Stochastic compressible Navier-Stokes equations under location uncertainty and its approximations for ocean modelling

Abstract: The aim of this paper is to provide a stochastic version under location uncertainty of the compressible Navier-Stokes equations. The modelling under location uncertainty setting is used here to derive a physically consistent stochastic dynamical system for compressible flows. It relies on an extended stochastic version of the Reynolds transport theorem involving stochastic source terms. In a similar way as in the deterministic case, this conservation theorem is applied to density, momentum and total energy in order to obtain a transport equation of the primitive variables, i.e. density, velocity and temperature. For the modelling of ocean dynamics, the transport of mass fraction of species, such as salinity, is also considered. We show that performing low Mach and Boussinesq approximations to this more general set of equations allows us to recover previous versions of incompressible stochastic Navier-Stokes equations and the stochastic Boussinesq equations, respectively. Finally, we provide some research directions on the use of this general set of equations in the perspective of relaxing the Boussinesq and hydrostatic approximation for ocean modelling.