From additive to transport noise in 2D fluid dynamics

Abstract

Additive noise in Partial Differential equations, in particular those of fluid mechanics, has relatively natural motivations. The aim of this work is showing that suitable multiscale arguments lead rigorously, from a model of fluid with additive noise, to transport type noise. The arguments apply both to small-scale random perturbations of the fluid acting on a large-scale passive scalar and to the action of the former on the large scales of the fluid itself. Our approach consists in studying the (stochastic) characteristics associated to small-scale random perturbations of the fluid, here modelled by stochastic 2D Euler equations with additive noise, and their convergence in the infinite scale separation limit.

Stochastic Primitive Equations with Horizontal Viscosity and Diffusivity

“…Of particular interest in fluid dynamics and also for geophysical flows is a noise of transport type which appears naturally when stochastic models are derived from Hamiltonian principles as proposed in [33] (see also [4] for a brief description) and yield a physically relevant randomization [2] with energy conservation. Recently, the importance of transport noise was discussed in the connection with unresolved small scales, see [17,18] and the references therein.…”

Enhanced dissipation for stochastic Navier-Stokes equations with transport noise

“…According to arguments by separation of scales [32,33], stochastic 2D fluid equations driven by multiplicative transport noise in Stratonovich form are suitable models in fluid dynamics, see also [37,17] for variational considerations. Here, the transport noise is assumed to be spatially divergence free, replacing the incompressible flows mentioned above.…”