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Vortex waves in deep water: Lagrange approach

Presented by: 
Anatoly Abrashkin Higher School of Economics, Moscow
Date: 
Thursday 10th August 2017 - 17:00 to 18:00
Venue: 
INI Seminar Room 1
Abstract: 
A lecture consists of two parts. The first one deals with a theory of weakly nonlinear vortex waves. The vorticity is set in the series expansion in the small parameter of wave steepness. Each term of this row is an arbitrary function of the vertical Lagrange coordinate. We study different types of the waves: the stationary waves on shear flow, the standing vortex waves and the spatial vortex waves in the low vorticity fluid. The perturbation theory up to the third order is analyzed. The nonlinear Schrödinger (NLS) equation describing weakly nonlinear wave packets in an infinity-depth fluid with non-uniform vorticity is obtained. The vorticity is assumed to be an arbitrary function of both Lagrangian coordinates and quadratic in the small parameter proportional to the wave’s steepness. The effects of vorticity are manifested in a shift of the wavenumber of the carrier wave and a changing of the coefficient in nonlinear term of the NLS equation. The modulated Gouyon waves are studied. There is a special case of the vortex waves for which the resulting non-linearity in the NLS equation vanishes. The Gerstner wave belongs among them. The second part of the lecture presents the theory of strongly nonlinear waves. A vortex model of a rogue wave formation at the background of uniform waves is proposed. It based on an exact analytical solution of equations of 2D hydrodynamics of an ideal incompressible fluid. A unique feature of flows of this class is the dependence of complex coordinate of liquid particle’s motion on two functions that may be arbitrary to a large extent. As a consequence the model may be used for the analysis of different forms of surface pressure as well as of liquid vorticity, i.e. when taking into account both these factors of air flow impact on the surface waves simultaneously. A process of formation of a rogue wave in the field of the Gerstner wave is studied. The physical parameters of the rogue wave and feasibility of the proposed scenario are disc.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons