Second order perturbation theory of two-scale systems in fluid dynamics

Abstract

In the present paper we study fast-slow systems of coupled equations from fluid dynamics, where the fast component is perturbed by additive noise. We prove that, under a suitable limit of infinite separation of scales, the slow component of the system converges in law to a solution of the initial equation perturbed by transport noise, and subject to the influence of an additional Itô–Stokes drift. The resulting limit equation is very similar to turbulent models derived heuristically. Our results apply to the Navier–Stokes equations in dimensions $d=2,3$; the surface quasi-geostrophic equations in dimension $d=2$; and the primitive equations in dimensions $d=2,3$.