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Modulation-resonance mechanism for surface waves in a two-layer fluid system

Published online by Cambridge University Press:  25 July 2019

Shixiao W. Jiang Open the ORCID record for Shixiao W. Jiang [Opens in a new window] ,
Gregor Kovačič Open the ORCID record for Gregor Kovačič [Opens in a new window] ,
Douglas Zhou Open the ORCID record for Douglas Zhou [Opens in a new window]  and
David Cai
Shixiao W. Jiang*
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802-6400, USA
Gregor Kovačič*
Affiliation:
Mathematical Sciences Department, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180, USA
Douglas Zhou*
Affiliation:
School of Mathematical Sciences, MOE-LSC, and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, PR China
David Cai
Affiliation:
School of Mathematical Sciences, MOE-LSC, and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, PR China Courant Institute of Mathematical Sciences and Center for Neural Science, New York University, New York, NY 10012, USA NYUAD Institute, New York University Abu Dhabi, P.O. Box 129188, Abu Dhabi, United Arab Emirates
*
†Email addresses for correspondence: suj235@psu.edu, kovacg@rpi.edu, zdz@sjtu.edu.cn
†Email addresses for correspondence: suj235@psu.edu, kovacg@rpi.edu, zdz@sjtu.edu.cn
†Email addresses for correspondence: suj235@psu.edu, kovacg@rpi.edu, zdz@sjtu.edu.cn
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Abstract

We propose a Boussinesq-type model to study the surface/interfacial wave manifestation of an underlying, slowly varying, long-wavelength baroclinic flow in a two-layer, density-stratified system. The results of our model show numerically that, under strong nonlinearity, surface waves, with their typical wavenumber being the resonant $k_{res}$, can be generated locally at the leading edge of the underlying, slowly varying, long-wavelength baroclinic flow. Here, the resonant $k_{res}$ satisfies the class 3 triad resonance condition among two short-mode waves and one long-mode wave in which all waves propagate in the same direction. Moreover, when the slope of the baroclinic flow is sufficiently small, only one spatially localized large-amplitude surface wave packet can be generated at the leading edge. This localized surface wave packet becomes high in amplitude and large in group velocity after the interaction with its surrounding waves. These results are qualitatively consistent with various experimental observations including resonant surface waves at the leading edge of an internal wave. Subsequently, we propose a mechanism, referred to as the modulation-resonance mechanism, underlying these surface phenomena, based on our numerical simulations. The proposed modulation-resonance mechanism combines the linear modulation, ray-based, theory for the spatiotemporal asymmetric behaviour of surface waves and the nonlinear class 3 triad resonance theory for the energy focusing of surface waves around the resonant wavenumber $k_{res}$ in Fourier space.

JFM classification

Geophysical and Geological Flows: Baroclinic flows Waves/Free-surface Flows: Surface gravity waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 875 , 25 September 2019 , pp. 807 - 841
Copyright
© 2019 Cambridge University Press 

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