Merge pull request #173 from FourierFlows/ncc/upsilon-v

Changes \upsilon -> v in Docs

docs/src/modules/barotropicqgql.md

@@ -14,13 +14,13 @@ where overline above denotes a zonal mean, $\overline{\phi}(y, t) = \int \phi(x,
 - 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.
 
 
-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
+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
 
-$$\underbrace{f_0 + \beta y}_{\text{planetary PV}} + \underbrace{(\partial_x \upsilon
+$$\underbrace{f_0 + \beta y}_{\text{planetary PV}} + \underbrace{(\partial_x v
 	- \partial_y u)}_{\text{relative vorticity}} +
 	\underbrace{\frac{f_0 h}{H}}_{\text{topographic PV}}.$$
 
-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:
+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:
 
 $$\partial_t \overline{\zeta} + \mathsf{J}(\overline{\psi}, \underbrace{\overline{\zeta} + \overline{\eta}}_{\equiv \overline{q}}) + \overline{\mathsf{J}(\psi', \underbrace{\zeta' + \eta'}_{\equiv q'})} = \underbrace{-\left[\mu + \nu(-1)^{n_\nu} \nabla^{2n_\nu}
 \right] \overline{\zeta} }_{\textrm{dissipation}} \ .$$
@@ -47,7 +47,7 @@ Thus:
 
 $$\mathcal{L} = \beta\frac{\mathrm{i}k_x}{k^2} - \mu - \nu k^{2n_\nu}\ ,$$
 $$\mathcal{N}(\widehat{\zeta}) = - \mathrm{i}k_x \mathrm{FFT}(u q)^{\textrm{QL}}-
-	\mathrm{i}k_y \mathrm{FFT}(\upsilon q)^{\textrm{QL}}\ .$$
+	\mathrm{i}k_y \mathrm{FFT}(v q)^{\textrm{QL}}\ .$$
 
 
 ## Examples

docs/src/modules/multilayerqg.md

@@ -4,7 +4,7 @@
 
 This module solves the layered quasi-geostrophic equations on a beta-plane of variable fluid 
 depth ``H - h(x, y)``. The flow in each layer is obtained through a streamfunction ``\psi_j`` as 
-``(u_j, \upsilon_j) = (-\partial_y \psi_j, \partial_x \psi_j)``, ``j = 1, \dots, n``, where ``n`` 
+``(u_j, v_j) = (-\partial_y \psi_j, \partial_x \psi_j)``, ``j = 1, \dots, n``, where ``n`` 
 is the number of fluid layers.
 
 The QGPV in each layer is

docs/src/modules/singlelayerqg.md

@@ -3,11 +3,11 @@
 ### Basic Equations
 
 This module solves the barotropic or equivalent barotropic quasi-geostrophic vorticity equation 
-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 
+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 
 fields can be obtained from the quasi-geostrophic potential vorticity (QGPV). Here the QGPV is
 
 ```math
-	\underbrace{f_0 + \beta y}_{\text{planetary PV}} + \underbrace{\partial_x \upsilon
+	\underbrace{f_0 + \beta y}_{\text{planetary PV}} + \underbrace{\partial_x v
 	- \partial_y u}_{\text{relative vorticity}} - \!\!
 	\underbrace{\frac{1}{\ell^2} \psi}_{\text{vortex stretching}} \!\! + 
 	\underbrace{\frac{f_0 h}{H}}_{\text{topographic PV}} \ ,
@@ -64,7 +64,7 @@ Thus:
 ```math
 \begin{aligned}
 \mathcal{L} & = \beta \frac{\mathrm{i} k_x}{k^2 + 1/\ell^2} - \mu - \nu k^{2n_\nu} \ , \\
-\mathcal{N}(\widehat{q}) & = - \mathrm{i} k_x \mathrm{FFT}[u (q+\eta)] - \mathrm{i} k_y \mathrm{FFT}[\upsilon (q+\eta)] \ .
+\mathcal{N}(\widehat{q}) & = - \mathrm{i} k_x \mathrm{FFT}[u (q+\eta)] - \mathrm{i} k_y \mathrm{FFT}[v (q+\eta)] \ .
 \end{aligned}
 ```
 

docs/src/modules/surfaceqg.md

@@ -7,7 +7,7 @@ buoyancy $b_s = b(x, y, z=0)$, as described in Capet et al., 2008. The buoyancy
 velocity at the surface are related through a streamfunction $\psi$ via:
 
