Variational closures for composite homogenised fluid flows

Abstract

Homogenisation theory has seen recent applications in deriving stochastic transport models for fluid dynamics. In this work, we first derive the stochastic Lagrange-to-Euler map that underpins stochastic transport noise in fluid dynamics as the homogenisation limit of a parameterised flow map decomposing into rapidly fluctuating and slow components. Specifically, we prove convergence of this parameterised flow 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 converges to a stochastic flow of diffeomorphisms that transforms the full flow dynamics into an SDE-governed stochastic flow through composition, while the slow component requires closure. Our second contribution formulates two distinct variational closures for the slow component of the homogenised flow that exploit the composite structure of the stochastic flow. For the first closure, the critical points of a new variational principle satisfy a system of random-coefficient PDEs, which can be transformed into a system of stochastic PDEs via the coadjoint action of the stochastic flow map obtained from homogenising the fluctuating component. We show that these equations coincide with the stochastic Euler-Poincaré equations previously derived in Holm, Proc. Royal Soc. (2015). For the second closure, we modify the assumptions on the slow component and the associated variational principle to derive averaged models inspired by previous work on mean flow dynamics such as the Generalised Lagrangian Mean.