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On Bragg resonances and wave triad interactions in two-layered shear flows

Published online by Cambridge University Press:  25 March 2019

Raunak Raj Open the ORCID record for Raunak Raj [Opens in a new window]  and
Anirban Guha Open the ORCID record for Anirban Guha [Opens in a new window]
Raunak Raj
Affiliation:
Environmental and Geophysical Fluids Group, Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, U.P. 208016, India
Anirban Guha*
Affiliation:
Environmental and Geophysical Fluids Group, Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, U.P. 208016, India
*
†Email address for correspondence: [email protected]
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Abstract

The standard resonance conditions for Bragg scattering as well as weakly nonlinear wave triads have been traditionally derived in the absence of any background velocity. In this paper, we have studied how these resonance conditions get modified when uniform, as well as various piecewise linear velocity profiles, are considered for two-layered shear flows. Background velocity can influence the resonance conditions in two ways: (i) by causing Doppler shifts, and (ii) by changing the intrinsic frequencies of the waves. For Bragg resonance, even a uniform velocity field changes the resonance condition. Velocity shear strongly influences the resonance conditions since, in addition to changing the intrinsic frequencies, it can cause unequal Doppler shifts between the surface, pycnocline and the bottom. Using multiple scale analysis and Fredholm alternative, we analytically obtain the equations governing both the Bragg resonance and the wave triads. We have also extended the higher-order spectral method, a highly efficient computational tool usually used to study triad and Bragg resonance problems, to incorporate the effect of piecewise linear velocity profile. A significant aspect, both on the theoretical and numerical fronts, has been extending the potential flow approximation, which is the basis of the study of these kinds of problems, to incorporate piecewise constant background shear.

JFM classification

Geophysical and Geological Flows: Coastal engineering Instability: Nonlinear instability Geophysical and Geological Flows: Stratified flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 867 , 25 May 2019 , pp. 482 - 515
Copyright
© 2019 Cambridge University Press 

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