Skip to content

Commit 9ccea29

Browse files
authoredDec 25, 2020
Merge pull request #173 from FourierFlows/ncc/upsilon-v
Changes \upsilon -> v in Docs
2 parents 8d4cf0f + d1793c7 commit 9ccea29

File tree

6 files changed

+19
-19
lines changed

6 files changed

+19
-19
lines changed
 

‎docs/src/modules/barotropicqgql.md

+4-4
Original file line numberDiff line numberDiff line change
@@ -14,13 +14,13 @@ where overline above denotes a zonal mean, $\overline{\phi}(y, t) = \int \phi(x,
1414
- Constantinou, N. C., Farrell, B. F., and Ioannou, P. J. (2014). [Emergence and equilibration of jets in beta-plane turbulence: applications of Stochastic Structural Stability Theory.](http://doi.org/10.1175/JAS-D-13-076.1) *J. Atmos. Sci.*, **71 (5)**, 1818-1842.
1515

1616

17-
As in the [BarotropicQG module](barotropicqg.md), the flow is obtained through a streamfunction $\psi$ as $(u, \upsilon) = (-\partial_y\psi, \partial_x\psi)$. All flow fields can be obtained from the quasi-geostrophic potential vorticity (QGPV). Here the QGPV is
17+
As in the [BarotropicQG module](barotropicqg.md), the flow is obtained through a streamfunction $\psi$ as $(u, v) = (-\partial_y\psi, \partial_x\psi)$. All flow fields can be obtained from the quasi-geostrophic potential vorticity (QGPV). Here the QGPV is
1818

19-
$$\underbrace{f_0 + \beta y}_{\text{planetary PV}} + \underbrace{(\partial_x \upsilon
19+
$$\underbrace{f_0 + \beta y}_{\text{planetary PV}} + \underbrace{(\partial_x v
2020
- \partial_y u)}_{\text{relative vorticity}} +
2121
\underbrace{\frac{f_0 h}{H}}_{\text{topographic PV}}.$$
2222

23-
The dynamical variable is the component of the vorticity of the flow normal to the plane of motion, $\zeta\equiv \partial_x \upsilon- \partial_y u = \nabla^2\psi$. Also, we denote the topographic PV with $\eta\equiv f_0 h/H$. After we apply the eddy-mean flow decomposition above, the QGPV dynamics are:
23+
The dynamical variable is the component of the vorticity of the flow normal to the plane of motion, $\zeta\equiv \partial_x v- \partial_y u = \nabla^2\psi$. Also, we denote the topographic PV with $\eta\equiv f_0 h/H$. After we apply the eddy-mean flow decomposition above, the QGPV dynamics are:
2424

2525
$$\partial_t \overline{\zeta} + \mathsf{J}(\overline{\psi}, \underbrace{\overline{\zeta} + \overline{\eta}}_{\equiv \overline{q}}) + \overline{\mathsf{J}(\psi', \underbrace{\zeta' + \eta'}_{\equiv q'})} = \underbrace{-\left[\mu + \nu(-1)^{n_\nu} \nabla^{2n_\nu}
2626
\right] \overline{\zeta} }_{\textrm{dissipation}} \ .$$
@@ -47,7 +47,7 @@ Thus:
4747

4848
$$\mathcal{L} = \beta\frac{\mathrm{i}k_x}{k^2} - \mu - \nu k^{2n_\nu}\ ,$$
4949
$$\mathcal{N}(\widehat{\zeta}) = - \mathrm{i}k_x \mathrm{FFT}(u q)^{\textrm{QL}}-
50-
\mathrm{i}k_y \mathrm{FFT}(\upsilon q)^{\textrm{QL}}\ .$$
50+
\mathrm{i}k_y \mathrm{FFT}(v q)^{\textrm{QL}}\ .$$
5151

5252

5353
## Examples

‎docs/src/modules/multilayerqg.md

+1-1
Original file line numberDiff line numberDiff line change
@@ -4,7 +4,7 @@
44

55
This module solves the layered quasi-geostrophic equations on a beta-plane of variable fluid
66
depth ``H - h(x, y)``. The flow in each layer is obtained through a streamfunction ``\psi_j`` as
7-
``(u_j, \upsilon_j) = (-\partial_y \psi_j, \partial_x \psi_j)``, ``j = 1, \dots, n``, where ``n``
7+
``(u_j, v_j) = (-\partial_y \psi_j, \partial_x \psi_j)``, ``j = 1, \dots, n``, where ``n``
88
is the number of fluid layers.
99

1010
The QGPV in each layer is

‎docs/src/modules/singlelayerqg.md

+3-3
Original file line numberDiff line numberDiff line change
@@ -3,11 +3,11 @@
33
### Basic Equations
44

