Balance and the Slow Quasimanifold: Some Explicit Results

The ultimate limitations of the balance, slow-manifold, and potential vorticity inversion concepts are investigated. These limitations are associated with the weak but nonvanishing spontaneous-adjustment emission, or Lighthill radiation, of inertia–gravity waves by unsteady, two-dimensional or layerwise-two-dimensional vortical flow (the wave emission mechanism sometimes being called ‘‘geostrophic’’ adjustment even though it need not take the flow toward geostrophic balance). Spontaneous-adjustment emission is studied in detail for the case of unbounded f -plane shallow-water flow, in which the potential vorticity anomalies are confined to a finite-sized region, but whose distribution within the region is otherwise completely general. The approach assumes that the Froude number F and Rossby number R satisfy F K 1 and R * 1 (implying, incidentally, that any balance would have to include gradient wind and other ageostrophic contributions). The method of matched asymptotic expansions is used to obtain a general mathematical description of spontaneous-adjustment emission in this parameter regime. Expansions are carried out to O(F4), which is a high enough order to describe not only the weakly emitted waves but also, explicitly, the correspondingly weak radiation reaction upon the vortical flow, accounting for the loss of vortical energy. Exact evolution on a slow manifold, in its usual strict sense, would be incompatible with the arrow of time introduced by this radiation reaction and energy loss. The magnitude O(F4) of the radiation reaction may thus be taken to measure the degree of ‘‘fuzziness’’ of the entity that must exist in place of the strict slow manifold. That entity must, presumably, be not a simple invariant manifold, but rather an O(F4)-thin, multileaved, fractal ‘‘stochastic layer’’ like those known for analogous but low-order coupled oscillator systems. It could more appropriately be called the ‘‘slow quasimanifold.’’