barotropicqg.md

BarotropicQG Module

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:

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

where $\mathsf{J}(a, b) = (\partial_x a)(\partial_y b)-(\partial_y a)(\partial_x b)$. On the right hand side, $f(x,y,t)$ is forcing, $\mu$ is 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:

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

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

Thus:

$$\mathcal{L} = \beta\frac{\mathrm{i}k_x}{k^2} - \mu - \nu k^{2n_\nu}\ ,$$ $$\mathcal{N}(\widehat{\zeta}) = - \mathrm{i}k_x \mathrm{FFT}(u q)- \mathrm{i}k_y \mathrm{FFT}(\upsilon q)\ .$$

Examples

• examples/barotropicqg_betadecay.jl: A script that simulates decaying quasi-geostrophic flow on a beta-plane demonstrating zonation.

• examples/barotropicqg_betaforced.jl: A script that simulates forced-dissipative quasi-geostrophic flow on a beta-plane demonstrating zonation. The forcing is temporally delta-correlated and its spatial structure is isotropic with power in a narrow annulus of total radius kf in wavenumber space.

• examples/barotropicqg_acc.jl: A script that simulates barotropic quasi-geostrophic flow above topography reproducing the results of the paper by

  Constantinou, N. C. (2018). A barotropic model of eddy saturation. J. Phys. Oceanogr., 48 (2), 397-411.