Nonlinear Dispersion in Wave-Current Interactions
Speaker
Darryl Holm, Imperial College
Time
2021-10-12 19:00 ~ 20:00, in 4 hours (Asia/Shanghai Time)
Venue
ZOOM-APP
ZOOM Info
ZOOM Conference ID: 856 1908 2397
Password: 098556
Abstract
Via a sequence of approximations of the Lagrangian in Hamilton's principle for dispersive nonlinear gravity waves we derive a hierarchy of Hamiltonian models for describing wave-current interaction (WCI) in nonlinear dispersive wave dynamics on free surfaces. A subclass of these WCI Hamiltonians admits \emph{emergent singular solutions} for certain initial conditions. These singular solutions are identified with a singular momentum map for left action of the diffeomorphisms on a semidirect-product Lie algebra. This semidirect-product Lie algebra comprises vector fields representing horizontal current velocity acting on scalar functions representing wave elevation. We use computational simulations to demonstrate the dynamical interactions of the emergent wavefront trains which are admitted by this special subclass of Hamiltonians for a variety of initial conditions. In particular, we investigate: (1) A variety of localised initial current configurations in still water whose subsequent propagation generates surface-elevation dynamics on an initially flat surface; and (2) The release of initially confined configurations of surface elevation in still water that generate dynamically interacting fronts of localised currents and wave trains. The results of these simulations show intricate wave-current interaction patterns whose structures are similar to those seen, for example, in Synthetic Aperture Radar (SAR) images taken from the space shuttle.
Bio
Darryl D Holm is a Professor of Applied Mathematics at Imperial College London and a Laboratory Fellow in the Computation and Computer Science Division at Los Alamos National Laboratory. His main research interests lie in nonlinear science – ranging from integrable to chaotic behavior, from solitons to turbulence, and then on to shape analysis. Most of his work is based on Lie symmetry reduction of Hamilton's principle, often by smooth invertible maps such as the relabelling invariance in Eulerian fluid dynamics. In these fields, he is particularly interested in emergent singular phenomena
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