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seaice.rst
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seaice.rst
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.. _sub_phys_pkg_seaice:
SEAICE Package
**************
Authors: Martin Losch, Dimitris Menemenlis, An Nguyen, Jean-Michel
Campin, Patrick Heimbach, Chris Hill, Jinlun Zhang, and Damien Ringeisen
.. _ssub_phys_pkg_seaice_intro:
Introduction
============
Package :filelink:`seaice <pkg/seaice>` provides a dynamic and thermodynamic
interactive sea ice model.
CPP options enable or disable different aspects of the package
(:numref:`ssub_phys_pkg_seaice_config`). Run-time options, flags, filenames and
field-related dates/times are set in ``data.seaice``
(:numref:`ssub_phys_pkg_seaice_runtime`). A description of key subroutines is
given in :numref:`ssub_phys_pkg_seaice_subroutines`. Available diagnostics
output is listed in :numref:`ssub_phys_pkg_seaice_diagnostics`.
.. _ssub_phys_pkg_seaice_config:
SEAICE configuration and compiling
==================================
Compile-time options
--------------------
As with all MITgcm packages, SEAICE can be turned on or off at compile time
(see :numref:`building_code`)
- using the ``packages.conf`` file by adding ``seaice`` to it
- or using :filelink:`genmake2 <tools/genmake2>` adding ``-enable=seaice`` or
``-disable=seaice`` switches
- **required packages and CPP options**:
:filelink:`seaice <pkg/seaice>` requires the external forcing package
:filelink:`pkg/exf` to be enabled; no additional CPP options are required.
Parts of the :filelink:`seaice <pkg/seaice>` code can be enabled or disabled at
compile time via CPP preprocessor flags. These options are set in
:filelink:`SEAICE_OPTIONS.h <pkg/seaice/SEAICE_OPTIONS.h>`.
:numref:`tab_phys_pkg_seaice_cpp` summarizes the most important ones. For more
options see :filelink:`SEAICE_OPTIONS.h <pkg/seaice/SEAICE_OPTIONS.h>`.
.. tabularcolumns:: |\Y{.375}|\Y{.1}|\Y{.55}|
.. csv-table:: Some of the most relevant CPP preprocessor flags in the :filelink:`seaice <pkg/seaice>` package.
:header: "CPP option", "Default", Description"
:widths: 30, 10, 60
:name: tab_phys_pkg_seaice_cpp
:varlink:`SEAICE_DEBUG`, #undef, enhance STDOUT for debugging
:varlink:`SEAICE_ALLOW_DYNAMICS`, #define, sea ice dynamics code
:varlink:`SEAICE_CGRID`, #define, LSR solver on C-grid (rather than original B-grid)
:varlink:`SEAICE_ALLOW_EVP`, #define, enable use of EVP rheology solver
:varlink:`SEAICE_ALLOW_JFNK`, #define, enable use of JFNK rheology solver
:varlink:`SEAICE_ALLOW_KRYLOV`, #define, enable use of Krylov rheology solver
:varlink:`SEAICE_ALLOW_TEM`, #undef, enable use of the truncated ellipse method (TEM) and coulombic yield curve
:varlink:`SEAICE_ALLOW_MCS`, #undef, enable use of Mohr-Coulomb yield curve with shear flow rule
:varlink:`SEAICE_ALLOW_MCE`, #undef, enable use of Mohr-Coulomb yield curve with elliptical plastic potential
:varlink:`SEAICE_ALLOW_TD`, #undef, enable use of teardrop and parabolic Lens yield curves with normal flow rules
:varlink:`SEAICE_LSR_ZEBRA`, #undef, use a coloring method for LSR solver
:varlink:`SEAICE_EXTERNAL_FLUXES`, #define, use :filelink:`pkg/exf`-computed fluxes as starting point
:varlink:`SEAICE_ZETA_SMOOTHREG`, #define, use differentiable regularization for viscosities
:varlink:`SEAICE_DELTA_SMOOTHREG`, #undef, use differentiable regularization for :math:`1/\Delta`
:varlink:`SEAICE_ALLOW_BOTTOMDRAG`, #undef, enable grounding parameterization for improved fastice in shallow seas
:varlink:`SEAICE_ITD`, #undef, run with dynamical sea Ice Thickness Distribution (ITD)
:varlink:`SEAICE_VARIABLE_SALINITY`, #undef, enable sea ice with variable salinity
:varlink:`SEAICE_CAP_ICELOAD`, #undef, enable to limit seaice load (:varlink:`siceLoad`) on the sea surface
:varlink:`ALLOW_SITRACER`, #undef, enable sea ice tracer package
:varlink:`SEAICE_BICE_STRESS`, #undef, B-grid only for backward compatiblity: turn on ice-stress on ocean
:varlink:`EXPLICIT_SSH_SLOPE`, #undef, B-grid only for backward compatiblity: use ETAN for tilt computations rather than geostrophic velocities
.. _ssub_phys_pkg_seaice_runtime:
Run-time parameters
===================
Run-time parameters (see :numref:`tab_phys_pkg_seaice_runtimeparms`) are set in
`data.seaice` (read in :filelink:`pkg/seaice/seaice_readparms.F`).
Enabling the package
--------------------
:filelink:`seaice <pkg/seaice>` package is switched on/off at run-time by
setting :varlink:`useSEAICE` ``= .TRUE.`` in ``data.pkg``.
General flags and parameters
----------------------------
:numref:`tab_phys_pkg_seaice_runtimeparms` lists most run-time parameters.
