GLM) theor

In this talk I will outline the geometric view of ideal fluid mechanics, as set out by V.I. Arnold in seminal papers in the 1960s. We will then look at applications to Lagrangian averaging in fluid flows, in particular the generalised Lagrangian mean (GLM) theory of Andrews & McIntyre from the 1970s, with recent developments by Soward & Roberts. This GLM framework is used to describe fluctuation and wave based phenomena in fluid flows in a way that preserves key structural elements of the ideal fluid equations. The original formulation of GLM theory, 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. It is also not immediately apparent why the theory works, and what choices may be made in setting up a system of Lagrangian averaging. In our recent work we have developed 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.

References:
Andrews, D.G. and McIntyre, M.E., An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech. 1978a, 89, 609–646.
Arnold, V.I. and Khesin, B.A., Topological Methods in Hydrodynamics, in Applied Mathematical Sciences, vol. 125, 1998 (New York: Springer-Verlag).
Gilbert, A.D. and Vanneste, J.,Geometric generalised Lagrangian-mean theories. J. Fluid Mech. 2018, 839, 95–134.
Soward, A.M. and Roberts, P.H., The hybrid Euler–Lagrange procedure using an extension of Moffatt’s method. J. Fluid Mech. 2010, 661, 45–72.

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Zoom link: https://us02web.zoom.us/j/81583752461?pwd=c2xXN0hoQVBVc2JLNmNYRHkxTzV4UT09

Meeting ID: 815 8375 2461
Passcode: 495149