Abstract
Homboe (Geophys. Publ., vol. 24, 1962, pp. 67112) postulated that resonant interaction between two or more progressive, linear interfacial waves produces exponentially growing instabilities in idealized (brokenline profiles), homogeneous or densitystratified, 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 nonnormal growth. We have demonstrated the mechanism behind linear shear instabilities by proposing a purely kinematic model consisting of two linear, Dopplershifted, 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 phaselock 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 reframed our model into a nonlinear autonomous dynamical system, the steadystate configuration of which corresponds to the resonant configuration of the wave interaction model. When interpreted in terms of the canonical normalmode theory, the steadystate/resonant configuration corresponds to the growing normal mode of the discrete spectrum. The instability mechanism occurring prior to reaching steady state is nonmodal, favouring rapid transient growth. Depending on the wavenumber and initial phaseshift, nonmodal 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 normalmode theory. Using our model we have derived the discrete spectrum nonmodal stability equations for three classical examples of shear instabilities: Rayleigh/KelvinHelmholtz, Holmboe and TaylorCaulfield. 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 normalmode theory.
Original language  English 

Pages (fromto)  336364 
Number of pages  29 
Journal  Journal of Fluid Mechanics 
Volume  755 
Early online date  18 Aug 2014 
DOIs  
Publication status  Published  25 Sep 2014 
Keywords
 instability
 shear layers
 waves/freesurface flows
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Guha, Anirban
 Civil Engineering  Lecturer in Fluid Mechanics (Teaching and Research)
Person: Academic