

      Resistive (Tearing) Instabilities, Stability of Non-static Equilibria, Nonlinear Stability Effects

      +

      + Course Motivation + # +

      +

      Plasma phenomena tend to be hard to treat because of the span of relevant scales. You have ions, electrons, and photons interacting through electromagnetic interactions. There is a tremendous variation in mass across species, which leads to a large span of both spatial and temporal scales. The species can interact through both short scale collisions and long range interactions through EM forces. In contrast, in normal gas dynamics you may consider only the short-scale interactions. As a consequence, we can describe dispersive plasma waves.

      +

      For comparison, remember in gas dynamics, the speed of sound is

      + + + + + \[\frac{\omega}{k} = v_s = \pdv{\omega}{k} \qquad \text{(gas)}\] + + +

      Here the phase velocity + \( \frac{\omega}{k} \) + + is equal to the phase velocity + \( \pdv{\omega}{k} \) + +. In a plasma, we can have non-linear dispersion relations in which the phase and group velocity are different.

      + + \[\frac{\omega}{k} = v(\omega, k) \neq \pdv{\omega}{k} \qquad \text{(plasma)} \] + + +

      The number of particles we typically deal with in a laboratory plasma is roughly on the order of a mole of particles

      + + \[\text{particles} \sim O(10^{23})\] + + +

      With long-range interactions, we have a combinatorial explosion of interacting particles! It is not possible to track individual particles at such a scale, so we need much simpler plasma descriptions. These are the plasma models we will discuss in the next chapter. When simplifying our models, we need to pay careful attention to the simplifications we are making because in general, inaccurate physics lead to incorrect conclusions.

      +
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      + + + + + + + + + + + + + + + + + diff --git a/docs/notes/UWAA558/02-plasma-models/index.html b/docs/notes/UWAA558/02-plasma-models/index.html new file mode 100644 index 00000000..f0eb86e2 --- /dev/null +++ b/docs/notes/UWAA558/02-plasma-models/index.html @@ -0,0 +1,1887 @@ + + + + + + + + + + + + +Plasma Models | My Notes + + + + + + + + + + + + + + +
      + + +
      +
      + +
      + + + Plasma Models + + +
      + + + + +
      + + + +

      + Plasma Models + # +

      +

      + Working towards MHD + # +

      +

      Let’s start from a full-particle description with the goal of reaching a continuum description (kinetic model). Then, we’ll look at the forces on the separate species and form a multi-fluid model, finally simplifying to a single-fluid MHD model.

      +

      The most important question to ask ourselves is “when is this model going to be useful?” The MHD model is the mathematical model for magnetized plasmas that are treated as a fluid. This means that we can define a fluid element (some lil' box of plasma) and define the physical properties (mass, density, magnetization, etc.) of the element. We need to make some assumptions about scale in order to do this. In terms of spatial scales, we abstract properties below a discrete scale + + + + \( a_0 \) + + into the properties of a fluid element

      + + \[\frac{a_0}{L} \rightarrow 0 \qquad a_0 = \text{discrete scale} \qquad L = \text{spatial scale of interest}\] + + +

      Length scales smaller than the discrete scale will not be properly captured by the model, so scales like the particle radius will be meaningless in our fluid model.

      +

      + Plasma Definition + # +

      +

      A plasma is a quasi-neutral gas of charged and neutral particles which exhibit collective behavior. The particles (electrons, ions, neutrals) interact through EM fields and collisions.

      +

      Mathematically you would think that plasmas could be treated as individual particles. Doing so gives an N-body problem with classical interactions through the Lorentz force (Coulomb interaction removed by assumption of quasi-neutrality) and binary collisions. Each particle + \( i \) + + has a well-defined mass + \( m_i \) + + and charge + \( q_i \) + + which do not change in time. The governing equations are

      +

      + \[\dv{\vec v_i}{t} = \frac{q_i}{m_i} (\vec E + \vec v_i \cross \vec B) + \sum_{j \neq i} \left[ \left. \dv{\vec v_{ij}}{t} \right|_{coll} (\vec r_i - \vec r_j) \right]\] + + + + \[\dv{\vec r_i}{t} = \vec v_i\] + +

      +

      The fields E and B are described by the Maxwell equations

      +

      + \[\pdv{B}{t} = - \curl E\] + + + + \[\frac{1}{c^2} \pdv{E}{t} = \curl B - \mu_0 \sum_i q_i v_i \delta (r - r_i)\] + + + + \[\div B = 0\] + + + + \[\epsilon_0 \div E = \sum_i q_i \delta (r - r_i)\] + +

      +

      + Klimontovich Equation + # +

      +

      Re-writing the force relations as a statement of conservation in phase space, we get Klimontovich equation for species + \( \alpha \) + +

      + + \[\dv{N_\alpha}{t} = 0 = \pdv{N_\alpha}{t} + \pdv{}{q} \cdot (\dot{q} N_\alpha) + \pdv{}{p} \cdot (\dot{p}N_\alpha)\] + + +

      The particle phase space is defined by + + \[N_\alpha(p, q) = \sum_i \delta(p - p_i) \delta(q - q_i)\] + + +where + \( p \) + + and + \( q \) + + are generalized momentum and position coordinates. The resulting + \( N(p) \) + + looks very spiky, with nonzero values only at the exact values inhabited by particles. Unfortunately, that means that only the tools of discrete mathematics are applicable to the distribution, and we’re forbidden from our favorite tool (calculus). To make the analysis possible, we can smooth over the discreteness by performing an ensemble average of the Klimontovich equation. This gives us a statistical description using smooth distribution functions:

      + + \[f(x, v, t) \qquad \frac{(\text{no. of particles})}{(\text{unit distance})^3(\text{unit velocity})^3}\] + + +

      That is, + \( f(x, v, t) \) + + is the number of particles at position + \( x \) + + with velocity + \( v \) + + at time + \( t \) + +. We also work with normalized distributions which give the probability of finding a particle.

      +

      By our definition, we can integrate to get the total number of particles at time + \( t \) + +.

      + + \[\int \int \dd x \dd v f(x, v, t) = N(t)\] + + +

      The ensemble averaging process works well if the the number of particles is very large

      + + \[N \gg 1\] + + +

      Figure 12.1

      +

      See a kinetic theory text (e.g. Krall and Trivelpiece) for a full description of the ensemble averaging process.

      +

      Now we can write our Klimontovich equation in terms of continuous quantities

      + + \[\pdv{f}{t} + \left[ \pdv{}{q} \cdot (\dot{q} f) + \pdv{}{p} \cdot (\dot{p}f) \right] = \text{(cross terms)}\] + + +

      The collision terms now have an infinite number of cross terms. We call this the BBGKY (Bogoliubov–Born–Green–Kirkwood–Yvon) hierarchy.

      +

      Expressing as the Boltzmann equation (generally called the Boltzmann-Maxwell equation, since the solution requires solving for the electromagnetic fields of the Maxwell equations)

      + + \[\pdv{f_\alpha}{t} + \vec v \cdot \pdv{f_\alpha}{\vec x} + \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} = \left. \pdv{f_\alpha}{t} \right|_{coll}\] + + +

      We leave the collision term in. A lot of the work of kinetic theory is coming up with an applicable form of the collision operator which is appropriate but still simple enough to solve. We often write the collision operator as the product of binary collisions

      + + \[ \left. \pdv{f_\alpha}{t} \right|_{coll} = \sum_\beta C_{\alpha \beta}\] + + +

      Notice that the terms + \( \vec v \cdot \pdv{f_\alpha}{\vec x} \) + + and + \( \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \) + + are advection equations, advecting in + \( \vec x \) + + and + \( \vec v \) + + respectively. If we ignore the collision term (set the RHS to zero) we have the Vlasov equation.

      +

      Note that the fields E and B at any location are generated from the charges and currents of the entire plasma volume, including externally applied fields. As a result, there’s an inherent integrating process taking into account the sources across the whole volume that leads to long-range smoothly varying forces. This is in contrast to the collisional effects, which by their nature lead to very short range abrupt forces. It makes sense to make a distinction between the long-range electromagnetic forces and the short range collisional forces.

      +

      Because the Boltzmann-Maxwell model is inherently 6-dimensional, it is a very challenging model to implement. The B-M model provides a complete description, but it is often too detailed to solve.

      +

      If we solve the Vlasov equation for two parallel opposite beams, for example, we see that the

      +

      As it turns out, the integral, centroid, and variance are all that are required to fully describe a Maxwellian distribution

      + + \[f_{M, \alpha} (\vec v) = n_0 \left( \frac{ m_\alpha }{2 \pi T_\alpha} \right)^{3/2} \text{exp}\left[ - \frac{ \frac{1}{2} m_\alpha (v - v_\alpha)^2}{T_\alpha} \right]\] + + +

      In other words, we can write the Maxwellian distribution as + \( f_M(n_0, \vec v_\alpha, T_\alpha ; \vec v) \) + +. We care about Maxwellian distributions so much in plasma physics because it is the solution to the Boltzmann equation for + \( \left. \pdv{f_\alpha}{t} \right|_{coll} = 0 \) + + (Vlasov equation). That’s not saying that there are no collisions, it is saying that there are so many collisions that the effect is isotropic and the overall force is zero.

      +

      Another feature of the Maxwellian distribution is + \( \vec v \cdot \pdv{f_\alpha}{\vec x} = 0 \) + +. This is a famous result called the Boltzmann H-theorem, and says that any initial distribution will relax (and very quickly) to a Maxwellian distribution.

      +

      By replacing our velocity distribution with the associated Maxwell distribution, we arrive at a Plasma Fluid Model

      +
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      + +
      + + + Plasma Fluid Model + + +
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      + Plasma Fluid Model + # +

      +

      We take velocity moments of each of the pieces of the kinetic model:

      +
        +
      • distribution function, + + + + \( f_\alpha \) + + + \( \rightarrow \) + + fluid variables
        • +
      • Boltzmann Equation + \( \rightarrow \) + + governing equations describing the evolution of the fluid variables.
        • +
      +

      Starting with the zeroth moment (integral) of the distribution function:

      + + \[\int f_\alpha (\vec x, \vec v, t) \dd \vec v = n_\alpha(\vec x, t)\] + + +

      1st Moment (momentum):

      + + \[m_\alpha \int \vec v \cdot f_\alpha(\vec x, \vec v, t) \dd \vec v = \vec p_\alpha (\vec x, t) = m_\alpha n_\alpha \vec v_\alpha\] + + +

      which is to say that the velocity is the 1st moment divided by the zeroth moment

      + + \[v_\alpha = \frac{\int \vec v f_\alpha}{\int f_\alpha}\] + + +

      2nd Moment:

      + + \[\int \vec v \vec v f_\alpha (\vec x, \vec v, t) \dd \vec v = \vec E_\alpha(\vec x, t) (\text{energy tensor})\] + + +

      We can simplify the 2nd moment by taking a reduced 2nd moment. This means that we’re going to insert a dot product

      + + \[\int \vec v \cdot \vec v f_\alpha (\vec x, \vec v, t) \dd \vec v\] + + +

      Before moving forward, we want to define a “random” velocity. Note that

      + + \[\int \vec v f_\alpha - \vec v_\alpha \int f_\alpha = 0 \rightarrow \int (\vec v - \vec v_\alpha) f_\alpha \dd \vec v = 0\] + + +

      We can define a random velocity + \( \vec w = \vec v - \vec v_\alpha \) + +. It is random in the sense that it is a fluctuation about the mean velocity, and when we integrate it we get zero. We can use this to define the energy tensor using the mean velocity to get a meaningful result. The pressure tensor is the second moment, using the random velocity

      + + \[\vec P_\alpha = m_\alpha \int \vec w \vec w f_\alpha \dd \vec v = P_\alpha \overline{I} + \overline{\Pi}_\alpha\] + + +

      where we’ve decomposed the pressure into an isotropic value + \( P_\alpha \) + + and what’s called the Braginskii stress tensor + \( \overline{\Pi}_\alpha \) + +. The average isotropic pressure is given by the reduced 2nd moment:

      + + \[P_\alpha = n_\alpha T_\alpha = \frac{1}{3} \int m_\alpha \vec w \cdot \vec w f_\alpha\] + + +

      where the factor of + \( 1/3 \) + + comes from the number of degrees of freedom in our system. It is related to the thermodynamic factor + \( \gamma \) + + where + \( \gamma = \frac{DOF + 2}{DOF} \) + +. Now we can define the temperature + \( T_\alpha \) + + as

      + + \[T_\alpha (\vec x, t) = \frac{1}{DOF} \frac{\int m_\alpha \vec w \cdot \vec w f_\alpha \dd \vec v}{\int f_\alpha \dd \vec v}\] + + +

      Now that we’ve got + \( n_\alpha \) + +, + \( v_\alpha \) + +, and + \( T_\alpha \) + + we have what we need to define a Maxwellian distribution. Higher moments would be required to describe non-Maxwellian distribution functions. For example, the 3rd moment is called the skewness of the distribution. The 4th moment is the kurtosis. So on and so forth. These give a measure of degree of departure from a Maxwellian distribution, in which case it is often more useful to talk about the excess kurtosis, where the excess kurtosis of a Maxwellian is defined to be zero. You can continue to calculate the moment expansion, and in general it requires an infinite number of moments to describe an arbitrary distribution function. Because of the Boltzmann H-theorem and the tendency of plasmas to quickly relax to Maxwellian, we can usually get away with using just the first three moments.

      +

      Now, what are the governing equations? We get these by taking moments of the Boltzmann equation. Let’s proceed carefully in sections, so we’ll integrate each piece of the BE in terms.

      +

      + 0th Moment of Boltzmann Equation (Conservation) + # +

      + + \[\int \pdv{f_\alpha}{t} \dd \vec v + \int \vec v \cdot \pdv{f_\alpha}{x} \dd v + \int \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = \int \left. \pdv{f_\alpha}{t} \right|_{coll} \dd \vec v\] + + +

      First, we take + \( \int \pdv{f_\alpha}{t} \dd \vec v \) + +. Because t and v are both independent variables, with an argument of sufficient smoothness we can reverse the order of integration and differentiation

      + + \[\int \pdv{f_\alpha}{t} \dd \vec v = \pdv{}{t}\int f_\alpha \dd v = \pdv{n_\alpha}{t}\] + + +

      For + \( \int \vec v \cdot \pdv{f_\alpha}{x} \dd v \) + + we can perform an integration by parts

      + + \[ \int \vec v \cdot \pdv{f_\alpha}{x} \dd v = \int \pdv{}{\vec x} \cdot ( \vec v f_\alpha) \dd \vec v - \int f_\alpha \pdv{}{\vec x} \cdot \vec v \dd \vec v \\ + = \int \pdv{}{\vec x} \cdot ( \vec v f_\alpha) \dd \vec v \\ + = \pdv{}{\vec x} \cdot \int \vec v f_\alpha \dd \vec v \\ + = \pdv{}{\vec x} \cdot (n_\alpha \vec v_\alpha) = \div (n_\alpha \vec v_\alpha)\] + + +

      Once again, we’ve switched the order of integration of + \( x \) + + and + \( v \) + +, which we can only do because we have specified that + \( f \) + + is a distribution function, and as such meets the criterion of sufficient smoothness.

      +

      For the last part, we can write it as a surface integral

      + + \[\int \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = \oint f_\alpha \frac{q_\alpha}{m_\alpha} ( \vec E + \vec v \cross \vec B) \cdot \dd S_v - \int \frac{q_\alpha}{m_\alpha} f_\alpha \pdv{}{\vec v} \cdot (\vec E + \vec v \cross \vec B) \dd \vec v\] + + +

      For + \( f_\alpha \) + + to be well-defined, we require + \( \lim_{v \rightarrow \infty} v^3 f_\alpha = 0 \) + + so the surface term vanishes and we’re left with

      + + \[\int \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = - \int \frac{q_\alpha}{m_\alpha} f_\alpha \pdv{}{\vec v} \cdot (\vec E + \vec v \cross \vec B) \dd \vec v \] + + +

      Distributing the divergence through,

      +

      + \[\pdv{}{\vec v} \cdot \vec E = 0\] + + +because + \( \vec E \) + + is independent of + \( \vec v \) + +.

      +

      + \[\pdv{}{\vec v} \cdot ( \vec v \cross \vec B) = 0\] + + +because + \( \vec v \cross \vec B \) + + is always orthogonal to + \( \vec v \) + +.

      +

      We’ve been working through this in normal vector notation for familiarity, but there is another notation known as Einstein Tensor Notation using the lovely Levi-Civita symbol + \( \epsilon_{ijk} \) + +.

      + + \[\epsilon_{ijk} = 1 \qquad \text{even permutations of i j k} \\ + = -1 \qquad \text{odd permutations of i j k} \\ += 0 \qquad \text{any repeated indexes}\] + + +

      We can write out vector products as products of indices and operators, and any repeated indices are implicitly summed:

      + + \[\vec v \cross \vec B = \epsilon_{ijk} v_j B_k = \sum_{jk} \epsilon_{ijk} v_j B_k\] + + +

      where + \( \epsilon_{ijk} \) + + is defined to be 1 for even permutations of ijk, -1 for negative permutations if ijk, and 0 for any repeated indices.

      +

      We can now write the derivative with respect to + \( \vec v \) + + as

      + + \[\pdv{}{\vec v} \cdot \vec v \cross \vec B = \partial _{v_i} \epsilon_{ijk} v_j B_k \\ += \epsilon_{ijk} (\partial_{v_i} v_j) B_k\] + + +

      We see that we’re taking the derivative of the j-th component of velocity with respect to the i-th component of velocity, and that’s how we can most easily point out that the quantity is zero without relying on properties of 3-vector products.

      +

      We’ll also want to use the divergence in Einstein tensor notation

      + + \[\div \vec A = \partial_i A_j \delta_{ij} = \partial_i A_i\] + + +

      Finally, we come back to the collision term in the zeroth moment of the B-M equation

      + + \[\sum_\beta \int C_{\alpha \beta} \dd \vec v = 0\] + + +

      We can say the collision term is zero by making a physical argument, rather than a mathematical one. We assert that collisions cannot create or destroy particles. This assumption is now baked into our equations going forward, but is not always true! Ionization, recombination, fusion reactions all create/destroy species.

      +

      Finally, we’ve got

      + + \[\pdv{n_\alpha}{t} + \div ( n_\alpha \vec v_\alpha) = 0 \qquad \text{continuity equation}\] + + +

      So by taking the 0th moment of the Boltzmann Equation, we’ve arrived at the continuity equation for + \( n_\alpha \) + + by introducing + \( \vec v_\alpha \) + +.

      +

      + 1st Moment of Boltzmann Equation + # +

      + + \[\int \vec v \pdv{f_\alpha}{t} \dd \vec v + \int \vec v \vec v \pdv{f_\alpha}{\vec x} \dd \vec v + \int \vec v \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = \int \left. \vec v \pdv{f_\alpha}{t} \right|_{coll}\] + + +

      Term 1:

      + + \[\int \vec v \pdv{f_\alpha}{t} \dd \vec v = \pdv{}{t} \int \vec v f_\alpha \dd \vec v - \int \pdv{\vec v}{t} f_\alpha \dd \vec v \\ += \pdv{}{t} \left( n_\alpha \vec v_\alpha \right)\] + + +

      Before we get to the second term, let’s have an aside about dyad math. An outer product + \( \vec A \vec B \) + + gives a second-rank tensor. In Cartesian coordinates, it looks like

      + + \[\vec A \vec B = \begin{bmatrix} +A_x B_x & A_x B_y & A_x B_z \\ +A_y B_x & A_y B_y & A_y B_z \\ +A_z B_x & A_z B_y & A_z B_z \\ + \end{bmatrix}\] + + +

      A useful property to do with dot products:

      + + \[\vec A \vec B \cdot \vec C = \vec A ( \vec B \cdot \vec C ) = (\vec A \vec B) \cdot \vec C\] + + + + \[( \vec V \cdot \grad) \vec B = \vec V \cdot ( \grad \vec B)\] + + +

      Back to term 2:

      + + \[\int \vec v \vec v \cdot \pdv{f_\alpha}{\vec x} \dd \vec v = \int \vec v \cdot \pdv{}{\vec x} ( \vec v f_\alpha) \dd \vec v - \int \vec v f_\alpha \pdv{}{\vec x} \cdot \vec v \dd \vec v \\ + = \int \pdv{}{\vec x} \cdot ( \vec v \vec v f_\alpha) \dd \vec v - \int \vec v f_\alpha \cdot \pdv{\vec v}{\vec x} \dd \vec v \\ + = \pdv{}{\vec x} \cdot \int \vec v \vec v f_\alpha \dd \vec v\] + + +

      Re-expanding in terms of random velocity + \( \vec v = \vec v_\alpha + \vec w \) + +

      + + \[= \pdv{}{\vec x} \cdot \int (\vec v_\alpha + \vec w) (\vec v_\alpha + \vec w) f_\alpha \dd \vec v \\ += \pdv{}{\vec x} \cdot \int (\vec v_\alpha \vec v_\alpha + \vec v_\alpha \vec w + \vec w \vec v_\alpha + \vec w \vec w) f_\alpha \dd \vec v \\ += \pdv{}{\vec x} \cdot \left[ \vec v_\alpha \vec v_\alpha \int f_\alpha \dd \vec v + \vec v_\alpha \int \vec w f_\alpha \dd \vec v + \left( \int \vec w f_\alpha \dd \vec v \right) \vec v_\alpha + \int \vec w \vec w f_\alpha f_\alpha \dd \vec v \right]\] + + +

      The middle two terms are zero by the very definition of + \( \vec v_\alpha \) + +. The random velocity + \( \vec w \) + + is defined such that the integral of the random velocity over all phase space is zero.

      + + \[= \pdv{}{\vec x} \cdot \left( \vec v_\alpha \vec v_\alpha n_\alpha \right) + \pdv{}{\vec x} \cdot \frac{ \vec P_\alpha}{m_\alpha} \\ += \div (n_\alpha \vec v_\alpha \vec v_\alpha) + \frac{1}{m_\alpha} \div \vec P_\alpha\] + + +

      Substituting in the pressure tensor,

      + + \[= \div ( n_\alpha \vec v_\alpha \vec v_\alpha ) + \frac{1}{m_\alpha} \left( \grad P_\alpha + \div \vec \Pi _\alpha \right)\] + + +

      Moving on to term 3:

      + + \[\int \vec v \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = \int \frac{q_\alpha}{m_\alpha} \pdv{}{\vec v} \cdot \left[( \vec E + \vec v \cross \vec B) f_\alpha \right] \vec v \dd \vec v - \int \frac{q_\alpha}{m_\alpha} f_\alpha \pdv{}{\vec v} \cdot \left[ \vec E + \vec v \cross \vec B \right] \vec v \dd \vec v \\ += \int \frac{q_\alpha}{m_\alpha} \pdv{}{\vec v} \cdot \left[ f_\alpha (\vec E + \vec v \cross \vec B) \vec v \right] \dd \vec v - \int \frac{q\alpha}{m\alpha} f ( \vec E + \vec v\alpha \cross \vec B) \cdot \pdv{\vec v}{\vec v} \dd \vec v \\ += \frac{q\alpha}{m\alpha} \oint f\alpha ( \vec E + \vec v\alpha \cross \vec B) \vec v \cdot \dd \vec S_v - \frac{q\alpha}{m\alpha} \left[ \vec E \int f \dd \vec v\alpha - \vec B \cross \int \vec v f\alpha \dd \vec v \right] \\ += - \frac{q\alpha}{m\alpha} n\alpha (\vec E + v\alpha \cross \vec B)\] + + +

      Finally, the 4th term gives

      + + \[ \int \left. \vec v \pdv{f_\alpha}{t} \right|_{coll} = \sum_\beta \int \vec v C_{\alpha \beta} \dd \vec v \\ += \int \vec v C_{\alpha \alpha} \dd \vec v + \sum_{\beta \neq \alpha} \int \vec v C_{\alpha \beta} \dd \vec v\] + + +

      Collisions of like particles + \( \alpha \) + + do not result in a net change of momentum of species + \( \alpha \) + +, so all we have left is the change in momentum due to collisions of unlike particles

      + + \[= \sum_{\beta \neq \alpha} \left[ \vec v_{\alpha} \int C_{\alpha \beta} \dd \vec v + \int \vec w C_{\alpha \beta} \dd \vec v \right]\] + + +

      For the same reason as before, we assert that collisions between particles + \( \alpha \) + + and + \( \beta \) + + do not lead to the creation or destruction of any species, so the 0th moment + \( \int C_{\alpha \beta} \dd \vec v = 0\) + +. This leads to the conclusion that only random motion contributes to momentum transfer, not + \( \vec v_\alpha \) + +. Viscosity and friction are good examples of similar physical processes where bulk velocity does not transfer momentum, but random motion does.

