A unified theory of balance in the extratropics

Many physical systems exhibit dynamics with vastly different time scales. Often the different motions interact only weakly and the slow dynamics is naturally constrained to a subspace of phase space, in the vicinity of a slow manifold. In geophysical fluid dynamics this reduction in phase space is called balance. Classically, balance is understood by way of the Rossby number $R$ or the Froude number $F$ ; either $R \,{\ll}\, 1$ or $F \,{\ll}\, 1$ . We examined the shallow-water equations and Boussinesq equations on an $f$ -plane and determined a dimensionless parameter $\epsilon$ , small values of which imply a time-scale separation. In terms of $R$ and $F$ , \[\epsilon = \frac{RF}{\sqrt{R^2 + F^2}}.\] We then developed a unified theory of (extratropical) balance based on $\epsilon$ that includes all cases of small $R$ and/or small $F$ . The leading-order systems are ensured to be Hamiltonian and turn out to be governed by the quasi-geostrophic potential-vorticity equation. However, the height field is not necessarily in geostrophic balance, so the leading-order dynamics are more general than in quasi-geostrophy. Thus the quasi-geostrophic potential-vorticity equation (as distinct from the quasi-geostrophic dynamics) is valid more generally than its traditional derivation would suggest. In the case of the Boussinesq equations, we have found that balanced dynamics generally implies hydrostatic balance without any assumption on the aspect ratio; only when the Froude number is not small and it is the Rossby number that guarantees a time-scale separation must we impose the requirement of a small aspect ratio to ensure hydrostatic balance.

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