Past exam of the geophysical fluid dynamics course of the University of Oxford / 2023 / 2 / b / Solution

The quasi-geostrophic potential-vorticity equation is the leading balanced form of potential-vorticity conservation for an inviscid rotating fluid with stable density stratification, small Rossby number and an adiabatic process for each fluid element. Its parent conservation law is Ertel potential vorticity conservation. Horizontal geostrophic balance, vertical hydrostatic balance, and the beta-plane approximation reduce it to advection of one scalar by the leading horizontal geostrophic flow.

One can see its origin by combining the vertical vorticity equation with the thermal equation. Set $D_g={\partial }_t+{\mathbf{u}}_g\cdot {\nabla }_h$, ${\mathbf{u}}_g=(-{\psi }_y,{\psi }_x)$, and ${\zeta }_g={\nabla }_h^2\psi$. The quasi-geostrophic approximation givesHere $b$ is the buoyancy perturbation relative to the reference stable density stratification, $N=N(z)$ its buoyancy frequency, and $w$ the small vertical velocity. The stretching of absolute vorticity in the first equation is offset by the change of stratification represented by the second. More precisely,The last dot product is zero because thermal-wind balance makes ${\partial }_z{\mathbf{u}}_g$ perpendicular to ${\nabla }_{h}b$. Adding therefore yieldsThe constant $f_0$ has no effect on the tendency, but its inclusion makes the connection with absolute vorticity clear. Vertical advection of $q$ is beyond the retained order; the vertical velocity has already supplied the stretching and thermodynamic terms.

For a simple balanced weather forecast, the essential cycle is diagnose, advect, invert:

This is an elliptic boundary value problem, not a local algebraic diagnosis. Its solution supplies the next geostrophic flow and, through $b=f_0{\psi }_z$, the next thermal field. Repeat the cycle to move the balanced weather systems forward.

Interior potential vorticity alone is insufficient: boundary thermal anomalies can induce flow even with zero interior anomaly, as in the Eady model. The method filters fast inertia-gravity waves and is useful for large-scale extratropical weather, while tropical dynamics, convection, moisture, friction and diabatic heating require additional physics. If vertical motion is wanted, the quasi-geostrophic omega equation supplies a further diagnostic calculation.