.. tabularcolumns:: |\Y{.275}|\Y{.20}|\Y{.525}|
.. table:: Run-time parameters and default values
:class: longtable
:name: tab_phys_pkg_seaice_runtimeparms
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| Name | Default value | Description |
+====================================+==============================+=========================================================================+
| :varlink:`SEAICEwriteState` | FALSE | write sea ice state to file |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEuseDYNAMICS` | TRUE | use dynamics |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEuseJFNK` | FALSE | use the JFNK-solver |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEuseTEM` | FALSE | use truncated ellipse method or coulombic yield curve |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEuseMCS` | FALSE | use the Mohr-Coulomb yield curve with shear flow rule |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEuseMCE` | FALSE | use the Mohr-Coulomb yield curve with elliptical plastic potential |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEuseTD` | FALSE | use the teardrop yield curve with normal flow rule |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEusePL` | FALSE | use the parabolic Lens yield curve with normal flow rule |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEuseStrImpCpl` | FALSE | use strength implicit coupling in LSR/JFNK |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEuseMetricTerms` | TRUE | use metric terms in dynamics |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEuseEVPpickup` | TRUE | use EVP pickups |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEuseFluxForm` | TRUE | use flux form for 2nd central difference advection scheme |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICErestoreUnderIce` | FALSE | enable restoring to climatology under ice |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEupdateOceanStress` | TRUE | update ocean surface stress accounting for sea ice cover |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEscaleSurfStress` | TRUE | scale atmosphere and ocean-surface stress on ice by concentration (AREA)|
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEaddSnowMass` | TRUE | in computing seaiceMass, add snow contribution |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`useHB87stressCoupling` | FALSE | turn on ice-ocean stress coupling following |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`usePW79thermodynamics` | TRUE | flag to turn off zero-layer-thermodynamics for testing |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEadvHeff` | TRUE | flag to turn off advection of scalar variable :varlink:`HEFF` |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEadvArea` | TRUE | flag to turn off advection of scalar variable :varlink:`AREA` |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEadvSnow` | TRUE | flag to turn off advection of scalar variable :varlink:`HSNOW` |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEadvSalt` | TRUE | flag to turn off advection of scalar variable :varlink:`HSALT` |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEadvScheme` | 77 | set advection scheme for seaice scalar state variables |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEuseFlooding` | TRUE | use flood-freeze algorithm |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_no_slip` | FALSE | use no-slip boundary conditions instead of free-slip |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_deltaTtherm` | :varlink:`dTtracerLev` (1) | time step for seaice thermodynamics (s) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_deltaTdyn` | :varlink:`dTtracerLev` (1) | time step for seaice dynamics (s) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_deltaTevp` | 0.0 | EVP sub-cycling time step (s); values :math:`>` 0 turn on EVP |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEuseEVPstar` | FALSE | use modified EVP\* instead of EVP, following :cite:`lemieux:12` |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEuseEVPrev` | FALSE | "revisited form" variation on EVP\*, following :cite:`bouillon:13` |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEnEVPstarSteps` | unset | number of modified EVP\* iterations |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_evpAlpha` | unset | EVP\* parameter (non-dim.), to replace |
| | | 2*\ :varlink:`SEAICE_evpTauRelax`\ /\ :varlink:`SEAICE_deltaTevp` |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_evpBeta` | unset | EVP\* parameter (non-dim.), to replace |
| | | :varlink:`SEAICE_deltaTdyn`\ /\ :varlink:`SEAICE_deltaTevp` |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEaEVPcoeff` | unset | largest stabilized frequency for adaptive EVP (non-dim.) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEaEVPcStar` | 4.0 | aEVP multiple of stability factor (non-dim.), see :cite:`kimmritz:16` |
| | | :math:`\alpha * \beta = c^\ast * \gamma` |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEaEVPalphaMin` | 5.0 | aEVP lower limit of alpha and beta (non-dim.), see :cite:`kimmritz:16` |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_elasticParm` | 0.33333333 | EVP parameter :math:`E_0` (non-dim.), sets relaxation timescale |
| | | :varlink:`SEAICE_evpTauRelax` = |
| | | :varlink:`SEAICE_elasticParm` * :varlink:`SEAICE_deltaTdyn` |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_evpTauRelax` | :varlink:`dTtracerLev` (1) * | relaxation time scale :math:`T` for EVP waves (s) |
| | :varlink:`SEAICE_elasticParm`| |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_OLx` | :varlink:`OLx` - 2 | overlap for LSR-solver or preconditioner, :math:`x`-dimension |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_OLy` | :varlink:`OLy` - 2 | overlap for LSR-solver or preconditioner, :math:`y`-dimension |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEnonLinIterMax` | 2/10 | maximum number of non-linear (outer loop) iterations (LSR/JFNK) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICElinearIterMax` | 1500/10 | maximum number of linear iterations (LSR/JFNK) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_JFNK_lsIter` | (off) | start line search after “lsIter” Newton iterations |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEnonLinTol` | 1.0E-05 | non-linear tolerance parameter for JFNK solver |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`JFNKgamma_lin_min` | 0.10 | minimum tolerance parameter for linear JFNK solver |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`JFNKgamma_lin_max` | 0.99 | maximum tolerance parameter for linear JFNK solver |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`JFNKres_tFac` | unset | tolerance parameter for FGMRES residual |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_JFNKepsilon` | 1.0E-06 | step size for the FD-gradient in s/r seaice_jacvec |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_dumpFreq` | dumpFreq | dump frequency (s) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_dump_mdsio` | TRUE | write snapshot using :filelink:`/pkg/mdsio` |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_dump_mnc` | FALSE | write snapshot using :filelink:`/pkg/mnc` |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_initialHEFF` | 0.0 | initial sea ice thickness averaged over grid cell (m) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_drag` | 1.0E-03 | air-ice drag coefficient (non-dim.) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`OCEAN_drag` | 1.0E-03 | air-ocean drag coefficient (non-dim.) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_waterDrag` | 5.5E-03 | water-ice drag coefficient (non-dim.) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_dryIceAlb` | 0.75 | winter sea ice albedo |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_wetIceAlb` | 0.66 | summer sea ice albedo |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_drySnowAlb` | 0.84 | dry snow albedo |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_wetSnowAlb` | 0.70 | wet snow albedo |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_waterAlbedo` | 0.10 | water albedo (not used if #define :varlink:`SEAICE_EXTERNAL_FLUXES`) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_strength` | 2.75E+04 | sea ice strength constant :math:`P^{\ast}` (N/m\ :sup:`2`) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_cStar` | 20.0 | sea ice strength constant :math:`C^{\ast}` (non-dim.) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_eccen` | 2.0 | VP rheology ellipse aspect ratio :math:`e` |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_eccfr` | = :varlink:`SEAICE_eccen` | sea ice plastic potential ellipse aspect ratio :math:`e_G` |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEmcMU` | 1.0 | slope of the Mohr-Coulomb yield curve |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEpressReplFac` | 1.0 | use replacement pressure (0.0-1.0) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_tensilFac` | 0.0 | tensile factor for the yield curve |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_rhoAir` | 1.3 (or | density of air (kg/m\ :sup:`3`) |