      +

      The momentum transfer from + \( \alpha \) + + to + \( \beta \) + + must equal transfer from + \( \beta \) + + to + \( \alpha \) + +. Momentum is globally conserved.

      + + \[\int \vec w C_{\alpha \beta} \dd \vec v = - \int \vec w C_{\beta \alpha} \dd \vec v\] + + +

      Now we can finally write out the full momentum equation

      + + \[\pdv{}{t} (n_\alpha \vec v_\alpha ) + \div (n_\alpha \vec v_\alpha \vec v_\alpha) + \frac{1}{m_\alpha} \div \vec P_\alpha - \frac{q_\alpha}{m_\alpha} n_\alpha ( \vec E + \vec v _\alpha \cross \vec B) = \sum_{\beta \neq \alpha} \int \vec w C_{\alpha \beta} \dd \vec v\] + + +

      The collision term is often represented as the momentum transfer vector + \( \vec R_{\alpha \beta} \) + + to + \( \alpha \) + + from + \( \beta \) + +.

      +

      + \[\vec R_{\alpha \beta} \equiv m_\alpha \int \vec w C_{\alpha \beta} \dd \vec v\] + + + + \[\pdv{}{t} (m_\alpha n_\alpha v_\alpha) + \div (m_\alpha n_\alpha \vec v_\alpha \vec v_\alpha) + \grad \vec P_{\alpha} + \div \vec \Pi_{\alpha} - q_\alpha n_\alpha (\vec E + \vec v _\alpha \cross \vec B) = \sum_{\beta} R_{\alpha \beta}\] + +

      +

      Once again, we’ve written the above in a conservation law form. The terms that aren’t strictly conservation terms are the source term + \( q_\alpha n_\alpha (\vec E + \vec v _\alpha \cross \vec B) \) + + and sink term + \( \sum_{\beta} R_{\alpha \beta} \) + +.

      +

      Introducing the mass density

      + + \[\rho_\alpha = m_\alpha n_\alpha \qquad \text{mass density}\] + + +

      + \[\pdv{}{t} (\rho_\alpha \vec v _\alpha) + \div (\rho_\alpha \vec v_\alpha v_\alpha + \ldots \\ += \rho_\alpha \pdv{\vec v_\alpha}{t} + \vec v_\alpha \left( \pdv{\rho_\alpha}{t} + \div \rho_\alpha \vec v_\alpha \right) + \rho_\alpha \vec v_\alpha \cdot \grad \vec v_\alpha + \ldots\] + + +The term in parentheses is just the continuity equation, which is zero, and what’s left is a more usual form of the momentum equation

      + + \[\text{Momentum Equation:}\\ +\rho_\alpha \left(\pdv{\vec v_\alpha}{t} + \vec v_\alpha \cdot \grad \vec v_\alpha \right) + \grad \vec P_\alpha + \div \vec \Pi_\alpha - q_\alpha n_\alpha (\vec E + \vec v_\alpha \cross \vec B) = \sum_{\beta \neq \alpha} \vec R_{\alpha \beta}\] + + +

      So by taking the 1st moment of the BE, we arrive at the momentum equation for + \( \vec v_\alpha \) + + and we have introduced + \( \vec P_\alpha \) + +.

      +

      + Momentum transfer (collisions) + # +

      +

      Looking back at + \( \vec R_{\alpha \beta} \) + +, it’s worth noting that the actual situation is complicated by the potential for multi-body collisions. While it is true that the probability of, e.g. three-body collisions is small, but there are many three-body collisional processes which are extremely important in plasma physics. Three-body recombination is the primary loss term in the ionization balance, for example. For now the simplest model is to only include binary collisions that result in small angel deflections. If we suppose we only have two species with distributions that look like the following:

      +

      Figure 12.2

      +

      then we expect collisions between species to tend to drag the distributions towards each other. We should expect the collision-based momentum transfer to be proportional to the distance between the distributions, i.e. the total current density + \( \vec j \) + +.

      +

      For an ion-electron plasma, the collision momentum transfer vector (to ions from electrons) is

      + + \[\vec R_{ie} = - n_e e \eta \vec j\] + + +

      where

      + + \[\eta = \frac{m_e \nu_{ei}}{n e^2} \qquad \text{resistivity}\] + + + + \[\vec j = n e (\vec v_i - \vec v_e) \qquad \text {current density for Z = 1}\] + + +

      + 2nd Moment of Boltzmann Equation + # +

      +

      Finally we’re on to our last expansion term. In general, we would say that the second moment would be

      + + \[\int \vec v \vec v \pdv{f_\alpha}{t} \dd \vec v = \pdv{}{t} \int \vec v \vec v f_\alpha \dd \vec v = \pdv{}{t} \vec E_\alpha / m_\alpha \rightarrow \pdv{}{t} \vec P_\alpha\] + + +

      Usually we think of the second moment as giving us energy, a scalar, rather than a tensor equation that we have here. Since the pressure tensor is symmetric, the energy equation will contain 6 independent quantities. Together with the continuity and velocity equations, we have 10 quantities in total, so we call this expansion a 10-moment expansion.

      +

      Instead of the full tensor equation, we’re actually going to compute the reduced 2nd moment by contracting + \( \vec v \cdot \vec v \) + +

      + + \[\int \vec v \cdot \vec v \pdv{f_\alpha}{t} \dd v\] + + +

      So, let’s get after it:

      + + \[\int v^2 \pdv{f_\alpha}{t} \dd \vec v + \int v^2 \vec v \cdot \pdv{f_\alpha}{\vec x} \dd \vec v + \int v^2 \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = \left. \int v^2 \pdv{f_\alpha}{t} \right|_{coll} \dd \vec v\] + + +

      Simplifying… Term 1:

      + + \[\int v^2 \pdv{f_\alpha}{t} \dd \vec v = \int \pdv{}{t} (v^2 f_\alpha) \dd \vec v - \int \pdv{v^2}{t} f_\alpha \dd \vec v \\ += \pdv{}{t} \int (\vec v_\alpha + \vec w) \cdot (\vec v_\alpha + \vec w) f_\alpha \dd \vec v \\ += \pdv{}{t} \left[ \int v_\alpha ^2 f_\alpha \dd \vec v + 2 \int \vec v_\alpha \cdot \vec w f_\alpha \dd \vec v + \int w^2 f_\alpha \dd \vec v \right] \\ += \pdv{}{t} \left[ v_\alpha ^2 \int f_\alpha \dd \vec v + 2 \vec v_\alpha \cdot \int \vec w f_\alpha \dd \vec v + 3 \frac{n_\alpha T_\alpha}{m_\alpha} \right] \\ += \pdv{}{t} \left[ n_\alpha v_\alpha ^2 + 3 \frac{n_\alpha T_\alpha}{m_\alpha} \right]\] + + +

      Term 2:

      + + \[\int v^2 \vec v \cdot \pdv{f_\alpha}{\vec x} \dd \vec v = \int \pdv{}{\vec x} \cdot (v^2 \vec v f_\alpha) \dd \vec v - \int f_\alpha \pdv{}{\vec x} \cdot (v^2 \vec v) \dd \vec v \\ += \pdv{}{\vec x} \cdot \int ( \vec v_\alpha + \vec w) \cdot (\vec v_\alpha + \vec w) \vec w f_\alpha \dd \vec v \\ += \pdv{}{\vec x} \cdot \int (v_\alpha ^2 + 2 \vec v_\alpha \cdot \vec w + w^2) \vec v f_\alpha \dd \vec v \\ += \pdv{}{\vec x} \cdot \left[ \int v_\alpha ^2 (\vec v_\alpha + \vec w) f_\alpha \dd \vec v + \int 2 \vec v_\alpha \cdot \vec w (\vec v_\alpha + \vec w) f_\alpha \dd \vec v + \int w^2 (\vec v_\alpha + \vec w) f_\alpha \dd \vec v \right] \\ += \pdv{}{\vec x} \cdot \left[ v_\alpha ^2 \vec v_\alpha \int f_\alpha \dd \vec v + v_\alpha ^2 \int \vec w f_\alpha \dd \vec v + 2 \vec v_\alpha \cdot \left(\int \vec w f_\alpha \dd \vec v \right) \vec v_\alpha \\ +\qquad + 2 \vec v_\alpha \cdot \int \vec w \vec w f_\alpha \dd \vec v + \vec v_\alpha \int w^2 f_\alpha \dd \vec v + \int \vec w w^2 f_\alpha \dd \vec v \right] \\ += \pdv{}{\vec x} \cdot (n_\alpha v_\alpha ^2 \vec v_\alpha) + \pdv{}{\vec x} \cdot 2 \vec v_\alpha \cdot \frac{\vec P_\alpha}{m_\alpha} + \pdv{}{\vec x} \cdot \left(\frac{3 n_\alpha T_\alpha \vec v_\alpha}{m_\alpha} \right) + \pdv{}{\vec x} \cdot \left( \frac{2 \vec h_\alpha}{m_\alpha} \right)\] + + +

      In keeping with the progression so far, the very last term would be something like a full 3rd moment + \( \int \vec w \vec w \vec w f_\alpha \dd \vec v \) + +. Instead, we have a “contracted 3rd moment” of the distribution + \( \vec h_\alpha \) + + which is actually the heat flux, or the random energy flux of random energy.

      +

      Term 3:

      + + \[\int v^2 \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v \\ + = \oint f_\alpha \frac{q_\alpha}{m_\alpha} v^2 ( \vec E + \vec v \cross \vec B) \cdot \dd \vec S_v - \int f_\alpha \frac{q_\alpha}{m_\alpha} \pdv{}{\vec v} \cdot \left[ v^2 ( \vec E + \vec v \cross \vec B) \right] \dd \vec v \\ += - \frac{q_\alpha}{m_\alpha} \int f_\alpha \left[ v^2 \left(\pdv{}{\vec v} \cdot \vec E \right) + \left(\vec E \cdot \pdv{}{\vec v} v^2 \right) \right] \dd \vec v \\ +\qquad - \frac{q_\alpha}{m_\alpha}\int f_\alpha \left[v^2 \pdv{}{\vec v} \cdot (\vec v \cross \vec B) + \vec v \cross \vec B \cdot \pdv{}{\vec v} v^2 \right] \dd \vec v \\ += - \frac{q_\alpha}{m_\alpha} \int f_\alpha (2 \vec E \cdot \vec v) \dd \vec v \\ += - \frac{2 q_\alpha}{m_\alpha} n_\alpha \vec v_\alpha \cdot \vec E\] + + +

      We’ve ended up with the work done on a fluid by the electric field. As expected, there is no work term associated with the bulk magnetic field.

      +

      Term 4:

      + + \[\left. \int v^2 \pdv{f_\alpha}{t} \right|_{coll} \dd \vec v = \int v^2 C_{\alpha \alpha} \dd \vec v + \sum _{\beta \neq \alpha} \int v^2 C_{\alpha \beta} \dd \vec v \] + + +

      We can make an energy conservation argument to get rid of the first term. The first term goes to 0 since collisions of like particles results in no net energy exchange.

      + + \[= \sum_{\beta \neq \alpha} \int ( \vec v_\alpha + \vec w) \cdot(\vec v_\alpha + \vec w) C_{\alpha \beta} \dd \vec v \\ += \sum_{\beta \neq \alpha}\left( v_\alpha ^2 \int C_{\alpha \beta} \dd \vec v + 2 \vec v_\alpha \cdot \int \vec w C_{\alpha \beta} \dd \vec v + \int w^2 C_{\alpha \beta} \dd \vec v \right)\] + + +

      Since collisions (we assume) don’t create or destroy particles, the first term + \( v_\alpha ^2 \int C_{\alpha \beta} \dd \vec v \) + + goes away. The second term + \( \vec v_\alpha \cdot \int \vec w C_{\alpha \beta} \dd \vec v \) + + we interpret as a frictional force due to the relative motion between + \( \alpha \) + + and + \( \beta \) + +

      + + \[= \sum_{\beta \neq \alpha} 2 \frac{\vec v_\alpha \cdot \vec R_{\alpha \beta}}{m_\alpha} + 2 \frac{Q_{\alpha \beta}}{m_\alpha}\] + + +

      where + \( Q_{\alpha \beta} \) + + is the heat exchange term (heat generation term in Braginskii). It describes the heat exchange by random collisions of unlike particles, analogous to viscous heating.

      + + \[Q_{\alpha \beta} = \frac{1}{2} \int m_\alpha w^2 C_{\alpha \beta} \dd \vec v\] + + +

      If we were to draw distribution functions for species + \( \alpha \) + + and + \( \beta \) + + (as shown), even though they have the same centroid they will interact by viscous heating. Species + \( \beta \) + + will heat up and + \( \alpha \) + + will cool

      +

      Figure 12.3

      +

      If we multiply by + \( \frac{1}{2} m_\alpha \) + + then the energy equation becomes

      + + \[\text{Energy Equation: } \\ \qquad \pdv{}{t} \left( \frac{3}{2} n_\alpha T_\alpha + \frac{1}{2} \rho_\alpha v_\alpha ^2 \right) + \\ + \div \left(\frac{3}{2} n_\alpha T_\alpha + \frac{1}{2} \rho_\alpha v_\alpha ^2 \right) \vec v_\alpha + \div (\vec v_\alpha \cdot \vec P_\alpha) + \div \vec h_\alpha - q_\alpha n_\alpha \vec v_\alpha \cdot \vec E \\ = \sum_{\beta \neq \alpha} (v_\alpha \cdot \vec R_{\alpha \beta} + Q_{\alpha \beta})\] + + +

      Notice that the total energy appears

      + + \[E_\alpha = \frac{3}{2} n_\alpha T_\alpha + \frac{1}{2} \rho_\alpha v_\alpha ^2\] + + +

      The factor of + \( \frac{3}{2} \) + + as usual comes from + \( \gamma \) + +

      + + \[\frac{3}{2} = \frac{1}{\gamma - 1} = \frac{1}{5/3 - 1}\] + + +

      where

      + + \[\gamma = \frac{DOF + 2}{DOF} = \frac{5}{3}\] + + +

      We can remove the kinetic portion of the energy equation by subtracting the product of the momentum equation with + \( \vec v_\alpha \) + +.

      + + \[\rho_\alpha \vec v_\alpha \cdot \pdv{v_\alpha}{t} = \frac{1}{2} \pdv{}{t} (\rho_\alpha v_\alpha ^2)\] + + +

      + \[\pdv{}{t} \left( \frac{3}{2} n_\alpha T_\alpha \right) + \div \left(\frac{3}{2} n_\alpha T_\alpha \vec v_\alpha \right) + \div \left( \vec v_\alpha \cdot \vec P_\alpha \right) - \vec v_\alpha \cdot (\div \vec P_\alpha) + \div \vec h_\alpha = \sum_{\beta \neq \alpha} Q_{\alpha \beta}\] + + + + \[\frac{3}{2} n_\alpha \left( \pdv{T_\alpha}{t} + \vec v_\alpha \cdot \div T_\alpha \right) + \frac{3}{2} T_\alpha \left[ \pdv{n_\alpha}{t} + \div (n_\alpha \vec v_\alpha) \right] + \vec P_\alpha \cdot \cdot \grad \vec v_\alpha + \div \vec h_\alpha = \sum_{\beta \neq \alpha} Q_{\alpha \beta}\] + + + + \[\frac{3}{2} n_\alpha \left( \pdv{T_\alpha}{t} + \vec v_\alpha \cdot \div T_\alpha \right) + \vec P_\alpha \cdot \cdot \grad \vec v_\alpha + \div \vec h_\alpha = \sum_{\beta \neq \alpha} Q_{\alpha \beta}\] + +

      +

      Where + \( \vec P_\alpha \cdot \cdot \grad \vec v_\alpha \) + + is a tensor contraction in two indices. It is a generalization of the dot product, i.e. + + \[\vec P_\alpha \cdot \cdot \grad \vec v_\alpha = \delta_{ik} \delta_{jl} P_{ij} \partial_k v_l \\ += P_{xx} \partial_x v_x + P_{xy} \partial_y v_x + \ldots\] + +

      + + \[\frac{3}{2} n_\alpha \left( \pdv{T_\alpha}{t} + \vec v_\alpha \cdot \grad T_\alpha \right) + P_\alpha \div \vec v_\alpha + \vec \Pi_\alpha \cdot \cdot \grad \vec v_\alpha + \div \vec h_\alpha = \sum_{\beta \neq \alpha} Q_{\alpha \beta}\] + + +

      Now the continuity, momentum, and energy equations describe the evolution of each fluid + \( \alpha \) + +. The only velocity information we have retained in taking moments is the centroid + \( \vec v_\alpha \) + + and the width + \( T_\alpha \) + +. This is the only information we will include in the 5-moment model (5N-moment plasma fluid model). The higher moments (describing skewness, kurtosis) are not evolved with the 5-moment model. There are fluid models that do evolve those moments, but we won’t touch on those here.

      +

      + Closure Problem + # +

      +

      We still need to “close” the fluid model (i.e. solving the Closure Problem). As we have been finding out, each additional moment of the Boltzmann Equation introduces the next-higher moment. In calculating the 0th moment, we introduce the 1st moment, etc. We need to address the closure problem by relating higher moment variables to lower moment variables. Applying a closure scheme (relating variables that are not evolved directly) is usually equivalent to making a statement on heat flow.

      +

      In our model, we are not evolving the heat flux + \( \vec h_\alpha \) + +, so we need to relate it to one of the other variables that we do evolve directly. We usually do that by writing down a conductivity relation

      + + \[\vec h_\alpha = - \kappa \grad T_\alpha\] + + +

      for some conductivity + \( \kappa \) + +.

      +

      We’re also only evolving the scalar pressure, so we need to relate the stress tensor + \( \vec \Pi_\alpha \) + + to the other variables. We can do that with a viscosity relation

      + + \[\vec \Pi_\alpha = \nu \grad \vec v_\alpha\] + + +

      for some viscosity + \( \nu \) + +. We call the introduced quantities + \( \kappa \) + + and + \( \nu \) + + the transport coefficients. They are usually derived by performing an expansion of the equations we’ve already discussed. They are what lead to “nondimensional” numbers that characterize the flow in our system.

      +

      We can also achieve closure by constraining portions of the system. In an isothermal system

      +

      + \[\text{isothermal: } \quad T = \text{const.}, p \propto n\] + + + + \[\text{adiabatic: } \quad p \propto n^\gamma\] + + + + \[\text{force-free, cold: } \quad p = 0\] + +

      +

      The expressions that lead us to a closure scheme are the equations of state. They are not time-dependent, they simply relate fluid parameters to other properties of the system.

      +

      Maxwell’s equations to couple the electromagnetic terms to the fluid variables. + + \[\pdv{\vec B}{t} = - \curl \vec E\] + + + + \[\epsilon_0 \pdv{\vec E}{t} = \frac{1}{\mu_0} \curl \vec B - \sum_{alpha} q_\alpha n_\alpha \vec v_\alpha \] + + + + \[\epsilon_0 \div \vec E = \sum_\alpha q_\alpha n_\alpha \] + + + + \[\div \vec B = 0\] + +

      +

      The fluid variables provide the source terms for Ampere’s law and Gauss' law, and we’ve already seen how the electromagnetic fields appear in the momentum equations as source terms for the fluid forces.

      +

      + Stopping after the 2nd moment (5N-Moment Plasma Fluid Model) + # +

      +

      What assumptions have we made by stopping here? It’s important to know what they are and to recognize what they mean for the fluid model.

      +
        +
      • Each species + \( \alpha \) + + is well-represented by a Maxwellian with a small perturbation. The pressure tensor is not strictly diagonal and the heat flux is not zero, so the small perturbations are what lead to the transport coefficients. This is due to the process by which we obtain the transport coefficients, called the Chapman-Enskog expansion.
        • +
      • Kinetic effects (stream instabilities, counter-flow instabilities, etc.) are not captured.
        • +
      +

      The variables of the 5-moment model that we do evolve are + + \[n, v_x, v_y, v_z, T\] + +

      +

      We can take additional moments. The 10N-moment equation evolves, in addition to the quantities in the 5N-moment model, we evolve all of the independent terms of the pressure tensor + + \[n, v_x, v_y, v_z, P_{xx}, P_{xy}, P_{xz} , P_{yy} , P_{yz} , P_{zz} \] + +

      +

      The 13-N moment model has everything from the 10N-moment model plus the heat flux tensor + \( h_x, h_y, h_z \) + +

      +
      + + + + + + + + +
      + +
      + + + + +
      + + +
      + + + + + + + + + + + + + + + + + diff --git a/docs/notes/UWAA558/04-two-fluid-plasma-model/index.html b/docs/notes/UWAA558/04-two-fluid-plasma-model/index.html new file mode 100644 index 00000000..f4f2e141 --- /dev/null +++ b/docs/notes/UWAA558/04-two-fluid-plasma-model/index.html @@ -0,0 +1,2975 @@ + + + + + + + + + + + + +Two-Fluid Plasma Model | My Notes + + + + + + + + + + + + + + +
      + + +
      +
      + +
      + + + Two-Fluid Plasma Model + + +
      + + + + +
      + + + +

      + Two-Fluid Plasma Model (ions-electrons) + # +

      +

      Restricting our multi-species fluid model to ions and electrons, what can we say about wave behavior in a magnetized 2-fluid plasma? Let’s start with a cold plasma approximation ( + + + + \( p = 0 \) + +) and neglect collisions. The momentum equation reduces to

      + + \[m_\alpha \left( \pdv{\vec v_\alpha}{t} + \vec v_\alpha \cdot \grad \vec v_\alpha \right) - q_\alpha (\vec E + \vec v_\alpha \cross \vec B) = 0\] + + +

      From here on out we can avoid some clutter (and wrist strain) by dropping the + \( \alpha \) + + subscripts and acknowledging that we have sets of equations for ions and electrons. Apply a perturbation to an equilibrium + \( g = g_0 + g_1 \) + +

      + + \[m \left(\pdv{\vec v_0}{t} + \vec v_0 \cdot \grad \vec v_0 + \pdv{\vec v_0}{t} + \vec v_1 \cdot \grad \vec v_0 + \vec v_0 \cdot \grad \vec v_1 + \vec v_1 \cdot \grad \vec v_1 \right) \\ +\qquad - q (E_0 + E_1) - q (\vec v_0 \cross \vec B_0 + \vec v_1 \cross \vec B_0 + \vec v_0 \cross \vec B_1 + \vec v_1 \cross \vec B_1) = 0\] + + +

      We can drop some terms because equilibrium has to satisfy the original equation. We can balance all of the subscript-0 terms and sum them to get zero.

      + + \[m \left( \pdv{\vec v_1}{t} + \vec v_1 \cdot \grad \vec v_0 + \vec v_0 \cdot \grad \vec v_1 + \vec v_1 \cdot \grad \vec v_1 \right) + q (- E_1 - \vec v_1 \cross \vec B_0 - \vec v_0 \cross \vec B_1 - \vec v_1 \cross \vec B_1)\] + + +

      Let’s now make the assumption that the perturbation is small, that is + \( g_1 \ll g_0 \) + +. That means that nonlinear products of perturbation terms are negligible (linearization process).

