Towards compatible finite element discretizations of stochastic rotating shallow water equations

We introduce a stochastic representation of the rotating shallow water equations and a corresponding structure preserving finite element discretization in Firedrake. The stochastic flow model follows from using a stochastic transport principle and a decomposition of the fluid flow into a large-scale component and a noise term that models the unresolved flow components. Similarly to the deterministic case, this stochastic model (denoted as modeling under location uncertainty (LU)) conserves the global energy of any realization. Consequently, it permits us to generate an ensemble of physically relevant random simulations with a good trade-off between the representation of the model error and the ensemble's spread. Applying a compatible finite element discretisation of the deterministic part of the equations combined with a standard weak finite element discretization of the stochastic terms, the resulting stochastic scheme preserves (spatially) the total energy. To address the enstrophy accumulation at the grid scale, we applied an anticipated potential vorticity method (APVM) to stabilize the stochastic scheme. Using this setup, we compare different realizations of noise parametrizations in the context of geophysical flow phenomena and study potential pathways to fully energy preserving stochastic discretizations.