Ch 16. Baroclinic instability

Overview

Ch. 15’s barotropic instability draws energy from horizontal shear in the mean flow’s kinetic energy. Mid-latitude storms and ocean mesoscale eddies draw energy from somewhere else entirely: the potential energy locked up in a sloping temperature surface — the same thermal-wind shear from ch. 5. This chapter builds the two classic models of that mechanism: the Eady problem, a boundary-value eigenvalue problem giving a closed growth-rate curve from stratification and shear alone, and the 2-layer (Phillips) model, whose nonlinear evolution shows what the instability actually does — grow, peak, break, and equilibrate. It is arguably the single most consequential instability in geophysical fluid dynamics.

The model

Notebook: notebooks/ch16_baroclinic-instability.py → exported to /nb/ch16_baroclinic-instability.html. Governing equations use the symbols in notation:

\sigma=\mu c_i,\qquad \mu=\frac{kNH}{f_0},\qquad q_i=\nabla^2\psi_i+F(\psi_j-\psi_i)+\beta y.

gfdlib.baroclinic solves both growth-rate problems as small ( $2\times2$ ) eigenvalue problems, derived directly rather than taken from a memorized closed form: eady_growth_rate matches the Eady interior solution to the linearized thermal boundary condition at $z=0,H$ , verified against the published benchmark ( $\mu_{max}\approx1.61$ , $\sigma_{max}\approx0.31$ , cutoff $\mu_c\approx2.399$ ). invert_2layer is a closed-form $2\times2$ Helmholtz solve; rhs_2layer is the nonlinear tendency for gfdlib.timestep.ifrk4_step, cross-checked against an independently derived linear eigenvalue to $<0.1\%$ once transients settle. A fixed mean shear with no other sink turned out to be an unlimited energy source — verified, not assumed: without bottom drag, nonlinear growth continued past any physically reasonable amplitude regardless of hyperviscosity, until overflow. rhs_2layer therefore includes the standard Phillips (1954) bottom Ekman drag term, which produces a genuine growth-peak-decay- equilibrate life cycle (confirmed by direct integration to $t=500$ at the notebook’s full resolution). That full run is ~ $10^5$ time steps — too heavy to integrate live in the browser — so it’s precomputed offline and shipped as data for Part B’s widget to scrub; a second, smaller live panel lets you test parameter changes yourself over the growth phase.

What to look for

The Eady curve’s shape is the forecast. In the Eady panel, note three features: growth peaks at $\mu\approx1.61$ (so the preferred wavelength is a fixed multiple, $\sim3.9$ , of the deformation radius — 1000-km storms in the atmosphere, 100-km eddies in the ocean, from one dimensionless number), the shortwave cutoff at $\mu_c\approx2.399$ (disturbances much smaller than $L_R$ cannot tap the APE), and no longwave cutoff.
The full life cycle, scrubbed. Drag through the precomputed run while watching the eddy-energetics panel: on its log axis the growth phase is a clean straight line with slope $2\sigma$ (energy is quadratic in amplitude — the factor of two is a classic trap), then the curve peaks, breaks, and equilibrates under the drag. During the linear phase EKE and EPE grow in lockstep at a fixed ratio — the growing mode’s fingerprint — with every joule drawn from the mean state’s APE, the energy pathway Ch. 15’s barotropic problem doesn’t have.
Phillips’ criterion, live. In the interactive panel, raise $\beta$ at fixed $F\,\Delta U$ : growth weakens and shuts off entirely near $\beta=F\,\Delta U$ , where the lower layer’s PV gradient $\beta-F\Delta U$ stops changing sign. The sign-change logic is Ch. 15’s Rayleigh-Kuo criterion again, wearing vertical-shear clothes.
Drag sets the equilibrium, not the growth. Vary the Ekman drag $r$ : the initial growth rate barely moves (the instability is an inviscid, interior mechanism), but the amplitude the life cycle equilibrates at changes dramatically — and at $r=0$ it doesn’t equilibrate at all, the failure mode “The model” section describes.

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 storm track — synoptic-scale (~1000 km) cyclones and anticyclones extracting available potential energy from the pole-to-equator temperature gradient via exactly this mechanism, with an e-folding time of order a day.
Ocean: mesoscale eddies (~50-100 km, an order of magnitude smaller, set by the much smaller oceanic deformation radius) extracting potential energy from sloping density surfaces at ocean fronts and along the edges of currents like the Gulf Stream, with e-folding times of weeks rather than days — the same instability, the same equations, wildly different scales.

Exercises

(analytic) Show that the Eady problem’s interior solution $\phi(z)=A\cosh(\mu z/H)+B\sinh(\mu z/H)$ follows from $q'=0$ with rigid lids, and that matching it to the linearized thermal boundary condition at $z=0,H$ gives a $2\times2$ generalized eigenvalue problem for $c$ rather than a scalar equation.
(computational) Modify ch16_baroclinic-instability.py’s B2 panel to sweep $\beta$ at fixed $F,\Delta U$ and locate the marginal-stability threshold numerically (where Phillips’ necessary condition, $\beta_2=\beta-F\Delta U$ changing sign, stops holding). Compare it to the analytic threshold $\beta=F\Delta U$ .
(exploratory) In B1’s precomputed life cycle, compare the wavenumber content of $q_1'$ early in the run (still near-sinusoidal, dominated by the seeded mode) to late in the run (post-breaking). What does the change tell you about how baroclinic eddies cascade energy once they reach finite amplitude?