      + + \[m \left( \pdv{\vec v_1}{t} + \vec v_0 \cdot \grad \vec v_1 + \vec v_1 \cdot \grad \vec v_0 \right) - q \vec E_1 - q (\vec v_1 \cross \vec B_0 + \vec v_0 \cross \vec B_1) = 0\] + + +

      Now, assume that the equilibrium is a static equilibrium, that is + \( \vec v_0 = 0 \) + +. If we decompose into components that are parallel and perpendicular to the equilibrium magnetic field + \( \vec B_0 \) + +, then

      +

      + \[\pdv{v_{1, \parallel}}{t} - \frac{q}{m} E_{1, \parallel} = 0\] + + + + \[\pdv{\vec v_{1, \perp}}{t} - \frac{q}{m} \left(\vec E_{1, \perp} + B_0 \vec v_{1, \perp} \cross \vu z \right) = 0\] + +

      +

      The parallel component + \( E_{1, \parallel} \) + + will lead us to the ordinary wave (O-wave). Consideration of the more general case with perpendicular components will lead to the X-wave.

      +

      The plasma velocity is related to the fields through the current density (Maxwell equations). Faraday’s law gives

      +

      + \[\pdv{\vec B}{t} = - \curl \vec E\] + + + + \[\rightarrow \curl \pdv{\vec B}{t} = - \curl \curl \vec E = - \grad (\div \vec E) + \nabla ^2 \vec E \] + +

      +

      Ampere’s law gives

      +

      + \[\epsilon_0 \pdv{\vec E}{t} = \frac{1}{\mu_0} \curl \vec B - \sum_{\alpha} q_\alpha n_\alpha \vec v_\alpha\] + + + + \[\epsilon_0 \pdv{^2 \vec E }{t ^2} = \frac{1}{\mu_0} \curl \pdv{\vec B}{t} \\ += \frac{1}{\mu_0} \left[ \nabla ^2 \vec E - \grad (\div \vec E) \right] - \sum_\alpha q_\alpha \pdv{}{t} (n_\alpha \vec v_\alpha) \] + +

      +

      Since this is a linear system, assume that the perturbed quantities have a wave-like structure. That is, the perturbed quantities + \( g_1 \) + + are proportional to + \( e^{i(\omega t + \vec k \cdot \vec r)} \) + +. This lets us transform the spatial and temporal derivatives into factors of + \( \omega \) + + and + \( \vec k \) + +

      + + \[- \epsilon_0 \omega ^2 \vec E_1 = - \frac{1}{\mu_0} \left[k^2 \vec E_1 - \vec k (\vec k \cdot \vec E_1) \right] + i \omega e n_0 \vec v_1\] + + +

      Let’s now consider only high frequency oscillations, assuming that only the electrons respond and the ions remain stationary. There’s nothing particularly complicated about including the ion response, this just lets us drop the + \( \alpha \) + + subscripts and focus on a single set of equations.

      + + \[- i \omega \frac{e n_0}{\epsilon_0} \vec v_1 = (\omega ^2 - c^2 k^2) \vec E_1 + c^2 \vec k (\vec k \cdot \vec E_1)\] + + +

      Now let’s apply the perturbed form to the linearized momentum equation

      + + \[\pdv{v_{1, \parallel}}{t} - \frac{q}{m} E_{1, \parallel} = 0 \\ +\rightarrow i \omega v_{1, \parallel} = - \frac{e}{m} E_{1, \parallel} \\ +\rightarrow i \omega \frac{e n_0}{\epsilon_0} v_{1, \parallel} = \frac{e^2 n_0}{\epsilon_0 m} E_{1, \parallel}\] + + +

      Combine the momentum equation and the Maxwell equations to eliminate + \( \vec E_1 \) + + and + \( \vec v_1 \) + +

      + + \[\frac{e^2 n_0}{\epsilon_0 m} E_{1, \parallel} = ( \omega ^2 - c^2 k^2 ) E_{1, \parallel} + c^2 k_{\parallel} \vec (k \cdot \vec E_1)\] + + +

      Consider different possibilities for the + \( \vec k \) + + vector. If it is along the magnetic field + \( \vec k = k_{\parallel} \vu{e}_\parallel \) + + (longitudinal wave) then

      + + \[\frac{e^2 n_0}{\epsilon_0 m} = \omega ^2 - c^2 k^2 + c^2 k^2 = \omega_{pe}^2\] + + +

      For + \( \vec k = k_{\perp} \vu e_\perp \) + + (transverse wave) then we get the dispersion relation for the O-wave

      + + \[\text{dispersion relation for O-waves:} \qquad \omega^2 - c^2 k^2 = \omega_p ^2\] + + +

      The electric field is in the same direction as the magnetic field + \( (\vec E_1 = \vec E_{1, \parallel}) \) + +, which means the O-wave is linearly polarized. At large + \( k \) + + we just have regular light waves, but as we turn the frequency downwards we see a cut-off at the plasma frequency:

      +

      Figure 12.4

      +

      It turns out that the dispersion relation for the X-wave has the same cut-off, but also has another branch with a resonance

      +

      Figure 12.5

      +

      The two-fluid plasma model is highly reduced from the full kinetic model, but it is still too complete to be useful when studying gross plasma behavior. Further reductions of the model are possible by making asymptotic assumptions:

      +

      + Low-frequency Asymptotic Assumption + # +

      +
        +
      • Eliminate high frequency, short wavelength phenomena by using pre-Maxwell field equations. Formally, this is + \( \epsilon_0 \rightarrow 0 \) + +.
        • +
      +

      The direct consequences of the low-frequency approximation are

      +

      + \[c^2 = \frac{1}{\epsilon_0 \mu_0} \qquad c \rightarrow \infty\] + + + + \[\omega_p ^2 = \frac{n e^2}{\epsilon_0 m} \qquad \omega_p \rightarrow \infty\] + + + + \[\lambda_D = \frac{v_T}{\omega_p} \rightarrow 0\] + +

      +

      This means that all phenomena will have + \( \omega \ll \omega_p \) + +, limiting the frequencies we can resolve to the ion plasma frequency. The characteristic speeds will be limited by the speed of light

      + + \[\frac{\omega}{k} \ll c\] + + +

      and all characteristic lengths will be much greater than the Debye length

      + + \[x_0 \gg \lambda_D\] + + +

      Looking at Gauss' law, + + \[\epsilon_0 \div \vec E = \sum_\alpha q_\alpha n_\alpha \rightarrow \sum_\alpha q_\alpha n_\alpha = 0\] + +

      +

      so we now have charge neutrality everywhere in the domain. For H plasma, locally we have + \( n_e = n_i \) + + everywhere.

      +

      Looking at Ampere’s law,

      + + \[\epsilon_0 \pdv{\vec E}{t} = \frac{1}{\mu_0} \curl \vec B - \sum_\alpha q_\alpha n_\alpha \vec v_\alpha = 0 \\ +\rightarrow \vec j = \frac{1}{\mu_0} \curl \vec B\] + + +

      Things we do not get from this approximation are + \( \vec E = 0 \) + + or + \( \pdv{\vec E}{t} = 0 \) + +. It does mean that plasma dynamics occur on a sufficiently large spatial scale that charge separation is small, and they occur on a sufficiently long temporal scale that electrons respond quickly.

      +

      + Tiny electron asymptotic assumption + # +

      +

      2nd approximation: neglect electron inertia in the momentum equation. Formally, we let the electron mass + \( m_e \rightarrow 0 \) + +

      +

      + \[\omega_{pe}^2 = \frac{n e^2}{\epsilon_0 m_e} \rightarrow \infty\] + + + + \[\omega_{c, e} = \frac{e B}{m_e} \rightarrow \infty\] + + + + \[v_{T, e} \rightarrow \infty\] + + +The Larmor radius goes to zero + + \[r_{l, e} = \frac{v_{T, e}}{\omega_{c, e}} \rightarrow 0\] + + +Importantly, as the gyroradius + \( r_{l, e} \) + + goes to 0 (because the thermal velocity goes as + \( \sqrt{m_e} \) + + and the cyclotron frequency goes as + \( m_e \) + +), this means that the electrons are tied to the magnetic field. +The skin depth is also small. + + \[\delta_e = \frac{c}{\omega_{p, e}} \rightarrow 0\] + +

      +

      So all phenomena that we capture must have + \( \omega \ll \omega_{p, e} \) + +, + \( \omega \ll \omega_{c, e} \) + +, and + \( x_0 \gg r_{L, e} \) + +, + \( x_0 \gg \delta_e \) + + .

      +

      The electron momentum equation becomes

      + + \[\grad P_e + \div \vec \Pi _e + e n_e (\vec E + \vec v \cross \vec B) = \sum_{\beta \neq \alpha} \vec R_{\alpha \beta}\] + + +

      The momentum equation is now a state equation, not an evolution equation. It now simply relates the dynamical variables to each other at any point in time.

      +

      Now, note that along magnetic field lines electrons can travel long distances at very fast (finite) speeds which can produce low frequency, long wavelength phenomena. Neglecting electron inertia implies that electrons respond instantaneously, meaning we cannot capture these modes. An example of such a phenomena is drift waves.

      +

      The characteristic speeds + \( c \) + + and + \( v_{T, e} \) + + have disappeared from the model. Remaining is + \( v_{T, i} \) + +. This means that the ion dynamics dictate the plasma evolution.

      +
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      + + + + + + + + + + + + + + + + + diff --git a/docs/notes/UWAA558/05-mhd-model/index.html b/docs/notes/UWAA558/05-mhd-model/index.html new file mode 100644 index 00000000..a080a0bd --- /dev/null +++ b/docs/notes/UWAA558/05-mhd-model/index.html @@ -0,0 +1,8090 @@ + + + + + + + + + + + + +Magnetohydrodynamic (MHD) Model | My Notes + + + + + + + + + + + + + + +
      + + +
      +
      + +
      + + + Magnetohydrodynamic (MHD) Model + + +
      + + + + +
      + + + +

      + Magnetohydrodynamic (MHD) Model + # +

      +

      Applying approximations to the two-fluid plasma model will allow us to arrive at a single-fluid (center-of-mass) description. The result is the ideal magnetohydrodynamic model (MHD).

      +

      First, define the MHD variables:

      + + + + + \[\text{mass density:} \qquad \rho = n_i m_i + n_i m_e\] + + + + \[\text{fluid velocity:} \qquad \vec v = \frac{n_i m_i \vec v_i + n_e m_e \vec v_e}{n_i m_i + n_e m_e}\] + + + + \[\text{current density:} \qquad \vec j = q_i n_i \vec v_i - e n_e \vec v_e \rightarrow \vec v_e = \frac{q_i n_i \vec v_i}{e n_e} - \vec j / e n_e\] + + + + \[\text{total pressure:} \qquad p = p_i + p_e = n_e T_e + n_i T_i\] + + + + \[\text{total temperature:} \qquad T = \frac{n_i T_i + n_e T_e}{(n_i + n_e)/2}\] + + +

      Now let’s begin applying asymptotic approximations. For the mass density, applying the first approx (charge neutrality) we have

      + + \[\rho \approx n (m_i + m_e)\] + + +

      Using approx 2 (vanishing electron mass)

      +

      + \[\rho \approx n m_i\] + + +where + + \[n = n_i \qquad n_e = Z n\] + +

      +

      The center-of-mass velocity (charge neutrality) gives

      + + \[\vec v \approx \frac{m_i \vec v_i + m_e Z \vec v_e}{m_i + Z m_e}\] + + +

      with small electron mass approximation:

      + + \[\vec v \approx \vec v_i\] + + +

      The current density is (charge neutrality approx)

      + + \[\vec j \approx Z e n (\vec v_i - \vec v_e) \qquad \vec v_e = \vec v_i - \vec j / Z e n\] + + +

      The pressure and total temperature are (with charge neutrality)

      + + \[P \approx n (T_i + Z T_e)\] + + + + \[T \approx \frac{T_i + Z T_e}{(1 + Z)/2}\] + + +

      + MHD Momentum Equation + # +

      +

      Now we combine the two-fluid equations with these asymptotic approximations to obtain the governing equations for the MHD variables. Multiplying the ion continuity equation by the ion mass gives

      + + \[\pdv{}{t} (m_i n_i) + \div (m_i n_i \vec v_i) = 0 \\ +\rightarrow \pdv{\rho}{t} + \div (\rho \vec v) = 0 \quad \text{(MHD continuity eq.)}\] + + +

      If we multiply the two-fluid continuity equations by the charge and sum them, we get

      + + \[\pdv{}{t}(q_i n_i - e n_e) + \div (q_i n_i \vec v_i - e n_e \vec v_e) = 0 \\ +\rightarrow \div \vec j = 0 \quad \text{(no accumulation of charge)}\] + + +

      If we add the electron and ion momentum equations and apply the small electron mass approximation, and recognizing that + \( \vec R_{ei} = - \vec R_{ie} = n e \eta \vec j \) + +

      +

      + \[\rho_i \left(\pdv{\vec i}{t} + \vec v_i \cdot \grad \vec v_i \right) + \grad (P_i + P_e) + \div (\vec \Pi _i + \vec \Pi_e) \\ +\qquad - (q_i n_i - e n_e) \vec E - (q_i n_i \vec v_i - e n_e \vec v_e) \cross \vec B = 0\] + + + + \[\text{MHD Momentum Equation} \\ +\rightarrow \rho \left(\pdv{\vec v}{t} + \vec v \cdot \grad \vec v \right) + \grad p - \vec j \cross \vec B = - \div (\vec \Pi _i + \vec \Pi _e)\] + +

      +

      !!! info “MHD Momentum Equation”

      +
      
+  \[    \rho \left(\pdv{\vec v}{t} + \vec v \cdot \grad \vec v \right) + \grad p - \vec j \cross \vec B = - \div (\vec \Pi _i + \vec \Pi _e)
+    \]
+
+
+
      +

      If we take the electron momentum equation, apply the small electron mass asymptotic approximation and introduce the current density, then we have

      +

      + MHD Ohm’s Law + # +

      + + \[e n_e (\vec E + \vec v_i \cross \vec B - \vec j \cross \vec B / Z e n) = - \grad P_e - \div \vec \Pi_e + \vec R_{ei}\] + + +

      Substituting + \( \vec R_{ei} = e n_e \eta \vec j \) + + + + \[\text{MHD Generalized Ohms Law}\\ +\vec E + \vec v \cross \vec B = \eta \vec j + \frac{1}{Z e n} \left(\vec j \cross \vec B - \grad p_e - \div \vec \Pi_e \right)\] + +

      +

      !!! info “MHD Generalized Ohms Law”

      +
      
+  \[    \vec E + \vec v \cross \vec B = \eta \vec j + \frac{1}{Z e n} \left(\vec j \cross \vec B - \grad p_e - \div \vec \Pi_e \right)
+    \]
+
+
+
      +

      + MHD Energy Equation + # +

      +

      The energy equation is found by adding the ion and electron energies, but first we need to manipulate them into a common form. The ion energy equation is

      + + \[\frac{1}{\gamma - 1} \left[ \pdv{}{t} (n_i T_i) + \div (n_i T_i \vec v_i) \right] + n_i T_i \div \vec v_i = RHS_i\] + + +

      where + \( RHS_i \equiv - \vec \Pi_i \cdot \cdot \grad \vec v_i - \div \vec h_i + Q_{ie} \) + +

      + + \[\frac{1}{\gamma - 1} \left( \pdv{p_i}{t} + \vec v_i \cdot \grad p_i \right) + \frac{\gamma}{\gamma - 1} p_i (\div \vec v_i) = RHS_i\] + + +

      Multiply by + \( \frac{\gamma - 1}{n_i ^\gamma} \) + + and define the total derivative with respect to the ion velocity as

      + + \[\dv{}{t} \equiv \pdv{}{t} + \vec v_i \cdot \grad\] + + + + \[\rightarrow \dv{}{t} \left( \frac{p_i}{n_i ^\gamma} \right) - p_i \dv{}{t} \left( \frac{1}{n_i ^\gamma} \right) + \frac{\gamma p_i}{n_i ^\gamma} \div \vec v_i = \frac{\gamma - 1}{n_i ^\gamma} RHS_i\] + + +

      Simplify further by recognizing that carrying out the total derivative gives

      + + \[\dv{}{t} \left( \frac{p_i}{n_i ^\gamma} \right) - \frac{\gamma p_i}{n_i ^{\gamma + 1}} \dv{}{t} n_i + \frac{\gamma p_i}{n_i ^\gamma} \div \vec v_i = \frac{\gamma - 1}{n_i ^\gamma} RHS_i\] + + +

      The continuity equation says + + \[\dv{}{t} n_i = - n_i \div \vec v_i\] + + +Multiply by + \( m_i ^{- \gamma} \) + + and use + \( \rho = n m_i \) + + and the resulting ion energy equation is

      + + \[\dv{}{t} \left( \frac{p_i}{\rho ^\gamma} \right) = \frac{\gamma - 1}{\rho ^\gamma} RHS_i\] + + +

      We want to get the electron energy equation in the same form. The steps are very similar:

      + + \[\dv{}{t} \left( \frac{p_e}{n_e ^\gamma} \right) + \vec v_e \cdot \grad \left( \frac{p_e}{n_e ^\gamma} \right) = \frac{\gamma - 1}{n_e ^\gamma} RHS_e\] + + +

      Since we’ve defined our center-of-mass reference frame to be that of the ions, we can not use the same total derivative

      + + \[\pdv{}{t} + \vec v_e \cdot \grad \neq \dv{}{t}\] + + +

      We now apply the charge neutrality approximation

      + + \[n_e = Z n \qquad \vec v_e = \vec v_i - \frac{1}{Z e n} \vec j\] + + + + \[\rightarrow \dv{}{t} \left( \frac{p_e}{Z^\gamma n^\gamma} \right) + \vec v_i \cdot \grad \left( \frac{p_e}{Z^\gamma n^\gamma} \right) = \frac{1}{Z e n} \vec j \cdot \grad \frac{p_e}{Z^\gamma n^\gamma} + \frac{\gamma - 1}{Z^\gamma n^\gamma} RHS_e\] + + +

      Multiply by + \( (Z / m_i)^\gamma \) + + and the result is

      + + \[\dv{}{t} \frac{p_e}{\rho ^\gamma} = \frac{1}{Z e n} \vec j \cdot \grad \frac{p_e}{\rho^\gamma} + \frac{\gamma - 1}{\rho^\gamma} RHS_e\] + + +

      Finally we can add the ion energy equation to the electron energy equation to get

      + + \[\dv{}{t} \left( \frac{p}{\rho^\gamma} \right) = \\ +\frac{\gamma - 1}{\rho^\gamma} \left[ Q_{ie} + Q_{ei} - \div (\vec h_i + \vec h_e) - \vec \pi_i \cdot \cdot \grad \vec v_i - \vec \Pi_e \cdot \cdot \grad \vec v_e \right] + \frac{\vec j}{Z e n} \cdot \grad \frac{p_e}{\rho^\gamma}\] + + +

      !!! info “MHD Energy Equation”

      +
      
+  \[    \dv{}{t} \left( \frac{p}{\rho^\gamma} \right) = \frac{\gamma - 1}{\rho^\gamma} \left[ Q_{ie} + Q_{ei} - \div (\vec h_i + \vec h_e) - \vec \pi_i \cdot \cdot \grad \vec v_i - \vec \Pi_e \cdot \cdot \grad \vec v_e \right] + \frac{\vec j}{Z e n} \cdot \grad \frac{p_e}{\rho^\gamma}
+    \]
+
+
+
      +

      Obviously, we’ve retained a number of terms that are specific to the behavior of the electrons. It is possible to incorporate the electron behavior by using a single-fluid MHD model with two temperatures + \( T_i \neq T_e \) + +. One can imagine a hierarchy of models, in which the most simplified is the single-fluid MHD model in which you evolve + \( \rho \) + +, + \( \vec v \) + +, and + \( T \) + +. Moving up a level, you have a MHD model with two temperatures in which you evolve + \( \rho \) + +, + \( \vec v \) + +, + \( T_i \) + +, and + \( T_e \) + +. Upwards from there you move back into the realm of multi-fluid models.

      +

      Now to relate the fields back to source terms. The low-frequency Maxwell’s equations are

      +

      + \[\pdv{\vec B}{t} = - \curl \vec E\] + + + + \[\vec j = \frac{1}{\mu_0} \curl \vec B\] + + + + \[\div \vec B = \div \vec E = 0\] + +

      +

      Like in the multi-fluid plasma model, we still need to close the system by expressing some of our variables using equations of state ( + \( \vec h \) + +, + \( \vec \Pi \) + +).

      +

      To simplify further, we can make some assumptions about heat flow

      +

      + \[\text{isothermal} \qquad \rightarrow p \propto n \qquad T = \text{const.} \qquad \gamma = 1\] + + + + \[\text{adiabatic} \qquad \rightarrow p \propto n^\gamma\] + + + + \[\text{cols plasma / force-free} \qquad \rightarrow p = \text{const.} \qquad \gamma = 0\] + +

      +

      + Ideal MHD Model + # +

      +

      The extended MHD equations are simpler than the two-fluid model, but they can still be quite complicated. We can often still get useful analysis from further reductions. The ideal MHD model is such a reduction that we can get by dropping (with justification) several terms from the extended model. We justify the simplifications by comparing the magnitude of the neglected terms to the terms that are retained.

      +

      Recall the characteristic speed is + \( v_{T, i} \) + +. If we say that the characteristic length plasma length is + \( L \) + +, then we can define characteristic time + \( \tau = L / v_{T, i} \) + +.

      +

      The derivation of the two-fluid plasma model assumed a Maxwellian velocity distribution. We need the velocity distribution to thermalize, reach local thermodynamic equilibrium, and become Maxwellian. This means that we need many collisions, in fact so many collisions occurring frequently enough that we can ignore collisional effects. There must then be many collisions during the characteristic time + \( \tau \) + +.

