Ch 18. Geostrophic (2D) turbulence

Overview

Rotation and stratification make large-scale atmospheric and oceanic flow approximately two-dimensional — and 2D turbulence is a different animal from the 3D turbulence of engineering. Because vortex stretching is absent, the flow conserves enstrophy $Z=\tfrac12\langle\zeta^2\rangle$ alongside energy $E=\tfrac12\langle|\nabla\psi|^2\rangle$ , and that second invariant forces energy upscale: small vortices merge into ever-larger ones (the inverse cascade), while enstrophy drains to small scales through filaments. On a rotating planet the upscale march does not continue forever — when eddies grow big enough to feel $\beta$ , Rossby-wave dynamics arrest the cascade near the Rhines scale $k_\beta\simeq\sqrt{\beta/2U}$ and reorganize the flow into zonal jets. This chapter’s notebook lets you watch both acts: the dual cascade, and its arrest.

The model

Notebook: notebooks/ch18_geostrophic-turbulence.py → exported to /nb/ch18_geostrophic-turbulence.html. Governing equations use the symbols in notation:

\frac{\partial \zeta}{\partial t} + J(\psi,\zeta) + \beta v = -\nu(-\nabla^2)^{n_\nu}\zeta, \qquad \nabla^2\psi = \zeta .

Pseudo-spectral, 2/3-rule dealiased, with an integrating factor that treats dissipation and the Rossby-wave propagator exactly (gfdlib.timestep.ifrk4_step). Presets: decaying McWilliams (1984) turbulence, a hyperviscous variant, and a $\beta$ -plane jet run.

What to look for

Merger all the way up. Run the decaying (McWilliams 1984) preset and scrub: a fine-grained random field organizes itself into a handful of large, long-lived vortices through repeated mergers — Ch. 7’s two-vortex merger applied recursively. This is the inverse cascade in physical space.
The two invariants part ways. Watch the energy and enstrophy curves together: energy stays nearly flat while enstrophy collapses. That asymmetry is the entire logic of 2D turbulence — dissipation (which acts at small scales) can reach the enstrophy but barely touches the energy, so the energy has nowhere to go but up-scale. Fjørtoft’s theorem (exercise 1) is this plot, proved.
The spectrum migrates. In the spectral panel, follow the energy peak: it marches steadily toward lower $k$ while a power-law tail feeds enstrophy toward the dissipation range. Compare the tail’s slope with the $k^{-3}$ enstrophy-cascade prediction — and note where (and why) coherent vortices make it steeper.
Jets from a knob. Switch to the β-plane preset: the isotropic inverse cascade proceeds until eddies reach the Rhines scale, then anisotropizes into east-west bands — watch the Hovmöller panel of $\bar u(y,t)$ develop persistent stripes. Sweep $\beta$ and check the jet spacing against $\pi\sqrt{2U/\beta}$ (exercise 3 makes this quantitative). Zero knobs were labeled “make jets”; $\beta$ plus an arrested cascade is sufficient.

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 banded winds of Jupiter and Saturn are the textbook Rhines-arrested inverse cascade; on Earth, the eddy-driven midlatitude jet is maintained by exactly this upscale momentum transfer from baroclinic eddies (Ch. 16 supplies the eddies).
Ocean: the Southern Ocean and the subtropical gyres carry multiple quasi-zonal jets (“striations”) with spacing near the local Rhines scale, and the mesoscale eddy field ( $L\sim100$ km) inverse-cascades energy toward the basin scale until $\beta$ — and bottom drag — intervene.

Exercises

(analytic) From conservation of $E$ and $Z$ , prove Fjørtoft’s theorem: if spectral energy spreads from a middle wavenumber $k_1$ to $k_0<k_1<k_2$ with $k_2/k_1 = k_1/k_0 = 2$ , more energy must go to $k_0$ than to $k_2$ . Where does the enstrophy go?
(computational) Modify ch18_geostrophic-turbulence.py to add linear (Ekman) drag $-r\zeta$ to the linear operator. How does the final vortex size — and the jet amplitude in the $\beta$ preset — depend on $r$ ?
(exploratory) Sweep $\beta$ at fixed energy: find the smallest $\beta$ at which the Hovmöller diagram of $\bar u(y,t)$ shows persistent stripes, and check the jet spacing against $\pi/k_\beta$ with $U=\sqrt{2E}$ .