Ch 23. Conservation laws & wave activity

Overview

Can waves accelerate the jet they ride on? For conservative, small-amplitude waves the answer is a theorem, and the answer is no: zonal-mean flow and wave pseudomomentum change in exact opposition, $\partial_t(\bar u+A)=0$ pointwise in latitude — the non-acceleration theorem. Waves are a loan of momentum, repaid in full unless the wave dies; only transience and dissipation (Ch. 13’s critical layers being the canonical executioner) convert the loan into a grant. This chapter derives the theorem in three short steps — Noether’s zonal-translation charge, the Taylor identity, the zonal-mean momentum equation — and then verifies it pointwise against the full nonlinear barotropic model.

The model

Notebook: notebooks/ch23_wave-activity.py → exported to /nb/ch23_wave-activity.html. Governing equations use the symbols in notation.

A small Rossby-wave packet on a sinusoidal shear $\bar U=U_0\cos y$ is evolved with Ch. 18’s full nonlinear pseudo-spectral solver; gfdlib.wavemean.wave_activity diagnoses $\bar u(y)$, $A(y)$, and $\bar q_y(y)$ from every snapshot. The residual $\Delta(\bar u+A)$ comes out below 1% of either term (also enforced in the test suite), and a hyperviscosity slider provides the theorem’s off switch.

Both fluids

• Atmosphere: the quasi-biennial oscillation is non-acceleration’s loophole running as an engine — waves launched by tropical convection break at slowly-descending critical lines, depositing momentum that reverses the stratospheric winds every ~14 months; sudden stratospheric warmings are the same ledger settled violently in midwinter. The Eliassen–Palm flux plotted daily in stratosphere diagnostics is exactly this chapter’s wave-activity flux, generalized to stratified flow.
• Ocean: eddy parameterizations built on PV fluxes (Ch. 19’s caution about up-gradient fluxes included) are constrained by the same identity — where eddies are steady and conservative, their momentum forcing of the mean gyres must vanish, however energetic they look.

Exercises

1. (analytic) Derive the Taylor identity $\overline{v'q'}=-\partial_y\overline{u'v'}$ from $q'=\nabla^2\psi'$, $u'=-\psi'_y$, $v'=\psi'_x$: expand $\overline{\psi'_x(\psi'_{xx}+\psi'_{yy})}$, and show every term is either a pure $x$-derivative of a product (zonal average zero) or $-\partial_y(\overline{u'v'})$.
2. (computational) Modify ch23_wave-activity.py to also plot the displacement form of wave activity, $\tfrac12\bar q_y\overline{\eta^2}$ with $\eta=-q'/\bar q_y$, and confirm it is identical to $A$ (it is algebraically the same object — this is a consistency check on the diagnostics, and a good way to internalize that $A$ measures contour displacement).
3. (exploratory) Using the $\nu$ slider as a dial rather than a switch, measure the final-time residual $\max_y|\Delta(\bar u+A)|$ as a function of $\nu$. Is the momentum “lost” to dissipation proportional to $\nu$ at small $\nu$, and where in $y$ does the mean flow actually end up changed?

Further reading

Andrews, D. G. & McIntyre, M. E. (1976), J. Atmos. Sci. 33, 2031–2048; Charney, J. G. & Drazin, P. G. (1961), J. Geophys. Res. 66, 83–109; Bühler, Waves and Mean Flows 2nd ed., ch. 4–5; Vallis, AOFD 2nd ed., ch. 10 (wave–mean-flow interaction), §10.4 (non- acceleration); Baldwin et al. (2001), Rev. Geophys. 39, 179–229 (the QBO).