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Geodesic motion on the groups of diffeomorphisms with $H^1$ metric as geometric generalised Lagrangian mean theory. (arXiv:1810.01377v1 [math-ph])
来源于:arXiv
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 diffeomorphism 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
diffeomo 查看全文>>