M. Oliver and S. Vasylkevych,
Geodesic motion on groups of diffeomorphisms with \(H^1\) metric as geometric generalised Lagrangian mean theory,
Geophys. Astrophys. Fluid Dyn. 113 (2019), 466-490.

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 diffeomorphisms 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 nearly isotropic. Similarly, the EPDiff equations arise as the Lagrangian averaged Burgers' equations using the same argument on the full diffeomorphism group. A serious drawback of this construction is that the assumptions of Lie transport of the fluctuation vector field and isotropy of fluctuations cannot persist except for an asymptotically vanishing interval of time. To remedy the problem of persistence of isotropy, we suggest adding strong mean-reverting stochastic term to the Taylor hypothesis and identify a scaling regime in which the inclusion of the stochastic term leads to the same averaged equations up to a constant.

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