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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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References

Baines, P. G. & Mitsudera, H. 1994 On the mechanism of shear flow instabilities. J. Fluid Mech. 276, 327–342.Google Scholar
Bakas, N. A. & Ioannou, P. J. 2009 Modal and nonmodal growths of inviscid planar perturbations in shear flows with a free surface. Phys. Fluids 21 (2), 024102.Google Scholar
Balmforth, N. J. & Sassi, R. 2000 A shocking display of synchrony. Physica D 143, 21–55.Google Scholar
Bretherton, F. P. 1966 Baroclinic instability and the short wavelength cut-off in terms of potential vorticity. Q. J. R. Meteorol. Soc. 92 (393), 335–345.Google Scholar
Cairns, R. A. 1979 The role of negative energy waves in some instabilities of parallel flows. J. Fluid Mech. 92, 1–14.CrossRefGoogle Scholar
Carpenter, J. R., Tedford, E. W., Heifetz, E. & Lawrence, G. A. 2013 Instability in stratified shear flow: review of a physical interpretation based on interacting waves. Appl. Mech. Rev. 64 (6), 060801 1–17.Google Scholar
Caulfield, C. P. 1994 Multiple linear instability of layered stratified shear flow. J. Fluid Mech. 258, 255–285.CrossRefGoogle Scholar
Caulfield, C. P., Peltier, W. R., Yoshida, S. & Ohtani, M. 1995 An experimental investigation of the instability of a shear flow with multilayered density stratification. Phys. Fluids 7, 3028–3041.Google Scholar
Constantinou, N. C. & Ioannou, P. J. 2011 Optimal excitation of two-dimensional Holmboe instabilities. Phys. Fluids 23 (7), 074102.Google Scholar
Davies, H. C. & Bishop, C. H. 1994 Eady edge waves and rapid development. J. Atmos. Sci. 51 (13), 1930–1946.Google Scholar
Drazin, P. G. & Howard, L. N. 1966 Hydrodynamic Stability of Parallel Flow of Inviscid Fluid. Academic.Google Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Farrell, B. 1984 Modal and non-modal baroclinic waves. J. Atmos. Sci. 41 (4), 668–673.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1996 Generalized stability theory. Part I: autonomous operators. J. Atmos. Sci. 53 (14), 2025–2040.Google Scholar
Goldstein, S. 1931 On the stability of superposed streams of fluids of different densities. Proc. R. Soc. Lond. A 132, 524–548.Google Scholar
Guha, A., Rahmani, M. & Lawrence, G. A. 2013 Evolution of a barotropic shear layer into elliptical vortices. Phys. Rev. E 87, 013020.Google Scholar
Harnik, N., Heifetz, E., Umurhan, O. M. & Lott, F. 2008 A buoyancy–vorticity wave interaction approach to stratified shear flow. J. Atmos. Sci. 65 (8), 2615–2630.Google Scholar
Heifetz, E., Bishop, C. H. & Alpert, P. 1999 Counter-propagating Rossby waves in the barotropic Rayleigh model of shear instability. Q. J. R. Meteorol. Soc. 125 (560), 2835–2853.Google Scholar
Heifetz, E., Bishop, C. H., Hoskins, B. J. & Methven, J. 2004 The counter-propagating Rossby-wave perspective on baroclinic instability. I: mathematical basis. Q. J. R. Meteorol. Soc. 130 (596), 211–231.Google Scholar
Heifetz, E. & Methven, J. 2005 Relating optimal growth to counterpropagating Rossby waves in shear instability. Phys. Fluids 17 (6), 064107.Google Scholar
Hill, D. F. & Foda, M. A. 1996 Subharmonic resonance of short internal standing waves by progressive surface waves. J. Fluid Mech. 321, 217–234.Google Scholar
Holmboe, J. 1962 On the behaviour of symmetric waves in stratified shear layers. Geophys. Publ. 24, 67–112.Google Scholar
Hoskins, B. J., McIntyre, M. E. & Robertson, A. W. 1985 On the use and significance of isentropic potential vorticity maps. Q. J. R. Meteorol. Soc. 111 (470), 877–946.Google Scholar
Howard, L. N. & Maslowe, S. A. 1973 Stability of stratified shear flows. Boundary-Layer Meteorol. 4 (1–4), 511–523.CrossRefGoogle Scholar
Jamali, M., Seymour, B. & Lawrence, G. A. 2003 Asymptotic analysis of a surface-interfacial wave interaction. Phys. Fluids 15 (1), 47–55.Google Scholar
Jazayeri, S. A. & Li, X. 2000 Nonlinear instability of plane liquid sheets. J. Fluid Mech. 406, 281–308.Google Scholar
Kundu, P. K. & Cohen, I. M. 2004 Fluid Mechanics. Elsevier.Google Scholar
Lawrence, G. A., Browand, F. K. & Redekopp, L. G. 1991 The stability of a sheared density interface. Phys. Fluids A 3 (10), 2360–2370.Google Scholar
Lindzen, R. S. 1988 Instability of plane parallel shear flow (toward a mechanistic picture of how it works). Pure Appl. Geophys. 126 (1), 103–121.Google Scholar
Miles, J. W. 1957 On the generation of surface waves by shear flows. J. Fluid Mech. 3, 185–204.Google Scholar
Orr, W. M. F. 1907 Stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Proc. R. Irish Acad. A 27, 9–138.Google Scholar
Pikovsky, A., Rosenblum, M. G. & Kurths, J. 2001 Synchronization, A Universal Concept in Nonlinear Sciences. Cambridge University Press.Google Scholar
Rabinovich, A., Umurhan, O. M., Harnik, N., Lott, F. & Heifetz, E. 2011 Vorticity inversion and action-at-a-distance instability in stably stratified shear flow. J. Fluid Mech. 670, 301–325.Google Scholar
Rayleigh, Lord 1880 On the stability, or instability, of certain fluid motions. Proc. Lond. Math. Soc. 12, 57–70.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Sutherland, B. R. 2010 Internal Gravity Waves. Cambridge University Press.Google Scholar
Taylor, G. I. 1931 Effect of variation in density on the stability of superposed streams of fluid. Proc. R. Soc. Lond. A 132, 499–523.Google Scholar
Tedford, E. W., Pieters, R. & Lawrence, G. A. 2009 Symmetric Holmboe instabilities in a laboratory exchange flow. J. Fluid Mech. 636, 137–153.Google Scholar
Thorpe, S. A. 1973 Experiments on instability and turbulence in a stratified shear flow. J. Fluid Mech. 61, 731–751.Google Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578–584.Google Scholar
Wen, F. 1995 Resonant generation of internal waves on the soft sea bed by a surface water wave. Phys. Fluids 7 (8), 1915–1922.Google Scholar