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Nelin. Dinam., 2011, Volume 7, Number 2, Pages 257–282 (Mi nd258)  

Resonant three-wave interaction of waves having a common critical layer

S. M. Churilov

Institute of Solar-Terrestrial Physics

Abstract: Within the context of the weakly nonlinear approach, the leading nonlinear contribution to the development of unstable disturbances in shear flows should be made by resonant three-wave interaction, i.e., the interaction of triplets of such waves that have a common critical layer (CL), and their wave vectors form a triangle. Surprisingly, the subharmonic resonance proves to be the only such interaction that has been studied so far. The reason for this is that in many cases, the requirement of having a common CL produces too rigid selection of waves which can participate in the interaction. We show that in a broad spectral range, Holmboe waves in sharply stratified shear flows can have a common CL, and examine the evolution of small ensembles consisting of several interrelated triads of those waves. To do this, the evolution equations are derived which describe the three-wave interaction and have the form of nonlinear integral equations. Analytical and numerical methods are both used to find their solutions in different cases, and it is shown that at the nonlinear stage disturbances increase, as a rule, explosively.

Keywords: shear flow, sharp density stratification, three-wave interactions, critical layer.

Full text: PDF file (1086 kB)
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UDC: 551.465:551.511
MSC: 76B70, 76E05, 76E20, 76E30
Received: 12.05.2011
Revised: 20.05.2011

Citation: S. M. Churilov, “Resonant three-wave interaction of waves having a common critical layer”, Nelin. Dinam., 7:2 (2011), 257–282

Citation in format AMSBIB
\Bibitem{Chu11}
\by S.~M.~Churilov
\paper Resonant three-wave interaction of waves having a common critical layer
\jour Nelin. Dinam.
\yr 2011
\vol 7
\issue 2
\pages 257--282
\mathnet{http://mi.mathnet.ru/nd258}


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  • Нелинейная динамика
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