A consistent stochastic large-scale representation of the Navier-Stokes equations

Abstract: In this paper we analyze the theoretical properties of a stochastic representation of the incompressible Navier-Stokes equations defined in the framework of the modeling under location uncertainty (LU). This setup built from a stochastic version of the Reynolds transport theorem incorporates a so-called transport noise and involves several specific additional features such as a large scale diffusion term, akin to classical subgrid models, and a modified advection term arising from the spatial inhomogeneity of the small-scale velocity components. This formalism has been numerically evaluated in a series of studies with a particular interest on geophysical flows approximations and data assimilation. In this work we focus more specifically on its theoretical analysis. We demonstrate, through classical arguments, the existence of martingale solutions for the stochastic Navier-Stokes equations in LU form. We show they are pathwise and unique for 2D flows. We then prove that if the noise intensity goes to zero, these solutions converge, up to a subsequence in dimension $3$, to a solution of the deterministic Navier-Stokes equation. similarly to the grid convergence property of well established large-eddies simulation strategies, this result allows us to give some guarantee on the interpretation of the LU Navier-Stokes equations as a consistent large-scale model of the deterministic Navier-Stokes equation.