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Lagrangian reduction and wave mean flow interaction. (arXiv:2301.04121v2 [math.AP] UPDATED)

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How does one derive models of dynamical feedback effects in multiscale,
multiphysics systems such as wave mean flow interaction (WMFI)? We shall
address this question for hybrid dynamical systems, whose motion can be
expressed as the composition of two or more Lie-group actions. Hybrid systems
abound in fluid dynamics. Examples include: the dynamics of complex fluids such
as liquid crystals; wind-driven waves propagating with the currents moving on
the sea surface; turbulence modelling in fluids and plasmas; and
classical-quantum hydrodynamic models in molecular chemistry. From among these
examples, the motivating question in this paper is: How do wind-driven waves
produce ocean surface currents? The paper first summarises the geometric
mechanics approach for deriving hybrid models of multiscale, multiphysics
motions in ideal fluid dynamics. It then illustrates this approach for WMFI in
the examples of 3D WKB waves and 2D wave amplitudes governed by the nonlinear
Schr\"odinger (NLS) equation propagating in the frame of motion of an ideal
incompressible inhomogeneous Euler fluid flow. The results for these examples
tell us that the fluid flow in WMFI does not create waves. However, feedback in
the opposite direction is possible, since 3D WKB and 2D NLS wave dynamics can
indeed create circulatory fluid flow.

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