| | :filelink:`pkg/exf` value) | |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_cpAir` | 1004.0 (or | specific heat of air (J/kg/K) |
| | :filelink:`pkg/exf` value) | |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_lhEvap` | 2.5E+06 (or | latent heat of evaporation (J/kg) |
| | :filelink:`pkg/exf` value) | |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_lhFusion` | 3.34E+05 (or | latent heat of fusion (J/kg) |
| | :filelink:`pkg/exf` value) | |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_dalton` | 1.75E-03 | ice-ocean transfer coefficient for latent and sensible heat (non-dim.) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`useMaykutSatVapPoly` | FALSE | use Maykut polynomial to compute saturation vapor pressure |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_iceConduct` | 2.16560E+00 | sea ice conductivity (W m\ :sup:`-1` K\ :sup:`-1`) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_snowConduct` | 3.10000E-01 | snow conductivity (W m\ :sup:`-1` K\ :sup:`-1`) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_emissivity` | 0.970018 (or | longwave ocean surface emissivity (non-dim.) |
| | :filelink:`pkg/exf` value) | |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_snowThick` | 0.15 | cutoff snow thickness to use snow albedo (m) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_shortwave` | 0.30 | ice penetration shortwave radiation factor (non-dim.) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_saltFrac` | 0.0 | salinity newly formed ice (as fraction of ocean surface salinity) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_frazilFrac` | 1.0 (or | frazil to sea ice conversion rate, as fraction |
| | computed from other parms) | (relative to the local freezing point of sea ice water) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEstressFactor` | 1.0 | scaling factor for ice area in computing total ocean stress (non-dim.) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`HeffFile` | unset | filename for initial sea ice eff. thickness field :varlink:`HEFF` (m) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`AreaFile` | unset | filename for initial fraction sea ice cover :varlink:`AREA` (non-dim.) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`HsnowFile` | unset | filename for initial eff. snow thickness field :varlink:`HSNOW` (m) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`HsaltFile` | unset | filename for initial eff. sea ice salinity field :varlink:`HSALT` |
| | | (g/m\ :sup:`2`) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`LSR_ERROR` | 1.0E-04 | sets accuracy of LSR solver |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`DIFF1` | 0.0 | parameter used in advect.F |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`HO` | 0.5 | lead closing parameter :math:`h_0` (m); demarcation thickness between |
| | | thick and thin ice which determines partition between vertical and |
| | | lateral ice growth |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`MIN_ATEMP` | -50.0 | minimum air temperature (:sup:`o`\ C) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`MIN_LWDOWN` | 60.0 | minimum downward longwave (W/m\ :sup:`2`) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`MIN_TICE` | -50.0 | minimum ice temperature (:sup:`o`\ C) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`IMAX_TICE` | 10 | number of iterations for ice surface temperature solution |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_EPS` | 1.0E-10 | a "small number" used in various routines |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_area_reg` | 1.0E-5 | minimum concentration to regularize ice thickness |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_hice_reg` | 0.05 | minimum ice thickness (m) for regularization |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_multDim` | 1 | number of ice categories for thermodynamics |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_useMultDimSnow` | TRUE | use same fixed pdf for snow as for multi-thickness-category ice |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
The following dynamical ice thickness distribution and ridging parameters in
:numref:`tab_phys_pkg_seaice_ridging` are only active with #define
:varlink:`SEAICE_ITD`. All parameters are non-dimensional unless indicated.
.. tabularcolumns:: |\Y{.275}|\Y{.20}|\Y{.525}|
.. table:: Thickness distribution and ridging parameters
:name: tab_phys_pkg_seaice_ridging
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| Name | Default value | Description |
+====================================+==============================+=========================================================================+
| :varlink:`useHibler79IceStrength` | TRUE | use :cite:`hibler:79` ice strength; do not use :cite:`rothrock:75` |
| | | with #define :varlink:`SEAICE_ITD` |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEsimpleRidging` | TRUE | use simple ridging a la :cite:`hibler:79` |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICE_cf` | 17.0 | scaling parameter of :cite:`rothrock:75` ice strength parameterization |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEpartFunc` | 0 | use partition function of :cite:`thorndike:75` |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEredistFunc` | 0 | use redistribution function of :cite:`hibler:80` |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEridgingIterMax` | 10 | maximum number of ridging sweeps |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEshearParm` | 0.5 | fraction of shear to be used for ridging |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEgStar` | 0.15 | max. ice conc. that participates in ridging :cite:`thorndike:75` |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEhStar` | 25.0 | ridging parameter for :cite:`thorndike:75`, :cite:`lipscomb:07` |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEaStar` | 0.05 | similar to :varlink:`SEAICEgStar` for |
| | | :cite:`lipscomb:07` participation function |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEmuRidging` | 3.0 | similar to :varlink:`SEAICEhStar` for |
| | | :cite:`lipscomb:07` ridging function |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEmaxRaft` | 1.0 | regularization parameter for rafting |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEsnowFracRidge` | 0.5 | fraction of snow that remains on ridged ice |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`SEAICEuseLinRemapITD` | TRUE | use linear remapping scheme of :cite:`lipscomb:01` |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`Hlimit` | unset | nITD+1-array of ice thickness category limits (m) |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
| :varlink:`Hlimit_c1`, | 3.0, | when :varlink:`Hlimit` is not set, then these parameters |
| :varlink:`Hlimit_c2`, | 15.0, | determine :varlink:`Hlimit` from a simple function |
| :varlink:`Hlimit_c3` | 3.0 | following :cite:`lipscomb:01` |
+------------------------------------+------------------------------+-------------------------------------------------------------------------+
.. _ssub_phys_pkg_seaice_descr:
Description
===========
The MITgcm sea ice model is based on a variant of the viscous-plastic (VP)
dynamic-thermodynamic sea ice model (Zhang and Hibler 1997 :cite:`zhang:97`)
first introduced in Hibler (1979) and Hibler (1980)
:cite:`hibler:79,hibler:80`. In order to adapt this model to the requirements
of coupled ice-ocean state estimation, many important aspects of the original
code have been modified and improved, see Losch et al. (2010) :cite:`losch:10`:
- the code has been rewritten for an Arakawa C-grid, both B- and C-grid
variants are available; the C-grid code allows for no-slip and free-slip
lateral boundary conditions;
- three different solution methods for solving the nonlinear momentum
equations have been adopted: LSOR (Zhang and Hibler 1997 :cite:`zhang:97`),
EVP (Hunke and Dukowicz 1997 :cite:`hunke:97`),
JFNK (Lemieux et al. 2010 :cite:`lemieux:10`, Losch et al. 2014
:cite:`losch:14`);
- ice-ocean stress can be formulated as in Hibler and Bryan (1987)
:cite:`hibler:87` or as in Campin et al. (2008) :cite:`campin:08`;
- ice variables are advected by sophisticated, conservative advection
schemes with flux limiting;
- growth and melt parameterizations have been refined and extended in
order to allow for more stable automatic differentiation of the code.
The sea ice model is tightly coupled to the ocean compontent of the
MITgcm. Heat, fresh water fluxes and surface stresses are computed from the
atmospheric state and, by default, modified by the ice model at every time
step.
The ice dynamics models that are most widely used for large-scale climate
studies are the viscous-plastic (VP) model (Hilber 1979 :cite:`hibler:79`), the
cavitating fluid (CF) model (Flato and Hibler 1992 :cite:`flato:92`), and the
elastic-viscous-plastic (EVP) model (Hunke and Dukowicz 1997 :cite:`hunke:97`).