55
This module solves the barotropic or equivalent barotropic quasi-geostrophic vorticity equation
6-
on a beta-plane of variable fluid depth ``H - h(x, y)``. The flow is obtained through a streamfunction ``\psi`` as ``(u, \upsilon) = (-\partial_y \psi, \partial_x \psi)``. All flow
6+
on a beta-plane of variable fluid depth ``H - h(x, y)``. The flow is obtained through a streamfunction ``\psi`` as ``(u, v) = (-\partial_y \psi, \partial_x \psi)``. All flow
77
fields can be obtained from the quasi-geostrophic potential vorticity (QGPV). Here the QGPV is
88

99
```math
10-
\underbrace{f_0 + \beta y}_{\text{planetary PV}} + \underbrace{\partial_x \upsilon
10+
\underbrace{f_0 + \beta y}_{\text{planetary PV}} + \underbrace{\partial_x v
1111
- \partial_y u}_{\text{relative vorticity}} - \!\!
1212
\underbrace{\frac{1}{\ell^2} \psi}_{\text{vortex stretching}} \!\! +
1313
\underbrace{\frac{f_0 h}{H}}_{\text{topographic PV}} \ ,
@@ -64,7 +64,7 @@ Thus:
6464
```math
6565
\begin{aligned}
6666
\mathcal{L} & = \beta \frac{\mathrm{i} k_x}{k^2 + 1/\ell^2} - \mu - \nu k^{2n_\nu} \ , \\
67-
\mathcal{N}(\widehat{q}) & = - \mathrm{i} k_x \mathrm{FFT}[u (q+\eta)] - \mathrm{i} k_y \mathrm{FFT}[\upsilon (q+\eta)] \ .
67+
\mathcal{N}(\widehat{q}) & = - \mathrm{i} k_x \mathrm{FFT}[u (q+\eta)] - \mathrm{i} k_y \mathrm{FFT}[v (q+\eta)] \ .
6868
\end{aligned}
6969
```
7070

‎docs/src/modules/surfaceqg.md

+3-3
Original file line numberDiff line numberDiff line change
@@ -7,7 +7,7 @@ buoyancy $b_s = b(x, y, z=0)$, as described in Capet et al., 2008. The buoyancy
77
velocity at the surface are related through a streamfunction $\psi$ via:
88

99
```math
10-
(u_s, \upsilon_s, b_s) = (-\partial_y \psi, \partial_x \psi, -\partial_z \psi) .
10+
(u_s, v_s, b_s) = (-\partial_y \psi, \partial_x \psi, -\partial_z \psi) .
1111
```
1212

1313
The SQG model evolves the surface buoyancy,
@@ -46,9 +46,9 @@ In doing so the Jacobian is computed in the conservative form: $\mathsf{J}(f,g)
4646
Thus:
4747
```math
4848
\begin{aligned}
49-
\widehat{u} &= \frac{\mathrm{i} k_y}{k} \widehat{b_s}, \qquad \widehat{\upsilon} = -\frac{\mathrm{i} k_x}{k} \widehat{b_s}, \\
49+
\widehat{u} &= \frac{\mathrm{i} k_y}{k} \widehat{b_s}, \qquad \widehat{v} = -\frac{\mathrm{i} k_x}{k} \widehat{b_s}, \\
5050
\mathcal{L} & = - \nu k^{2n_\nu},\\
51-
\mathcal{N}(\widehat{b_s}) & = - \mathrm{i} k_x \mathrm{FFT}(u b) - \mathrm{i} k_y \mathrm{FFT}(\upsilon b) .
51+
\mathcal{N}(\widehat{b_s}) & = - \mathrm{i} k_x \mathrm{FFT}(u b) - \mathrm{i} k_y \mathrm{FFT}(v b) .
5252
\end{aligned}
5353
```
5454

‎docs/src/modules/twodnavierstokes.md

+6-6
Original file line numberDiff line numberDiff line change
@@ -4,9 +4,9 @@
44
### Basic Equations
55

66
This module solves two-dimensional incompressible turbulence. The flow is given
7-
through a streamfunction $\psi$ as $(u,\upsilon) = (-\partial_y\psi, \partial_x\psi)$.
7+
through a streamfunction $\psi$ as $(u,v) = (-\partial_y\psi, \partial_x\psi)$.
88
The dynamical variable used here is the component of the vorticity of the flow
9-
normal to the plane of motion, $\zeta=\partial_x \upsilon- \partial_y u = \nabla^2\psi$.
9+
normal to the plane of motion, $\zeta=\partial_x v- \partial_y u = \nabla^2\psi$.
1010
The equation solved by the module is:
