Geometric generalised Lagrangian mean theories

Abstract

Many fluctuation-driven phenomena in fluids can be analysed effectively using the generalised Lagrangian mean (GLM) theory of Andrews & McIntyre (1978). This theory relies on particle-following averaging to incorporate the constraints imposed by the material conservations. It relies implicitly on an Euclidean structure; as a result, it does not have a geometrically intrinsic interpretation and suffers from undesirable features, including the divergence of the Lagrangian-mean velocity for incompressible fluids. Motivated by this, we develop a geometric generalisation of GLM that we formulate intrinsically. The theory applies to arbitrary Riemannian manifolds; it also establishes a clear distinction between results that stem directly from geometric consistency and those that depend on particular choices. We show that the Lagrangian mean momentum -- the average of the pull-back of the momentum one-form -- obeys a simple equation which guarantees the conservation of Kelvin's circulation, irrespective of the mean-flow definition. We discuss four possible definitions of the mean flow: a direct extension of standard GLM, a definition based on optimal transportation, a definition based on a geodesic distance in the group of volume-preserving diffeomorphisms, and the glm definition proposed by Soward & Roberts (2010). Assuming small-amplitude perturbations, we carry out order-by-order calculations to obtain explicit expressions for the mean flow and pseudomomentum at leading order. We also show how the wave-action conservation of GLM extends to the geometric setting. To make the paper self-contained, we introduce the tools of differential geometry and main ideas of geometric fluid dynamics on which we rely. We mostly focus on the Euler equations for incompressible inviscid fluids but sketch out extensions to the rotating-stratified Boussinesq, compressible Euler and magnetohydrodynamic equations.