Compared to the VP model, the CF model does not allow ice shear in calculating
ice motion, stress, and deformation. EVP models approximate VP by adding an
elastic term to the equations for easier adaptation to parallel
computers. Because of its higher accuracy in plastic solution and relatively
simpler formulation, compared to the EVP model, we decided to use the VP model
as the default dynamic component of our ice model. To do this we extended the
line successive over relaxation (LSOR) method of Zhang and Hibler (1997)
:cite:`zhang:97` for use in a parallel configuration. An EVP model and a
free-drift implementation can be selected with run-time flags.
.. _para_phys_pkg_seaice_thsice:
Compatibility with ice-thermodynamics package :filelink:`pkg/thsice`
--------------------------------------------------------------------
By default :filelink:`pkg/seaice` includes the original so-called zero-layer
thermodynamics with a snow cover as in the appendix of Semtner (1976)
:cite:`semtner:76`. The zero-layer thermodynamic model assumes that ice does
not store heat and, therefore, tends to exaggerate the seasonal variability in
ice thickness. This exaggeration can be significantly reduced by using Winton's
(Winton 2000 :cite:`winton:00`) three-layer thermodynamic model that permits
heat storage in ice.
The Winton (2000) sea-ice thermodynamics have been ported to MITgcm; they
currently reside under :filelink:`pkg/thsice`, described in
:numref:`sub_phys_pkg_thsice`. It is fully compatible with the packages
:filelink:`seaice <pkg/seaice>` and :filelink:`exf <pkg/exf>`. When turned on
together with :filelink:`seaice <pkg/seaice>`, the zero-layer thermodynamics
are replaced by the Winton thermodynamics. In order to use package
:filelink:`seaice <pkg/seaice>` with the thermodynamics of
:filelink:`pkg/thsice`, compile both packages and turn both package on in
``data.pkg``; see an example in
:filelink:`verification/global_ocean.cs32x15/input.icedyn`. Note, that once
:filelink:`thsice <pkg/thsice>` is turned on, the variables and diagnostics
associated to the default thermodynamics are meaningless, and the diagnostics
of :filelink:`thsice <pkg/thsice>` must be used instead.
.. _para_phys_pkg_seaice_surfaceforcing:
Surface forcing
---------------
The sea ice model requires the following input fields: 10 m winds, 2 m air
temperature and specific humidity, downward longwave and shortwave radiations,
precipitation, evaporation, and river and glacier runoff. The sea ice model
also requires surface temperature from the ocean model and the top level
horizontal velocity. Output fields are surface wind stress, evaporation minus
precipitation minus runoff, net surface heat flux, and net shortwave flux. The
sea-ice model is global: in ice-free regions bulk formulae (by default computed
in package :filelink:`exf <pkg/exf>`) are used to estimate oceanic forcing from
the atmospheric fields.
.. _para_phys_pkg_seaice_dynamics:
Dynamics
--------
The momentum equation of the sea-ice model is
.. math::
m \frac{D\mathbf{u}}{Dt} = -mf\mathbf{k}\times\mathbf{u} +
\mathbf{\tau}_\mathrm{air} + \mathbf{\tau}_\mathrm{ocean}
- m \nabla{\phi(0)} + \mathbf{F}
:label: eq_momseaice
where :math:`m=m_{i}+m_{s}` is the ice and snow mass per unit area;
:math:`\mathbf{u}=u\mathbf{i}+v\mathbf{j}` is the ice velocity vector;
:math:`\mathbf{i}`, :math:`\mathbf{j}`, and :math:`\mathbf{k}` are unit vectors
in the :math:`x`, :math:`y`, and :math:`z` directions, respectively; :math:`f`
is the Coriolis parameter; :math:`\mathbf{\tau}_\mathrm{air}` and
:math:`\mathbf{\tau}_\mathrm{ocean}` are the wind-ice and ocean-ice stresses,
respectively; :math:`g` is the gravity accelation; :math:`\nabla\phi(0)` is the
gradient (or tilt) of the sea surface height; :math:`\phi(0) = g\eta +
p_{a}/\rho_{0} + mg/\rho_{0}` is the sea surface height potential in response
to ocean dynamics (:math:`g\eta`), to atmospheric pressure loading
(:math:`p_{a}/\rho_{0}`, where :math:`\rho_{0}` is a reference density) and a
term due to snow and ice loading ; and :math:`\mathbf{F}=\nabla\cdot\sigma` is
the divergence of the internal ice stress tensor :math:`\sigma_{ij}`.
Advection of sea-ice momentum is neglected. The wind and ice-ocean stress terms
are given by
.. math::
\begin{aligned}
\mathbf{\tau}_\mathrm{air} = & \rho_\mathrm{air} C_\mathrm{air}
|\mathbf{U}_\mathrm{air} -\mathbf{u}| R_\mathrm{air}
(\mathbf{U}_\mathrm{air} - \mathbf{u}) \\
\mathbf{\tau}_\mathrm{ocean} = & \rho_\mathrm{ocean}C_\mathrm{ocean}
|\mathbf{U}_\mathrm{ocean}-\mathbf{u}|
R_\mathrm{ocean}(\mathbf{U}_\mathrm{ocean} - \mathbf{u})
\end{aligned}
where :math:`\mathbf{U}_\mathrm{air/ocean}` are the surface winds of the
atmosphere and surface currents of the ocean, respectively;
:math:`C_\mathrm{air/ocean}` are air and ocean drag coefficients;
:math:`\rho_\mathrm{air/ocean}` are reference densities; and
:math:`R_\mathrm{air/ocean}` are rotation matrices that act on the wind/current
vectors.
.. _para_phys_pkg_seaice_VPrheology:
Viscous-Plastic (VP) Rheology
-----------------------------
For an isotropic system the stress tensor :math:`\sigma_{ij}` (:math:`i,j=1,2`)
can be related to the ice strain rate and strength by a nonlinear
viscous-plastic (VP) constitutive law:
.. math::
\sigma_{ij}=2\eta(\dot{\epsilon}_{ij},P)\dot{\epsilon}_{ij}
+ \left[\zeta(\dot{\epsilon}_{ij},P) -
\eta(\dot{\epsilon}_{ij},P)\right]\dot{\epsilon}_{kk}\delta_{ij}
- \frac{P}{2}\delta_{ij}
:label: eq_vpequation
The ice strain rate is given by
.. math::
\dot{\epsilon}_{ij} = \frac{1}{2}\left(
\frac{\partial{u_{i}}}{\partial{x_{j}}} +
\frac{\partial{u_{j}}}{\partial{x_{i}}}\right)
The maximum ice pressure :math:`P_{\max}` (variable :varlink:`PRESS0` in the
code), a measure of ice strength, depends on both thickness :math:`h` and
compactness (concentration) :math:`c`:
.. math::
:label: eq_icestrength
P_{\max} = P^{\ast}c\,h\,\exp\{-C^{\ast}\cdot(1-c)\},
with the constants :math:`P^{\ast}` (run-time parameter
:varlink:`SEAICE_strength`) and :math:`C^{\ast}` (run-time parameter
:varlink:`SEAICE_cStar`). By default, :math:`P` (variable :varlink:`PRESS` in
the code) is the replacement pressure
.. math::
:label: eq_pressrepl
P = (1-k_t)\,P_{\max} \left( (1 - f_{r})
+ f_{r} \frac{\Delta}{\Delta_{reg}} \right)
where :math:`f_{r}` is run-time parameter :varlink:`SEAICEpressReplFac`
(default = 1.0), and :math:`\Delta_{reg}` is a regularized form of
:math:`\Delta = \left[ \left(\dot{\epsilon}_{11}+\dot{\epsilon}_{22}\right)^2 +
e^{-2}\left( \left(\dot{\epsilon}_{11}-\dot{\epsilon}_{22} \right)^2 +
\dot{\epsilon}_{12}^2 \right) \right]^{\frac{1}{2}}`, for example
:math:`\Delta_{reg} = \max(\Delta,\Delta_{\min})`.