      +

      For ions to be thermalized,

      + + \[\frac{\tau_{ii}}{\tau} \ll 1\] + + +

      And similarly for electrons

      + + \[\frac{\tau_{e}}{\tau} \ll 1\] + + +

      The continuity equation remains unchanged from the extended MHD model

      +

      !!! info “Ideal MHD Continuity Equation”

      +
      
+  \[    \pdv{\rho}{t} + \div (\rho \vec v) = 0
+    \]
+
+    
+
      +

      The momentum equation is

      +

      + \[\rho \left( \pdv{\vec v}{t} + \vec v \cdot \grad \vec v \right) + \grad p - \vec j \cross \vec B = - \div (\vec \Pi_e + \vec \Pi_e)\] + + +Drop the anisotropic pressure

      +

      !!! info “Ideal MHD Momentum Equation”

      +
      
+  \[    \rho \left( \pdv{\vec v}{t} + \vec v \cdot \grad \vec v \right) + \grad p - \vec j \cross \vec B = 0
+    \]
+
+
+
      +

      The generalized Ohm’s law is

      + + \[\vec E + \vec v \cross \vec B = \eta \vec j + \frac{1}{Z e n} \left( \vec j \cross B - \grad p_e - \div \vec \Pi_e \right)\] + + +

      We’re going to drop the entire right hand side

      +

      !!! info “Ideal MHD Generalized Ohm’s Law”

      +
      
+  \[    \vec E + \vec v \cross \vec B = 0
+    \]
+
+
+
      +

      For the energy equation we have

      + + \[\dv{}{t} \left( \frac{p}{\rho^\gamma} \right) = \frac{\gamma - 1}{\rho^\gamma} [Q_{ie} + Q_{ei} - \div (\vec h_i + \vec h_e) - \vec \Pi_i \cdot \cdot \grad \vec v_i - \vec \Pi_e \cdot \cdot \grad \vec v_e] + \frac{\vec j}{Z e n} \cdot \grad \frac{p_e}{\rho^\gamma}\] + + +

      We neglect the entire right-hand side

      +

      !!! info “Ideal MHD Energy Equation”

      +
      
+  \[    \pdv{\rho}{t} + \div (p \vec v) = (1 - \gamma) p \div \vec v 
+    \]
+
+
+
      +

      + Collision/Pressure terms + # +

      +

      If we assume that the ions and electrons are in thermal equilibrium + \( T_i = T_e \) + +, we can relate the collision times

      + + \[\tau_{ee} : \tau_{ii} : \tau_{ei} = 1 : \left( \frac{m_i}{m_e} \right) ^{1/2} : \frac{m_i}{m_e}\] + + +

      The collision times are specifically collisional relaxation times of the Boltzmann equation

      + + \[\left. \pdv{f}{t} \right|_{coll} = \frac{f - f_{\text{Maxwellian}}}{\tau_{\alpha \beta}}\] + + +

      For electrons, the thermalization condition is much stricter for the ions

      + + \[\frac{\tau_{ee}}{\tau} = \left( \frac{m_e}{m_i} \right)^{1/2} \frac{\tau_{ii}}{\tau} \ll 1\] + + +

      Neglect the anisotropic pressure tensor in the momentum and generalized Ohm’s law, + \( \div \vec \Pi \) + +. + \( \vec \Pi \) + + is primarily the shear stress tensor. The ion thermal speed gives us a characteristic velocity for the plasma, so we use it to characterize the shear stress

      + + \[\vec \Pi_{i, max} \sim 2 \mu \left( \pdv{u}{x} - \frac{1}{3} \div \vec v \right) \sim \mu \frac{v_{T, i}}{L}\] + + +

      Standard treatments of the viscosity (Braginskii, etc.) show that viscosity scales with the number density, temperature, and collision time

      + + \[\mu \sim n T_i \tau_{ii} \sim p_i \tau_{ii}\] + + + + \[\Pi_{i, max} \sim p_i \frac{\tau_{ii} v_{T, i}}{L}\] + + +

      The specific term we want to get rid of is + \( \div \vec \Pi \) + +, so let’s compare it to a term we want to keep + \( \grad P \) + +

      + + \[\frac{\div \vec \Pi}{\grad p} \sim \frac{p_i \tau_{ii} v_{T, i} / L^2}{p_i / L} \sim \frac{\tau_{ii}}{\tau}\] + + +

      So, to neglect the anisotropic pressure term in the momentum equation, once again we require

      + + \[\frac{\tau_{ii}}{\tau} \ll 1\] + + +

      In other words, as long as the plasma is collision-dominated, we can drop the ion anisotropic pressure term. What about associated the electron term? If you can assume + \( T_i \approx T_e \) + +. Then + \( p_i \approx p_e \) + + for a neutral plasma, and

      + + \[\frac{\div \vec \Pi_e}{\grad p} \sim \frac{p_i \tau_{ee} v_{T, i} / L^2}{p_i / L} \sim \frac{\tau_{ee}}{\tau} \ll 1\] + + +

      + Magnetic terms + # +

      +

      In the generalized Ohm’s law, the diamagnetic drift term is

      + + \[\frac{\grad p_e}{Z e n} \sim \frac{n T_e / L}{Z en} \sim \frac{T_i / L}{Ze} \sim \frac{m_i v_{T, i}^2}{L Z e}\] + + +

      Compare + \( \grad p_e / Z e n \) + + to a term that we’re going to keep, which is the dynamo term + \( \vec v \cross \vec B \) + +

      + + \[\frac{ \grad p_e / Z en}{|\vec v \cross \vec B|} \sim \frac{m_i v_{T, i}^2}{LZe}{v_{T, i} B} \sim \frac{m_i}{ZeB}\frac{v_{T, i}}{L} = \frac{v_{T, i}}{\omega_{c, i}} \frac{1}{L} \sim \frac{r_{L, i}}{L} \ll 1\] + + +

      So to neglect the diamagnetic drift term, we need the plasma to be well-magnetized. This means the Larmor radius must be much less than the plasma characteristic length + \( r_{L, i} \ll L \) + +

      +

      Now what can we do with the Hall term + \( \frac{\vec j \cross \vec B}{Zen} \) + +. For a static plasma (or one with subsonic flows):

      + + \[\rho (\pdv{\vec v}{t} + \vec v \cdot \grad \vec v) - \grad p - \vec j \cross \vec B = 0\] + + +

      so by “subsonic” we mean that the static pressure is much larger than the dynamic pressure and we can discard the + \( \vec v \) + + terms.

      + + \[\vec j \cross \vec B \approx \grad p\] + + +

      Comparing the Hall term to the dynamo term + \( \vec v \cross \vec B \) + + also gives the same requirement

      + + \[\frac{r_{L, i}}{L} \ll 1\] + + +

      How well do some real plasmas hold up to the requirements of ideal MHD? Consider field nulls in a Z-pinch or an FRC, and weakly magnetized plasmas (such as those in a Hall thruster). Clearly, the magnetization requirement does not hold up at all points in space, and ideal MHD does not necessarily apply across the whole domain.

      +

      + Resistivity + # +

      +

      We neglect resistivity and the resistive electric field + \( \eta \vec j \) + + in the generalized Ohm’s law

      + + \[\eta \vec j \sim \eta \frac{\grad p}{B} \sim \frac{ \eta n T}{LB}\] + + + + \[\eta = \frac{m_e \nu_{ei}}{n e^2} = \frac{m_e}{n e^2 \tau_{ei}} = \frac{m_e ^2/ m_i}{n e^2 \tau_{ee}}\] + + + + \[\rightarrow \eta \vec j \sim \frac{m_e ^2}{e^2 L B} \frac{v_{T, i} ^2}{\tau_{ee}}\] + + +

      How does it compare to + \( \vec v \cross \vec B \) + +

      + + \[\frac{\eta \vec j}{|\vec v \cross \vec B|} \sim \left(\frac{m_e}{m_i} \right)^2 \frac{m_i}{e ^2 B^2} \frac{v_{T, i} ^2}{L^2} \frac{L}{v_{T, i} \tau_{ee}} \sim \left( \frac{m_e}{m_i}\right)^2 \frac{v_{T, i} ^2}{\omega_{c, i} ^2} \frac{1}{L^2} \sim \left( \frac{m_e}{m_i} \right) ^{3/2} \left( \frac{r_{L, i}}{L} \right) ^{2} \frac{\tau}{\tau_{ii}} \ll 1\] + + +

      This now places a lower limit on + \( \tau_{ii} \) + +; to neglect resistivity + \( \tau_{ii} \) + + cannot be too low

      + + \[\left( \frac{m_e}{m_i} \right) ^{3/2} \left( \frac{r_{L, i}}{L} \right) ^{2} \ll \frac{\tau_{ii}}{\tau} \ll 1\] + + +

      + Heating sources + # +

      +

      Neglect the collisional heating sources + \( Q_{ei} \) + +, + \( Q_{ie} \) + + in the energy equation. We do that by assuming that they are equal and opposite, which only happens when the temperatures are equal. In other words, we are again assuming local thermodynamic equilibrium between electrons and ions.

      + + \[1 \gg \frac{\tau_{ei}}{\tau} = \left( \frac{m_i}{m_e} \right)^{1/2} \frac{\tau_{ii}}{\tau} \rightarrow \frac{\tau_{ii}}{\tau} \ll \left( \frac{m_e}{m_i} \right) ^{1/2}\] + + +

      This is a much more restrictive condition than ion collisionality. Alternatively, we could track two temperature independently.

      +

      We also neglect the heat flux terms + \( \div (\vec h_i + \vec h_e) \) + + in the energy equation. Consider parallel (to the magnetic field) heat conduction which dominates since + \( \kappa_\perp \ll \kappa_\parallel \) + +, so

      + + \[\div \vec h \approx \grad_\parallel \cdot (\kappa_\parallel \grad_\parallel T)\] + + + + \[\grad_\parallel \cdot \left[ ( \kappa_{\parallel, i} + \kappa_{\parallel, e}) \grad_\parallel T \right]\] + + + + \[\kappa_{\parallel, e} \sim \frac{n T_e}{m_e} \tau_{ee} \qquad \text{ and } \qquad \kappa_{\parallel, i} \sim \frac{n T_i}{m_i} \tau_{ii}\] + + + + \[\frac{\kappa_{\parallel, e}}{\kappa_{\parallel, i}} \sim \frac{\tau_{ee}}{\tau_{ii}} \frac{m_i}{m_e} \sim \left( \frac{m_i}{m_e} \right) ^{1/2}\] + + +

      Compare the thermal conductivity to the rate of pressure change + \( \pdv{p}{t} \) + +

      + + \[\frac{\grad_\parallel \cdot (\kappa_{\parallel, e} \grad_\parallel T)}{\pdv{p}{t}} \sim \frac{n T^2 \tau_{ee}/m_e L^2}{nT/\tau} \sim \left( \frac{m_i}{m_e} \right)^{1/2} \frac{\tau_{ii}}{\tau} \ll 1\] + + +

      We’re back to the same requirement that we have ion-electron local thermodynamic equilibrium.

      +

      + Anisotropic pressure terms in energy equation + # +

      +

      Neglect the anisotropic pressure terms in the energy equation. That is, + \( \vec \Pi_i \cdot \cdot \grad \vec v_i \) + + and + \( \vec \Pi_e \cdot \cdot \grad \vec v_e \) + +. Skipping ahead, the result is

      +

      + \[\frac{\tau_{ii}}{\tau} \ll 1 \] + + + + \[\left( \frac{m_e}{m_i} \right)^{1/2} \frac{\tau_{ii}}{\tau} \frac{r_{L, i}}{L} \ll 1\] + +

      +

      + Electron convection term + # +

      +

      Lastly, we neglect the electron convection term in the energy equation + \( \frac{\vec j}{Zen} \cdot \grad \frac{P_e}{\rho^\gamma} \) + +. The result is

      + + \[\frac{r_{L, i}}{L} \ll 1\] + + +

      + Conservation Law Form of MHD + # +

      + + \[\pdv{}{t} q + \div \vec f = 0\] + + +

      We can express momentum in conservation law form:

      + + \[\pdv{}{t} (\rho \vec v) + \div \left[ \rho \vec v \vec v - \frac{ \vec B \vec B}{\mu_0} + (p + \frac{B^2}{2 \mu_0})\vec 1 \right] = 0\] + + +

      The conservation law form for the magnetic field looks like

      + + \[\pdv{ \vec B}{t} + \div \left[ \vec v \vec B - \vec B \vec v \right ] = 0\] + + +

      And of course the energy equation for

      + + \[E = \frac{1}{\gamma - 1} p + \frac{1}{2} \rho v^2 + \frac{ B^2}{2 \mu_0}\] + + + + \[\pdv{E}{t} + \div \left[ \left( E + p + \frac{B^2}{2 \mu_0} \right) - \left( \frac{ \vec B \cdot \vec v}{\mu_0} \right) \vec B \right] = 0\] + + +

      Conservation law forms are particularly useful when considering equilibrium steady-state force balance. This means that in steady-state equilibrium we have

      + + \[\div \left[ \rho \vec v \vec v - \frac{ \vec B \vec B}{\mu_0} + \left( p + \frac{B^2}{2 \mu_0} \right) \vec 1 \right] = 0\] + + +

      We can use this relationship and integrate over various volumes to determine the relationship between the various force balance terms

      +

      + Examples of equilibrium plasma confinement + # +

      +

      For a plasma in static equilibrium + \( \vec v = 0 \rightarrow \vec j \cross \vec B = \grad p \) + +

      + + \[ \div \vec T = 0 = \div \left( \right) + \] + + + + \[\vec j \cross \vec B = \frac{1}{\mu_0} (\curl \vec B) \cross \vec B = \frac{1}{\mu_0} (\vec B \cdot \grad \vec B - \frac{1}{2} \grad B^2) \\ += \frac{1}{\mu_0} ( B^2 \vu e_B \cdot \grad \vu e_B + \frac{1}{2} \vu e_B \vu e_B \cdot \grad B^2 - \frac{\grad B^2}{2} )\] + + +

      where + \( \vu e_B = \vec B / B \) + +

      +

      + \( \vu e_B \cdot \grad \vu e_B \) + + is the curvature of + \( \vec B \) + +. Write it like a curvature

      + + \[\vec K = - \vu r / R_c\] + + +

      + \[\vu e_B \vu e_B \cdot \grad B^2\] + + +is gradient of + \( B^2 \) + + that is parallel to B. Multiply by + \( e_B \) + + gives the component of gradient along + \( e_B \) + +. The difference between that and + \( \grad B^2 / 2 \) + + gives you the perpendicular gradient

      + + \[\vec j \cross \vec B = \frac{1}{\mu_0} (B^2 \vec \kappa - \frac{1}{2} \grad _\perp B^2) = \grad p = \grad_\perp p\] + + +

      identify + \( B^2 \vec \kappa \) + + is magnetic tension resulting from having a bent magnetic field line. + \( \frac{1}{2} \grad_\perp B^2 \) + + is magnetic pressure. They have to balance the plasma pressure at equilibrium.

      +

      For example, consider a cylindrical plasma that’s in equilibrium with a helical magnetic field + + \[\vec B = B_\theta (r) \vu \theta + B_z (r) \vu z\] + +

      +

      How is plasma pressure profile determined by the different components of the magnetic field? If we want to maximize the amount of pressure we confine, what should be maximized/minimized?

      + + \[\frac{B^2}{\mu_0} \vu \kappa = \grad _\perp ( p + \frac{B^2}{2 \mu_0} ) \] + + +

      + \( B_z \) + + is straight and has no curvature, so the only magnetic tension comes from + \( B_\theta \) + +, so the magnetic tension from + \( B_\theta \) + + must balance the total pressure.

      +

      Figure 12.6

      +

      The role of + \( B_z \) + + is displacing plasma pressure. The utility in defining + \( \beta \) + + as

      + + \[\beta = \frac{\text{plasma pressure}}{\text{magnetic pressure}}\] + + +

      + Conditions of Ideal MHD Validity + # +

      +

      The conditions for ideal MHD to be valid are

      +
        +
      1. High Ion Collisionality: + \( \frac{\tau_{ii}}{\tau} \ll 1 \) + +
        2. +
      2. Small ion Larmor radius: + \( \frac{r_{L, i}}{L} \ll 1 \) + +
        3. +
      3. Low resistivity: + \( \left(\frac{m_e}{m_i} \right)^{3/2} \left( \frac{r_{L, i}}{L} \right)^2 \ll 1 \) + +
        4. +
      +

      For a given plasma in force balance, we can relate the plasma pressure to the magnetic pressure

      + + \[\beta = \frac{n T}{B^2 / 2 \mu_0} = 4 \times 10^{-16} \frac{n_{cm^{-3}} T_{keV}}{B^2 _{T}}\] + + +

      Ion collision time is (Spitzer collisionality)

      +

      + \[\tau_{ii} = 2.09 \times 10^{7} \frac{T_{eV} ^{3/2} \mu ^{1/2}}{\ln \Lambda n_{cm ^{-3}}} \left[\text{s}\right]\] + + +where + \( \mu \equiv m_i / m_p \) + +.

      +

      Putting the conditions for ideal MHD in terms of + \( \beta \) + + and + \( \tau_{ii} \) + +,

      +
        +
      1. High collisionality: + + \[\frac{\tau_{ii}}{\tau} = 2.14 \times 10^{12} \frac{T^2}{n L} \ll 1 \] + + + + \[\rightarrow T_{eV} \ll 6.8 \times 10^{-7} L_{cm} ^{1/2} n_{cm^{-3}} ^{1/2}\] + +
        2. +
      2. Small gyroradius: + + \[\frac{r_{L, i}}{L} = 2.3 \times 10^7 \frac{1}{Z L} \sqrt{ \frac{\mu B}{n}} \ll 1\] + + + + \[\rightarrow n_{cm^{-3}} \gg 5.3 \times 10^{14} \frac{\mu B_{T}}{Z^2 L_{cm} ^2}\] + +
        3. +
      3. Low resistivity: + + \[\left( \frac{m_e}{m_i} \right) ^{3/2} \left( \frac{r_{L, i}}{L} \right) ^2 \frac{\tau}{\tau_{ii}} = 5.65 \frac{\mu B}{Z^2 L T^2}\] + + + + \[\rightarrow T_{eV} \gg 2.4 \frac{\mu ^{3/2} \beta^{1/2}}{Z L _{cm} ^{1.2}}\] + +
        4. +
      +

      Figure 12.7

      +

      + Perpendicular MHD + # +

      +

      For most configurations for magnetic fusion confinement, we are able to satisfy 2. and 3. but often have densities much too low to meet the high collisionality constraint. However, in practice ideal MHD does accurately model macroscopic behavior of many plasmas. At the same time, magnetized / fusion plasmas are often largely collisionless. We can understand why by re-writing the collisionality requirement as

      + + \[1 \gg \frac{\tau_{ii}}{\tau} = \frac{\tau_{ii} v_{T, i}}{\tau v_{T, i}} \sim \frac{\lambda}{L}\] + + +

      where + \( \lambda \) + + is the mean free path. The ratio + \( \lambda / L \) + + is also known as the Knudsen number. In magnetized plasmas, the mean free path is often very long, but the path between collisions can cover a great distance only by following magnetic field lines. Motion + \( \perp \) + + to the magnetic field is constrained by + \( r_{L, i} \) + +. This suggests an approach wherein we divide the plasma model into a 1-D kinetic model to be solved along the magnetic field and a 2-D MHD model + \( \perp \) + + to the magnetic field.

      +

      We consider diffusivity (terms like viscosity, conductivity) parallel and perpendicular to + \( \vec B \) + + + + \[k_\parallel \sim \frac{\lambda ^2}{\tau_{ii}}\] + + + + \[k_\perp \sim \frac{r_{L, i} ^2}{\tau_{ii}}\] + + + + \[\frac{k_\parallel}{k_\perp} \sim \left( \frac{\lambda}{r_{L, i}} \right)^2 \sim (\omega_{c, i} \tau_{ii} )^2\] + +

      +

      This changes one of our conditions for validity. Specifically, we can write

      + + \[\frac{r_{L, i}}{L} \frac{1}{ \omega_{c, i} \tau_{i i}} = 254 \frac{\mu \beta}{Z^2 L_{cm} T_{eV} ^2}\] + + +

      or

      + + \[T_{eV} \gg 16 \frac{\mu ^{1/2} \beta^{1/2}}{Z L^{1/2}}\] + + +

      This is now only slightly more restrictive than the low-resistivity condition. In fact, most of the plasmas that we looked at before (large tokamaks & toruses, propulsion systems) which had temperatures above the validity range now fall comfortably within the region of validity. The perpendicular MHD model is equivalent to a collisionless model, giving a much wider applicability than the collisional MHD model.

      +

      What about the parallel component? Our collisionality condition still isn’t valid parallel to the field. Basically, we ignore the parallel component, which is the same as assuming that + \( \rho \) + +, + \( T \) + +, + \( p \) + + are constant along magnetic field lines, with + \( \div \vec v = 0 \) + + . Let’s write out the expressions for collision-less MHD and for ideal MHD and compare:

      + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
      Collisionless MHDIdeal MHD
      Continuity + \( \pdv{\rho}{t} = 0 \) + + + \( \pdv{\rho}{t} = - \rho \div \vec v \) + +
      Momentum + \( \rho \dv{\vec v_\perp}{t} = \vec j \cross \vec B - \grad_\perp p \) + + + \( \rho \dv{\vec v}{t} = \vec j \cross \vec B - \grad p \) + +
      Parallel constraint + \( \vec B \cdot \grad \frac{v_\parallel}{B} = - \div \vec v _\perp \) + +
      Energy + \( \dv{p}{t} = 0 \) + + + \( \dv{p}{t} = - \gamma p \div \vec v \) + +
      +

      Collisionless MHD reproduces many of the effects of ideal MHD but has a wider region of validity. Corollary: ideal MHD is accurate beyond its region of validity, unless results lead to parallel gradients. For example, we know that MHD is not valid when representing confinement of a plasma confined in a magnetic mirror, which is an inherently kinetic phenomenon. But ideal MHD can generate parallel gradients within its region of validity, and we need to be careful. Ideal MHD does not require different models + \( \parallel \) + + and + \( \perp \) + + to the magnetic field, and is therefore preferred. We will continue to use ideal MHD outside of its region of validity.

      +
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      + + + + + + + + + + + + + + + + + diff --git a/docs/notes/UWAA558/06-boundary-conditions/index.html b/docs/notes/UWAA558/06-boundary-conditions/index.html new file mode 100644 index 00000000..4b51c84f --- /dev/null +++ b/docs/notes/UWAA558/06-boundary-conditions/index.html @@ -0,0 +1,2209 @@ + + + + + + + + + + + + +Boundary Conditions | My Notes + + + + + + + + + + + + + + +
      + + +
      +
      + +
      + + + Boundary Conditions + + +
      + + + + +
      + + + +

      + Boundary Conditions + # +

      +

      Mathematically, a well-posed problem requires both governing equations and a complete set of boundary conditions (the Cauchy data for the problem). The most common boundary conditions we use are perfectly conducting walls (flux surfaces) or a vacuum region.

      +

      + Perfectly Conducting Wall + # +

      +

      For the case where the plasma extends out to a perfectly conducting (impermeable) wall. Perfectly conducting walls do not support tangential electric field:

      + + + + + \[\left. \vec E_t \right|_{wall} = 0 \quad \rightarrow \quad \left. \vu n \cross \vec E \right| _{wall} = 0\] + + +

      Applying Faraday’s law at the wall,

      +

      + \[\left. \vu n \cdot \pdv{\vec B}{t} \right|_{wall} = \left. - \vu n \cdot \curl \vec E \right|_{wall} = \left. \div (\vu n \cross \vec E) \right| _{wall} = 0\] + + + + \[\pdv{}{t} \vu n \cdot \vec B |_{wall} = 0\] + + +If initially there is no normal magnetic field, then + + \[\vu n \cdot \vec B|_{wall} = 0 \quad \text{if initially true}\] + +

      +

      And of course, for an impermeable wall,

      + + \[\vu n \cdot \vec v |_{wall} = 0\] + + +

      Is this a sufficient set of boundary conditions? Think back to the governing equations in conservation form + + \[\pdv{}{t} \vec Q + \div \vec F = 0\] + + +The boundary conditions come into play when defining + \( \vec F \) + + at the boundary. In particular, we need to know what + \( \dd \vec S \cdot \vec F |_{wall} \) + + is. In our governing equations, this will involve + \( \vec E \) + +, + \( \vec B \) + +, and + \( \vec v \) + +.

      +

      + Insulating Boundary + # +

      +

      As a slight modification, an insulating boundary can have a tangential electric field. Consider a simple geometry of parallel electrodes with an insulating wall between them.

      +

      Figure 12.8

      +

      From Ohm’s law + + \[\vec E + \vec v \cross \vec B = 0\] + + +so the only way an electric field tangential to the wall can exist is if + \( \vu n \cdot \vec v \neq 0 \) + +.

      +

      For either a perfectly conducting or an insulating boundary, the other variables are arbitrary: + \( \rho \) + + , + \( p \) + +, + \( \vec v_t \) + +, + \( \vec B_t \) + +.

      +

      + Vacuum Region + # +

      +

      The plasma (radius + \( R_p \) + +) is supported by a region of vacuum out to a perfectly conducting wall at some radius + \( R_w \) + +. We assume that there is no plasma in the vacuum region. The governing equations in vacuum are just Maxwell’s equations

      + + \[\curl \vec B_{vac} = 0 \qquad \text{and} \qquad \div \vec B_{vac} = 0\] + + +

      At the wall, + + \[\vu n \cross \vec E |_{wall} = 0\] + + + + \[\left. \vu n \cdot \pdv{\vec B}{t} \right|_{wall} = 0\] + +

      +

      What happens at the plasma-vacuum interface? We need to specify jump conditions and continuity conditions. Let’s use square brackets to signify a jump:

      + + \[\left[ X \right] = \left. X \right|_{R_p + dr} - \left. X \right|_{R_p - dr}\] + + +

      The normal magnetic field has to be continuous.