1111

1212
$$\partial_t \zeta + \mathsf{J}(\psi, \zeta) = \underbrace{-\left[\mu(-1)^{n_\mu} \nabla^{2n_\mu}
@@ -31,7 +31,7 @@ Thus:
3131

3232
$$\mathcal{L} = -\mu k^{-2n_\mu} - \nu k^{2n_\nu}\ ,$$
3333
$$\mathcal{N}(\widehat{\zeta}) = - \mathrm{i}k_x \mathrm{FFT}(u \zeta)-
34-
\mathrm{i}k_y \mathrm{FFT}(\upsilon \zeta) + \widehat{f}\ .$$
34+
\mathrm{i}k_y \mathrm{FFT}(v \zeta) + \widehat{f}\ .$$
3535

3636

3737
### AbstractTypes and Functions
@@ -53,11 +53,11 @@ For the forced case ($f\ne 0$) parameters AbstractType is build with `ForcedPara
5353
For the unforced case ($f=0$) variables AbstractType is build with `Vars` and it includes:
5454
- `zeta`: Array of Floats; relative vorticity.
5555
- `u`: Array of Floats; $x$-velocity, $u$.
56-
- `v`: Array of Floats; $y$-velocity, $\upsilon$.
56+
- `v`: Array of Floats; $y$-velocity, $v$.
5757
- `sol`: Array of Complex; the solution, $\widehat{\zeta}$.
5858
- `zetah`: Array of Complex; the Fourier transform $\widehat{\zeta}$.
5959
- `uh`: Array of Complex; the Fourier transform $\widehat{u}$.
60-
- `vh`: Array of Complex; the Fourier transform $\widehat{\upsilon}$.
60+
- `vh`: Array of Complex; the Fourier transform $\widehat{v}$.
6161

6262
For the forced case ($f\ne 0$) variables AbstractType is build with `ForcedVars`. It includes all variables in `Vars` and additionally:
6363
- `Fh`: Array of Complex; the Fourier transform $\widehat{f}$.
@@ -73,7 +73,7 @@ The nonlinear term $\mathcal{N}(\widehat{\zeta})$ is computed via functions:
7373

7474
- `calcN_forced!`: computes $- \widehat{\mathsf{J}(\psi, \zeta)}$ via `calcN_advection!` and then adds to it the forcing $\widehat{f}$ computed via `calcF!` function. Also saves the solution $\widehat{\zeta}$ of the previous time-step in array `prevsol`.
7575

76-
- `updatevars!`: uses `sol` to compute $\zeta$, $u$, $\upsilon$, $\widehat{u}$, and $\widehat{\upsilon}$ and stores them into corresponding arrays of `Vars`/`ForcedVars`.
76+
- `updatevars!`: uses `sol` to compute $\zeta$, $u$, $v$, $\widehat{u}$, and $\widehat{v}$ and stores them into corresponding arrays of `Vars`/`ForcedVars`.
7777

7878

7979
## Examples

‎src/multilayerqg.jl

+2-2
Original file line numberDiff line numberDiff line change
@@ -612,14 +612,14 @@ verticalfluxes``_{3/2},...,``verticalfluxes``_{n-1/2}``, where ``n`` is the tota
612612
613613
The lateral eddy fluxes whithin the ``j``-th fluid layer are
614614
```math
615-
\\textrm{lateralfluxes}_j = \\frac{H_j}{H} \\int U_j \\, \\upsilon_j \\, \\partial_y u_j
615+
\\textrm{lateralfluxes}_j = \\frac{H_j}{H} \\int U_j \\, v_j \\, \\partial_y u_j
616616
\\frac{\\mathrm{d}^2 \\boldsymbol{x}}{L_x L_y} \\ , \\quad j = 1, \\dots, n \\ ,
617617
```
618618
while the vertical eddy fluxes at the ``j+1/2``-th fluid interface (i.e., interface between
619619
the ``j``-th and ``(j+1)``-th fluid layer) are
620620
```math
621621
\\textrm{verticalfluxes}_{j+1/2} = \\int \\frac{f_0^2}{g'_{j+1/2} H} (U_j - U_{j+1}) \\,
622-
\\upsilon_{j+1} \\, \\psi_{j} \\frac{\\mathrm{d}^2 \\boldsymbol{x}}{L_x L_y} \\ , \\quad
622+
v_{j+1} \\, \\psi_{j} \\frac{\\mathrm{d}^2 \\boldsymbol{x}}{L_x L_y} \\ , \\quad
623623
j = 1 , \\dots , n-1.
624624
```
625625
"""

0 commit comments

Comments
 (0)
Please sign in to comment.