Ch 6. Rotating shallow water & geostrophic adjustment

Overview

Perturb a rotating fluid away from rest and two things happen at once: fast inertia-gravity waves radiate the imbalance outward, and rotation traps a piece of the disturbance behind as a standing, balanced flow. This is geostrophic adjustment, and it is the mechanism by which almost every localized disturbance in the atmosphere or ocean — a convective burst, a wind-stress event, an eddy shed from an unstable current — settles into the slowly evolving balanced motion that the rest of this book is about. The deformation radius $L_R=c/f_0$ is the scale that decides the outcome: disturbances much smaller than $L_R$ disperse almost entirely as gravity waves, while disturbances much larger than $L_R$ survive almost unchanged as geostrophic flow. What is conserved throughout — and what therefore predicts the balanced end state before you integrate a single time step — is the linear potential vorticity $q=\zeta-\hat\eta$ .

The model

Notebook: notebooks/ch06_geostrophic-adjustment.py → exported to /nb/ch06_geostrophic-adjustment.html. Governing equations use the symbols in notation:

\hat\eta_t+\hat u_x+\hat v_y=0,\qquad \hat u_t-\hat f\hat v=-\hat\eta_x,\qquad \hat v_t+\hat f\hat u=-\hat\eta_y, \qquad \hat f=1+\hat\beta\,(y-y_0).

Pseudo-spectral on a doubly-periodic domain (gfdlib.spectral.Grid), classical RK4 (gfdlib.timestep.rk4 — the system is exactly linear, so no integrating factor is needed), with the new gfdlib.shallowwater module supplying the RHS and two potential-vorticity primitives: potential_vorticity (diagnose $q=\zeta-\hat\eta$ ) and invert_pv (solve $(\nabla^2-1)\eta_{bal}=q$ for the balanced height field carrying a given PV). Three presets: a small-scale burst ( $\sigma\ll L_R$ ), a large-scale surge ( $\sigma\gg L_R$ ), and a $\beta$ -plane run that shows the balanced remnant drift westward as a Rossby wave.

What to look for

Waves win at small scales. Run the small-scale burst preset and scrub the frame slider from the start: rings of inertia-gravity waves carry essentially the whole height anomaly away (at the nondimensional wave speed $c=1$ — check the ring radius against elapsed time), and the final $\eta$ is nearly flat. The overlay of $\eta_{bal}=$ invert_pv $(q_0)$ predicted that end state before the run started — compare them at the last frame.
Balance wins at large scales. Switch to the large-scale surge preset: now most of the height anomaly survives, and what changes instead is the wind field, which spins up into a rim current around the anomaly. Same equations, same initial shape — only $\sigma/L_R$ differs.
Energetics of the remnant. Watch the energy panel in the large-scale case: even when balance “wins,” at most one third of the released potential energy ends up in the balanced flow — the rest radiates. Rossby’s classic result, visible as the gap between the curves.
The remnant is not stuck. In the β-plane preset the balanced remnant itself drifts westward — adjustment hands the disturbance off to Rossby-wave dynamics (Ch. 8–9), which is where its story continues.

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: a convective outflow or a jet-stream meander is released unbalanced; within a day it sheds inertia-gravity waves and leaves behind a geostrophically balanced anomaly — the everyday mechanism that keeps synoptic weather systems in near-balance despite continuous forcing.
Ocean: wind-driven upwelling or a detrainment event from an unstable boundary current perturbs the thermocline; geostrophic adjustment converts part of that perturbation directly into a mesoscale eddy, while the rest escapes as fast barotropic and internal gravity waves.

Exercises

(analytic) Starting from $q=\zeta-\hat\eta$ and the momentum/continuity equations, derive $q_t=-\hat\beta\hat v$ . Confirm it reduces to exact pointwise conservation on the $f$ -plane.
(computational) Modify ch06_geostrophic-adjustment.py to add linear (Rayleigh) drag $-r\hat u,-r\hat v$ to the momentum equations. Does $q$ still evolve as $q_t=-\hat\beta\hat v$ , or does drag act as an additional PV source? Check against the modified notebook’s own diagnostic.
(exploratory) Sweep $\sigma/L_R$ finely between the “small” and “large” presets and plot the fraction of $\eta_0$ ’s peak amplitude that survives in $\eta_{bal}=$ invert_pv(q0). At what $\sigma/L_R$ does the survival fraction cross 50%?