      + + \[[\vu n \cdot \vec B]_{R_p} = 0\] + + +

      The tangential magnetic field jump is given by the surface current density at the jump

      + + \[\left[ \vu n \cross \vec B \right] _{R_p} = \mu_0 \vec K \] + + +

      Integrating + \( \grad_\perp (p + \frac{B^2}{2 \mu_0}) = \frac{B^2}{\mu_0} \vec \kappa \) + + over a differential volume across the surface gives

      + + \[\left[ p + \frac{B^2}{2 \mu_0} \right] _{R_p} = 0\] + + +

      The plasma shape is determined self-consistently by the wall shape and surface current. This is a free-boundary problem. Another option is to specify the plasma shape, and then determine the required wall shape. This is a fixed-boundary problem.

      +

      The most realistic case includes externally applied magnetic fields coming from source coils, perhaps computed by Biot-Savart law. The vacuum magnetic field is then + \( \vec B_{vac} = \vec B_{ext} + \vec B_{plasma} \) + +. The crazy coil shapes in the stellarator design come from the 3D geometry computations solving this problem.

      +

      + Conservation of Magnetic Flux (“Frozen-In” Flux) + # +

      +

      Locally, + \( \vec E + \vec v \cross \vec B = 0 \) + + with Faraday’s law + + \[\pdv{B}{t} = - \curl \vec E = - B \div \vec v + \vec B \cdot \grad \vec v - \vec v \cdot \grad B\] + + +From the continuity equation, + + \[\pdv{\rho}{t} + \vec v \cdot \grad \rho = - \rho \div \vec v\] + + +Combining we find that + + \[\dv{\vec B}{t} = \frac{\vec B}{\rho} \dv{\rho}{t} + \vec B \cdot \grad \vec v\] + + + + \[\rightarrow \dv{}{t} \left( \frac{\vec B}{\rho} \right) = \frac{\vec B}{\rho} \cdot \grad \vec v\] + +

      +

      This says that the field and plasma density move together. Locally, if the magnetic field increases then mass density increases, such that the ratio + \( \vec B / \rho \) + + remains constant. In the direction parallel to the magnetic field we have a term that involves field line twisting, which is a bit more complicated, but in the perpendicular direction + + \[\dv{}{t} \left( \frac{\vec B}{\rho} \right) _\perp = 0\] + +

      +

      If we consider globally the magnetic flux through a moving surface S at velocity + \( \vec u \) + +. The magnetic flux penetrating the surface is + + \[\Psi = \int \vec B \cdot \dd \vec S\] + + +or + + \[\dv{\Psi}{t} = \int \dv{\vec B}{t} \cdot \vu n \dd S\] + + + + \[= \int \pdv{\vec B}{t}\cdot \vu n \dd S + \oint \vec B \cross \vec u \dd \vec l\] + + +Using Faraday’s law

      +

      + \[\dv{\Psi}{t} = \int - \curl \vec E \cdot \vu n \dd S + \oint \vec B \cross \vec u \cdot \dd \vec l\] + + + + \[= \oint (- \vec E + \vec B \cross \vec u) \cdot \dd \vec l\] + + +Using the electric field from Ohm’s law + + \[\dv{\Psi}{t} = \oint(\vec v - \vec u) \cross \vec B \cdot \dd \vec l\] + + +This tells us that if the surface moves with the plasma + \( \vec u = \vec v \) + + then + + \[\dv{\Psi}{t} = 0\] + + +the flux through the surface is constant, and the flux is a constant of the topology. This is a direct consequence of ideal MHD. If we add even a small amount of resistivity, we dramatically alter the results in a process called “tearing” where the magnetic field “tears” and reconnects with itself.

      +
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      + + + + +
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      + + + + + + + + + + + + + + + + + diff --git a/docs/notes/UWAA558/07-equilibrium-for-fusion/index.html b/docs/notes/UWAA558/07-equilibrium-for-fusion/index.html new file mode 100644 index 00000000..69732c95 --- /dev/null +++ b/docs/notes/UWAA558/07-equilibrium-for-fusion/index.html @@ -0,0 +1,4404 @@ + + + + + + + + + + + + +Equilibrium for Fusion | My Notes + + + + + + + + + + + + + + +
      + + +
      +
      + +
      + + + Equilibrium for Fusion + + +
      + + + + +
      + + + +

      + Equilibrium for Fusion ( + + + + \( \beta \) + +) + # +

      +

      For a fusion device we would like to determine a magnetic configuration that confines plasma while it fuses. At fusion temperatures, the power required to maintain the equilibrium will be substantial. For a device to be useful, the power required to sustain the equilibrium must be less than the power released from fusion. Important loss terms for a confined plasma are transport (thermal conduction primarily) and radiation terms. The scaling factors are + \( P_{Brem} \sim n^2 T^{1/2} \) + + and + \( P_{cycl} \sim n^2 T^2 \) + + for radiation, and + \( P_L \sim \frac{3nT}{\tau_E} \) + + for thermal losses.

      +

      We know that the fusion source term will primarily come from the DT fusion reaction

      + + \[\text{D} + \text{T} \rightarrow \text{He}^4 (3.5\, MeV) + \text{n} (14.1\, MeV)\] + + +

      The primary fusion reaction releases an + \( \alpha \) + +-particle and a high-energy neutron. The concept of ignition is that the neutron leaves the plasma, and the + \( \alpha \) + + (with energy + \( E_\alpha = 3.5 MeV) \) + + remains to heat the plasma.

      + + \[P_\alpha = \frac{1}{4} n^2 \langle \sigma v \rangle E_\alpha \qquad \text{(assuming} \quad n_D = n_T = n/2 \text{)}\] + + + + \[P_\alpha > P_L \quad \rightarrow \quad n \tau_E > \frac{12 T}{E_\alpha \langle \sigma v \rangle}\] + + +

      To sustain fusion, we set the fusion heating term above the thermal loss term. The reaction cross-section + \( \sigma \) + + can be maximized to give the Lawson criterion

      + + \[n \tau_E > 10^{14} s / cm^3\] + + +

      The Lawson criterion only applies at fusion temperatures, but it is a useful parameter even outside of ignition since it gives a ratio of fusion power to lost power

      + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
      + \( T_i (keV) \) + + + \( \langle \sigma v \rangle (cm^3 / s) \) + +Required + \( n \tau_E (s / cm^3) \) + +
      1 + \( 7 \cdot 10^{-21} \) + + + \( 5 \cdot 10^{17} \) + +
      5 + \( 1.4 \cdot 10^{-17} \) + + + \( 1.2 \cdot 10^{15} \) + +
      20 + \( 4.3 \cdot 10^{-16} \) + + + \( 1.6 \cdot 10^{14} \) + +
      60 + \( 8.7 \cdot 10^{-16} \) + + + \( 2.4 \cdot 10^{14} \) + +
      +

      We can see that the required + \( n \tau_E \) + + actually has a minimum around + \( 20 keV \) + + (at least, as far as the data in the table goes). Even though the maximum cross-section is at a much higher temperature, what we’re really concerned with is the ratio of the fusion source term to the thermal loss term, which is linear in temperature.

      +

      MHD equilibrium does not place a limit on the density + \( n \) + +. Instead, it places a limit on + \( \beta \) + + in order to achieve equilibrium force-balance + \( (\beta = 1) \) + +

      + + \[\beta = \frac{n (T_e + T_i)}{B^2 / 2 \mu_0} \rightarrow n = \frac{ \beta B^2}{4 \mu_0 T}\] + + +

      In this form, we can more clearly see what our options are to achieve MHD equilibrium. Some devices (large-scale tokamaks) are able to achieve the requisite confinement time at a low + \( \beta \) + + by making use of very strong magnetic fields. Other devices are able to make use of more modest magnetic fields by working at a higher + \( \beta \) + +.

      +

      Therefore,

      + + \[\tau_E > \frac{1}{\beta B^2} \frac{48 \mu_0}{E_\alpha} \frac{T^2}{\langle \sigma v \rangle}\] + + +

      The term + \( \frac{T^2}{\langle \sigma v \rangle} \) + + has a minimum at + \( 10-20 keV \) + +. At 15 keV and a magnetic field of + \( 5T \) + + (many actual components cannot reasonably exceed such magnetic fields) then

      + + \[\tau_E > \frac{0.1}{\beta} \text{s}\] + + +

      For a large-scale toroidal device with + \( \beta = 1\% \) + +, the confinement time + \( \tau_E > 10s \) + +. If we consider a common diffusivity (how fast energy will leave due to thermal conductivity) + \( D_E \approx 1 m^2 / s\) + +, so for a characteristic radius + \( a \) + +

      + + \[\tau_E \approx \frac{a^2}{4 D_E} \rightarrow a > 6.3 \text{m}\] + + +

      This gives you a sense of why low- + \( \beta \) + + devices need to be so large. Instead, if we consider + \( \beta \sim 50\% \) + +, + \( \tau_E > 0.2 \text{s} \) + + and

      + + \[\beta \sim 50\% \rightarrow a > 0.9 \text{m}\] + + +

      When you consider that the cost of a device (to first order) scales with the volume of the device, achieving a high + \( \beta \) + + is very important for fusion equilibrium. However, when we consider MHD stability we are generally forced into lower + \( \beta \) + + to avoid destructive instabilities. Configuration optimization is the process of balancing this trade-off.

      +

      + Virial Theorem + # +

      +

      Application of the virial theorem to energy balance for the stress tensor + \( \vec T \) + + tells us that MHD equilibria must be supported by externally supplied currents. Many times you’ll hear of theoretical designs for compact toroid devices which can maintain stability under their own currents, but they are the MHD stability equivalent of a perpetual motion machine. A compact toroid cannot exist unsupported.

      +

      Writing static equilibrium:

      + + \[\div \left[ - \frac{ \vec B \vec B}{\mu_0} + \left( p + \frac{B^2}{2 \mu_0}\right) \vec I \right] = \div \vec T = 0\] + + +

      If we define the direction of the magnetic field to be + \( \vu e _B = \vu z \) + + then

      +

      + \[\vec T = p_\perp ( \vec I - \vu e_B \vu e_B ) + p_\parallel \vu e_B \vu e_B \\ += \begin{bmatrix} p_\perp & 0 & 0 \\ 0 & p_\perp & 0 \\ 0 & 0 & p_\parallel \end{bmatrix}\] + + +where

      + + \[p_\perp = p + \frac{B^2}{2 \mu_0}\] + + +

      and + + \[p_\parallel = p - \frac{B^2}{2 \mu_0}\] + +

      +

      A gradient vector identity gives

      + + \[\div (\vec r \cdot \vec T) = \vec r \cdot ( \div \vec T) + \vec T \cdot \cdot \grad \vec r\] + + +

      Integrating this expression over a volume and assuming that the volume contains a confined MHD equilibrium that is self-contained and self-supported:

      +

      Figure 12.10

      + + \[\int_V \div ( \vec r \cdot \vec T) \dd V = \int_V (\vec r \cdot \overbrace{\cancel{(\div \vec T)}}^{\text{MHD equil.}} + \vec T \cdot \cdot \grad \vec r) \dd V\] + + + + \[\grad \vec r = \vec I\] + + +

      so + + \[\vec T \cdot \cdot \grad \vec r = p_\perp + p_\perp + p_\parallel \\ += 3p + \frac{B^2}{2 \mu_0}\] + +

      + + \[\int_V (3p + \frac{B^2}{2 \mu_0} ) \dd V = \int _V \div ( \vec r \cdot \vec T) \dd V = \oint _S (\vec r \cdot \vec T) \cdot \vu n \dd S \\ += \oint _S \left[ \vec r \cdot \vec I p_\perp + \vec r \cdot \vu e_B \vu e_B (p_\parallel - p_\perp) \right]\cdot \vu n \dd S \\ +=\oint \left[ \left( \cancel{p} + \frac{B^2}{2 \mu_0} \right) \vu r \cdot \vu n - \frac{B^2}{\mu_0} (\vec r \cdot \vu e_B)(\vu e_B \cdot \vu n) \right] \dd S\] + + +

      Beyond where the plasma is contained, the pressure does not contribute + \( p = 0 \) + +. If all current sources are contained in the configuration, the magnetic field + \( \sim 1/r^3 \) + + for a dipole, + \( \sim 1/r^4 \) + + for a quadrupole, etc. Therefore the right-hand side will fall off like

      +

      + \[RHS \propto \oint_S B^2 r \dd S \propto \left( \frac{1}{r^3} \right) ^2 r r^2 \propto \frac{1}{r^3} \text{(dipole)}\] + + +so + \( RHS \rightarrow 0 \) + + as + \( r \rightarrow \infty \) + +. But what about the left-hand side? Both of the terms in the volume integral are positive definite, so the LHS must be positive finite and the equality can’t possibly hold. The assumption that the plasma is self-contained must be invalid. This tells us that we must have external currents.

      +

      + Magnetic Flux Surfaces + # +

      +

      The vast difference in thermal conductivity parallel and perpendicular to the magnetic field in a plasma confinement configuration leads to an avoidance of any open field lines. Magnetic equilibria are generally toroidal to eliminate end losses from open configurations. In general fusion confinement devices, magnetic field lines lie on a set of closed nested toroidal surfaces. This means that we can no longer describe any equilibria in a solely 1D geometry. The minor radius is no longer the only important scale length.

      +

      From + \( \vec j \cross \vec B = \grad p \) + +, we know that the pressure gradient is perpendicular to + \( \vec j \) + + and + \( \vec B \) + +, and therefore both + \( \vec B \) + + and + \( \vec j \) + + lie on surfaces of uniform pressure. We call these toroidal surfaces either magnetic surfaces or flux surfaces. We can use these surfaces to build a 1-dimensional description.

      +

      Figure 12.11

      +

      As a brief aside, some geometrical vocabulary will be useful when describing toroidal geometry. A toroid is any surface of revolution with a hole in the middle. A torus is the particular case of a toroid in which the revolved figure is a circle.

      +

      We will define our global toroidal coordinate system to consist of the major axis + \( (z) \) + +, the distance from the major axis + \( (R) \) + +, and the azimuthal angle around the major axis + \( (\phi) \) + +.

      +

      Figure 12.13

      +

      We will also make use of a poloidal coordinate system measured by minor radius (distance from the minor axis) + \( r \) + + and the poloidal angle from the minor axis + \( \theta \) + +. We will generally refer to a point on the torus relative to the major axis + \( (R, z, \phi) \) + +, or relative to the minor axis + \( (r, \theta, \phi) \) + +, or in spherical coordinates + \( (R, \theta, \phi) \) + +. The major radius + \( R_0 \) + + is the distance from the major axis to the minor axis. The minor radius + \( a \) + + is the characteristic distance from the minor axis to the exterior of the revolved figure. Usually we will find symmetry under + \( \phi \) + +.

      +

      The aspect ratio of a torus is the ratio of the major radius to the minor radius.

      + + \[A = \frac{R_0}{a}\] + + +

      When we talk about a “toroidal surface,” we mean a cross-section of the toroidal rotation. When we talk about a “poloidal surface” we mean a surface which is coplanar with the minor axis:

      +

      Figure 12.14

      +

      The poloidal flux is determined by the size of the poloidal surface and the poloidal magnetic field:

      + + \[\Psi _p = \int_{S_p} \vec B \cdot \dd \vec S\] + + +

      and the toroidal flux is determined by the size of a toroidal surface and the toroidal magnetic field:

      + + \[\Psi_t = \int_{S_t} \vec B \cdot \dd \vec S\] + + +

      Considering the poloidal flux, we can see that if we expand the size of the surface towards the minor radius, the flux will increase until eventually we come to a point where the flux begins to decrease. The position of this maximum is called the magnetic axis, which does not necessarily correspond to the minor axis. In fact, it is generally displaced from the minor axis.

      +

      Figure 12.15

      +

      To refer back to something more familiar, we’ll define the same terms for a cylindrical geometry + \((r, \theta, z) \) + +. An axial surface corresponds with a toroidal surface, and an azimuthal surface corresponds with a poloidal surface:

      +

      Figure 12.16

      +

      If we consider the trajectory of a single field line, what sorts of surfaces will it trace out? What surface will contain the field line? As it turns out, there are three options:

      +
        +
      1. Rational surface - the field line closes on itself, and it does so after a finite number of revolutions. One way to quickly visualize such a surface is to draw a Poincaré puncture plot. Choose a toroidal plane and plot a point wherever the field line punctures the surface. A Poincaré puncture plot of a rational surface contains a finite number of points and no continuous curves.
        2. +
      2. Ergodic surface - the field line completely covers an entire surface, which is to say the field line punctures any toroidal surface an infinite number of times. In other words, it never closes on itself and defines an irrational curve.
        3. +
      3. Stochastic region - In this case, there is no definite surface and the field line fills a volume.
        4. +
      +

      Figure 12.17

      +

      Generally rational surfaces and ergodic surfaces are largely equivalent, but by introducing a small amount of resistivity a rational surface can lead to magnetic islands. One can imagine the addition of resistivity equivalent to allowing a small degree of motion of the magnetic field lines. In an ergodic surface, a flux surface is defined by a single (irrational) field line. If it moves toward itself in one location it will necessarily move away from itself in another location. But in a rational surface, different field lines can lie on the same constant pressure surface and will tend to move towards each other. By concentrating into magnetic islands, the flux surfaces are now more closely spaced, and the pressure gradient increases (a bad thing!)

      +

      Figure 12.18

      +

      Surface quantities: Since pressure, current (not current density!), and flux (not field!) are constant along a flux surface, it is convenient to use flux + \( \Psi_p \) + + as a coordinate. A particular poloidal flux itself uniquely determines a poloidal surface with constant pressure and current. The flux surface quantities are + \( p, \, \Psi_p, \, \Psi_t, \, I_p = \int_{S_p} \vec j \cdot \dd \vec S, \, I_t = \int_{S_t} \vec j \cdot \dd \vec S\) + +.

      +

      Surface quantities are not independent. The poloidal current + \( I_p \) + + affects the toroidal field + \( B_t \) + + and toroidal flux + \( \Psi_t \) + +. The toroidal current affects the poloidal field + \( B_p \) + + and poloidal flux + \( \Psi_p \) + +.

      + + \[B_t = \vec B \cdot \vu \phi\] + + + + \[\vec B_p = B_\theta \vu \theta + B_z \vu z \\ += B_r \vu R + B_z \vu z\] + + +

      + Toroidal Force Balance + # +

      +

      Toroidal equilibria solves the end losses of linear equilibrium, but generates a new force which must be balanced. This is a result of the virial theorem.

      +

      + Poloidal Fields (Tire Tube Pressure Force) + # +

      +

      If you think of a flexible bike tire being inflated, as the pressure within the inner tube increases, the major radius will increase! Why is this? The pressure within the tire is isotropic. As we pump up the tire, the pressure increase causes a force imbalance with the atmospheric pressure that causes the tube to expand. To simplify, we can consider a square tube with inner “radius” + \( a \) + +. The radial force scales with the pressure over the outer and inner surfaces

      + + \[F_r = p S_{outer} - p S_{inner} \\ +\sim p (R_0 + a) a - p (R_0 - a) a \sim 2 a^2 p\] + + +

      !!! error “Missing content here on how to balance radial forces for purely poloidal fields (axisymmetric) with a conducting wall”

      +

      Alternatively, we can use vertical external field coils to increase the field strength near the outer wall. They will also tend to center the plasma between the coils, so they have the advantage of preventing the plasma from drifting upwards or downwards

      +

      Figure 12.19

      +

      + Toroidal Fields + # +

      +

      Now let’s consider driving a toroidal field in the plasma using a central coil on the major axis. Driving a toroidal field + \( B_\phi \) + + will also create a poloidal current + \( j_\theta \) + +. We know that the driven field will decay as + \( 1/R \) + + through the plasma. In actuality, the relationship is not perfectly + \( 1/R \) + + because of the generated poloidal current, which will modify the field within the plasma. Depending on the orientation of + \( j_\theta \) + +, it could either increase or decrease the poloidal field within the plasma. These are two different operations by which the plasma interacts with the magnetic field. When the induced plasma current tends to increase the field, we have a paramagnetic effect. In the opposite case (decrease) there is a diamagnetic effect. Generally, plasmas are diamagnetic, but there are certain situations where they become paramagnetic.

      +

      Figure 12.20

      +

      Here again, we have a larger field on the in-bore side than on the out-bore side, so there will be a force imbalance tending to push the plasma towards larger radii. We might consider surrounding the plasma with a conducting wall, as we did previously, but we run into a difficulty determining a current distribution in the wall which would balance the radial effect. Any current distribution must be circular; because of the geometry of a torus such a current distribution would have the same force contribution on both the inside and outside edges. There is no way to stabilize the distribution with purely toroidal fields.

      +
      + + + + + + + + +
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      + + + + +
      + + +
      + + + + + + + + + + + + + + + + + diff --git a/docs/notes/UWAA558/08-1d-equilibria/index.html b/docs/notes/UWAA558/08-1d-equilibria/index.html new file mode 100644 index 00000000..231353e3 --- /dev/null +++ b/docs/notes/UWAA558/08-1d-equilibria/index.html @@ -0,0 +1,5970 @@ + + + + + + + + + + + + +1-D Equilibria | My Notes + + + + + + + + + + + + + + +
      + + +
      +
      + +
      + + + 1-D Equilibria + + +
      + + + + +
      + + + +

      + 1-Dimensional Equilibria + # +

      +

      + The + + + + \( \theta \) + +-pinch + # +

      +

      In a + \( \theta \) + + pinch, we have an applied axial field generated by a driven azimuthal current distribution. The way these usually work is that you begin with a plasma generated by some pre-ionization process and zero field. Then you crank up the current to drive an azimuthal current in the plasma (in the opposite direction as the external current).

      + + \[\vec j \cross \vec B = \grad p \quad \rightarrow \quad j_\theta B_z = \dv{p}{r}\] + + + + \[j_\theta = - \frac{1}{\mu_0} \dv{B_z}{r}\] + + +

      + \[\dv{p}{r} = - \frac{1}{\mu_0} B_z \dv{B_z}{r} = - \dv{}{r} \left( \frac{B_z ^2}{2 \mu_0} \right)\] + + + + \[p + \frac{B_z ^2}{2 \mu_0} = \text{constant} = \frac{B_0 ^2}{2 \mu_0}\] + +

      +

      Figure 12.21

      +

      At equilibrium, the magnetic pressure balances the plasma pressure. If we say that the pressure is

      + + \[p = p_0 e^{- r^2 / a^2}\] + + +

      with + \( p_0 \) + + the pressure on-axis, then we can solve for the axial field

      + + \[B_z = B_0 (1 - B_0 e^{-r^2/a^2})^{1/2}\] + + +

      We can define the peak + \( \beta \) + + to be the ratio of the on-axis pressure to the maximum magnetic field

      + + \[\beta_0 = \frac{p_0}{B_0 ^2 / 2 \mu_0}\] + + +

      By definition, the peak + \( \beta \) + + will always be + \( \leq 1 \) + +. We can define + \( \langle \beta \rangle \) + +

      + + \[\langle \beta \rangle = \frac{ \langle p \rangle }{B_a ^2 / 2 \mu_0}\] + + +

      where + \( B_a \) + + is a characteristic field value, typically taken to be at the plasma edge.

      +

      + \[\langle \beta \rangle = \frac{2 \mu_0}{B_0 ^2 \pi a^2} \int _0 ^a 2 \pi r p \, \dd r\] + + + + \[= \frac{2}{a^2} \int_0 ^a \frac{ rp}{B_0 ^2/2 \mu_0} \dd r = \frac{2}{a^2} \int_0 ^a \left( 1 - \frac{B_z ^2}{B_0 ^2} \right) r \dd r\] + +

      +

      In this form, we can see that because + \( B_z \) + + will everywhere be less than + \( B_0 \) + +, we can increase + \( \langle \beta \rangle \) + + by driving + \( B_z \) + + as low as possible. In this particular example, + \( \langle \beta \rangle / \beta_0 = 63\% \) + +.

      +

      + Z-pinch + # +

      +

      In the case of a Z-pinch, we only have an applied axial current.