The tensile strength factor :math:`k_t` (run-time parameter
:varlink:`SEAICE_tensilFac`) determines the ice tensile strength :math:`T =
k_t\cdot P_{\max}`, as defined by König Beatty and Holland (2010)
:cite:`konig:10`. :varlink:`SEAICE_tensilFac` is zero by default.
Different VP rheologies can be used to model sea ice dynamics. The different
rheologies are characterized by different definitions of the bulk and shear
viscosities :math:`\zeta` and :math:`\eta` in :eq:`eq_vpequation`. The
following :numref:`tab_phys_pkg_seaice_rheologies` is a summary of the
available choices with recommended (sensible) parameter values. All the
rheologies presented here depend on the ice strength :math:`P`
:eq:`eq_pressrepl`.
.. tabularcolumns:: |\Y{.275}|\Y{.450}|\Y{.275}|
.. table:: Overview over availabe sea ice viscous-plastic rheologies
:class: longtable
:name: tab_phys_pkg_seaice_rheologies
+---------------------------------------+---------------------------------------+----------------------------------------------------+
| Name | CPP flags | Run-time flags (recommended value) |
+=======================================+=======================================+====================================================+
| :ref:`rheologies_ellnfr` | None (default) | - :varlink:`SEAICE_eccen` (= 2.0) |
| | | - :varlink:`SEAICE_tensilFac` (= 0.0) |
+---------------------------------------+---------------------------------------+----------------------------------------------------+
| :ref:`rheologies_ellnnfr` | None | - :varlink:`SEAICE_eccen` (= 2.0) |
| | | - :varlink:`SEAICE_eccfr` (< 2.0) |
| | | - :varlink:`SEAICE_tensilFac` (= 0.0) |
+---------------------------------------+---------------------------------------+----------------------------------------------------+
| :ref:`rheologies_TEM` | :varlink:`SEAICE_ALLOW_TEM` | - :varlink:`SEAICEuseTEM` (=.TRUE.) |
| | | - :varlink:`SEAICE_eccen` (= 1.4) |
| | | - :varlink:`SEAICE_eccfr` (< 1.4) |
| | | - :varlink:`SEAICE_tensilFac` (= 0.05) |
| | | - :varlink:`SEAICEmcMU` (= 0.6 to 0.8) |
+---------------------------------------+---------------------------------------+----------------------------------------------------+
| :ref:`rheologies_MCE` | :varlink:`SEAICE_ALLOW_MCE` | - :varlink:`SEAICEuseMCE` (=.TRUE.) |
| | | - :varlink:`SEAICE_eccen` (= 1.4) |
| | | - :varlink:`SEAICE_eccfr` (< 1.4) |
| | | - :varlink:`SEAICE_tensilFac` (= 0.05) |
| | | - :varlink:`SEAICEmcMU` (= 0.6 to 0.8) |
+---------------------------------------+---------------------------------------+----------------------------------------------------+
| :ref:`rheologies_MCS` | :varlink:`SEAICE_ALLOW_MCS` | - :varlink:`SEAICEuseMCS` (=.TRUE.) |
| | | - :varlink:`SEAICE_tensilFac` (= 0.05) |
| | | - :varlink:`SEAICEmcMU` (= 0.6 to 0.8) |
+---------------------------------------+---------------------------------------+----------------------------------------------------+
| :ref:`rheologies_TD` | :varlink:`SEAICE_ALLOW_TD` | - :varlink:`SEAICEuseTD` (=.TRUE.) |
| | | - :varlink:`SEAICE_tensilFac` (= 0.025) |
+---------------------------------------+---------------------------------------+----------------------------------------------------+
| :ref:`rheologies_PL` | :varlink:`SEAICE_ALLOW_TD` | - :varlink:`SEAICEusePL` (=.TRUE.) |
| | | - :varlink:`SEAICE_tensilFac` (= 0.025) |
+---------------------------------------+---------------------------------------+----------------------------------------------------+
**Note:** With the exception of the default rheology and the TEM (with
:varlink:`SEAICEmcMU` : :math:`\mu=1.0`), these rheologies are not implemented
in EVP (:numref:`para_phys_pkg_seaice_EVPdynamics`).
.. _rheologies_ellnfr:
Elliptical yield curve with normal flow rule
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The default rheology in the sea ice module of the MITgcm implements the widely
used elliptical yield curve with a normal flow rule :cite:`hibler:79`. For
this yield curve, the nonlinear bulk and shear viscosities :math:`\zeta` and
:math:`\eta` are functions of ice strain rate invariants and ice strength such
that the principal components of the stress lie on an elliptical yield curve
with the ratio of major to minor axis :math:`e = 2.0` (run-time parameter
:varlink:`SEAICE_eccen`); they are given by:
.. math::
\begin{aligned}
\zeta =& \min\left(\frac{(1+k_t)P_{\max}}{2\max(\Delta,\Delta_{\min})},
\zeta_{\max}\right) \\
\eta =& \frac{\zeta}{e^2}
\end{aligned}
:label: eq_zetareg
with the abbreviation
.. math::
\Delta = \left[
\left(\dot{\epsilon}_{11}+\dot{\epsilon}_{22}\right)^2
+ e^{-2}\left( \left(\dot{\epsilon}_{11}-\dot{\epsilon}_{22} \right)^2
+ \dot{\epsilon}_{12}^2 \right)
\right]^{\frac{1}{2}}
The bulk viscosities are bounded above by imposing both a minimum
:math:`\Delta_{\min}` (for numerical reasons, run-time parameter
:varlink:`SEAICE_deltaMin` is set to a default value of
:math:`10^{-10}\,\text{s}^{-1}`, the value of :varlink:`SEAICE_EPS`) and a
maximum :math:`\zeta_{\max} = P_{\max}/(2\Delta^\ast)`, where
:math:`\Delta^\ast=(2\times10^4/5\times10^{12})\,\text{s}^{-1}` :math:`=
2\times10^{-9}\,\text{s}^{-1}`. Obviously, this corresponds to regularizing
:math:`\Delta` with the typical value of :varlink:`SEAICE_deltaMin` :math:`=
2\times10^{-9}`. Clearly, some of this regularization is redundant. (There is
also the option of bounding :math:`\zeta` from below by setting run-time
parameter :varlink:`SEAICE_zetaMin` :math:`>0`, but this is generally not
recommended). For stress tensor computation the replacement pressure :math:`P =
2\,\Delta\zeta` is used so that the stress state always lies on the elliptic
yield curve by definition.
Defining the CPP-flag :varlink:`SEAICE_ZETA_SMOOTHREG` in
:filelink:`SEAICE_OPTIONS.h <pkg/seaice/SEAICE_OPTIONS.h>` before compiling
replaces the method for bounding :math:`\zeta` by a smooth (differentiable)
expression:
.. math::
\begin{split}
\zeta &= \zeta_{\max}\tanh\left(\frac{(1+k_t)P_{\max}}{2\,
\min(\Delta,\Delta_{\min}) \,\zeta_{\max}}\right)\\
&= \frac{(1+k_t)P_{\max}}{2\Delta^\ast}
\tanh\left(\frac{\Delta^\ast}{\min(\Delta,\Delta_{\min})}\right)
\end{split}
:label: eq_zetaregsmooth
where :math:`\Delta_{\min}=10^{-20}\,\text{s}^{-1}` should be chosen to avoid
divisions by zero.
In this default formulation the yield curve does not allow isotropic tensile
stress, that is, sea ice can be "pulled apart" without any effort. Setting the
parameter :math:`k_t` (:varlink:`SEAICE_tensilFac`) to a small value larger
than zero, extends the yield curve into a region where the divergence of the
stress :math:`\sigma_{11}+\sigma_{22} > 0` to allow some tensile stress.
Besides this commonly used default rheology, a number of a alternative
rheologies are implemented. Some of these are experiemental and should be used
with caution.
.. _rheologies_ellnnfr:
Elliptical yield curve with non-normal flow rule
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Defining the run-time parameter :varlink:`SEAICE_eccfr` with a value different
from :varlink:`SEAICE_eccen` allows one to use an elliptical yield curve with a
non-normal flow rule as described in Ringeisen et al. (2020)
:cite:`ringeisen:20`. In this case the viscosities are functions of
:math:`e_F` (:varlink:`SEAICE_eccen`) and :math:`e_G`
(:varlink:`SEAICE_eccfr`):
.. math::
\begin{aligned}
\zeta &= \frac{P_{\max}(1+k_t)}{2\Delta} \\
\eta &= \frac{\zeta}{e_G^2} = \frac{P_{\max}(1+k_t)}{2e_G^2\Delta}
\end{aligned}
with the abbreviation
.. math::
\Delta = \sqrt{(\dot{\epsilon}_{11}-\dot{\epsilon}_{22})^2
+\frac{e_F^2}{e_G^4}((\dot{\epsilon}_{11}
-\dot{\epsilon}_{22})^2+4\dot{\epsilon}_{12}^2)}.
Note that if :math:`e_G=e_F=e`, these formulae reduce to the normal flow rule.
.. _rheologies_TEM:
Truncated ellipse method (TEM) for elliptical yield curve
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
In the so-called truncated ellipse method, the shear viscosity :math:`\eta` is
capped to suppress any tensile stress:
.. math::
\eta = \min\left(\frac{\zeta}{e^2},
\frac{\frac{(1+k_t)\,P_{\max}}{2}-\zeta(\dot{\epsilon}_{11}+\dot{\epsilon}_{22})}
{\sqrt{\max(\Delta_{\min}^{2},(\dot{\epsilon}_{11}-\dot{\epsilon}_{22})^2
+4\dot{\epsilon}_{12}^2})}\right).
:label: eq_etatem
To enable this method, set ``#define`` :varlink:`SEAICE_ALLOW_TEM` in
:filelink:`SEAICE_OPTIONS.h <pkg/seaice/SEAICE_OPTIONS.h>` and turn it on with
:varlink:`SEAICEuseTEM` ``=.TRUE.`` in ``data.seaice``. This parameter
combination implies the default of :varlink:`SEAICEmcMU` :math:`= 1.0`.
Instead of an ellipse that is truncated by constant slope coulombic limbs, this
yield curve can also be seen as a Mohr-Coulomb yield curve with elliptical flow
rule that is truncated for high :math:`P` by an ellipse. As a consequence, the
Mohr-Coulomb slope :varlink:`SEAICEmcMU` can be set in ``data.seaice`` to
values :math:`\ne 1.0`. This defines a coulombic yield curve similar to the
ones shown in Hibler and Schulson (2000) :cite:`hibler:00` and Ringeisen et
al. (2019) :cite:`ringeisen:19`.
For this rheology, it is recommended to use a non-zero tensile strength, so set
:varlink:`SEAICE_tensilFac` :math:`=k_{t}>0` in ``data.seaice``, e.g., :math:`=
0.05` or 5%.
.. _rheologies_MCE:
Mohr-Coulomb yield curve with elliptical plastic potential
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
To use a Mohr-Coulomb rheology, set ``#define`` :varlink:`SEAICE_ALLOW_MCE` in
:filelink:`SEAICE_OPTIONS.h <pkg/seaice/SEAICE_OPTIONS.h>` and
:varlink:`SEAICEuseMCE` ``= .TRUE.`` in ``data.seaice``. This Mohr-Coulomb
yield curve uses an elliptical plastic potential to define the flow rule. The
slope of the Mohr-Coulomb yield curve is defined by :varlink:`SEAICEmcMU` in
``data.seaice``, and the plastic potential ellipse aspect ratio is set by
:varlink:`SEAICE_eccfr` in ``data.seaice``. For details of this rheology, see
https://doi.org/10.26092/elib/380, Chapter 2.
For this rheology, it is recommended to use a non-zero tensile strength, so set
:varlink:`SEAICE_tensilFac` :math:`>0` in ``data.seaice``, e.g., :math:`= 0.05`
or 5%.
.. _rheologies_MCS:
Mohr-Coulomb yield curve with shear flow rule
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
To use the specifc Mohr-Coulomb rheology as defined first by Ip et al. (1991)
:cite:`ip:91`, set ``#define`` :varlink:`SEAICE_ALLOW_MCS` in
:filelink:`SEAICE_OPTIONS.h <pkg/seaice/SEAICE_OPTIONS.h>` and
:varlink:`SEAICEuseMCS` ``= .TRUE.`` in ``data.seaice``. The slope of the
Mohr-Coulomb yield curve is defined by :varlink:`SEAICEmcMU` in
``data.seaice``. For details of this rheology, including the tensile strength,
see https://doi.org/10.26092/elib/380, Chapter 2.
