Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics.

C J Cotter,G A Gottwald,D D Holm

doi:10.1098/rspa.2017.0388

Abstract

In Holm (Holm 2015 Proc. R. Soc. A 471, 20140963. (doi:10.1098/rspa.2014.0963)), stochastic fluid equations were derived by employing a variational principle with an assumed stochastic Lagrangian particle dynamics. Here we show that the same stochastic Lagrangian dynamics naturally arises in a multi-scale decomposition of the deterministic Lagrangian flow map into a slow large-scale mean and a rapidly fluctuating small-scale map. We employ homogenization theory to derive effective slow stochastic particle dynamics for the resolved mean part, thereby obtaining stochastic fluid partial equations in the Eulerian formulation. To justify the application of rigorous homogenization theory, we assume mildly chaotic fast small-scale dynamics, as well as a centring condition. The latter requires that the mean of the fluctuating deviations is small, when pulled back to the mean flow.

Highlights

When studying complex turbulent flows or astrophysical and geophysical fluids, in which physical processes occur over a wide range of spatial and temporal scales, we are faced with the inevitable problem that our limited computational resources will eventually force us to under-represent processes that occur below certain temporal and spatial scales
The diffusive limit we have described relies on homogenization theory
We made a chaoticity assumption, whereby the temporal derivative of ζ evolves according to some unspecified chaotic dynamics

Summary

Introduction

When studying complex turbulent flows or astrophysical and geophysical fluids, in which physical processes occur over a wide range of spatial and temporal scales, we are faced with the inevitable problem that our limited computational resources will eventually force us to under-represent processes that occur below certain temporal and spatial scales. Stochastic partial differential fluid equations have been proposed to model the influence of unresolved scales on the resolved scales of interest [1,2,3,4,5] These novel approaches introduce stochasticity into the flow map for the Lagrangian particle trajectories, the noise in the Lagrange-to-Euler map produces a random Eulerian vector field. The aim of this paper is to establish conditions under which the stochastic vector field Ansatz in equation (2.18) may be derived by applying the method of homogenization [7,13], for the purpose of gaining insight into the situations where such a model can be used in fluid dynamics

Stochastic Lagrangian multi-time dynamics

Diffusive limit of stochastic Lagrangian dynamics

Summary and discussion