      + + \[\vec j = j_z (r) \vu z\] + + +

      For force balance

      +

      + \[\vec j \cross \vec B = \grad p \quad \rightarrow \quad - j_z B_\theta = \dv{p}{r}\] + + + + \[j_z = \frac{1}{\mu_0} \frac{1}{r} \dv{}{r} ( r B_\theta) \] + +

      + + \[- \dv{p}{r} = \frac{B_\theta}{\mu_0 r} \dv{}{r} ( r B_\theta)\] + + +

      If we find it convenient we can separate this into a magnetic pressure term

      + + \[- \dv{}{r} \left( p + \frac{B_\theta ^2}{2 \mu_0} \right) = \frac{B_\theta ^2}{\mu_0 r}\] + + +

      + Bennett Profile + # +

      +

      An example of an achievable distribution is the Bennett profile, which has a diffuse form

      +

      + \[B_\theta = \frac{\mu_0 I}{2 \pi} \frac{r}{r^2 + a^2}\] + + + + \[j_z = \frac{I}{\pi} \frac{a^2}{(r^2 + a^2) ^2}\] + +

      + + \[p = \frac{\mu_0 I^2}{8 \pi ^2} \frac{a^2}{(r^2 + a^2)^2}\] + + +

      Interestingly, + \( j \propto p \) + +. For a uniform temperature, + \( j \propto n \) + +. Since current density is the product of + \( \vec v \) + + and + \( n \) + +, this says that we have a uniform drift velocity and all particles are drifting with the same velocity at all points along the profile. If we consider what the equilibrium profile looks like for a Bennett profile:

      +

      Figure 12.22

      +

      So for + \( r < a \) + + we have magnetic tension and pressure which balance the plasma pressure. For + \( r \geq a \) + + we have magnetic tension which balances both plasma pressure and magnetic pressure.

      +

      The Z-pinch + \( \langle \beta \rangle \) + +

      + + \[\langle \beta \rangle \equiv \frac{ \langle p \rangle}{B_a ^2 / 2 \mu_0} \\ += \frac{2 \mu_0}{B_a ^2 \pi a^2} \int_0 ^a 2 \pi r p \dd r\] + + +

      If we multiply the force balance by + \( r^2 \) + + and integrate

      + + \[\int_0 ^a r^2 \dv{p}{r} \dd r + \frac{1}{\mu_0} \int_0 ^a r B_\theta \dv{}{r} (r B_\theta) \dd r = 0 \\ +0 = \left[ r^2 p \right] _0 ^a - \int_0 ^a p \dd (r^2) + \left[ \frac{(r B_\theta)^2}{2 \mu_0} \right] _0 ^a\] + + +

      If we have a discrete pinch such that + \( p(a) = 0 \) + + then the first term vanishes.

      + + \[\int_0 ^a 2 r p \dd r = \frac{(a B_a)^2}{2 \mu_0}\] + + +

      If we substitute our definition of + \( \langle \beta \rangle \) + +, we find + \( \langle \beta \rangle = 1 \) + +. For a diffuse pinch in which + \( p(a) \neq 0 \) + + we end up with + \( \langle \beta \rangle \leq 1 \) + + and we have a wall-supported plasma. Ideal confinement ( + \( \langle \beta \rangle = 1 \) + +) is a very nice property and is what makes the Z-pinch configuration so interesting.

      +

      + Stability Considerations + # +

      +

      Instability results if there exists a plasma displacement that leads to a lower energy state. There are several ways to provide stability in the context of MHD. The two most common are magnetic shear and magnetic well.

      +

      + Magnetic Shear + # +

      +

      In ideal MHD, magnetic field lines can not break or tear. Let’s consider some flux surface containing field lines + \( \vec B_3 \) + +. Behind it, we have another flux surface containing field lines + \( B_2 \) + + which are not parallel to + \( \vec B_3 \) + +, and the same for + \( \vec B_1 \) + +.

      +

      Figure 12.23

      +

      Because the field lines are a different angles to each other, these flux surfaces can not interpenetrate. In other words, if the flux surface pressures are + \( P_1 > P_2 > P_3 \) + +, we can maintain the pressure gradient and prevent the flux surfaces from moving each other. What prevents the surfaces from achieving a lower energy state is the magnetic shear between flux surfaces.

      +

      Without shear, the surfaces can interpenetrate and exchange positions. In the case of a toroidal geometry, magnetic shear is defined by the rotational transform + \( \iota \) + +, or by the safety factor

      + + \[q = \frac{2 \pi}{\iota} \] + + +

      Generally speaking, + \( q \) + + is generally referenced for tokamaks and + \( \iota \) + + is referenced for stellarators. Another way of picturing the safety factor in a toroidal geometry is

      + + \[q \equiv \frac{\text{no. of windings long way}}{\text{no. of windings short way}} \\ += \frac{ \dv{\psi_t}{V}}{\dv{\phi_p}{V}} = \dv{\phi_t}{\phi_p} \\ += \frac{n}{m} = \frac{\text{toroidal transits}}{\text{poloidal transits}}\] + + +

      In a cylindrical (1D) geometry it is just

      + + \[q = \frac{\text{longitudinal transits}}{\text{azimuthal transits}}\] + + +

      Let’s calculate the safety factor for a toroidal geometry:

      +

      Figure 12.24

      + + \[\dv{\phi_p}{V} = \frac{ B_\theta 2 \pi R \dd r}{2 \pi R_0 2 \pi r \dd r} \\ += \frac{ B_\theta}{2 \pi r} \frac{R}{R_0}\] + + + + \[\dv{\phi_t}{V} = \frac{B_\phi 2 \pi r \dd r}{2 \pi R_0 2 \pi r \dd r} \\ += \frac{B_\phi}{2 \pi R_0}\] + + + + \[q = \frac{r B_\phi}{R B_\theta}\] + + +

      In a cylindrical geometry the analysis is even simpler

      + + \[q = 2 \pi \frac{r B_z}{L B_\theta}\] + + +

      As a note, it would appear that + \( q \rightarrow 0 \) + + at the magnetic axis as + \( r \rightarrow 0 \) + +, but in general + \( B_\theta \rightarrow 0 \) + + as well, and the safety factor is generally bounded at + \( r \rightarrow 0 \) + +

      +

      + \( q \) + + is a flux surface quantity.

      +

      We care about magnetic shear. How does that relate to the safety factor? Magnetic shear is defined as

      + + \[s \equiv 2 \frac{\dd q / q}{\dd V /V} = 2 \dv{\ln (q)}{\ln(V)}\] + + +

      Even a uniform + \( B_z \) + + or + \( B_\theta \) + + produces a finite magnetic shear because of the way that + \( r \) + + and + \( B_\theta \) + + change. The safety factor is often considered synonymous with magnetic shear, and often we don’t even compute + \( s \) + +.

      +

      Shear is generally a stabilizing effect. Interchange between flux surfaces can be prevented/inhibited by shear, or by making it energetically unfavorable. Shear tends to stabilize current-driven instabilities.

      +

      + Magnetic Well + # +

      +

      As before, we can consider two adjacent flux surfaces + \( B_1, P_1 \) + + and + \( B_2, P_2 \) + +. If + \( B_2 > B_1 \) + + and + \( P_2 > P_1 \) + +, the interchange is energetically favorable. But if + \( B_2 > B_1 \) + + and + \( P_2 < P_1 \) + + then the interchange may be unfavorable without any magnetic shear.

      +

      Consider a plasma confined by an externally applied magnetic field generated by a coil + \( I \) + +

      +

      Figure 12.25

      +

      On the left side, the magnetic field gradient is in the same direction as the plasma pressure gradient, which is a destabilizing configuration. Flux surfaces are able to interchange easily, and the magnetic field is described as having bad curvature. On the right side, the gradients are in the same direction and the magnetic field has a good curvature.

      +

      We can define the “wellness” + \( W \) + + as

      +

      + \[W \equiv \frac{ \text{total pressure change relative to mag. pressure}}{\text{relative volume change}} \\ += \frac{\dd \langle p + B^2/2 \mu_0 \rangle / \langle B^2/2 \mu_0 \rangle}{\dd V / V}\] + + +where the angle brackets indicate a quantity integrated along a field line + + \[\langle Q \rangle \equiv \frac{\int_0 ^L \frac{ Q \dd l}{|B|}}{\int_0 ^l \frac{\dd l}{|B|}}\] + +

      +

      For a stabilizing effect, the wellness must be greater than 0. This means that the magnetic pressure must increase faster than the pressure decreases to prevent pressure-driven instabilities.

      +

      Since + \( W \) + + is evaluated along a field line, it is also a surface quantity.

      +

      + Application to 1D Equilibria + # +

      +

      + \( \theta \) + +-pinch: Since + \( B_\theta = 0 \) + +, + \( q \rightarrow \infty \) + +, which really just means + \( q \) + + is not well defined for a + \( \theta \) + +-pinch. If we consider some small + \( \delta B_\theta \) + +, we get a very large + \( q \) + +. From a magnetic shear perspective, a + \( \theta \) + +-pinch has very large values of shear and very good stability properties.

      +

      The wellness is

      +

      + \[W = \frac{V}{\langle B^2 \rangle} \dv{}{V} \langle 2 \mu_0 p + B^2 \rangle \\ + = \frac{\pi r^2 L}{B_z ^2} \frac{1}{2 \pi r L} \dv{}{r} (2 \mu_0 p + B_z ^2) \\ + = \frac{\mu_0 r}{B_z ^2} \dv{}{r} \left( p + \frac{B_z ^2}{2 \mu_0} \right) = 0 \] + + +so a + \( \theta \) + +-pinch has neutral magnetic well.

      +

      Vacuum case:

      +

      + \[W = \frac{\mu_0 r}{ B_z ^2} \dv{}{r} \left( \frac{ B_z ^2}{2 \mu_0} \right) = 0\] + + +So vacuum magnetic fields also have neutral wellness. This leads to a general result sometimes referred to as “a plasma cannot dig its own well.” In other words, by introducing plasma to a magnetic configuration, it cannot make the configuration more stable than it was. Plasmas make stability more challenging, not less.

      +

      Z-pinch:

      +

      Since + \( B_z = 0 \) + +, + \( q = 0 \) + + and there is no magnetic shear. Even for a small value of + \( \delta B_z \) + + you still get a small + \( q \) + +. The magnetic well properties of a Z-pinch are

      + + \[W = \frac{V}{\langle B^2 \rangle} \dv{}{V} \langle 2 \mu_0 p + B^2 \rangle \\ += \frac{\mu_0 r}{B_\theta ^2} \dv{}{r} \left( p + \frac{B_\theta ^2}{2 \mu_0} \right) \\ += \frac{\mu_0 r}{B_\theta ^2} \left( - \frac{ B_\theta ^2}{ \mu_0 r} \right) = -1\] + + +

      Recall that + \( W > 0 \) + + for stability, so the Z-pinch has negative magnetic well and provides no pressure stability.

      +

      In summary,

      +
        +
      • Both + \( \theta \) + +- and Z-pinch have high + \( \beta \) + +
        • +
      • + \( \theta \) + +-pinch is stable
        • +
      • Z-pinch is unstable
        • +
      • End losses in a + \( \theta \) + + pinch enormous since + \( k_\parallel \gg k_\perp \) + +
        • +
      +

      + Screw Pinch + # +

      +

      A natural extension is to combine a moderate toroidal field and a moderate poloidal field to produce a screw pinch configuration.

      +

      Figure 12.26

      +

      + \[\curl \vec B = \mu_0 \vec j\] + + + + \[\rightarrow j_\theta = - \frac{1}{\mu_0 } \dv{B_z}{r} \] + + + + \[j_z = \frac{1}{\mu_0 r} \dv{}{r} (r B_\theta)\] + +

      +

      For static MHD equilibrium

      +

      + \[\vec j \cross \vec B = \grad p\] + + + + \[\dv{}{r} \left( p + \frac{ B_\theta ^2 + B_z ^2}{2 \mu_0} \right) = - \frac{B_\theta ^2}{\mu_0 r}\] + +

      +

      We can define a toroidal + \( \beta \) + + where + \( B_0 = B_z (a) \) + + + + \[\beta_t = \frac{\langle p \rangle }{B_0 ^2 / 2 \mu_0} \\ += \frac{2 \mu_0}{ B_0 ^2} \frac{1}{\pi a^2} \int _0 ^a 2 \pi r p \dd r\] + +

      +

      and in the poloidal direction with + \( B_{\theta, a} = B_\theta (a) = \frac{\mu_0 I}{2 \pi a} \) + +

      + + \[\beta_p = \frac{\langle p \rangle}{B_{\theta, a} ^2 / 2 \mu_0} \\ += \frac{8 \pi ^2 a^2}{ \mu_0 I_0 ^2} \left( \frac{1}{\pi a^2} \int _0 ^a 2 \pi r p \dd r \right) \\ += \frac{16 \pi ^2}{\mu_0 I_0 ^2} \int_0 ^a r p \dd r\] + + +

      To proceed we can multiply the force balance by + \( r^2 \) + + and integrate

      + + \[\underbrace{\int_0 ^a r^2 \pdv{p}{r} \dd r}_{(1)} + \underbrace{\int _0 ^a \pdv{}{r} \left( \frac{ B_\theta ^2 + B_z ^2 }{2 \mu_0} \right) \dd r}_{(2)} + \underbrace{\int_0 ^a r^2 \frac{B_\theta ^2}{\mu_0 r} \dd r}_{(3)} = 0\] + + + + \[(1) = \int_0 ^a r^2 \dd p = \left. r^2 p \right|_0 ^a - \int_0 ^a p \dd (r^2) = - \int_0 ^a 2 r p \dd r\] + + + + \[(2) = \int_0 ^a r^2 \dd \left( \frac{B_\theta ^2 + B_z ^2}{2 \mu_0 } \right) \\ += \left. r^2 \left( \frac{ B_\theta ^2 + B_z ^2}{2 \mu_0 } \right) \right|_0 ^a - \int_0 ^a r \left( \frac{ B_\theta ^2 + B_z ^2}{\mu_0} \right) \dd r\] + + + + \[(3) = \int_0 ^a r \frac{B_\theta ^2}{\mu_0} \dd r\] + + +

      Combining we have

      + + \[- \int_0 ^a 2 r p \dd r + \frac{a^2 B_{\theta, a}^2}{2 \mu_0} + \overbrace{\frac{a^2 B_0 ^2}{2 \mu_0}}^{B_0 = B_z(r = a)} - \int_0 ^a r \frac{B_z ^2}{\mu_0} \dd r = 0 \\ +- \int_0 ^a 2 r p \dd r + \frac{ \mu_0 I_0 ^2}{8 \pi ^2} + \int_0 ^a r \left( \frac{ B_\theta ^2 - B_z ^2}{\mu_0} \right) \dd r = 0\] + + +

      Dividing + \( \int_0 ^a 2 r p \dd r \) + + gives

      +

      + \[\left[ \frac{16 \pi^2}{\mu_0 I^2} \int_0 ^a r p \dd r \right] ^{-1} + \left[\frac{4 \mu_0}{B_0 ^2 a^2} \int_0 ^a r p \dd r \right] ^{-1} \frac{2}{a^2} \int_0 ^a \left(1 - \frac{B_z ^2}{B_0 ^2} \right) r \dd r = 1\] + + +or

      +

      + \[\frac{1}{\beta_p} + \frac{\alpha_t}{\beta_t} = 1\] + + +where + + \[\alpha_t = \frac{2}{a^2} \int_0 ^a \left( 1 - \frac{B_z^2}{B_0 ^2} \right) r \dd r\] + + +is the diamagnetism.

      + + \[\beta _p = \left( 1 - \frac{\alpha_t}{\beta_t} \right) ^{-1}\] + + +

      If we have a diamagnetic current, then + \( \alpha > 0 \) + +. This maximizes confinement, since we have confinement in the azimuthal field, as well as the axial field. The limit where you have a skin current such that + \( B_z = 0 \) + + inside the plasma results in the best confinement and + \( \alpha _t = 1 \) + +.

      +

      Looking at the safety factor,

      + + \[q = \frac{2 \pi r B_z}{L B_\theta}\] + + +

      If we look at the edge + \( r = a \) + +, + + \[q_a = \frac{2 \pi a B_0}{L B_{\theta, a}}\] + +

      +

      As it turns out, this value of the edge safety factor is critically important, and for stability we require that + \( q_a > 1 \) + +.

      +

      The magnetic shear of a screw pinch is

      + + \[s = 2 \frac{V}{q} \dv{q}{V} \\ +V = \pi r^2 L \\ +\dd V = 2 \pi L r \dd r = \frac{2 V}{r} \dd r \\ +s = \frac{r}{q} \dv{q}{r}\] + + +

      The shear can be adjusted by changing the applied axial field.

      +

      The magnetic well is

      + + \[W = \frac{V}{B^2} \dv{(2 \mu_0 p + B^2)}{V} \\ += \frac{\mu_0 r}{B^2} \dv{}{r} \left( p + \frac{B^2}{2 \mu_0} \right) \\ += \frac{\mu_0 r}{B^2} \left( - \frac{B_\theta ^2}{\mu_0 r} \right) \\ += - \frac{B_\theta ^2}{B^2} \\ += - \frac{B_\theta ^2}{B_\theta ^2 + B_z ^2} \\ += - \left( 1 + \frac{B_z ^2}{B_\theta ^2} \right) ^{-1}\] + + +

      So the well is always less than zero, but adding + \( B_z \) + + improves the well.

      +

      By combining the properties of + \( \theta \) + +-pinch and Z-pinch, we are able to sacrifice some + \( \beta \) + + to achieve better stability properties. Of course, we have not addressed the end losses in any way; to do that, we need to connect the ends.

      +
      + + + + + + + + +
      + +
      + + + + +
      + + +
      + + + + + + + + + + + + + + + + + diff --git a/docs/notes/UWAA558/09-2d-equilibria/index.html b/docs/notes/UWAA558/09-2d-equilibria/index.html new file mode 100644 index 00000000..68a867df --- /dev/null +++ b/docs/notes/UWAA558/09-2d-equilibria/index.html @@ -0,0 +1,3135 @@ + + + + + + + + + + + + +2D Equilibria | My Notes + + + + + + + + + + + + + + +
      + + +
      +
      + +
      + + + 2D Equilibria + + +
      + + + + +
      + + + +

      + 2D Equilibria + # +

      +

      Let’s connect the ends of our 1D equilibria. Doing so is what gives us inherently toroidal configurations. From the 1-dimensional picture:

      +

      Figure 12.27

      + + + + + \[\vec j \cross \vec B = j_\theta B_z - j_z B_\theta \\ += \grad p = \dv{p}{r}\] + + +

      we move to an axisymmetric 2-dimensional torus, replacing our cylindrical coordinate system with a toroidal one

      +

      Figure 12.28

      + + \[\vec j \cross \vec B = \vec j_\theta \cross \vec B_\phi + \vec j_\phi \cross \vec B_\theta = \grad p\] + + +

      Eventually, the toroidal force balance will lead to the Grad-Shafranov Equation, which tells us how we can solve for a general equilibrium that solves + \( \vec j \cross \vec B = \grad p \) + +.

      +

      Let’s consider how we might achieve such a configuration. A toroidal magnetic field can be achieved by driving current through a poloidal coil. A more complicated problem is how to drive toroidal current. In general this is done by means of a transformer, where the plasma itself is the secondary circuit. Driving a time-varying current through the primary induces a toroidal current through the plasma. This is called a transformer drive for current.

      +

      + Grad-Shafranov equation + # +

      +

      Computing + \( j_\theta \) + + and + \( B_\phi \) + + can be computationally difficult in a toroidal geometry, so let’s do some work towards simplifying our force balance expression. The toroidal magnetic vector potential is defined as

      + + \[\vec B_\theta = \curl \vec A_\phi\] + + +

      If we integrate + \( B_\theta \) + + over a poloidal surface, Stokes' theorem gives

      + + \[\int _{S_p} \curl \vec A_\phi \cdot \dd \vec S = \oint \vec A_\phi \cdot \dd \vec l \\ += \int _{S_p} B_\theta \cdot \dd \vec S = \Psi _p \] + + +

      If the equilibrium is axisymmetric, + \( A_\phi \) + + must be uniform along + \( \dd l \) + +, so

      + + \[A_\phi \vu \phi \cdot \oint \dd \vec l = A_\phi 2 \pi R = \Psi _p \\ +\rightarrow A_\phi = \frac{ \Psi_p}{R} \vu \phi\] + + +

      where we absorb the factor of + \( 2 \pi \) + + into the poloidal flux + \( \Psi _p \) + +. After some manipulation, we can relate + \( B_\theta \) + + to the poloidal flux

      + + \[\vec B_\theta = \curl \vec A_\phi = - \frac{ \vu R}{R} \pdv{\Psi}{z} + \frac{\vu z}{R} \pdv{\Psi}{R}\] + + + + \[\mu_0 j_\phi \cross B_\theta = (\curl \vec B_\theta) \cross \vec B_\theta \\ += \curl \vec B_\theta \cross \left( \grad \Psi \cross \frac{\vu \phi}{R} \right) \\ += - \left[ \pdv{}{R} \left( \frac{1}{R} \pdv{\Psi}{R} \right) + \frac{1}{R} \pdv{\Psi ^2}{z^2} \right] \cdot \left[\frac{ \grad \Psi}{R} ( \vu \phi \cdot \vu \phi) - \frac{ \phi}{R} \cancel{(\grad \Psi \cdot \vu \phi)} \right]\] + + +

      That gives the first component of + \( \grad p \) + +, now let’s do the other one

      + + \[\mu_0 \vec j_\theta \cross \vec B_\phi = ( \curl \vec B_\phi) \cross \vec B_\phi \\ + = \left[ - \vu R \pdv{B_\phi}{z} + \vu z \frac{1}{R} \pdv{}{R} ( R B_\phi) \right] \cross \vec B_\phi \\ + = - \frac{B_\phi}{R} \left[ \vu R \pdv{}{R} (R B_\phi) + \vu z \pdv{}{z} (R B_\phi) \right] \\ + = - \frac{B_\phi}{R} \grad (R B_\phi)\] + + +

      Finally, since pressure is a flux surface quantity we can write

      + + \[\grad p = \dv{p}{\Psi} \grad \Psi = p' \grad \Psi\] + + +

      The toroidal force balance now looks like

      + + \[\mu_0 p' \grad \Psi = - \frac{1}{R} \left( \pdv{}{R} \frac{1}{R} \pdv{\Psi}{R} + \frac{1}{R} \pdv{^2 \Psi}{z^2} \right) \grad \Psi - \frac{B_\phi}{R} \grad(R B_\phi)\] + + +

      We notice that the only vector quantities here are + \( \grad \Psi \) + + and + \( \grad (R B_\phi) \) + +, so + \( \grad (R B_\phi) \) + + must be parallel to + \( \grad \Psi \) + + and is a flux surface quantity. We can define our new flux surface quantity as

      + + \[F(\Phi) \equiv R B_\phi = \frac{\mu_0 I_\theta}{2 \pi} = \frac{\mu_0}{2 \pi} \int_{S_p} \vec j_\theta \cdot \dd \vec S\] + + + + \[\grad F = \dv{F}{\Psi} \grad \Psi = F' \grad \Psi\] + + +

      Now each term in the toroidal force balance has a factor of + \( \grad \Psi \) + + attached. Let’s multiply through by + \( R^2 \) + + and factor out the gradient to arrive at the Grad-Shafranov equation:

      + + \[R^2 \mu_0 p' = - \Delta ^\star \Psi - F F'\] + + +

      where + + \[\Delta ^\star \equiv R \pdv{}{R} \frac{1}{R} \pdv{}{R} + \pdv{^2}{z^2}\] + +

      +

      To solve the Grad-Shafranov equation, you solve for + \( \Psi(R, z) \) + +, which determines + \( p(\Psi) \) + + and + \( F(\Psi) \) + +, which directly gives you + \( p(R, z) \) + + and + \( F(R, z) \) + + and completely defines the equilibrium.

