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A wave interaction approach to studying non-modal homogeneous and stratified shear instabilities

Published online by Cambridge University Press:  18 August 2014

Anirban Guha  and
Gregory A. Lawrence
Anirban Guha*
Affiliation:
Institute of Applied Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada Department of Civil Engineering, University of British Columbia, Vancouver, BC, V6T 1Z4, Canada
Gregory A. Lawrence
Affiliation:
Department of Civil Engineering, University of British Columbia, Vancouver, BC, V6T 1Z4, Canada
*
†Present address: Atmospheric and Oceanic Sciences, University of California Los Angeles, Los Angeles, CA 90095-1565, USA. Email address for correspondence: [email protected]
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Abstract

Homboe (Geophys. Publ., vol. 24, 1962, pp. 67–112) postulated that resonant interaction between two or more progressive, linear interfacial waves produces exponentially growing instabilities in idealized (broken-line profiles), homogeneous or density-stratified, inviscid shear layers. Here we have generalized Holmboe’s mechanistic picture of linear shear instabilities by (i) not initially specifying the wave type, and (ii) providing the option for non-normal growth. We have demonstrated the mechanism behind linear shear instabilities by proposing a purely kinematic model consisting of two linear, Doppler-shifted, progressive interfacial waves moving in opposite directions. Moreover, we have found a necessary and sufficient (N&S) condition for the existence of exponentially growing instabilities in idealized shear flows. The two interfacial waves, starting from arbitrary initial conditions, eventually phase-lock and resonate (grow exponentially), provided the N&S condition is satisfied. The theoretical underpinning of our wave interaction model is analogous to that of synchronization between two coupled harmonic oscillators. We have re-framed our model into a nonlinear autonomous dynamical system, the steady-state configuration of which corresponds to the resonant configuration of the wave interaction model. When interpreted in terms of the canonical normal-mode theory, the steady-state/resonant configuration corresponds to the growing normal mode of the discrete spectrum. The instability mechanism occurring prior to reaching steady state is non-modal, favouring rapid transient growth. Depending on the wavenumber and initial phase-shift, non-modal gain can exceed the corresponding modal gain by many orders of magnitude. Instability is also observed in the parameter space which is deemed stable by the normal-mode theory. Using our model we have derived the discrete spectrum non-modal stability equations for three classical examples of shear instabilities: Rayleigh/Kelvin–Helmholtz, Holmboe and Taylor–Caulfield. We have shown that the N&S condition provides a range of unstable wavenumbers for each instability type, and this range matches the predictions of the normal-mode theory.

JFM classification

Instability: Instability Wakes/Jets: Shear layers Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 755 , 25 September 2014 , pp. 336 - 364
Copyright
© 2014 Cambridge University Press 

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