Statistical properties of an enstrophy conserving discretisation for the stochastic quasi-geostrophic equation

TL;DR

This paper introduces a finite element discretisation for the stochastic quasi-geostrophic equation that conserves key statistical properties, aligning well with theoretical Gibbs distributions across various scenarios.

Contribution

It presents a novel finite element discretisation that preserves enstrophy and mean potential vorticity for the stochastic quasi-geostrophic equation, and analyzes its statistical mechanics.

Findings

Discretisation conserves first two moments of potential vorticity.

Statistical properties of the discretisation match Gibbs distribution.

Agreement observed across diverse set-ups.

Abstract

A framework of variational principles for stochastic fluid dynamics was presented by Holm (2015), and these stochastic equations were also derived by Cotter et al. (2017). We present a conforming finite element discretisation for the stochastic quasi-geostrophic equation that was derived from this framework. The discretisation preserves the first two moments of potential vorticity, i.e. the mean potential vorticity and the enstrophy. Following the work of Dubinkina and Frank (2007), who investigated the statistical mechanics of discretisations of the deterministic quasi-geostrophic equation, we investigate the statistical mechanics of our discretisation of the stochastic quasi-geostrophic equation. We compare the statistical properties of our discretisation with the Gibbs distribution under assumption of these conserved quantities, finding that there is agreement between the statistics under a wide range of set-ups.