  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.
