Statistical mechanical equilibria of the shallow water equations

Abstract

Statistical mechanical equilibria of the shallow water equations, valid in the limit of weak flow, are found to imply that quasi-geostrophic flow is unstable in the statistical mechanical limit. In equilibrium, most of the energy ends up in short-scale gravity-inertial waves implying an energy transfer to short scales during relaxation, regardless of the nature of the initial state. The situation is therefore quite unlike that described by the quasi-geostrophic version of the equations in which energy flow to short scales is prohibited. The transfer occurs via the divergent part of the flow which, in turn, appears to draw upon the energy of the large scale rotational modes. Following Sadourny, it is conjectured that this mechanism persists in forced-dissipative flows at large Reynolds numbers, leading to a weak direct energy cascade.