      +

      You can solve for the other terms as well. Since + \( \vec B_\theta = \frac{\grad \Psi}{R} \cross \vu \phi \) + +

      + + \[\vec j_\phi = - \frac{1}{\mu_0 R} \Delta ^\star \Psi \vu \phi\] + + +

      and since + \( \vec B_\phi = \frac{F}{R} \vu \phi \) + +

      + + \[\vec j_\theta = - \frac{1}{\mu_0 R} \grad (R B_\phi) \cross \vu \phi\] + + +

      For the G-S equation to be solvable, you need to specify the equilibrium by specifying + \( p(\Phi) \) + + and + \( F(\Phi) \) + +. In practice, this is usually done by making experimental measurements to determine + \( p \) + + and + \( F \) + +. A common code that does this is called EFIT, which takes the boundary conditions of the magnetic field and measurements of temperature, density to perform a least-squares fit to solve the G-S equation.

      +

      In general, the Grad-Shafranov equation leads to a matrix equation

      + + \[\overline \vec A \vec \Psi + \vec f(\Psi) = \vec g\] + + +

      Depending on the conditions we place on + \( \Psi \) + +, + \( \vec f(\Psi) \) + + can be a nonlinear function.

      +

      + Solutions to the Grad-Shafranov equation + # +

      +

      In the limit that + \( \vec j \parallel \vec B \) + +, then + \( \vec j \cross \vec B = 0 = \grad p \rightarrow p' = 0 \) + +. These are called force-free states. In the G-S equation, the pressure term vanishes and we’re left with

      + + \[\Delta ^\star \Psi + F F' = 0\] + + +

      Spheromaks and RFPs are examples of nearly force-free states in which the current is nearly parallel to the magnetic field. Notice that in completely force-free states, + \( \langle \beta \rangle = 0 \) + +.

      +

      Another interesting limit is the case where + \( F F' \gg \Delta ^\star \Psi \) + +. Now we have + + \[\grad p \approx \vec j_\theta \cross \vec B_\phi\] + + +which looks like a + \( \theta \) + +-pinch which has been connected at the ends. Remember from the previous section that we can not maintain radial force balance with purely toroidal fields, so the toroidal current is not zero (hence the + \( \approx \) + +) but is just high enough to maintain radial force balance. This sort of configuration is called a high- + \( \beta \) + + tokamak.

      +

      The other limit is + \( F F' \ll \Delta ^\star \Psi \) + + + + \[\grad p \approx \vec j_\phi \cross \vec B_\theta\] + + +which looks like an end-connected z-pinch. This configuration is usually called an Ohmically heated Tokamak, and the majority of currently operating tokamaks operate this way. As we know, a purely poloidal field has very bad stability properties, so + \( \vec B_\phi \) + + needs to be added to provide stability. The toroidal + \( \beta \) + + is very small + + \[\beta _t \ll 1 \qquad \beta _p \approx 1\] + +

      +

      + Stability Considerations + # +

      +

      The same stability factors exist in 2D equilibria that we found for 1D equilibria:

      +

      Magnetic shear - the safety factor + \( q(\Psi) = \frac{\Delta \phi}{\Delta \theta} \) + + for + \( \Delta \theta = 2 \pi \) + +. We can calculate + \( q \) + + more easily by integrating along a flux surface in the poloidal plane:

      + + \[q(\Psi) = \frac{1}{2 \pi} \int_0 ^{\Delta \phi} \dd \phi \\ + = \frac{1}{2 \pi} \int _0 ^{2 \pi} \dv{\phi}{\theta} \dd \theta \\ + = \frac{1}{2 \pi} \int_0 ^{2 \pi} \dd \theta \left. \frac{r B_\phi}{R B_\theta} \right|_{\text{along flux surf.}} \\ + = \frac{F(\Psi)}{2 \pi} \oint \frac{ \dd l_p}{R^2 B_\theta} \qquad \dd l_p = r \dd \theta\] + + +

      Magnetic well: similarly we can get the magnetic well factor by integrating around a flux surface in the poloidal plane

      + + \[\langle Q \rangle = \frac{\oint \frac{Q \dd l_p}{B_\theta}}{\oint \frac{ \dd l_p}{B_\theta}}\] + + +

      + Shafranov Shift + # +

      +

      Remember that when we had an equilibrium which had a toroidal current and a corresponding poloidal magnetic field, and a poloidal magnetic field, then radial force balance will tend to shift the configuration outwards away from the major axis and a conducting wall or external coil will be required to maintain the equilibrium. The radial force balance is really achieved by + \( \vec j_\phi \cross B_p \) + +

      +

      As we move towards the magnetic axis, + \( B_p \rightarrow 0 \) + + by definition. With less poloidal field to balance the radial force imbalance, there is more radial expansion. This means that inner portion of the plasma (inner flux surfaces) must shift radially further to achieve radial force balance.

      +

      Figure 12.29

      +

      The shift increases with plasma pressure. This effect is further enhanced if we have low poloidal fields, for example in the high- + \( \beta \) + + tokamak configurations.

      +

      Low aspect ratios also enhance the effect. Recall that the radial force imbalance increases with smaller aspect ratio, leading to a larger shift.

      +
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      + + + + + + + + + + + + + + + + + diff --git a/docs/notes/UWAA558/10-equilibrium-of-3d-configurations/index.html b/docs/notes/UWAA558/10-equilibrium-of-3d-configurations/index.html new file mode 100644 index 00000000..4bbc64c1 --- /dev/null +++ b/docs/notes/UWAA558/10-equilibrium-of-3d-configurations/index.html @@ -0,0 +1,763 @@ + + + + + + + + + + + + +Equilibrium of 3D Configurations | My Notes + + + + + + + + + + + + + + +
      + + +
      +
      + +
      + + + Equilibrium of 3D Configurations + + +
      + + + + +
      + + + +

      + Equilibrium of 3D Configurations + # +

      +

      In 3 dimensions, we lose the axisymmetry that allowed us to reach the Grad-Shefranov equation and we need to solve the full momentum equation in three dimensions. This is not something that we can actually do in this class, and the existing codes that do this are quite sophisticated.

      +

      Some general features of 3D equilibria are:

      +
        +
      1. No net toroidal current. This means that they tend to be steady-state configurations.
        2. +
      2. Radial confinement is accomplished by toroidal fields, as in the end-connected + + + + \( \theta \) + +-pinch. As we saw, toroidal fields cannot provide radial confinement in a purely axisymmetric configuration, but radial variation with + \( \phi \) + + can provide confinement.
        3. +
      3. Toroidal effect (radial force balance) is generated by helical magnetic fields. You can do an expansion of the magnetic field into a toroidal component, and a helical component that traces out a twisted shape as you move around the torus. These twisted shapes are what lead to radial confinement.
        4. +
      +

      + ELMO Bumpy Torus (EBT) + # +

      +

      In contrast to most other 3D configurations, even though the EBT is a 3D equilibrium, it has no helical windings.

      +

      Figure 12.30

      +

      Since there are no helical windings, we have to provide radial stability in another way. In the EBT configuration, you also drive hot poloidal electron rings (driven by electron cyclotron resonance) to provide both stability and heating.

      +

      + Stellarator + # +

      +

      The stellarator configuration is composed of a number of helical current lines (generated by helical coils with alternating currents), and a net toroidal field driven by poloidal coils. The direction of the currents alternate, for a total of + \( 2l \) + + current lines.

      +

      Figure 12.31

      +

      The result is a net magnetic field with a ratio such that + \( B_{\text{helical}} \gg B_\phi \) + +

      +

      Stellarators raise some very complicated engineering challenges both in the design and construction of the complicated geometry. It is also very difficult to maintain no net current within the plasma, especially during start-up. As you add plasma, you raise from zero + \( \beta \) + + to a finite + \( \beta \) + +, introducing things like bootstrap currents that need to be balanced.

      +

      + Torsatron + # +

      +

      Similar to a stellarator, the torsatron does not have alternating currents. All of the helical current lines are in the same direction. There are also no toroidal field coils.

      +

      The engineering is slightly simpler, but it is slightly less efficient at generating the helical magnetic field.

      +

      The flux surfaces in stellarators and torsatrons have geometrical cross-sections depending on the number + \( l \) + + of helical current lines. About the current lines, the flux surfaces are nearly circular. The flux surfaces within the plasma volume are determined by the separatrix of the helical coil fields.

      +

      Figure 12.32

      +
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      + + + + + + + + + + + + + + + + + diff --git a/docs/notes/UWAA558/11-mhd-stability/index.html b/docs/notes/UWAA558/11-mhd-stability/index.html new file mode 100644 index 00000000..55637ef4 --- /dev/null +++ b/docs/notes/UWAA558/11-mhd-stability/index.html @@ -0,0 +1,3163 @@ + + + + + + + + + + + + +MHD Stability | My Notes + + + + + + + + + + + + + + +
      + + +
      +
      + +
      + + + MHD Stability + + +
      + + + + +
      + + + +

      + MHD Stability + # +

      +

      Equilibrium is simply a balance of forces that results in a steady state. Beyond equilibrium, stability is the tendency of a perturbation to return to equilibrium, rather than increasing. We are very interested in analyzing the stability of MHD equilibria, including the plasma dynamics, so we need to use the complete ideal MHD model. The MHD equations are non-linear, which means that any evolution/dynamics are also going to be non-linear. We can define the initial deviation from equilibrium to be a linear phenomenon. As usual, we perform this linearization by letting + + + + \( Q(r, t) = Q_0 + Q_1(r, t) \) + + with + \( Q_1 \) + + being a small first-order perturbation. Since the equilibrium is both time and space independent, the general form of the perturbation is

      + + \[Q_1(r, t) = \vu Q_1 e^{-i (\omega t - \vec k \cdot \vec r)}\] + + +

      + \[\grad p_0 = \vec j_0 \cross \vec B_0\] + + + + \[p = p_0 + p_1 \qquad \rho = \rho_0 + \rho_1\] + + + + \[\vec j = \vec j_0 + \vec j_1 \qquad \vec B = \vec B_0 + \vec B_1\] + +

      +

      and for a static equilibrium + + \[\vec v = \vec v_1\] + +

      +

      In our momentum equations of the perturbed quantities, we assume that the static equilibrium holds, so most of the equilibrium terms drop out. We can define a velocity displacement + \( \vec \xi = \int_0 ^t \vec v_1 \dd t \) + +. As we integrate the field and pressure in time,

      +

      + \[\pdv{B_1}{t} = \curl (\vec v_1 \cross \vec B_0)\] + + + + \[\int \pdv{B_1}{t} = \vec B_1 = \curl \int \vec v_1 \cross \vec B_0 \dd t \\ + = \curl (\vec \xi \cross \vec B_0)\] + +

      +

      If we do the same for the pressure equation, we get

      +

      + \[p_1 = - \vec \xi \cdot \grad p_0 - \Gamma p_0 \div \vec \xi\] + + +where + \( \Gamma \) + + is the ratio of specific heats, to avoid confusion with typical perturbation growth rate + \( \gamma \) + +.

      +

      If we combine all of these together, substituting into the momentum equation, we can express the perturbation entirely in terms of + \( \vec \xi \) + + and the equilibrium properties:

      + + \[\rho_0 \pdv{ ^2 \vec \xi }{t^2} = \grad (\vec \xi \cdot \grad p_0 + \Gamma p_0 \div \vec \xi)\\ + + \frac{1}{\mu_0} \left[(\curl \vec B_0) \cross \curl (\vec \xi \cross \vec B_0) \right] \\ + + \frac{1}{\mu_0} \left[ \curl \curl ( \vec \xi \cross \vec B_0) \cross \vec B_0 \right]\] + + +

      We define the right-hand-side as the linearized forcing function of our equilibrium + + \[\rho_0 \pdv{ ^2 \vec \xi }{t^2} = \vec F(\vec \xi _i , p_0, \vec B_0)\] + +

      +

      For a linear force function, we can also write it in terms of a spring constant tensor

      + + \[\rho_0 \pdv{ ^2 \vec \xi }{t^2} = \vec F(\vec \xi) = - \overline \vec K \cdot \vec \xi\] + + +

      We can determine the stability behavior of a configuration by specifying an initial condition

      +

      + \[\vec \xi (t = 0) = 0 \qquad \text{and} \qquad \left. \pdv{\xi}{t} \right| _{t = 0} = f(\vec r)\] + + +and boundary conditions. A boundary condition may be a rigid wall + + \[\vec \xi \cdot \vu n |_{wall} = 0\] + +

      +

      One way we can tell whether a given solution is unstable is to assume a variation of the form + + \[\vec \xi \propto e^{-i \omega t}\] + + +If + \( \omega^2 > 0 \) + +, the displacement will oscillate in time without growth, and if + \( \omega^2 < 0 \) + + then the displacement will grow. In other words, if + \( \omega \) + + is real, then the mode is stable, and if + \( \omega \) + + is imaginary then the mode is unstable. The eigenvalue equation to be solved is

      +

      + \[- \omega ^2 \rho_0 \vec \xi = \vec F(\vec \xi)\] + + +which we can write as a matrix equation + + \[\overline \vec A\, \overline X = \lambda \overline X\] + + + + \[\frac{1}{\rho_0} \vec F (\vec \xi) = - \omega ^2 \vec \xi\] + +

      +

      For any arbitrary linear forcing function, we might get an infinite number of eigenvalues. How do we know which ones to look at? It turns out that the linearized force function + \( \vec F(\vec \xi) \) + + has the property of being self-adjoint, so + + \[\int \vec \eta \cdot \vec F(\vec \xi) \dd V = \int \xi \cdot \vec F( \vec \eta) \dd V\] + + +where + \( \vec \eta \) + + and + \( \vec \xi \) + + are arbitrary displacements that satisfy the same boundary conditions. If + \( \vec F \) + + is self-adjoint, then the system is Hermitian, which guarantees that we get real eigenvalues ( + \( \omega^2 \) + +) , orthogonal eigenfunctions, and most importantly we are guaranteed to have an ordered spectrum of eigenvalues. That is to say + \( \omega_0 ^2 < \omega _1 ^2 < \omega _2 ^2 < \ldots \) + +. This means that the eigenvalue of the lowest mode is guaranteed to be the most negative, and therefore dictates the stability of the system. If the lowest eigenvalue is negative, then the system is necessarily unstable, and if the lowest eigenvalue is positive, then we are guaranteed that all modes are stable.

      +

      Because + \( \vec F \) + + is self-adjoint, we can make use of the energy principle to write the variation in the sum of the kinetic and potential energy as:

      +

      + \[0 = \dv{}{t} \left[ \frac{1}{2} \int \rho_0 \left( \pdv{\vec \xi}{t} \right) ^2 \dd V - \frac{1}{2} \int \vec \xi \cdot \vec F ( \vec \xi) \dd V \right]\] + + +The kinetic energy term will always be positive, so we can formulate the stability based on the potential energy, often called a + \( \delta W \) + + approach + + \[\delta W = - \frac{1}{2} \int \vec \xi \cdot \vec F ( \vec \xi) \dd V\] + + +is the change in potential energy due to a displacement + \( \xi \) + +. If the potential energy decreases due to a displacement + \( \xi \) + +, then the kinetic energy must necessarily increase, so + \( \delta W < 0 \) + + indicates instability.

      +

      We can write the change in kinetic energy for our normal mode decomposition as + + \[\delta T = \frac{1}{2} \int \rho _0 \left( \pdv{\xi}{t} \right) ^2 \dd V = - \frac{1}{2} \omega ^2 \int \rho _0 \vec \xi ^ \star \cdot \vec \xi \dd V \\ += - \delta W = \frac{1}{2} \int \vec \xi ^\star \cdot \vec F(\vec \xi) \dd V\] + + + + \[\omega^2 = \frac{- \int \xi ^\star \cdot \vec F \dd V}{\int \rho_0 \xi ^\star \cdot \xi \dd V} = \frac{\delta W}{\frac{1}{2} \int \rho_0 \xi ^\star \cdot \xi \dd V}\] + +

      +

      The denominator is strictly positive, so the sign of + \( \omega^2 \) + + is determined by the sign of + \( \delta W \) + +

      + + \[\delta W < 0 \rightarrow \omega^2 < 0 \rightarrow \text{unstable} \\ +\delta W > 0 \rightarrow \omega^2 > 0 \rightarrow \text{stable}\] + + +

      Analyzing the form of + \( \delta W \) + + (within the plasma volume)

      + + \[\delta W = \frac{1}{2} \int_{plasma} \dd V \Gamma p_0(\div \vec \xi) ^2 + \vec \xi \cdot \grad p_0 (\div \vec \xi) \qquad \qquad \\ +\qquad \qquad + \frac{1}{\mu_0} \left[ \curl ( \vec \xi \cross \vec B_0) \right]^2 \\ +\qquad \qquad - \frac{1}{\mu_0} \left[\vec \xi \cross ( \curl \vec B_0) \right] \cdot \left[ \curl ( \vec \xi \cross \vec B_0) \right]\] + + +

      Generally speaking, the plasma volume does not extend to infinity, and we care very much about the boundary. The total + \( \delta W \) + + is the sum of that in the plasma volume + \( \delta W_F \) + +, the surface + \( \delta W_S \) + +, and the vacuum region + \( \delta W_V \) + +. The vacuum term looks like

      + + \[\delta W_V = \frac{1}{2} \int _{vac} \dd V \frac{ (\curl ( \vec \xi \cross \vec B_0))^2}{\mu_0} = \int_{vac} \dd V \frac{\vec B_1 ^2}{\mu_0} > 0\] + + +

      so the vacuum term is always positive, and has a stabilizing influence. The surface contribution offsets this

      + + \[\delta W_S = \frac{1}{2} \oint \dd S ( \vu n \cdot \vec \xi) ^2 \left[ \left[ \grad \left( p_0 + \frac{B_0^2}{2 \mu_0} \right) \right] \right] \cdot \vu n\] + + +

      Instabilities can be characterized as:

      +
        +
      • Internal/fixed boundary + \( \delta W = \delta W_F \) + +
        • +
      • External/free boundary + \( \delta W = \delta W_F + \delta W_S + \delta W_V \) + +
        • +
      +

      The plasma portion can be re-written slightly as

      + + \[\delta W_F = \frac{1}{2} \int \dd V \frac{ |B_{1, \perp}|^2}{\mu_0} \quad \leftarrow \text{Shear Alfven} \\ + + \mu_0 \left| \frac{B_{1, \parallel}}{\mu_0} - \frac{B_0 \xi \cdot \grad p_0}{B_0} ^2 \right|^2 \quad \leftarrow \text{Fast magnetosonic} \\ + + \Gamma p_0 |\div \xi|^2 \quad \leftarrow \text{Acoustic}\\ + + \frac{\vec j_0 \cdot \vec B_0}{B_0 ^2} (\vec B_0 \cross \vec \xi) \cdot \vec B_1 \quad \leftarrow \text{Current-driven (kink)} \\ +- 2 ( \vec \xi \cdot \grad p_0)(\vec \xi \cdot \vec \kappa) \quad \leftarrow \text{pressure-driven (interchange/balooning)}\] + + +

      where + \( \vec \kappa \) + + is the curvature vector + \( \vu e_B \cdot \grad \vu e_B \) + +. If we look at each of these terms, the first three terms are all going to be stabilizing effects, which means that all instability is going to come from the last two terms, the current-driven instability term and the pressure-driven instability term.

      +

      Going back to the screw pinch, + + \[\dv{p}{r} = j_\theta B_z - j_z B_\theta\] + + +we have current in the same direction as magnetic field ( + \( j_\theta \) + + with + \( B_\theta \) + + and + \( j_z \) + + with + \( B_z \) + +), so kink instabilities are possible. We also have a pressure gradient, so interchange instabilities are also possible.

      +

      As a concrete example, look at the pressure driven instability term in a Z-pinch. + + \[\kappa = - \frac{ \vu r}{r}\] + + + + \[\vec \xi = \xi _r \vu r\] + + + + \[\delta W_{F, pressure} = \int \dd V \xi _r \dv{p_0}{r} \frac{\xi_r}{r} \\ + = \int \dd V \frac{2 \xi _r ^2}{r} \dv{p_0}{r}\] + +

      +

      In a Z-pinch, it is always the case that + \( \dv{p_0}{r} < 0 \) + +. As shown by Kadomtsev (1965) it turns out that these modes can be stabilized by adding + \( B_z \) + +, but this also introduces kink modes.

      +

      Going back to our stabilizing quantities of wellness and shear, current-driven instabilities are generally managed through shear, and pressure-driven instabilities are stabilized by well.

