Rotating shallow water dynamics: Extra invariant and the formation of zonal jets

Abstract

We show that rotating shallow water dynamics possesses an approximate (adiabatic-type) positive quadratic invariant, which exists not only at midlatitudes (where its analog in the quasigeostrophic equation has been previously investigated), but near the equator as well (where the quasigeostrophic equation is inapplicable). When deriving the extra invariant we find two kinds of small denominators: (i) those due to the triad resonances (as in the case of the quasigeostrophic equation) and (ii) those due to the equatorial limit, when the Rossby radius of deformation becomes infinite. We show that both kinds of small denominators can be canceled. The presence of the extra invariant can lead to the generation of zonal jets. We find that this tendency should be especially pronounced near the equator. A similar invariant occurs in magnetically confined fusion plasmas and can lead to the emergence of zonal flows.