Statistical properties of an enstrophy conserving discretisation for the stochastic quasi-geostrophic equation
For researchers in stochastic fluid dynamics and numerical methods, this work provides a discretisation that conserves key statistical moments, enabling more accurate long-time simulations.
The authors present a conforming finite element discretisation for the stochastic quasi-geostrophic equation that preserves mean potential vorticity and enstrophy, and show that its statistical properties agree with the Gibbs distribution under a wide range of setups.
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.