      +
      + + + + + + + + +
      + +
      + + + + +
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      + + + + + + + + + + + + + + + + + diff --git a/docs/notes/UWAA558/formulary/index.html b/docs/notes/UWAA558/formulary/index.html new file mode 100644 index 00000000..1d5f39ed --- /dev/null +++ b/docs/notes/UWAA558/formulary/index.html @@ -0,0 +1,3890 @@ + + + + + + + + + + + + +Formulary | My Notes + + + + + + + + + + + + + + +
      + + +
      +
      + +
      + + + Formulary + + +
      + + + + +
      + + + +
      + + + + \( \) + + +

      + Formulary + # +

      +
      +

      “Kinetic Description”

      + + \[\dv{\vec v}{t} = \frac{q_i}{m_i} (\vec E + \vec v_i \cross \vec B) + \sum_{j \neq i} \left[ \left. \dv{\vec v_{ij}}{t} \right|_{coll} \delta(\vec r_i - \vec r_j) \right]\] + +

      + + \[\pdv{\vec B}{t} = - \curl \vec E\] + +

      + + \[\frac{1}{c^2} \pdv{\vec E}{t} = \curl \vec B - \mu_0 \sum_i q_i \vec v_i \delta (\vec r - \vec r_i)\] + +

      + + \[\div \vec{B} = 0\] + +

      + + \[\div \vec E = \frac{1}{\epsilon_0} \sum_i q_i \delta (\vec r - \vec r_i)\] + +

      +

      Klimontovich equation:

      + + \[\dv{N}{t} = 0 = \pdv{N}{t} + \pdv{}{q_i} \cdot (\dot{q_i} N) \\ +N \equiv \sum_i \delta (p - p_i) \delta(q - q_i)\] + +

      + +
      + +
      +

      “Plasma Fluid Description”

      +

      Boltzmann Equation

      + + \[\pdv{f_\alpha}{t} + \vec v \cdot \pdv{f_\alpha}{t} + \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} = \left. \pdv{f_\alpha}{t} \right|_{coll} = \sum_{\beta \neq \alpha} C_{\alpha \beta}\] + +

      +

      Maxwellian distribution:

      + + \[f_\alpha (\vec v) = n_\alpha \left( \frac{m_\alpha}{2 \pi T} \right)^{3/2} e^{- \frac{m_\alpha(\vec v - \vec v_\alpha)^2}{2T}}\] + + +Moments of fluid model (moments of distribution \( \rightarrow \) moments of Boltzmann equation:

      + + \[\text{Continuity:} \qquad n_\alpha = \int f_\alpha \dd \vec v \\ +\quad \rightarrow \pdv{n_\alpha}{t} + \div (n_\alpha \vec v_\alpha) = 0\] + +

      + + \[\text{Momentum:} \qquad n_\alpha \vec v_\alpha = \int \vec v f_\alpha \dd v \\ +\quad \rightarrow \quad \pdv{}{t} (n_\alpha \vec v_\alpha ) + \div (n_\alpha \vec v_\alpha \vec v_\alpha) + \frac{1}{m_\alpha} \div \vec P_\alpha - \frac{q_\alpha}{m_\alpha} n_\alpha ( \vec E + \vec v _\alpha \cross \vec B) = \sum_{\beta \neq \alpha} \int \vec w C_{\alpha \beta} \dd \vec v\] + +

      + + \[\rightarrow \rho_\alpha \left(\pdv{\vec v_\alpha}{t} + \vec v_\alpha \cdot \grad \vec v_\alpha \right) + \grad \vec P_\alpha + \div \vec \Pi_\alpha - q_\alpha n_\alpha (\vec E + \vec v_\alpha \cross \vec B) = \sum_{\beta \neq \alpha} \vec R_{\alpha \beta}\] + +

      + + \[\text{Energy:} \qquad \int \vec v \vec v \pdv{f_\alpha}{t} \dd \vec v = \pdv{}{t} \int \vec v \vec v f_\alpha \dd \vec v = \pdv{}{t} \vec E_\alpha / m_\alpha \rightarrow \pdv{}{t} \vec P_\alpha\] + +

      + + \[\rightarrow \quad \frac{3}{2} n_\alpha \left( \pdv{T_\alpha}{t} + \vec v_\alpha \cdot \grad T_\alpha \right) + P_\alpha \div \vec v_\alpha + \vec \Pi_\alpha \cdot \cdot \grad \vec v_\alpha + \div \vec h_\alpha = \sum_{\beta \neq \alpha} Q_{\alpha \beta}\] + + +Closure relations

      + + \[\vec h_\alpha = - \kappa \grad T_\alpha\] + +

      + + \[\overline \Pi_ \alpha = \nu \grad \vec v_\alpha\] + +

      + +
      + +

      + Ideal MHD + # +

      +
      +

      “Ideal MHD”

      +

      Continuity:

      + + \[\pdv{\rho}{t} + \div (\rho \vec v) = 0\] + + +Momentum:

      + + \[\rho \left( \pdv{\vec v}{t} + \vec v \cdot \grad \vec v \right) + \grad p - \vec j \cross \vec B = 0\] + + +Generalized Ohm’s Law

      + + \[\vec E + \vec v \cross \vec B = \frac{1}{Zen}\cancel{(\vec j \cross \vec B - \grad p_e)} = 0\] + + +Energy

      + + \[\dv{}{t} \left( \frac{p}{\rho^\gamma} \right) = 0\] + +

      + +
      + +
      +

      “Lawson Criterion”

      + + \[n \tau_E > 10^4 s/m^3\] + +

      + +
      + +
      +

      “Conservation Law Form of Ideal MHD”

      +

      Continuity:

      + + \[\pdv{\rho}{t} + \div (\rho \vec v) = 0\] + + +Momentum:

      + + \[\pdv{(\rho \vec v)}{t} + \div \left[ \rho \vec v \vec v - \frac{\vec B \vec B}{\mu_0} + \left( p + \frac{B^2}{2 \mu_0} \right) \overline{I} \right] = 0\] + + +Energy:

      + + \[\pdv{\epsilon}{t} + \div \left[ \left( \epsilon + p + \frac{B^2}{2 \mu_0} \right) \vec v - (\vec B \cdot \vec v) \frac{\vec B}{\mu_0} \right] = 0\] + +

      + + \[\pdv{\vec B}{t} + \div ( \vec v \vec B - \vec B \vec v) = 0\] + + +where

      + + \[\epsilon = \frac{1}{\gamma - 1} p + \frac{1}{2} \rho v^2 + \frac{B^2}{2\mu_0}\] + +

      +

      Static Equilibrium:

      + + \[\vec j \cross \vec B = \grad p\] + +

      + + \[\frac{B^2}{\mu_0} \vec K = \grad_\perp (p + \frac{B^2}{2 \mu_0})\] + +

      + + \[\vec K \equiv \frac{\vec B}{|B|} \cdot \grad \frac{ \vec B}{|B|}\] + +

      +

      Conservation of flux:

      + + \[\vec E + \vec v \cross \vec B = 0\] + +

      + + \[\pdv{\vec B}{t} = - \curl \vec E\] + +

      + + \[\rightarrow \dv{}{t} \left( \frac{\vec B}{\rho} \right) = \frac{\vec B}{\rho} \cdot \grad \vec v\] + +

      + +
      + +

      + 1D Equilibria + # +

      +
      +

      "\( \theta \)-pinch"

      + + \[B_\theta = 0\] + +

      + + \[j_\theta B_z = \dv{p}{r}\] + +

      + + \[j_\theta = - \frac{1}{\mu_0} \dv{B_z}{r}\] + +

      + + \[\rightarrow p + \frac{B_z ^2}{2 \mu_0} = \frac{B_0 ^2}{2 \mu_0}\] + +

      + + \[\langle \beta \rangle = \frac{2}{a^2} \int_0 ^a \frac{r p}{B_0 ^2 / 2 \mu_0} \dd r\] + +

      + + \[q = \infty\] + +

      + + \[W = \frac{\mu_0 r}{B_z ^2} \dv{}{r} \left( p + \frac{B_z ^2}{2 \mu_0} \right) = 0\] + +

      + +
      + +
      +

      “Z-pinch”

      + + \[B_z =0\] + +

      + + \[\grad p = \dv{p}{r} = - j_z B_\theta\] + +

      + + \[- \dv{}{r} \left( p + \frac{B_\theta ^2}{2 \mu_0} \right) = \frac{B_\theta ^2}{\mu_0 r}\] + +

      + + \[\langle \beta \rangle = \frac{2 \mu_0}{B_0 ^2 \pi a^2} \int _0 ^a 2 \pi r p \dd r = 1 \quad \text{ if } \quad p(a) = 0\] + +

      + + \[q = S = 0\] + +

      + + \[W = 1\] + +

      + +
      + +
      +

      “Screw pinch”

      + + \[\dv{}{r} \left( p + \frac{B^2}{2 \mu_0} \right) = - \frac{B_\theta ^2}{\mu_0 r}\] + +

      + + \[\beta_t = \frac{2 \mu_0}{B_z (a) ^2} \left( \frac{1}{\pi a^2} \int_0 ^a 2 \pi r p \dd r \right)\] + +

      + + \[\beta_p = \left( 1 - \frac{\alpha_t}{\beta _t} \right)^{-1} \qquad \alpha_t \equiv \frac{2}{a^2} \int_0 ^a \left(1 - \frac{B_z ^2}{B_0 ^2} \right) r \dd r\] + +

      + + \[q = \frac{2 \pi r B_z}{L B_\theta}\] + +

      + + \[q_a = \frac{4 \pi ^2 a^2 B_0}{\mu_0 I_a}\] + +

      + + \[S = \frac{r}{q} \dv{q}{r}\] + +

      + + \[W = - \frac{B_\theta ^2}{B_\theta ^2 + B_z ^2}\] + +

      + +
      + +
      +

      “Stability”

      +

      Shear:

      + + \[S = 2 \frac{ dq / q}{dV / V} = 2 \frac{d \ln q}{d \ln V}\] + +

      + + \[q = \frac{\text{\# long windings}}{\text{\# short windings}} = \dv{\psi_t}{\psi_p}\] + + +Shear for toroid

      + + \[q = \frac{r B_\phi}{R B_\theta}\] + + +Shear for cylinder

      + + \[q = 2 \pi \frac{r B_z}{L B_\theta}\] + + +Well

      + + \[W = \frac{ d \langle p + B^2 / 2 \mu_0 \rangle / \langle B^2 / 2 \mu_0 \rangle}{dV / V}\] + + +For stabilization, \( B^2 / 2 \mu_0 \) should increase faster than \( p \) decreases

      + +
      + +

      + 2D Equilibria + # +

      +
      +

      “Grad-Shafranov Equation: Static toroidal equilibrium”

      + + \[\grad p = \vec j_\theta \cross \vec B_\phi + \vec j_\phi \cross \vec B_\theta\] + +

      + + \[A_\phi = \frac{\phi}{R} \vu \phi\] + +

      + + \[\phi = \frac{\phi_p}{2 \pi}\] + +

      + + \[\vec B_\theta = - \frac{\vu R}{R} \pdv{\psi}{z} + \frac{ \vu z}{R} \pdv{\psi}{R} = \frac{ \grad \psi}{R} \cross \vu \phi\] + +

      + + \[F \equiv R B_\phi\] + +

      + + \[\Delta ^\star \equiv R \pdv{}{R} \frac{1}{R} \pdv{}{R} + \pdv{^2}{z^2}\] + +

      + + \[\Delta ^\star \psi = \pdv{^2 \psi}{z^2} + \pdv{^2 \psi}{R^2} - \frac{1}{R} \pdv{\psi}{R}\] + +

      + + \[\vec j_\phi = - \frac{1}{\mu_0 R} \Delta ^\star \psi \vu \phi\] + +

      + + \[\vec j_\theta = \frac{1}{\mu_0 R} \grad (F) \cross \vu \phi\] + +

      + + \[R^2 \mu_0 \pdv{p}{\psi} = - \Delta ^\star \psi - F \pdv{F}{\psi}\] + +

      + + \[q(\psi) = \frac{F(\psi)}{2 \pi} \oint_{p} \frac{r \dd \theta}{R^2 B_\theta}\] + + +Limits:

      + + \[\text{Force-free:} \qquad \vec j \parallel \vec B\] + +

      + + \[\rightarrow \Delta ^\star \psi + F F' = 0\] + +

      + + \[\text{Connected $\theta$ pinch:} \qquad FF' \gg \Delta ^\star \psi\] + +

      + + \[\rightarrow \grad p \approx \vec j_\theta \cross \vec B_\phi\] + +

      + + \[\text{Connected Z-pinch:} \qquad FF' \ll \Delta ^\star \phi\] + +

      + + \[\rightarrow \grad p \approx j_\phi \cross B_\theta\] + +

      + +
      + +

      + MHD Stability + # +

      +
      +

      “Linear stability”

      + + \[\pdv{\rho_1}{t} = - \vec v_1 \grad \rho_0 - \rho_0 \div \vec v_1\] + +

      + + \[\pdv{\vec B_1}{t} = \curl ( \vec v_1 \cross \vec B_0)\] + +

      + + \[\rho_0 \pdv{\vec v_1}{t} = - \grad p_1 + \vec j_0 \cross \vec B_1 - \vec j_1 \cross \vec B_0\] + +

      + + \[\pdv{p_1}{t} = - \vec v_1 \cdot \grad p_0 - \gamma p_0 \div \vec v_1\] + + +For linear perturbation \( \vec \xi = \int_0 ^t \vec v_1 \dd t \) the momentum equation becomes

      + + \[\rho_0 \pdv{^2 \xi}{t^2} = \vec F(\xi)\] + + +where

      + + \[F(\xi) = \grad (\xi \cdot \grad p_0 + \gamma p_0 \div \xi) + \frac{1}{\mu_0} \left[ ( \curl \vec B_0) \cross \curl (\xi \cross \vec B_0) + \curl \curl (\xi \cross \vec B_0) \cross \vec B_0 \right]\] + +

      +

      Eigenvalues of \( \frac{1}{\rho_0} \vec F (\xi) = \omega^2 \xi \) are real and ordered. Only need to check \( n=0 \) to determine stability/instability of configuration.

      + +
      + +
      +

      "\( \delta W \) Approach"

      +

      \( \delta W = \) change in potential energy due to a displacement \( \xi \)

      + + \[\delta W < 0 \rightarrow \text{instability}\] + +

      + + \[\delta W = - \frac{1}{2} \int \xi \cdot F(\xi) \dd V = \delta W_F + \delta W_S + \delta W_V\] + + +Surface term:

      + + \[\delta W_s = \frac{1}{2} \oint \dd S (\vu n \cdot \xi) ^2 \left( \vu n \cdot \grad p_0 + \left[ \vu n \cdot \grad \frac{B_0 ^2}{2 \mu_0} \right]_{jump} \right)\] + + +Vacuum term:

      + + \[\delta W_V = \int_{vac} \dd V \frac{B_1 ^2}{\mu_0}\] + + +Plasma (free) term:

      + + \[\delta W_F = \frac{1}{2} \int \dd V \frac{ |B_{1, \perp}|^2}{\mu_0} \quad \leftarrow \text{Shear Alfven} \\ ++ \mu_0 \left| \frac{B_{1, \parallel}}{\mu_0} - \frac{B_0 \xi \cdot \grad p_0}{B_0} ^2 \right|^2 \quad \leftarrow \text{Fast magnetosonic} \\ ++ \Gamma p_0 |\div \xi|^2 \quad \leftarrow \text{Acoustic}\\ ++ \frac{\vec j_0 \cdot \vec B_0}{B_0 ^2} (\vec B_0 \cross \vec \xi) \cdot \vec B_1 \quad \leftarrow \text{Current-driven (kink)} \\ +- 2 ( \vec \xi \cdot \grad p_0)(\vec \xi \cdot \vec \kappa) \quad \leftarrow \text{pressure-driven (interchange/balooning)}\] + + +Shear Alfven, fast magnetosonic, and acoustic modes are stabilizing. Current-driven and pressure-driven modes can lead to instability.

      + +
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Then, we&rsquo;ll look at the forces on the separate species and form a multi-fluid model, finally simplifying to a single-fluid MHD model. +The most important question to ask ourselves is &ldquo;when is this model going to be useful?&rdquo; The MHD model is the mathematical model for magnetized plasmas that are treated as a fluid. + + + + Plasma Fluid Model + https://peppyhare.github.io/r/notes/UWAA558/03-plasma-fluid-model/ + Mon, 01 Jan 0001 00:00:00 +0000 + + https://peppyhare.github.io/r/notes/UWAA558/03-plasma-fluid-model/ + Plasma Fluid Model # We take velocity moments of each of the pieces of the kinetic model: + distribution function, \( f_\alpha \) function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: "$$", right: "$$", display: true }, { left: "\\[", right: "\\]", display: true }, { left: "$", right: "$", display: false }, { left: "\\(", right: "\\)", display: false }], macros: { "\\lcm": " + + + + Two-Fluid Plasma Model + https://peppyhare.github.io/r/notes/UWAA558/04-two-fluid-plasma-model/ + Mon, 01 Jan 0001 00:00:00 +0000 + + https://peppyhare.github.io/r/notes/UWAA558/04-two-fluid-plasma-model/ + Two-Fluid Plasma Model (ions-electrons) # Restricting our multi-species fluid model to ions and electrons, what can we say about wave behavior in a magnetized 2-fluid plasma? Let&rsquo;s start with a cold plasma approximation ( \( p = 0 \) function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: "$$", right: "$$", display: true }, { left: "\\[", right: "\\]", display: true }, { left: "$", right: " + + + + Magnetohydrodynamic (MHD) Model + https://peppyhare.github.io/r/notes/UWAA558/05-mhd-model/ + Mon, 01 Jan 0001 00:00:00 +0000 + + https://peppyhare.github.io/r/notes/UWAA558/05-mhd-model/ + Magnetohydrodynamic (MHD) Model # Applying approximations to the two-fluid plasma model will allow us to arrive at a single-fluid (center-of-mass) description. The result is the ideal magnetohydrodynamic model (MHD). +First, define the MHD variables: + \[\text{mass density:} \qquad \rho = n_i m_i &#43; n_i m_e\] function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: "$$", right: "$$", display: true }, { left: "\\[", right: "\\]", display: true }, { left: " + + + + Boundary Conditions + https://peppyhare.github.io/r/notes/UWAA558/06-boundary-conditions/ + Mon, 01 Jan 0001 00:00:00 +0000 + + https://peppyhare.github.io/r/notes/UWAA558/06-boundary-conditions/ + Boundary Conditions # Mathematically, a well-posed problem requires both governing equations and a complete set of boundary conditions (the Cauchy data for the problem). The most common boundary conditions we use are perfectly conducting walls (flux surfaces) or a vacuum region. +Perfectly Conducting Wall # For the case where the plasma extends out to a perfectly conducting (impermeable) wall. Perfectly conducting walls do not support tangential electric field: + + + + Equilibrium for Fusion + https://peppyhare.github.io/r/notes/UWAA558/07-equilibrium-for-fusion/ + Mon, 01 Jan 0001 00:00:00 +0000 + + https://peppyhare.github.io/r/notes/UWAA558/07-equilibrium-for-fusion/ + Equilibrium for Fusion ( \( \beta \) function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: "$$", right: "$$", display: true }, { left: "\\[", right: "\\]", display: true }, { left: "$", right: "$", display: false }, { left: "\\(", right: "\\)", display: false }], macros: { "\\lcm": "\\mathop{\\mathrm{lcm}}", "\\sen": '\\text{sen}\\,', "\\dd": "\\mathop{\\mathrm{d} #1}", "\\abs": "\\lvert #1 \\rvert", "\\dd": "\\text{d}", "\\cross": "\\times", "\\pdv": "\\frac{\\partial #1}{\\partial #2}" + + + + 1-D Equilibria + https://peppyhare.github.io/r/notes/UWAA558/08-1d-equilibria/ + Mon, 01 Jan 0001 00:00:00 +0000 + + https://peppyhare.github.io/r/notes/UWAA558/08-1d-equilibria/ + 1-Dimensional Equilibria # The \( \theta \) function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: "$$", right: "$$", display: true }, { left: "\\[", right: "\\]", display: true }, { left: "$", right: "$", display: false }, { left: "\\(", right: "\\)", display: false }], macros: { "\\lcm": "\\mathop{\\mathrm{lcm}}", "\\sen": '\\text{sen}\\,', "\\dd": "\\mathop{\\mathrm{d} #1}", "\\abs": "\\lvert #1 \\rvert", "\\dd": "\\text{d}", "\\cross": "\\times", "\\pdv": "\\frac{\\partial #1}{\\partial #2}" + + + + 2D Equilibria + https://peppyhare.github.io/r/notes/UWAA558/09-2d-equilibria/ + Mon, 01 Jan 0001 00:00:00 +0000 + + https://peppyhare.github.io/r/notes/UWAA558/09-2d-equilibria/ + 2D Equilibria # Let&rsquo;s connect the ends of our 1D equilibria. Doing so is what gives us inherently toroidal configurations. From the 1-dimensional picture: + \[\vec j \cross \vec B = j_\theta B_z - j_z B_\theta \\ = \grad p = \dv{p}{r}\] function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: "$$", right: "$$", display: true }, { left: "\\[", right: "\\]", display: true }, { left: " + + + + Equilibrium of 3D Configurations + https://peppyhare.github.io/r/notes/UWAA558/10-equilibrium-of-3d-configurations/ + Mon, 01 Jan 0001 00:00:00 +0000 + + https://peppyhare.github.io/r/notes/UWAA558/10-equilibrium-of-3d-configurations/ + Equilibrium of 3D Configurations # In 3 dimensions, we lose the axisymmetry that allowed us to reach the Grad-Shefranov equation and we need to solve the full momentum equation in three dimensions. This is not something that we can actually do in this class, and the existing codes that do this are quite sophisticated. +Some general features of 3D equilibria are: + No net toroidal current. This means that they tend to be steady-state configurations. + + + + MHD Stability + https://peppyhare.github.io/r/notes/UWAA558/11-mhd-stability/ + Mon, 01 Jan 0001 00:00:00 +0000 + + https://peppyhare.github.io/r/notes/UWAA558/11-mhd-stability/ + MHD Stability # Equilibrium is simply a balance of forces that results in a steady state. Beyond equilibrium, stability is the tendency of a perturbation to return to equilibrium, rather than increasing. We are very interested in analyzing the stability of MHD equilibria, including the plasma dynamics, so we need to use the complete ideal MHD model. The MHD equations are non-linear, which means that any evolution/dynamics are also going to be non-linear. + + + + Formulary + https://peppyhare.github.io/r/notes/UWAA558/formulary/ + Mon, 01 Jan 0001 00:00:00 +0000 + + https://peppyhare.github.io/r/notes/UWAA558/formulary/ + \( \) function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: "$$", right: "$$", display: true }, { left: "\\[", right: "\\]", display: true }, { left: "$", right: "$", display: false }, { left: "\\(", right: "\\)", display: false }], macros: { "\\lcm": "\\mathop{\\mathrm{lcm}}", "\\sen": '\\text{sen}\\,', "\\dd": "\\mathop{\\mathrm{d} #1}", "\\abs": "\\lvert #1 \\rvert", "\\dd": "\\text{d}", "\\cross": "\\times", "\\pdv": "\\frac{\\partial #1}{\\partial #2}", "\\curl": "\\nabla \\cross #1" + + + + diff --git a/docs/notes/index.html b/docs/notes/index.html new file mode 100644 index 00000000..e715f72e --- /dev/null +++ b/docs/notes/index.html @@ -0,0 +1,469 @@ + + + + + + + + + + + + +Notes | My Notes + + + + + + + + + + + + + + + +
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      + Development of a Collisionless Plasma Kinetic Solver and an Investigation of One-Dimensional Plasma Waves and Instabilities + # +

      +

      Shielded potential of a test electron:

      + + + + + \[\phi(r) = \frac{-e}{4 \pi \epsilon_0 r} e ^{- r / \lambda_D}\] + + +

      where the Debye length is + \( \lambda_D = \sqrt{\frac{\epsilon_0 T_e}{ n_e e}} \) + +. The mean free path between large-angle collisions is estimated as

      +

      + \[\lambda_{mfp} \sim \frac{\epsilon_0 T_e ^2}{\phi_e n_e \log ( \Lambda)}\] + + +where + \(\phi_e = e^2 / 4 \pi \epsilon_0\) + + are the constants from the Coulomb force law.

      +

      Smooth out the discreteness of particles via spatial average over small volumes:

      + + \[\rho \rightarrow \langle \rho_c \rangle \qquad \vec E \rightarrow \langle \vec E \rangle + \delta \vec E\] + + +

      The mean field + \(\langle \vec E \rangle\) + + is responsible for collective modes of plasma motion. Estimate the collisionality of the plasma by comparing the length scales + \(\lambda_{mfp} / \lambda_D\) + +

      + + \[\frac{\lambda_{mfp}}{\lambda_D} \sim \frac{T_e ^{3/2}}{n_e ^{1/2}}\] + + +

      Plasma is seen to become collisionless as the temperature becomes high or the plasma becomes more rarified.

      +

      + Phase space mechanics + # +

      +

      To arrive at a kinetic equation governing the collisionless mechanics, consider the one-dimensional motion of a single particle

      + + \[\dot{x} = v \qquad \dot v = F(x)\] + + +

      and define the phase space coordinates as + \(\vec{\dot r} = \vec F \equiv [ v, F(x) ]\) + +. The flux vector is similar to the velocity field of a fluid flow. The streamlines of + \(\vec F\) + + are the streamlines which a particle will follow if the flux is constant in time. Phase flow is always analogous to that of an incompressible fluid because the flux divergence is zero:

      + + \[\div [v, F(x)] = \pdv{v}{x} + \pdv{F(x)}{v}\] + + +

      If the phase fluid density is given by a function + \(f(x, v, t)\) + + where + \(t\) + + is the time parameter, because any instantiation of a particle can not leave the phase plane, the probability density will be conserved. We can write a conservation law: + + \[\pdv{f(x, v, t)}{t} = - \div ( f (x, v, t) \vec F)\] + + +and due to the flow’s incompressibility + + \[\div (f \vec F) = f ( \div \vec F) + \vec F \cdot \grad f = \vec F \cdot \grad f \\ +\rightarrow \pdv{f}{t} = - \left[ v, F(x) \right] \cdot \left[ \pdv{f}{x}, \pdv{f}{v} \right] \\ +\rightarrow \pdv{f}{t} + v \pdv{f}{x} + F(x) \pdv{f}{v} = 0\] + +

      +
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      diff --git a/docs/ru/categories/index.xml b/docs/ru/categories/index.xml index 08a88ee5..1635c190 100644 --- a/docs/ru/categories/index.xml +++ b/docs/ru/categories/index.xml @@ -4,10 +4,6 @@ Categories on My Notes https://peppyhare.github.io/r/ru/categories/ Recent content in Categories on My Notes - Hugo -- gohugo.io - - - - + Hugo -- gohugo.io - \ No newline at end of file + diff --git a/docs/ru/index.html b/docs/ru/index.html index d6df3ae5..5c56b609 100644 --- a/docs/ru/index.html +++ b/docs/ru/index.html @@ -2,7 +2,7 @@ - + @@ -18,7 +18,7 @@ - + @@ -62,6 +62,14 @@

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