## Geometric generalised Lagrangian mean theories

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##### Date

2018-01-25##### Author

Gilbert, AD

Vanneste, J

##### Date issued

2018-01-25

##### Journal

Journal of Fluid Mechanics

##### Type

Article

##### Language

en

##### Publisher

Cambridge University Press (CUP)

##### Rights

© 2018 Cambridge University Press.

##### Embargo

2018-07-25

##### Reason for embargo

Under embargo until 25 July 2018 in compliance with publisher policy.

##### Abstract

Many fluctuation-driven phenomena in fluids can be analysed effectively using the generalised
Lagrangian mean (GLM) theory of Andrews & McIntyre (1978a). This finiteamplitude
theory relies on particle-following averaging to incorporate the constraints
imposed by the material conservation of certain quantities in inviscid regimes. Its original
formulation, in terms of Cartesian coordinates, relies implicitly on an assumed Euclidean
structure; as a result, it does not have a geometrically intrinsic, coordinate-free interpretation
on curved manifolds, and suffers from undesirable features. Motivated by this,
we develop a geometric generalisation of GLM that we formulate intrinsically using
coordinate-free notation. One benefit is that the theory applies to arbitrary Riemannian
manifolds; another is that it establishes a clear distinction between results that stem
directly from geometric consistency and those that depend on particular choices.
Starting from a decomposition of an ensemble of flow maps into mean and perturbation,
we define the Lagrangian-mean momentum as the average of the pull-back of the momentum
one-form by the perturbation flow maps. We show that it obeys a simple equation
which guarantees the conservation of Kelvin’s circulation, irrespective of the specific
definition of the mean flow map. The Lagrangian-mean momentum is the integrand in
Kelvin’s circulation and distinct from the mean velocity (the time derivative of the mean
flow map) which advects the contour of integration. A pseudomomentum consistent with
that in GLM can then be defined by subtracting the Lagrangian-mean momentum from
the one-form obtained from the mean velocity using the manifold’s metric.
The definition of the mean flow map is based on choices made for reasons of convenience
or aesthetics. We discuss four possible definitions: 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 orderby-order
calculations to obtain explicit expressions for the mean velocity and Lagrangianmean
momentum 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 in some detail the tools of differential
geometry and main ideas of geometric fluid dynamics on which we rely. These include
variational formulations which we use for alternative derivations of some key results.
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. We illustrate our results with an application to the interaction of
inertia-gravity waves with balanced mean flows in rotating–stratified fluids.

##### Description

This is the author accepted manuscript. The final version is available from Cambridge University Press (CUP) via the DOI in this record

##### Citation

Vol. 839, pp. 95-134.

##### ISSN

0022-1120