VIRTUAL SEMINAR SERIES -- INTEGRABLE SYSTEMS: Ring waves, singular solutions and hypergeometric functions

Karima Khusnutdinova (Loughborough University)

Wednesday 20th January 13:30-14:30 Zoom seminar hosted by ICMS

Abstract

There exists a linear modal decomposition (separation of variables) in the far-field set of Euler equations describing ring waves in a stratified fluid over a parallel shear flow, more complicated than the known decomposition for plane waves. We used it to derive a 2+1D cylindrical Korteweg - de Vries (cKdV)-type model for the amplitudes of the waves, generalising R.S. Johnson's result for surface waves in a homogeneous fluid. Here, we consider a two-layered fluid with a rather general family of depth-dependent upper-layer currents (e.g. a river inflow, an exchange flow in a strait, or a wind-generated current). In the rigid-lid approximation, we analyse the set of modal equations and find the necessary singular solution of the nonlinear first-order ordinary differential equation responsible for the adjustment of the speed of the long interfacial ring waves in different directions (which can be viewed as a generalisation of the linear dispersion relation) in terms of the hypergeometric function. This allows us to obtain an analytical description of the wavefronts and modal functions of these 3D waves. We will also discuss a ring waves' generalisation of L.V. Ovsyannikov's long-wave instability criterion for plane interfacial waves on a piecewise-constant current, which on physical level manifests itself in the counter-intuitive squeezing of the wavefront of the interfacial ring wave.

 

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