Ch 7. Vorticity and potential vorticity

Overview

Potential vorticity is the single most useful diagnostic quantity in this subject because of one property: invertibility. Knowing $q(x,y)$ everywhere — plus a balance condition and boundary conditions — fixes the entire velocity field, uniquely, with no other information needed. And because $q$ is also materially conserved, that same fact tells you how the flow evolves: invert for the velocity, advect $q$ with it, invert again. This chapter builds that two-step algorithm from the vorticity equation and uses it to explain two classic results: why two vortices merge or orbit forever depending on their separation, and why regions where eddies have mixed PV into a staircase of flat, well-mixed bands develop sharp jets exactly at the risers between bands — not within the quiet plateaus.

The model

Notebook: notebooks/ch07_vorticity-pv.py → exported to /nb/ch07_vorticity-pv.html. Governing equations use the symbols in notation:

q_t+J(\psi,q)=0,\qquad \zeta=\nabla^2\psi=q-\beta y.

Reuses the exact pseudo-spectral machinery from Ch. 18 (gfdlib.spectral.Grid, gfdlib.timestep.ifrk4_step) — this chapter’s new contribution is gfdlib.pv, which builds initial PV fields: a periodic-safe Gaussian blob for placing vortices, and an idealized alternating-band staircase. A fully reactive “invert, don’t integrate” panel shows the balance step alone (no time-stepping); a second, Run-gated panel evolves the same field forward to show the resulting dynamics.

What to look for

Inversion is instant — use it first. The upper panel needs no Run button: it takes the PV field you built and solves for $\psi$ and the winds. Drag the separation slider and watch the velocity field between two like-signed vortices reorganize — everything you see is diagnosed from $q$ alone, which is the invertibility principle doing its work.
Find the merger threshold. In the merger preset, run the evolution at a few separations: well-separated vortices orbit each other essentially forever, while below a critical separation (a few core radii) they wrap around each other and merge into one core, throwing off filament arms. The transition is surprisingly sharp — this is the same vortex-merger physics that drives Ch. 18’s inverse cascade.
The dipole travels. The opposite-signed pair preset produces the one configuration that self-propels: each vortex advects the other in the same direction. Confirm the speed falls as you increase the separation.
Jets live at the risers. In the staircase preset, look at the zonal-mean wind: it peaks exactly at the sharp PV jumps between the flat, well-mixed bands, not inside the plateaus. Then use the reactive panel (no Run needed) to vary the number of bands and watch exercise 3’s question take shape.

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: the tropopause’s PV field is routinely mapped and inverted operationally — “PV thinking” lets forecasters read cyclogenesis directly off a PV map, and jet streams are themselves regions of sharp PV gradient exactly as in this chapter’s staircase.
Ocean: mesoscale eddies are, to a good approximation, isolated PV anomalies that interact by mutual advection just like this notebook’s vortex pairs; multiple oceanic jets (e.g. in the Antarctic Circumpolar Current) are believed to be maintained by the same PV-staircase mechanism demonstrated here.

Exercises

(analytic) Show that $Dq/Dt=0$ combined with $\zeta=\nabla^2\psi$ is a closed system — that is, that no other equation or unknown is needed to advance $q$ forward in time. This closure is what “invertibility” buys you.
(computational) Modify ch07_vorticity-pv.py’s merger preset to use three like-signed vortices arranged in a triangle instead of two. Do they merge pairwise, or all at once?
(exploratory) In the staircase preset, use the “invert, don’t integrate” panel (no need to press Run) to find how the jet strength scales with the number of bands at fixed amplitude — does adding more, narrower bands make each jet stronger or weaker?