Ch 13. Wave–mean-flow interaction

Overview

Two ways a linear wave leaves a permanent mark on the mean state it rides on. A floating buoy does not end up exactly where it started once a wave train passes, despite the wave’s Eulerian-mean velocity being exactly zero at every fixed point — it Stokes-drifts, a direct consequence of the wave field being evaluated at the parcel’s true, displaced position. And a Rossby wave riding on a sheared mean flow cannot reach the latitude where its Doppler-shifted phase speed matches the flow: its wavenumber diverges as it approaches that critical layer, setting up exactly the structure that (once dissipation or finite amplitude enters) lets it deposit momentum into the mean flow there.

The model

Notebook: notebooks/ch13_wave-mean.py → exported to /nb/ch13_wave-mean.html. Governing equations use the symbols in notation.

u_S=\frac{A^2k}{2\omega},\qquad \omega(y,k,l)=U(y)k-\frac{\beta k}{k^2+l^2},\qquad \frac{dy}{dt}=\frac{2\beta kl}{(k^2+l^2)^2},\quad \frac{dl}{dt}=-k\frac{dU}{dy}.

gfdlib.wavemean.wave_velocity/stokes_drift derive and verify the Stokes formula against direct particle advection (gfdlib.timestep.rk4, reusing Ch. 1’s machinery); ray_rhs_shear extends Ch. 9’s ray tracing to a Doppler-shifted background shear, verified to conserve $\omega$ exactly along the ray.

What to look for

The drift is real, and it is quadratic. In Part A, watch a particle ride the wave: it oscillates, but each cycle ends slightly downstream of where it began. Read off the measured drift and check it against $u_S=A^2k/2\omega$ ; then double the amplitude $A$ and confirm the drift quadruples. A linear wave produces a mean effect only at second order — this factor of four is that statement made tangible.
The Eulerian mean is exactly zero. The wave is $u'=A\cos(kx-\omega t)$ , which averages to nothing at every fixed point over a period — yet the particle undeniably moves. The entire drift lives in the correlation between where the parcel is and what the wave is doing there. Lagrangian and Eulerian means are different animals.
The ray runs out of road. In Part B, launch a ray into the shear: $l$ grows linearly (at the rate $-k\,dU/dy$ — check the slope), the group velocity collapses as $1/l^3$ , and the ray asymptotes to its critical latitude $y_c$ without ever arriving. Move $\Lambda$ and watch $y_c$ move with it: the wall sits exactly where the intrinsic frequency $\omega-U(y)k$ is Doppler-shifted to zero.
Same wall, both directions. Launch with $l_0>0$ and $l_0<0$ from the same point (exercise 3 continues this): rays can leave a shear region freely but approach their critical layer only asymptotically — which is why, once dissipation acts, the momentum they carry is deposited there and nowhere else. That one-way delivery is the QBO mechanism in the “Both fluids” section below.

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

Ocean: Stokes drift is directly measured by surface drifters and is a leading-order term in how floating debris and larvae are transported — distinct from, and typically larger than, the mean Eulerian current at the surface.
Atmosphere: critical-layer absorption of upward-propagating waves is the accepted driving mechanism of the stratospheric quasi-biennial oscillation (QBO) — waves deposit momentum preferentially where their phase speed matches the slowly-descending mean wind, reversing that wind and shifting the critical layer downward in a self-sustaining cycle.

Exercises

(analytic) Redo the Stokes-drift derivation for $u'(x,t)=A\sin(kx-\omega t)$ instead of $\cos$ — confirm you get the identical result $A^2k/(2\omega)$ (the phase offset cannot matter, since $\langle\sin^2\rangle=\langle\cos^2\rangle=1/2$ ).
(computational) Modify ch13_wave-mean.py’s Part B to use a $\tanh$ -shaped shear $U(y)=U_0\tanh(y/L)$ instead of linear shear, and locate the critical layer numerically (where does $l$ start diverging?) rather than from a closed-form $y_c$ .
(exploratory) In Part B, launch several rays with different $l_0$ (some positive, some negative) from the same $y_0$ . Do they all approach the same $y_c$ , or does $y_c$ depend on $l_0$ through $\omega_0$ ? Explain why launching with the “wrong sign” of $l_0$ might send a ray toward a completely different (or no) critical layer.