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SUMMARY:Karima Khusnutdinova (University Loughborough)
DTSTART:20200610T110000Z
DTEND:20200610T120000Z
DTSTAMP:20260423T005845Z
UID:mmandim/3
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/mmandim/3/">
 Near-integrable models for long surface and internal ring waves in stratif
 ied shear flows</a>\nby Karima Khusnutdinova (University Loughborough) as 
 part of Mathematical models and integration methods\n\n\nAbstract\nIn this
  talk I will first overview some general results concerning the effects of
  the parallel shear flow on long weakly-nonlinear surface and internal rin
 g waves in a stratified fluid (e.g.\, oceanic internal waves generated in 
 narrow straits and river-sea interaction zones)\, generalising the results
  for surface waves in a homogeneous fluid [1]. We showed that despite the 
 clashing geometries of the waves and the shear flow\, there exists a linea
 r modal decomposition (separation of variables) in the far-field set of Eu
 ler equations describing the waves in a stratified fluid\, more complicate
 d than the known decomposition for plane waves [2\,3]. We used it to descr
 ibe the wavefronts of surface and internal waves\, and to derive a 2D cyli
 ndrical Korteweg - de Vries (cKdV)-type model for the amplitudes of the wa
 ves. The distortion of the wavefronts is described explicitly by construct
 ing the singular solution (envelope of the general solution) of a respecti
 ve nonlinear first-order differential equation. \n\nNext\, we consider a t
 wo-layer fluid with a rather general depth-dependent upper-layer current (
 e.g. a river inflow\, or a wind-generated current). In the rigid-lid appro
 ximation\, we 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 wave in different directions in terms 
 of the hypergeometric function [4]. This allows us to obtain an analytical
  description of the wavefronts and vertical structure of the ring waves fo
 r a large family of the current profiles and to illustrate their dependenc
 e on the  density jump and the type and the strength of the current. We wi
 ll also discuss a 2D generalisation of the long-wave instability criterion
  for plane interfacial waves on a piecewise-constant current [4]\, which o
 n physical level manifests itself in the counter-intuitive squeezing of th
 e wavefront of the interfacial ring wave.\n\nREFERENCES\n\n1. R.S. Johnson
 \, Ring waves on the surface of shear flows: a linear and nonlinear theory
 \, J. Fluid Mech.\, 215\, 1638-1660 (1990).\n\n2. K.R. Khusnutdinova\, X. 
 Zhang\, Long ring waves in a stratified fluid over a shear flow\, J. Fluid
  Mech.\, 794\, 17-44 (2016).\n\n3. K.R. Khusnutdinova\, X. Zhang\, Nonline
 ar ring waves in a two-layer fluid\, Physica D\, 333\, 208-221 (2016).\n\n
 4. K.R. Khusnutdinova\, Long internal ring waves in a two-layer fluid with
  an upper-layer current\, submitted (2020). \n\n5. L.V. Ovsyannikov\, Two-
 layer 'shallow water' model\, J. Appl. Math. Tech. Phys. 20\, 127-135 (197
 9).\n\nZoom limk: https://us04web.zoom.us/j/75476385312?pwd=dTU0U0VQMTN5Vl
 FOMVVHNmhaS1pCZz09\n
LOCATION:https://researchseminars.org/talk/mmandim/3/
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