Ch 8. The quasi-geostrophic approximation

Overview

Quasi-geostrophy is what you get when you take rotating shallow water seriously in the limit every large-scale flow actually lives in: small Rossby number. The leading-order flow is exactly geostrophic; the whole content of the dynamics is in the small ageostrophic correction, and QG theory shows that correction is completely captured by conserving a single scalar, the quasi-geostrophic potential vorticity. This chapter derives that equation from Ch. 6’s RSW, and uses an isolated vortex to show what’s new relative to Ch. 7’s barotropic PV: a deformation-radius stretching term that screens vortices, caps the Rossby-wave dispersion relation, and sends an isolated eddy drifting westward while it sheds a trailing wake — the beta-gyre.

The model

Notebook: notebooks/ch08_qg-approximation.py → exported to /nb/ch08_qg-approximation.html. Governing equations use the symbols in notation:

q_t+J(\psi,q)=0,\qquad q=\nabla^2\psi-\frac{\psi}{L_R^2}+\beta y.

Reuses Ch. 18’s pseudo-spectral machinery and Ch. 6’s Helmholtz PV inversion directly — nondimensionalized by $L_R$ , the QGPV balance operator $(\nabla^2-1)\psi=q-\beta y$ is the same operator as the shallow-water PV inversion, just now driving a fully nonlinear evolution. No new inversion code exists or is needed; gfdlib.qg supplies only the modified Rossby-wave dispersion relation.

What to look for

Screening. Set $\beta=0$ and look at the initial streamfunction: a vortex of radius $\sigma\lesssim L_R$ has its far-field velocity cut off exponentially beyond $L_R$ (the $-\psi/L_R^2$ term turns the inversion from a long-range logarithm into a short-range Yukawa-like kernel). Compare against Ch. 7, where the same blob’s influence extended across the whole domain.
Westward drift from nothing. Turn $\beta$ up: an isolated, perfectly symmetric vortex begins to translate westward without being pushed. Watch the first frames closely — advection of planetary vorticity builds an antisymmetric secondary dipole (the beta-gyre) across the vortex, and it is that dipole which carries the parent along at a speed of order $\beta L_R^2$ .
The wake. As the vortex drifts it sheds a trailing Rossby-wave wake. Check the dispersion-relation panel: the wake’s dominant wavelength sits near the wavenumber whose westward phase speed matches the vortex’s own drift — the wave the vortex can resonate with.
The capped dispersion curve. In the dispersion panel, note that $|\omega|$ has a maximum near $k\sim1/L_R$ : unlike Ch. 7’s barotropic waves, QG Rossby waves cannot propagate faster than $\beta L_R^2$ , and long waves all bunch at that speed — why oceanic eddies of many sizes drift westward at nearly the same rate.

Prefer to run it yourself? Download as a Jupyter notebook — it installs its own dependencies in the first cell. The sliders are static in Jupyter; edit value= and re-run to change a parameter.

Both fluids

Atmosphere: synoptic-scale weather systems are quasi-geostrophic to good approximation — QG theory (via the omega equation, not covered here) is still the standard first-pass tool for diagnosing where a system will intensify.
Ocean: mesoscale eddies (the ocean’s weather, $L\sim50$ –150 km) are observed to drift predominantly westward at a speed set by $\beta L_R^2$ — precisely the beta-gyre mechanism this notebook reproduces from a single isolated vortex.

Exercises

(analytic) Starting from $q=\nabla^2\psi-\psi/L_R^2+\beta y$ , show that in the limit $L_R\to\infty$ (or equivalently rescaling lengths $\gg L_R$ ) QG reduces exactly to Ch. 7’s barotropic PV. What physically has been assumed away in that limit?
(computational) Modify ch08_qg-approximation.py to place two vortices of opposite sign, each individually beta-drifting. Do they still form a self-propelling dipole (Ch. 7), or does the beta-drift dominate?
(exploratory) Using the dispersion-relation panel, predict which vortex radius $\sigma$ (relative to $L_R$ ) should shed the most visible wake, then confirm it in the snapshot browser.