Ch 24. Balanced models & the slow manifold

Overview

The book’s closing retrospective, with a runnable demonstration at its center. Twenty-three chapters solved reduced equations while the fluid obeys none of them exactly — and it worked because rotating, stratified flow keeps its fast (inertia-gravity) and slow (PV-controlled) lives separate. A state initialized with zero fast content lives on the slow manifold, and every balanced model in this book is the dynamics restricted to it: geostrophy at $O(1)$, QG at $O(Ro)$, the planetary balances at basin scale. The demonstration is Richardson’s 1922 forecast disaster in miniature: one height field, three wind initializations, identical PV to machine precision — and divergence ringing in exact proportion to the projection off the manifold.

The model

Notebook: notebooks/ch24_balanced-models.py → exported to /nb/ch24_balanced-models.html. Governing equations use the symbols in notation.

Ch. 6’s rotating-shallow-water machinery unchanged, plus one new one-line diagnostic (gfdlib.shallowwater.divergence): the slow mode has $\delta=0$ identically while inertia-gravity waves are made of it, so RMS divergence directly meters the distance from the slow manifold.

Both fluids

• Atmosphere: balanced initialization descends directly from Richardson’s failure — every operational forecast system since has projected its analysis toward the slow manifold (nonlinear normal-mode initialization historically; digital filters and careful data assimilation today) before integrating.
• Ocean: state estimation faces the same problem with a slower clock — an ocean reanalysis initialized off-balance rings with near-inertial waves and spurious convection; and the “spontaneous imbalance” caveat (the slow manifold leaks gravity waves at $\sim e^{-c/Ro}$) is an active research topic precisely because ocean fronts reach $Ro\sim1$, where the leak stops being exponential and becomes Ch. 12’s internal-wave generation in earnest.

Exercises

1. (analytic) Show that for the linear $f$-plane system, the geostrophic initialization $u=-\eta_y$, $v=\eta_x$ is an exact steady state (substitute into gfdlib.shallowwater.rhs’s equations), and that an arbitrary initial state’s fast-mode amplitude — hence its later divergence ringing — is a linear functional of the initialization error, giving the notebook’s exact $(1-\alpha)$ scaling.
2. (computational) Modify ch24_balanced-models.py to add a fourth run initialized with the PV-consistent balanced state shallowwater.invert_pv(q0) (Ch. 6’s adjustment end-state) instead of the raw $\eta_0$ — for a small-scale bump ($\sigma<L_R$), how different are “balancing the winds to the observed height” and “balancing everything to the observed PV”?
3. (exploratory) Turn on $\beta$ (edit the coriolis call): the slow mode is no longer exactly steady (Rossby waves live on the slow manifold and move). Can you still cleanly separate slow evolution from fast ringing in the divergence record — and what does that tell you about why divergence, rather than any time derivative, was the right diagnostic all along?

Further reading

Lorenz, E. N. (1986), J. Atmos. Sci. 43, 1547–1557 (“On the existence of a slow manifold”); Leith, C. E. (1980), J. Atmos. Sci. 37, 958–968 (nonlinear normal-mode initialization); Vanneste, J. & Yavneh, I. (2004), J. Atmos. Sci. 61, 211–223 (exponentially small imbalance); Vallis, AOFD 2nd ed., §5.9 (the slow manifold and balance); Richardson, L. F. (1922), Weather Prediction by Numerical Process (the origin story).