Ch 9. Rossby waves

Overview

A Rossby wave’s group velocity depends on its own wavenumber, so a wave packet moving through a slowly varying background doesn’t travel in a straight line — its wavenumber refracts as it goes, and it follows a curved ray, exactly as light refracts through a lens. This chapter builds that ray theory from the dispersion relation and group velocity you’ve already used (ch. 18, ch. 8), then takes it somewhere the rest of this book hasn’t gone: onto the sphere itself. With the right background flow, the rays this notebook traces are exact great circles — a genuine theorem (Hoskins & Karoly 1981), not a plotting trick, and the mechanism behind how real atmospheric teleconnections (like the Pacific–North America pattern excited by El Niño) arc across the globe.

The model

Notebook: notebooks/ch09_rossby-waves.py → exported to /nb/ch09_rossby-waves.html. Governing equations use the symbols in notation:

\frac{d\mathbf x}{dt}=\frac{\partial\omega}{\partial\mathbf k},\qquad \frac{d\mathbf k}{dt}=-\frac{\partial\omega}{\partial\mathbf x}.

gfdlib.rossby deliberately avoids a hand-derived symbolic ray equation: ray_rhs differentiates the single scalar dispersion_omega by central differences, so correctness rests on one formula rather than a chain of algebra, and great_circle_deviation checks the result against an independent, coordinate-free geometric criterion — no PDE, no time-stepping in the usual sense, just gfdlib.timestep.rk4 on a 3-variable ray. Cheap enough that the whole notebook is reactive, with no Run button.

What to look for

The arc, not the straight line. Launch a fan of rays from $\phi_0=20°$ : each one climbs poleward, turns at its own turning latitude (where its meridional wavenumber passes through zero), and arcs back toward the equator. No ray crosses its turning latitude — refraction by the planetary vorticity gradient bends it back, exactly as a lens bends light back toward the dense medium.
Great circles, verified. The great_circle_deviation readout stays at machine precision for solid-body rotation — the Hoskins–Karoly theorem holding exactly, not approximately, in front of you. (Then see exercise 2 for how quickly it breaks when the background flow isn’t solid-body.)
Rotation strength sets the reach. Increase $\Omega_s$ : the stationary wavenumber $K_s$ falls everywhere (stronger westerlies can hold longer waves stationary), and since a ray turns where $K_s(\phi)$ has dropped to its own zonal wavenumber, lowering the whole $K_s$ profile slides each turning latitude equatorward. Confirm it in the fan: the same rays that grazed the pole at small $\Omega_s$ turn low at large $\Omega_s$ .
Where rays bunch, anomalies live. Notice the rays launched at slightly different angles converge near their turning latitudes (the ray density is highest there). Ray theory says wave amplitude accumulates where rays crowd — one reason observed teleconnection centers of action sit at preferred latitudes rather than smearing along the whole path.

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: stationary Rossby wave trains excited by tropical convection anomalies propagate poleward along great-circle-like arcs before curving back equatorward — the observed teleconnection patterns linking ENSO to midlatitude weather are, to good approximation, segments of exactly the rays this notebook traces.
Ocean: oceanic Rossby waves (much slower, non-dispersive westward phase propagation dominates over the ray-refraction effects shown here) carry the memory of wind-stress anomalies across entire ocean basins, setting the timescale for basin-scale adjustment.

Exercises

(analytic) From $\omega=-\beta k/(k^2+l^2)$ , derive $\mathbf c_g=\beta(k^2-l^2,2kl)/(k^2+l^2)^2$ and show $\mathbf c_g\cdot\mathbf k>0$ always — group velocity always has an eastward component relative to the phase, even though phase itself always moves westward.
(computational) Modify ch09_rossby-waves.py to use $U(\phi)=U_0\cos^2\phi$ instead of solid-body rotation. Confirm with the great-circle deviation diagnostic that the rays are no longer exact great circles, and describe qualitatively how they differ.
(exploratory) Find the launch angle (at fixed $\phi_0,\Omega_s$ ) that sends a ray closest to the pole without crossing it. How does that critical angle change as you increase $\Omega_s$ ?