Merge pull request #41 from FourierFlows/FixTwoDTurbDocs

Fixes typos and notation in TwoDTurb docs

docs/src/modules/twodturb.md

@@ -1,8 +1,5 @@
 # TwoDTurb Module
 
-```math
-\newcommand{\J}{\mathsf{J}}
-```
 
 ### Basic Equations
 
@@ -12,10 +9,10 @@ The dynamical variable used here is the component of the vorticity of the flow
 normal to the plane of motion, $\zeta=\partial_x \upsilon- \partial_y u = \nabla^2\psi$.
 The equation solved by the module is:
 
-$$\partial_t \zeta + \J(\psi, \zeta) = \underbrace{-\left[\mu(-1)^{n_\mu} \nabla^{2n_\mu}
+$$\partial_t \zeta + \mathsf{J}(\psi, \zeta) = \underbrace{-\left[\mu(-1)^{n_\mu} \nabla^{2n_\mu}
 +\nu(-1)^{n_\nu} \nabla^{2n_\nu}\right] \zeta}_{\textrm{dissipation}} + f\ .$$
 
-where $\J(a, b) = (\partial_x a)(\partial_y b)-(\partial_y a)(\partial_x b)$. On
+where $\mathsf{J}(a, b) = (\partial_x a)(\partial_y b)-(\partial_y a)(\partial_x b)$. On
 the right hand side, $f(x,y,t)$ is forcing, $\mu$ is hypoviscosity, and $\nu$ is
 hyperviscosity. Plain old linear drag corresponds to $n_{\mu}=0$, while normal
 viscosity corresponds to $n_{\nu}=1$.
@@ -24,10 +21,10 @@ viscosity corresponds to $n_{\nu}=1$.
 
 The equation is time-stepped forward in Fourier space:
 
-$$\partial_t \widehat{\zeta} = - \widehat{J(\psi, \zeta)} -\left(\mu k^{2n_\mu}
+$$\partial_t \widehat{\zeta} = - \widehat{\mathsf{J}(\psi, \zeta)} -\left(\mu k^{2n_\mu}
 +\nu k^{2n_\nu}\right) \widehat{\zeta}  + \widehat{f}\ .$$
 
-In doing so the Jacobian is computed in the conservative form: $\J(a,b) =
+In doing so the Jacobian is computed in the conservative form: $\mathsf{J}(a,b) =
 \partial_y [ (\partial_x a) b] -\partial_x[ (\partial_y a) b]$.
 
 Thus:
@@ -43,9 +40,9 @@ $$\mathcal{N}(\widehat{\zeta}) = - \mathrm{i}k_x \mathrm{FFT}(u \zeta)-
 
 For the unforced case ($f=0$) parameters AbstractType is build with `Params` and it includes:
 - `ν`:   Float; viscosity or hyperviscosity coefficient.
-- `nν`: Integer$>0$; the order of viscosity $n_\nu$. Case $n_\nu=1$ give normal viscosity.
+- `nν`: Integer$>0$; the order of viscosity $n_\nu$. Case $n_\nu=1$ gives normal viscosity.
 - `μ`: Float; bottom drag or hypoviscosity coefficient.
-- `nμ`: Integer$\ge 0$; the order of hypodrag $n_\mu$. Case $n_\mu=0$ give plain linear drag $\mu$.
+- `nμ`: Integer$\ge 0$; the order of hypodrag $n_\mu$. Case $n_\mu=0$ gives plain linear drag $\mu$.
 
 For the forced case ($f\ne 0$) parameters AbstractType is build with `ForcedParams`. It includes all parameters in `Params` and additionally:
 - `calcF!`: Function that calculates the forcing $\widehat{f}$
@@ -72,9 +69,9 @@ For the forced case ($f\ne 0$) variables AbstractType is build with `ForcedVars`
 
 The nonlinear term $\mathcal{N}(\widehat{\zeta})$ is computed via functions:
 
-- `calcN_advection!`: computes $- \widehat{J(\psi, \zeta)}$ and stores it in array `N`.
+- `calcN_advection!`: computes $- \widehat{\mathsf{J}(\psi, \zeta)}$ and stores it in array `N`.
 
-- `calcN_forced!`: computes $- \widehat{J(\psi, \zeta)}$ 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`.
+- `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`.