Equilibrium statistical mechanics and energy partition for the shallow water model

TL;DR

This paper applies large deviation theory to the shallow water model to compute equilibrium states, revealing how energy is partitioned between large-scale vortical flows and small-scale fluctuations, with implications for geophysical phenomena.

Contribution

It introduces a discretized model for the shallow water system and demonstrates how small scale fluctuations influence the selection of equilibrium states, extending phenomenological approaches.

Findings

Energy partition between large-scale vortices and small-scale waves is characterized.

Microcanonical equilibrium states depend on small scale fluctuations and system parameters.

Explicit equilibria are computed in the quasi-geostrophic limit, illustrating the theory.

Abstract

The aim of this paper is to use large deviation theory in order to compute the entropy of macrostates for the microcanonical measure of the shallow water system. The main prediction of this full statistical mechanics computation is the energy partition between a large scale vortical flow and small scale fluctuations related to inertia-gravity waves. We introduce for that purpose a discretized model of the continuous shallow water system, and compute the corresponding statistical equilibria. We argue that microcanonical equilibrium states of the discretized model in the continuous limit are equilibrium states of the actual shallow water system. We show that the presence of small scale fluctuations selects a subclass of equilibria among the states that were previously computed by phenomenological approaches that were neglecting such fluctuations. In the limit of weak height fluctuations, the equilibrium state can be interpreted as two subsystems in thermal contact: one subsystem corresponds to the large scale vortical flow, the other subsystem corresponds to small scale height and velocity fluctuations. It is shown that either a non-zero circulation or rotation and bottom topography are required to sustain a non-zero large scale flow at equilibrium. Explicit computation of the equilibria and their energy partition is presented in the quasi-geostrophic limit for the energy-enstrophy ensemble. The possible role of small scale dissipation and shocks is discussed. A geophysical application to the Zapiola anticyclone is presented.