 ```math
-(u_s, \upsilon_s, b_s) = (-\partial_y \psi, \partial_x \psi, -\partial_z \psi) .
+(u_s, v_s, b_s) = (-\partial_y \psi, \partial_x \psi, -\partial_z \psi) .
 ```
 
 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)
 Thus:
 ```math
 \begin{aligned}
-\widehat{u} &= \frac{\mathrm{i} k_y}{k} \widehat{b_s}, \qquad \widehat{\upsilon} = -\frac{\mathrm{i} k_x}{k} \widehat{b_s}, \\
+\widehat{u} &= \frac{\mathrm{i} k_y}{k} \widehat{b_s}, \qquad \widehat{v} = -\frac{\mathrm{i} k_x}{k} \widehat{b_s}, \\
 \mathcal{L} & = - \nu k^{2n_\nu},\\
-\mathcal{N}(\widehat{b_s}) & = - \mathrm{i} k_x \mathrm{FFT}(u b) - \mathrm{i} k_y \mathrm{FFT}(\upsilon b) .
+\mathcal{N}(\widehat{b_s}) & = - \mathrm{i} k_x \mathrm{FFT}(u b) - \mathrm{i} k_y \mathrm{FFT}(v b) .
 \end{aligned}
 ```
 

docs/src/modules/twodnavierstokes.md

@@ -4,9 +4,9 @@
 ### Basic Equations
 
 This module solves two-dimensional incompressible turbulence. The flow is given
-through a streamfunction $\psi$ as $(u,\upsilon) = (-\partial_y\psi, \partial_x\psi)$.
+through a streamfunction $\psi$ as $(u,v) = (-\partial_y\psi, \partial_x\psi)$.
 The dynamical variable used here is the component of the vorticity of the flow
-normal to the plane of motion, $\zeta=\partial_x \upsilon- \partial_y u = \nabla^2\psi$.
+normal to the plane of motion, $\zeta=\partial_x v- \partial_y u = \nabla^2\psi$.
 The equation solved by the module is:
 
 $$\partial_t \zeta + \mathsf{J}(\psi, \zeta) = \underbrace{-\left[\mu(-1)^{n_\mu} \nabla^{2n_\mu}
@@ -31,7 +31,7 @@ Thus:
 
 $$\mathcal{L} = -\mu k^{-2n_\mu} - \nu k^{2n_\nu}\ ,$$
 $$\mathcal{N}(\widehat{\zeta}) = - \mathrm{i}k_x \mathrm{FFT}(u \zeta)-
-	\mathrm{i}k_y \mathrm{FFT}(\upsilon \zeta) + \widehat{f}\ .$$
+	\mathrm{i}k_y \mathrm{FFT}(v \zeta) + \widehat{f}\ .$$
 
 
 ### AbstractTypes and Functions
@@ -53,11 +53,11 @@ For the forced case ($f\ne 0$) parameters AbstractType is build with `ForcedPara
 For the unforced case ($f=0$) variables AbstractType is build with `Vars` and it includes:
 - `zeta`: Array of Floats; relative vorticity.
 - `u`: Array of Floats; $x$-velocity, $u$.
-- `v`: Array of Floats; $y$-velocity, $\upsilon$.
+- `v`: Array of Floats; $y$-velocity, $v$.
 - `sol`: Array of Complex; the solution, $\widehat{\zeta}$.
 - `zetah`: Array of Complex; the Fourier transform $\widehat{\zeta}$.
 - `uh`: Array of Complex; the Fourier transform $\widehat{u}$.
-- `vh`: Array of Complex; the Fourier transform $\widehat{\upsilon}$.
+- `vh`: Array of Complex; the Fourier transform $\widehat{v}$.
 
 For the forced case ($f\ne 0$) variables AbstractType is build with `ForcedVars`. It includes all variables in `Vars` and additionally:
 - `Fh`: Array of Complex; the Fourier transform $\widehat{f}$.
@@ -73,7 +73,7 @@ The nonlinear term $\mathcal{N}(\widehat{\zeta})$ is computed via functions:
 
 - `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`.
 
-- `updatevars!`: uses `sol` to compute $\zeta$, $u$, $\upsilon$, $\widehat{u}$, and $\widehat{\upsilon}$ and stores them into corresponding arrays of `Vars`/`ForcedVars`.
+- `updatevars!`: uses `sol` to compute $\zeta$, $u$, $v$, $\widehat{u}$, and $\widehat{v}$ and stores them into corresponding arrays of `Vars`/`ForcedVars`.
 
 
 ## Examples

src/multilayerqg.jl

@@ -612,14 +612,14 @@ verticalfluxes``_{3/2},...,``verticalfluxes``_{n-1/2}``, where ``n`` is the tota
 
 The lateral eddy fluxes whithin the ``j``-th fluid layer are
 ```math
-\\textrm{lateralfluxes}_j = \\frac{H_j}{H} \\int U_j \\, \\upsilon_j \\, \\partial_y u_j 
+\\textrm{lateralfluxes}_j = \\frac{H_j}{H} \\int U_j \\, v_j \\, \\partial_y u_j 
 \\frac{\\mathrm{d}^2 \\boldsymbol{x}}{L_x L_y} \\ , \\quad j = 1, \\dots, n \\ ,
 ```
 while the vertical eddy fluxes at the ``j+1/2``-th fluid interface  (i.e., interface between 
 the ``j``-th and ``(j+1)``-th fluid layer) are
 ```math
 \\textrm{verticalfluxes}_{j+1/2} = \\int \\frac{f_0^2}{g'_{j+1/2} H} (U_j - U_{j+1}) \\, 
-\\upsilon_{j+1} \\, \\psi_{j} \\frac{\\mathrm{d}^2 \\boldsymbol{x}}{L_x L_y} \\ , \\quad 
+v_{j+1} \\, \\psi_{j} \\frac{\\mathrm{d}^2 \\boldsymbol{x}}{L_x L_y} \\ , \\quad 
 j = 1 , \\dots , n-1.
 ```
 """