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A generalized action-angle representation of wave interaction in stratified shear flows

Published online by Cambridge University Press:  17 November 2017

Eyal Heifetz  and
Anirban Guha Open the ORCID record for Anirban Guha [Opens in a new window]
Eyal Heifetz
Affiliation:
Department of Geophysics, School of Earth Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Anirban Guha*
Affiliation:
Mechanical Engineering Department, Indian Institute of Technology Kanpur, U.P. 208016, India
*
†Email address for correspondence: [email protected]
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Abstract

In this paper we express the linearized dynamics of interacting interfacial waves in stratified shear flows in the compact form of action-angle Hamilton’s equations. The pseudo-energy serves as the Hamiltonian of the system, the action coordinates are the contribution of the interfacial waves to the wave action and the angles are the phases of the interfacial waves. The term ‘generalized action angle’ aims to emphasize that the action of each wave is generally time dependent and this allows for instability. An attempt is made to relate this formalism to the action at a distance resonance instability mechanism between counter-propagating vorticity waves via the global conservations of pseudo-energy and pseudo-momentum.

JFM classification

Mathematical Foundations: Hamiltonian theory Instability: Instability Wakes/Jets: Shear layers
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 834 , 10 January 2018 , pp. 220 - 236
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Copyright
© 2017 Cambridge University Press 

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References

Arnol’d, V. I. 2013 Mathematical Methods of Classical Mechanics, vol. 60. Springer Science & Business Media.Google Scholar
Baines, P. G. & Mitsudera, H. 1994 On the mechanism of shear flow instabilities. J. Fluid Mech. 276, 327–342.Google Scholar
Biancofiore, L., Gallaire, F. & Heifetz, E. 2015 Interaction between counterpropagating Rossby waves and capillary waves in planar shear flows. Phys. Fluids 27 (4), 044104.Google Scholar
Biancofiore, L., Heifetz, E., Hoepffner, J. & Gallaire, F. 2017 The role of the surface tension in planar shear flows. Phys. Rev. Fluids 2, 103901.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
Bühler, O. 2009 Waves and Mean Flows. Cambridge University Press.CrossRefGoogle Scholar
Carpenter, J. R., Guha, A. & Heifetz, E. 2017 A physical interpretation of the wind-wave instability as interacting waves. J. Phys. Oceanogr. 47 (6), 1441–1455.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), 060801–17.Google Scholar
Caulfield, C. P. 1994 Multiple linear instability of layered stratified shear flow. J. Fluid Mech. 258, 255–285.Google 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
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability, 2nd edn. Cambridge University Press.Google Scholar
Guha, A. & Lawrence, G. A. 2014 A wave interaction approach to studying non-modal homogeneous and stratified shear instabilities. J. Fluid Mech. 755, 336–364.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., 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., 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., Harnik, N. & Tamarin, T. 2009 Canonical Hamiltonian representation of pseudoenergy in shear flows using counter-propagating Rossby waves. Q. J. R. Meteorol. Soc. 135 (645), 2161–2167.Google Scholar
Heifetz, E. & Mak, J. 2015 Stratified shear flow instabilities in the non-boussinesq regime. Phys. Fluids 27 (8), 086601.Google Scholar
Heifetz, E., Mak, J., Nycander, J. & Umurhan, O. M. 2015 Interacting vorticity waves as an instability mechanism for magnetohydrodynamic shear instabilities. J. Fluid Mech. 767, 199–225.CrossRefGoogle Scholar
Heifetz, E. & Methven, J. 2005 Relating optimal growth to counterpropagating Rossby waves in shear instability. Phys. Fluids 17 (6), 064107.Google Scholar
Held, I. M. 1985 Pseudomomentum and the orthogonality of modes in shear flows. J. Atmos. Sci. 42 (21), 2280–2288.Google Scholar
Holmboe, J. 1962 On the behavior of symmetric waves in stratified shear layers. Geofys. 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. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10 (4), 509–512.Google Scholar
Kundu, P. K. & Cohen, I. M. 2004 Fluid Mechanics. Elsevier.Google Scholar
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10 (4), 496–508.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
Salmon, R. 1988 Hamiltonian fluid mechanics. Annu. Rev. Fluid Mech. 20 (1), 225–256.Google Scholar
Shepherd, T. G. 1990 Symmetries, conservation laws, and hamiltonian structure in geophysical fluid dynamics. Adv. Geophys. 32, 287–338.CrossRefGoogle 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
Yellin-Bergovoy, R., Heifetz, E. & Umurhan, O. M. 2017 Physical mechanism of centrifugal-gravity wave resonant instability in swirling free surface rotating polygons. Phys. Rev. Fluids 2, 104801.Google Scholar