## CABARET and its application to geophysical fluid dynamics

PEQUOD solves the multi-layer quasi-geostrophic equations in a rectangular domain. The model has two primary modes of operation: basin mode and channel mode. Typically, the basin mode is used as a simple model for a wind driven double gyre, with energy input to the system via upper layer wind forcing. The channel mode is used as a simple model for baroclinic zonal jet formation, with the energy input to the system via lateral buoyancy forcing.

The quasi-geostrophic potential vorticity equation (1) is solved subject to forcing and dissipation

 (1)

where is the quasi-geostrophic potential vorticity (hereafter referred to as the "potential vorticity").The LHS in equation (1) is the material derivative, and is any forcing and dissipation. Equation (1) is then represented in a vertically layered form, which leads to a series of decoupled Helmholtz equations.

In PEQUOD two separate finite-difference approaches may be used to solve the layered quasi-geostrophic potential vorticity equation, however, this work will only be concerned with the CABARET approach. For the advection of relative potential vorticity, the scheme consists of three steps: predictor, extrapolator and corrector. The predictor and extrapolator step are performed using centred differencing in space. The extrapolator step is performed using upwinding of the advection term with a non-Total Variation Diminishing (TVD) flux limiter to bound the approximation. This scheme is second order accurate provided the method used to compute the fluxes is (at least) second order accurate.

Potential vorticity inversion for to yield the stream function is performed via the resulting linear elliptic problem being inverted using a direct solver. A discrete sine transform [7] is performed in the meridional direction, yielding a series of sparse one-dimensional Helmholtz problems in the zonal direction that can, for example, be inverted via Gaussian elimination. An inverse discrete sine transform of the resulting solutions of the one-dimensional problems completes the inversion.

Phil Ridley 2012-10-01