For this rheology, it is recommended to use a non-zero tensile strength, so set
:varlink:`SEAICE_tensilFac` :math:`>0` in ``data.seaice``, e.g., :math:`= 0.05`
or 5%.
**WARNING: This rheology is known to be unstable. Use with caution!**
.. _rheologies_TD:
Teardrop yield curve with normal flow rule
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The teardrop rheology was first described in Zhang and Rothrock (2005)
:cite:`zha:05`. Here we implement a slightly modified version (See
https://doi.org/10.26092/elib/380, Chapter 2).
To use this rheology, set ``#define`` :varlink:`SEAICE_ALLOW_TEARDROP` in
:filelink:`SEAICE_OPTIONS.h <pkg/seaice/SEAICE_OPTIONS.h>` and
:varlink:`SEAICEuseTD` ``= .TRUE.`` in ``data.seaice``. The size of the yield
curve can be modified by changing the tensile strength, using
:varlink:`SEAICE_tensFac` in ``data.seaice``.
For this rheology, it is recommended to use a non-zero tensile strength, so set
:varlink:`SEAICE_tensilFac` :math:`>0` in ``data.seaice``, e.g., :math:`=
0.025` or 2.5%.
.. _rheologies_PL:
Parabolic lens yield curve with normal flow rule
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The parabolic lens rheology was first described in Zhang and Rothrock (2005)
:cite:`zha:05`. Here we implement a slightly modified version (See
https://doi.org/10.26092/elib/380, Chapter 2).
To use this rheology, set ``#define`` :varlink:`SEAICE_ALLOW_TEARDROP` in
:filelink:`SEAICE_OPTIONS.h <pkg/seaice/SEAICE_OPTIONS.h>` and
:varlink:`SEAICEusePL` ``= .TRUE.`` in ``data.seaice``. The size of the yield
curve can be modified by changing the tensile strength, using
:varlink:`SEAICE_tensFac` in ``data.seaice``.
For this rheology, it is recommended to use a non-zero tensile strength, so set
:varlink:`SEAICE_tensilFac` :math:`>0` in ``data.seaice``, e.g., :math:`=
0.025` or 2.5%.
.. _para_phys_pkg_seaice_LSRJFNK:
LSR and JFNK solver
-------------------
In matrix notation, the discretized momentum equations can be written as
.. math::
:label: eq_matrixmom
\mathbf{A}(\mathbf{x})\,\mathbf{x} = \mathbf{b}(\mathbf{x}).
The solution vector :math:`\mathbf{x}` consists of the two velocity components
:math:`u` and :math:`v` that contain the velocity variables at all grid points
and at one time level. The standard (and default) method for solving
Eq. :eq:`eq_matrixmom` in the sea ice component of MITgcm is an iterative
Picard solver: in the :math:`k`-th iteration a linearized form
:math:`\mathbf{A}(\mathbf{x}^{k-1})\,\mathbf{x}^{k} =
\mathbf{b}(\mathbf{x}^{k-1})` is solved (in the case of MITgcm it is a Line
Successive (over) Relaxation (LSR) algorithm). Picard solvers converge slowly,
but in practice the iteration is generally terminated after only a few
nonlinear steps and the calculation continues with the next time level. This
method is the default method in MITgcm. The number of nonlinear iteration steps
or pseudo-time steps can be controlled by the run-time parameter
:varlink:`SEAICEnonLinIterMax` (default is 2).
In order to overcome the poor convergence of the Picard-solver, Lemieux et
al. (2010) :cite:`lemieux:10` introduced a Jacobian-free Newton-Krylov solver
for the sea ice momentum equations. This solver is also implemented in MITgcm
(see Losch et al. 2014 :cite:`losch:14`). The Newton method transforms
minimizing the residual :math:`\mathbf{F}(\mathbf{x}) =
\mathbf{A}(\mathbf{x})\,\mathbf{x} - \mathbf{b}(\mathbf{x})` to finding the
roots of a multivariate Taylor expansion of the residual :math:`\mathbf{F}`
around the previous (:math:`k-1`) estimate :math:`\mathbf{x}^{k-1}`:
.. math::
\mathbf{F}(\mathbf{x}^{k-1}+\delta\mathbf{x}^{k}) =
\mathbf{F}(\mathbf{x}^{k-1}) + \mathbf{F}'(\mathbf{x}^{k-1})
\,\delta\mathbf{x}^{k}
:label: eq_jfnktaylor
with the Jacobian :math:`\mathbf{J}\equiv\mathbf{F}'`. The root
:math:`\mathbf{F}(\mathbf{x}^{k-1}+\delta\mathbf{x}^{k})=0` is found by solving
.. math::
\mathbf{J}(\mathbf{x}^{k-1})\,\delta\mathbf{x}^{k} =
-\mathbf{F}(\mathbf{x}^{k-1})
:label: eq_jfnklin
for :math:`\delta\mathbf{x}^{k}`. The next (:math:`k`-th) estimate is given by
:math:`\mathbf{x}^{k}=\mathbf{x}^{k-1}+a\,\delta\mathbf{x}^{k}`. In order to
avoid overshoots the factor :math:`a` is iteratively reduced in a line search
(:math:`a=1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots`) until
:math:`\|\mathbf{F}(\mathbf{x}^k)\| < \|\mathbf{F}(\mathbf{x}^{k-1})\|`, where
:math:`\|\cdot\|=\int\cdot\,dx^2` is the :math:`L_2`-norm. In practice, the
line search is stopped at :math:`a=\frac{1}{8}`. The line search starts after
:varlink:`SEAICE_JFNK_lsIter` nonlinear Newton iterations (off by default).
Forming the Jacobian :math:`\mathbf{J}` explicitly is often avoided as “too
error prone and time consuming”. Instead, Krylov methods only require the
action of :math:`\mathbf{J}` on an arbitrary vector :math:`\mathbf{w}` and
hence allow a matrix free algorithm for solving :eq:`eq_jfnklin`. The action of
:math:`\mathbf{J}` can be approximated by a first-order Taylor series
expansion:
.. math::
\mathbf{J}(\mathbf{x}^{k-1})\,\mathbf{w} \approx
\frac{\mathbf{F}(\mathbf{x}^{k-1}+\epsilon\mathbf{w})
- \mathbf{F}(\mathbf{x}^{k-1})} \epsilon
:label: eq_jfnkjacvecfd
or computed exactly with the help of automatic differentiation (AD)
tools. :varlink:`SEAICE_JFNKepsilon` sets the step size :math:`\epsilon`.
