Ch 17. Symmetric & other instabilities

Overview

Chs. 15 and 16 covered the two mechanisms that dominate the large-scale storm track and ocean mesoscale. This chapter is a survey of three faster, more local mechanisms that matter at fronts and in stratified shear layers: inertial instability (unbalanced rotation), symmetric instability (a slantwise hybrid of inertial and buoyant), and Kelvin-Helmholtz instability (stratified shear, no rotation needed at all). All three turn out to reduce to comparing a Richardson number against a Rossby number — one map, three mechanisms, and a genuinely rigorous theorem (Miles-Howard) governing the third.

The model

Notebook: notebooks/ch17_other-instabilities.py → exported to /nb/ch17_other-instabilities.html. Governing equations use the symbols in notation:

q=(f-\partial U/\partial y)N^2-f(\partial U/\partial z)^2,\qquad (U-c)^2(\phi''-k^2\phi)-U''(U-c)\phi+N^2\phi=0.

gfdlib.symmetric derives the Ertel-PV sign criterion for symmetric and inertial instability directly (not from a memorized formula); at zero relative vorticity it reduces to the textbook value $\mathrm{Ri}=1$ , and for $(1+\mathrm{Ro})<0$ the front is unconditionally unstable at every $\mathrm{Ri}$ — pure inertial instability. gfdlib.instability. taylor_goldstein_growth_rate solves the Taylor-Goldstein equation as a natively linear $2n\times2n$ generalized eigenvalue problem (keeping the buoyancy perturbation as an independent unknown, rather than eliminating it into a harder-to-trust quadratic pencil); at zero stratification it reduces exactly to ch. 15’s barotropic solver. Validating it against the rigorous Miles-Howard theorem ( $\mathrm{Ri}\geq1/4$ everywhere $\Rightarrow$ stable) caught a real numerical issue: initial growth-rate curves didn’t vanish cleanly above the threshold, traced (not dismissed) to genuine finite-difference truncation error that is worst exactly at the delicate $\mathrm{Ri}=1/4$ point, where the equation’s critical-layer indicial roots coalesce — a classically hard point for simple discretizations. The notebook flags this honestly rather than hiding it.

What to look for

One map, three regimes. Move the marker around the Ro–Ri plane: along $\mathrm{Ro}=0$ the stability boundary sits exactly at $\mathrm{Ri}=1$ (symmetric instability of a balanced front); push $\mathrm{Ro}$ below $-1$ and the front is unstable at every Ri — anticyclonic shear has overwhelmed planetary rotation and the instability is purely inertial. The whole map is one sign condition on the Ertel PV, $q<0$ .
Why $\mathrm{Ri}=1$ , not $1/4$ . Note the factor-of-four gap between the symmetric threshold ( $\mathrm{Ri}<1$ ) and the Kelvin-Helmholtz one ( $\mathrm{Ri}<1/4$ ): slantwise parcels sliding along tilted absolute-momentum surfaces can extract energy in conditions where purely vertical overturning cannot. Exercise 3 asks you to articulate the extra degree of freedom.
Miles-Howard with the seams showing. In the Taylor-Goldstein panel, sweep $J$ upward: the growth-rate curve shrinks and dies by $J\approx1/4$ , as the theorem demands. Near the threshold, though, the computed curve doesn’t vanish perfectly cleanly — the notebook documents why (the critical layer’s indicial roots coalesce exactly at $\mathrm{Ri}=1/4$ , the hardest possible point for a finite-difference scheme) rather than tuning the plot until it looked right. Numerical evidence has error bars; this is what honest ones look like.
A familiar limit. Set $J=0$ and compare the Taylor-Goldstein curve to Ch. 15’s $\tanh$ -layer result: same equation, same peak, same $k\delta\approx0.445$ . Stratification is a continuous deformation of the barotropic problem, not a different subject.

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: symmetric instability is the mechanism behind banded precipitation structures in fronts and rapidly-intensifying cyclones — slantwise convection along sloping absolute-momentum surfaces, distinct from ordinary upright thunderstorm convection. Kelvin-Helmholtz billows form in strongly-sheared, weakly-stratified layers such as jet streaks and clear-air-turbulence zones.
Ocean: symmetric instability is a major mechanism restratifying mixed-layer fronts after wintertime convection, competing with submesoscale baroclinic instability (ch. 16) to set the depth and structure of the mixed layer. Kelvin-Helmholtz billows are directly observed at the base of the equatorial undercurrent and in energetic tidal-mixing fronts, where they are a leading mechanism for diapycnal mixing.

Exercises

(analytic) Starting from the Ertel PV $q=(f-\partial U/\partial y)N^2-f(\partial U/\partial z)^2$ and the thermal-wind relation $f\partial U/\partial z=-\partial b/\partial y$ , derive the nondimensional form $q/(fN^2)=(1+\mathrm{Ro})-1/\mathrm{Ri}$ used in gfdlib.symmetric.
(computational) Modify ch17_other-instabilities.py’s Part B to plot the peak growth rate (not the whole curve) as a function of $J$ swept from $0$ to $0.5$ . Confirm it decreases monotonically and reaches zero by around $J\approx0.4$ , consistent with — but, given the numerical caveats documented in the notebook, not a razor-sharp proof of — the Miles-Howard $\mathrm{Ri}=1/4$ threshold.
(exploratory) Compare the two Richardson-number thresholds head to head: symmetric instability at $\mathrm{Ro}=0$ requires $\mathrm{Ri}<1$ ; Kelvin-Helmholtz requires $\mathrm{Ri}<1/4$ . Why does a rotating, geostrophically-balanced parcel tolerate four times less shear before destabilizing than a non-rotating one? What extra degree of freedom does slantwise motion have that pure vertical overturning does not?