Variational closures for composite homogenised fluid flows

Abstract: Homogenisation theory has seen recent application in the derivation of stochastic transport models for fluids dynamics. In this work, we first construct the Lagrange-to-Euler map of an ideal fluid flow, modelled by a stochastic flow of diffeomorphism, as the deterministic homogenized limit of a parameterized flow map that decomposes into rapidly fluctuating and slow components. Specifically, we prove the convergence of this parameterized map to a scale-separated limit under the assumptions of a weak-invariance principle for the rapidly fluctuating component and path continuity for the slow component. In this limit, the rapidly fluctuating component becomes a stochastic flow of diffeomorphism that can introduce stochastic transport into the dynamics of the slow component, which remains to be closed. The second contribution of this work formulates two distinct types of variational closures for the slow component of the homogenized flow whilst utilising the composite structure of the full stochastic flow. In one case, the critical points of the new principle satisfy a system of random partial differential equations equivalent to a system of stochastic partial differential equations (SPDEs) under stochastic transport via a transformation by the stochastic flow. Additionally, we show these equations are stochastic Euler-Poincar\'e equations previously derived in Holm, Proc. Royal Soc. (2015). In the other case, we modify the assumptions on the slow component of composite flow and the associated variational principle to derive averaged models inspired by previous work such as Generalised Lagrangian Mean (GLM) and Lagrangian Average Euler Poincar\'e (LAEP).