Rotating Shallow Water Dynamics: Extra Invariant and the Formation of Zonal Jets

TL;DR

This paper demonstrates the existence of an approximate invariant in rotating shallow water dynamics, applicable near the equator, which can explain the formation of zonal jets and has parallels in plasma physics.

Contribution

It introduces a new adiabatic invariant in shallow water equations near the equator, extending previous mid-latitude results and explaining jet formation mechanisms.

Findings

Existence of an extra invariant in shallow water dynamics near the equator.

Cancellation of small denominators due to triad resonances and equatorial limits.

Potential for zonal jet formation driven by the invariant.

Abstract

We show that rotating shallow water dynamics possesses an approximate (adiabatic-type) positive quadratic invariant, which exists not only at mid-latitudes (where its analogue in the quasigeostrophic equation has been previously investigated), but near the equator as well (where the quasigeostrophic equation is inapplicable). Deriving the extra invariant, we find "small denominators" of two kinds: (1) due to the triad resonances (as in the case of the quasigeostrophic equation) and (2) due to the equatorial limit, when the Rossby radius of deformation becomes infinite. We show that the "small denominators" of both kinds 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. Similar invariant occurs in magnetically confined fusion plasmas and can lead to the emergence of zonal flows.