Nonlinear dispersion in wave-current interactions

Via a sequence of approximations the Lagrangian in Hamilton's principle for dispersive nonlinear gravity waves we derive hierarchy Hamiltonian models describing wave-current interaction (WCI) wave dynamics on free surfaces. A subclass these WCI Hamiltonians admits emergent singular solutions certain initial conditions. These solutions are identified with momentum map left action diffeomorphisms semidirect-product Lie algebra. This algebra comprises vector fields representing horizontal current velocity acting scalar functions elevation. We use computational simulations to demonstrate dynamical interactions emergent wavefront trains which admitted by this special variety style='text-indent:20px;'>In paper, investigate: style='text-indent:20px;'>(1) localised configurations still water whose subsequent propagation generates surface-elevation an initially flat surface; and style='text-indent:20px;'>(2) The release confined surface elevation that generate dynamically interacting fronts currents trains. results show intricate patterns structures similar those seen, example, Synthetic Aperture Radar (SAR) images taken from space shuttle.