Geometric approaches to Lagrangian averaging

Andrew D. Gilbert, Jacques Vanneste

Research output: Contribution to journalReview articlepeer-review

Abstract

Lagrangian averaging theories, most notably the Generalised Lagrangian Mean (GLM) theory of Andrews and McIntyre (1978a), have been primarily developed in Euclidean space and Cartesian coordinates. We re-interpret these theories using a geometric, coordinate-free formulation. This gives central roles to the flow map, its decomposition into mean and perturbation maps, and the momentum 1-form dual to the velocity vector. In this interpretation, the Lagrangian mean of any tensorial quantity is obtained by averaging its pull back to the mean configuration. Crucially, the mean velocity is not a Lagrangian mean in this sense. It can be defined in a variety of ways, leading to alternative Lagrangian mean formulations that include GLM and Soward and Roberts’s (2010) glm. These formulations share key features which the geometric approach uncovers. We derive governing equations both for the mean flow and for wave activities constraining the dynamics of the pertubations. The presentation focusses on the Boussinesq model for inviscid rotating stratified flows and reviews the necessary tools of differential geometry.
Original languageEnglish
JournalAnnual Review of Fluid Mechanics
Early online date15 Aug 2024
DOIs
Publication statusE-pub ahead of print - 15 Aug 2024

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