Ch 15. Barotropic instability

Overview

Whether a shear flow spontaneously rolls up into vortices or just sits there, however hard you perturb it, is decided entirely by the curvature of its own velocity profile. Rayleigh’s inflection-point criterion — later extended to a rotating planet by Kuo — says instability is impossible unless the meridional gradient of absolute vorticity changes sign somewhere in the flow. This chapter turns that criterion into a genuine growth-rate calculator (a new kind of numerics for this book: a matrix eigenvalue problem, not a time-stepping PDE), then hands the fastest-growing mode to the same nonlinear machinery from Ch. 18 to watch it roll up into a chain of Kelvin-Helmholtz vortices.

The model

Notebook: notebooks/ch15_barotropic-instability.py → exported to /nb/ch15_barotropic-instability.html. Governing equations use the symbols in notation:

(U-c)(\phi''-k^2\phi)+(\beta-U'')\phi=0.

Part A solves this as a generalized eigenvalue problem, rewritten $B^{-1}A$ and diagonalized with plain numpy.linalg.eigvals — no SciPy. Part B reuses gfdlib.spectral.Grid and gfdlib.timestep.ifrk4_step unchanged from ch18, with a new initial condition, gfdlib.instability.double_shear_layer. The solver is independently verified against two facts that don’t depend on this code being right in advance: a profile with no inflection point gives exactly zero growth at every wavenumber tested, and the classical $\tanh(y/\delta)$ shear layer’s growth-rate peak lands at $k\delta\approx0.445$ , matching the published value (Michalke 1964, $k\delta\approx0.4446$ ).

What to look for

A theorem you can poke. In Part A, select the cosine bump profile: no inflection point, and the growth-rate curve is exactly zero at every wavenumber — not small, zero. Rayleigh’s criterion is a hard prohibition. Switch to the $\tanh$ shear layer and the curve lifts off, peaking at $k\delta\approx0.445$ (Michalke’s classical value, reproduced live).
$\beta$ as a stabilizer. With the $\mathrm{sech}^2$ jet selected, raise $\beta$ : the necessary condition involves $\beta-U''$ , so a large enough planetary vorticity gradient removes the sign change and the growth-rate curve collapses. Find the threshold — that number is exercise 3.
The growth-rate curve knows the answer first. Part A’s curve marks its peak wavenumber; that wavelength sets how many vortices the roll-up will produce per shear layer. Run Part B with the seeded wavenumber $n$ matched to that prediction, then deliberately mismatched — the flow finds its preferred scale either way, but watch how much longer the mismatched seed takes, and by what route (the seeded mode grows first, then hands over).
Linear birth, nonlinear life. In Part B’s snapshot browser, mark the transition: exponential growth of a wiggle (linear theory’s domain), then braid roll-up into discrete vortices, then vortex pairing/merger (Ch. 7’s physics taking over). Linear theory predicts only the beginning — but it chooses which beginning.

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 polar-front jet and subtropical jet both meander and shed cutoff eddies via barotropic instability acting on their horizontal shear — the same inflection-point mechanism as this notebook’s shear layer, just embedded in a jet rather than isolated.
Ocean: western boundary currents (the Gulf Stream, Kuroshio) are strongly barotropically unstable, shedding rings and meanders at a wavelength set by the same growth-rate-curve logic computed here.

Exercises

(analytic) Derive the Rayleigh-Kuo equation from the linearized barotropic vorticity equation, starting from $\zeta_t+u\zeta_x+v\zeta_y+ \beta v=0$ linearized about $U(y)$ . Confirm your equation matches the classical $\beta=0$ (Rayleigh) form in Drazin & Reid.
(computational) Modify ch15_barotropic-instability.py’s Part A to add a fourth profile, a piecewise-linear jet (a well-known textbook case with an exact analytic growth-rate formula). Compare your numerical curve to the analytic one.
(exploratory) In Part A, find the value of $\beta$ (for the jet profile) at which the growth-rate curve’s maximum first drops to zero — the flow’s stabilization threshold.