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Annual Review of Fluid Mechanics

Volume 57, 2025

Geometric Approaches to Lagrangian Averaging

  • Andrew D. Gilbert1 and Jacques Vanneste2
  • 1Department of Mathematics and Statistics, University of Exeter, Exeter, United Kingdom 2School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh, United Kingdom; email: [email protected]
  • Vol. 57:117-140 (Volume publication date January 2025)
  • First published as a Review in Advance on August 15, 2024
  • Copyright © 2025 by the author(s).
    This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. See credit lines of images or other third-party material in this article for license information.

Abstract

Lagrangian averaging theories, most notably the generalized Lagrangian mean (GLM) theory of Andrews and McIntyre, have been primarily developed in Euclidean space and Cartesian coordinates. We reinterpret 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 volume-preserving version. These formulations share key features that the geometric approach uncovers. We derive governing equations both for the mean flow and for wave activities constraining the dynamics of the perturbations. The presentation focuses on the Boussinesq model for inviscid rotating stratified flows and reviews the necessary tools of differential geometry.

Keyword(s): flow map, generalized Lagrangian mean, pseudomomentum, wave activity, wave–mean flow interactions
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2025-01-22
2025-06-17
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Geometric Approaches to Lagrangian Averaging
Annual Review of Fluid Mechanics 57, 117 (2025); https://doi.org/10.1146/annurev-fluid-030524-095913
/content/journals/10.1146/annurev-fluid-030524-095913
/content/journals/10.1146/annurev-fluid-030524-095913
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