We use the Flexible Generalized Minimum RESidual (FMGRES) method with
right-hand side preconditioning to solve :eq:`eq_jfnklin` iteratively starting
from a first guess of :math:`\delta\mathbf{x}^{k}_{0} = 0`. For the
preconditioning matrix :math:`\mathbf{P}` we choose a simplified form of the
system matrix :math:`\mathbf{A}(\mathbf{x}^{k-1})` where
:math:`\mathbf{x}^{k-1}` is the estimate of the previous Newton step
:math:`k-1`. The transformed equation :eq:`eq_jfnklin` becomes
.. math::
\mathbf{J}(\mathbf{x}^{k-1})\,\mathbf{P}^{-1}\delta\mathbf{z} =
-\mathbf{F}(\mathbf{x}^{k-1}), \quad\text{with} \quad
\delta{\mathbf{z}} = \mathbf{P}\delta\mathbf{x}^{k}
:label: eq_jfnklinpc
The Krylov method iteratively improves the approximate solution to
:eq:`eq_jfnklinpc` in subspace (:math:`\mathbf{r}_0`,
:math:`\mathbf{J}\mathbf{P}^{-1}\mathbf{r}_0`,
:math:`(\mathbf{J}\mathbf{P}^{-1})^2\mathbf{r}_0`, :math:`\dots`,
:math:`(\mathbf{J}\mathbf{P}^{-1})^m\mathbf{r}_0`) with increasing :math:`m`;
:math:`\mathbf{r}_0 = -\mathbf{F}(\mathbf{x}^{k-1})
-\mathbf{J}(\mathbf{x}^{k-1})\,\delta\mathbf{x}^{k}_{0}` is the initial
residual of :eq:`eq_jfnklin`;
:math:`\mathbf{r}_0=-\mathbf{F}(\mathbf{x}^{k-1})` with the first guess
:math:`\delta\mathbf{x}^{k}_{0}=0`. We allow a Krylov-subspace of dimension \
:math:`m=50` and we do allow restarts for more than 50 Krylov iterations. The
preconditioning operation involves applying :math:`\mathbf{P}^{-1}` to the
basis vectors :math:`\mathbf{v}_0, \mathbf{v}_1, \mathbf{v}_2, \ldots,
\mathbf{v}_m` of the Krylov subspace. This operation is approximated by solving
the linear system :math:`\mathbf{P}\,\mathbf{w}=\mathbf{v}_i`. Because
:math:`\mathbf{P} \approx \mathbf{A}(\mathbf{x}^{k-1})`, we can use the
LSR-algorithm already implemented in the Picard solver. Each preconditioning
operation uses a fixed number of 10 LSR-iterations avoiding any termination
criterion. More details and results can be found in Losch et al. (2014)
:cite:`losch:14`).
To use the JFNK-solver set :varlink:`SEAICEuseJFNK` ``= .TRUE.,`` in the
namelist file ``data.seaice``; ``#define`` :varlink:`SEAICE_ALLOW_JFNK` in
:filelink:`SEAICE_OPTIONS.h <pkg/seaice/SEAICE_OPTIONS.h>` and we recommend
using a smooth regularization of :math:`\zeta` by ``#define``
:varlink:`SEAICE_ZETA_SMOOTHREG` (see above) for better convergence. The
nonlinear Newton iteration is terminated when the :math:`L_2`-norm of the
residual is reduced by :math:`\gamma_{\mathrm{nl}}` (run-time parameter
:varlink:`SEAICEnonLinTol` ``= 1.E-4,`` will already lead to expensive
simulations) with respect to the initial norm:
:math:`\|\mathbf{F}(\mathbf{x}^k)\| <
\gamma_{\mathrm{nl}}\|\mathbf{F}(\mathbf{x}^0)\|`. Within a nonlinear
iteration, the linear FGMRES solver is terminated when the residual is smaller
than :math:`\gamma_k\|\mathbf{F}(\mathbf{x}^{k-1})\|` where :math:`\gamma_k` is
determined by
.. math::
\gamma_k =
\begin{cases}
\gamma_0 &\text{for $\|\mathbf{F}(\mathbf{x}^{k-1})\| \geq r$}, \\
\max\left(\gamma_{\min},
\frac{\|\mathbf{F}(\mathbf{x}^{k-1})\|}
{\|\mathbf{F}(\mathbf{x}^{k-2})\|}\right)
&\text{for $\|\mathbf{F}(\mathbf{x}^{k-1})\| < r$,}
\end{cases}
:label: eq_jfnkgammalin
so that the linear tolerance parameter :math:`\gamma_k` decreases with the
nonlinear Newton step as the nonlinear solution is approached. This inexact
Newton method is generally more robust and computationally more efficient than
exact methods. Typical parameter choices are :math:`\gamma_0 =`
:varlink:`JFNKgamma_lin_max` :math:`= 0.99`, :math:`\gamma_{\min} =`
:varlink:`JFNKgamma_lin_min` :math:`= 0.1`, and :math:`r =`
:varlink:`JFNKres_tFac` :math:`\times\|\mathbf{F}(\mathbf{x}^{0})\|` with
:varlink:`JFNKres_tFac` :math:`= 0.5`. We recommend a maximum number of
nonlinear iterations :varlink:`SEAICEnewtonIterMax` :math:`= 100` and a maximum
number of Krylov iterations :varlink:`SEAICEkrylovIterMax` :math:`= 50`,
because the Krylov subspace has a fixed dimension of 50 (but restarts are
allowed for :varlink:`SEAICEkrylovIterMax` :math:`> 50`).
Setting :varlink:`SEAICEuseStrImpCpl` ``= .TRUE.,`` turns on “strength implicit
coupling” (see Hutchings et al. 2004 :cite:`hutchings:04`) in the LSR-solver
and in the LSR-preconditioner for the JFNK-solver. In this mode, the different
contributions of the stress divergence terms are reordered so as to increase
the diagonal dominance of the system matrix. Unfortunately, the convergence
rate of the LSR solver is increased only slightly, while the JFNK-convergence
appears to be unaffected.
.. _para_phys_pkg_seaice_EVPdynamics:
Elastic-Viscous-Plastic (EVP) Dynamics
--------------------------------------
Hunke and Dukowicz (1997) :cite:`hunke:97` introduced an elastic contribution
to the strain rate in order to regularize :eq:`eq_vpequation` in such a way
that the resulting elastic-viscous-plastic (EVP) and VP models are identical at
steady state,
.. math::
\frac{1}{E}\frac{\partial\sigma_{ij}}{\partial{t}} +
\frac{1}{2\eta}\sigma_{ij}
+ \frac{\eta - \zeta}{4\zeta\eta}\sigma_{kk}\delta_{ij}
+ \frac{P}{4\zeta}\delta_{ij}
= \dot{\epsilon}_{ij}.
:label: eq_evpequation
The EVP-model uses an explicit time stepping scheme with a short timestep.
According to the recommendation in Hunke and Dukowicz (1997) :cite:`hunke:97`,