BarotropicQG Module

\[\newcommand{\J}{\mathsf{J}}\]

Basic Equations

This module solves the quasi-geostrophic barotropic 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 fields can be obtained from the quasi-geostrophic potential vorticity (QGPV). Here the QGPV is

\[\underbrace{f_0 + \beta y}_{\text{planetary PV}} + \underbrace{(\partial_x \upsilon - \partial_y u)}_{\text{relative vorticity}} + \underbrace{\frac{f_0 h}{H}}_{\text{topographic PV}}.\]

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 $\eta\equiv f_0 h/H$. Thus, the equation solved by the module is:

\[\partial_t \zeta + \J(\psi, \underbrace{\zeta + \eta}_{\equiv q}) + \beta\partial_x\psi = \underbrace{-\left[\mu + \nu(-1)^{n_\nu} \nabla^{2n_\nu} \right] \zeta }_{\textrm{dissipation}} + f\ .\]

where $\J(a, b) = (\partial_x a)(\partial_y b)-(\partial_y a)(\partial_x b)$. On the right hand side, $f(x,y,t)$ is forcing, $\mu$ is linear drag, and $\nu$ is hyperviscosity. Plain old viscosity corresponds to $n_{\nu}=1$. The sum of relative vorticity and topographic PV is denoted with $q\equiv\zeta+\eta$.

Implementation

The equation is time-stepped forward in Fourier space:

\[\partial_t \widehat{\zeta} = - \widehat{\J(\psi, q)} +\beta\frac{\mathrm{i}k_x}{k^2}\widehat{\zeta} -\left(\mu +\nu k^{2n_\nu}\right) \widehat{\zeta} + \widehat{f}\ .\]

In doing so the Jacobian is computed in the conservative form: $\J(f,g) = \partial_y [ (\partial_x f) g] -\partial_x[ (\partial_y f) g]$.

Thus:

\[\mathcal{L} = \beta\frac{\mathrm{i}k_x}{k^2} - \mu - \nu k^{2n_\nu}\ ,\]

\[\mathcal{N}(\widehat{\zeta}) = - \mathrm{i}k_x \mathrm{FFT}(u q)- \mathrm{i}k_y \mathrm{FFT}(\upsilon q)\ .\]