Variational closures for composite homogenised fluid flows

Abstract

The Stochastic Advection by Lie Transport is a variational formulation of stochastic fluid dynamics introduced to model the effects of unresolved scales, whilst preserving the geometric structure of ideal fluid flows. In this work, we show that the SALT equations can arise from the decomposition of the fluid flow map into its mean and fluctuating components. The fluctuating component is realised as a prescribed stochastic diffeomorphism that introduces stochastic transport into the system and we construct it using homogenisation theory. The dynamics of the mean component are derived from a variational principle utilising particular forms of variations that preserve the composite structure of the flow. Using a new variational principle, we show that SALT equations can arise from random Lagrangians and are equivalent to random coefficient PDEs. We also demonstrate how to modify the composite flow and the associated variational principle to derive models inspired by the Lagrangian Averaged Euler-Poincare (LAEP) theory.