Balance and the slow quasimanifold: Some explicit results

The ultimate limitations of the balance, slow-manifold, and potential vorti city inversion concepts are investigated. These limitations are associated with the weak but nonvanishing spontaneous-adjustment emission, dr Lighthil l radiation, of inertia-gravity waves by unsteady, two-dimensional or layer wise-two-dimensional vortical flow (the wave emission mechanism sometimes b eing 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 p otential vorticity anomalies are confined to a finite-sized region, but who se distribution within the region is otherwise completely general. The appr oach assumes that the Froude number F and Rossby number R satisfy F much le ss than 1 and R greater than or equal to 1 (implying, incidentally, that an y balance would have to include gradient wind and other ageostrophic contri butions). The method of matched asymptotic expansions is used to obtain a g eneral mathematical description of spontaneous-adjustment emission in this parameter regime. Expansions are carried out to O(F-4), which is a high eno ugh order to describe not only the weakly emitted waves but also, explicitl y, the correspondingly weak radiation reaction upon the vortical Row, accou nting 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 int roduced by this radiation reaction and energy loss. The magnitude O(F-4) of the radiation reaction may thus be taken to measure the degree of "fuzzine ss" of the entity that must exist in place of the strict slow manifold. Tha t entity must, presumably, be not a simple invariant manifold, but rather a n O(F-4)-thin, multileaved, fractal "stochastic layer" like those known for analogous but low-order coupled oscillator systems. It could more appropri ately be called the "slow quasimanifold."