Geodesic motion on the groups of diffeomorphisms with $H^1$ metric as geometric generalised Lagrangian mean theory

TL;DR

This paper develops a geometric framework for Lagrangian averaging of fluid flows on diffeomorphism groups, deriving Euler-$ abla$ and EPDiff equations as mean flow equations under specific assumptions.

Contribution

It generalizes the derivation of averaged fluid equations using intrinsic geometric methods on diffeomorphism groups, connecting them to Lagrangian mean theory.

Findings

Euler-$ abla$ equations as Lagrangian averages on volume-preserving diffeomorphisms

EPDiff equations as averages on full diffeomorphism group

Unified geometric derivation of fluid equations from mean Lagrangian

Abstract

Generalized Lagrangian mean theories are used to analyze the interactions between mean flows and fluctuations, where the decomposition is based on a Lagrangian description of the flow. A systematic geometric framework was recently developed by Gilbert and Vanneste (J. Fluid Mech., 2018) who cast the decomposition in terms of intrinsic operations on the group of volume preserving diffeomorphism or on the full diffeomorphism group. In this setting, the mean of an ensemble of maps can be defined as the Riemannian center of mass on either of these groups. We apply this decomposition in the context of Lagrangian averaging where equations of motion for the mean flow arise via a variational principle from a mean Lagrangian, obtained from the kinetic energy Lagrangian of ideal fluid flow via a small amplitude expansion for the fluctuations. We show that the Euler-$\alpha$ equations arise as Lagrangian averaged Euler equations when using the $L^2$-geodesic mean on the volume preserving diffeomorphism group of a manifold without boundaries, imposing a `Taylor hypothesis', which states that first order fluctuations are transported as a vector field by the mean flow, and assuming that fluctuations are statistically isotropic. Similarly, the EPDiff equations arise as the Lagrangian averaged Burgers' equations using the same argument on the full diffeomorphism group. These results generalize an earlier observation by Oliver (Proc. R. Soc. A, 2017) to manifolds in geometrically fully intrinsic terms.