Skip to main content
×
Home
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 9
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Wunsch, Scott and Keller, Kurt 2013. Unstable modes of a sheared pycnocline above a stratified layer. Dynamics of Atmospheres and Oceans, Vol. 60, p. 1.


    Constantinou, Navid C. and Ioannou, Petros J. 2011. Optimal excitation of two dimensional Holmboe instabilities. Physics of Fluids, Vol. 23, Issue. 7, p. 074102.


    Heifetz, E. and Mak, J. 2015. Stratified shear flow instabilities in the non-Boussinesq regime. Physics of Fluids, Vol. 27, Issue. 8, p. 086601.


    Biancofiore, L. Gallaire, F. and Heifetz, E. 2015. Interaction between counterpropagating Rossby waves and capillary waves in planar shear flows. Physics of Fluids, Vol. 27, Issue. 4, p. 044104.


    Biancofiore, L. and Gallaire, F. 2012. Counterpropagating Rossby waves in confined plane wakes. Physics of Fluids, Vol. 24, Issue. 7, p. 074102.


    Yellin-Bergovoy, Ron Heifetz, Eyal and Umurhan, Orkan M. 2016. On the mechanism of self gravitating Rossby interfacial waves in proto-stellar accretion discs. Geophysical & Astrophysical Fluid Dynamics, Vol. 110, Issue. 3, p. 274.


    ODA, Mayuko and KANEHISA, Hirotada 2015. Interaction between Rossby and Gravity Waves in a Simple Analytical Model. Journal of the Meteorological Society of Japan. Ser. II, Vol. 93, Issue. 4, p. 425.


    Tamarin, Talia Heifetz, Eyal Umurhan, Orkan M. and Yellin, Ron 2015. On the nonnormal–nonlinear interaction mechanism between counter-propagating Rossby waves. Theoretical and Computational Fluid Dynamics, Vol. 29, Issue. 3, p. 205.


    Carpenter, Jeffrey R. Tedford, Edmund W. Heifetz, Eyal and Lawrence, Gregory A. 2013. Instability in Stratified Shear Flow: Review of a Physical Interpretation Based on Interacting Waves. Applied Mechanics Reviews, Vol. 64, Issue. 6, p. 060801.


    ×
  • Journal of Fluid Mechanics, Volume 670
  • March 2011, pp. 301-325

Vorticity inversion and action-at-a-distance instability in stably stratified shear flow

  • A. RABINOVICH (a1), O. M. UMURHAN (a2) (a3), N. HARNIK (a1), F. LOTT (a4) and E. HEIFETZ (a1)
  • DOI: http://dx.doi.org/10.1017/S002211201000529X
  • Published online: 14 January 2011
Abstract

The somewhat counter-intuitive effect of how stratification destabilizes shear flows and the rationalization of the Miles–Howard stability criterion are re-examined in what we believe to be the simplest example of action-at-a-distance interaction between ‘buoyancy–vorticity gravity wave kernels’. The set-up consists of an infinite uniform shear Couette flow in which the Rayleigh–Fjørtoft necessary conditions for shear flow instability are not satisfied. When two stably stratified density jumps are added, the flow may however become unstable. At each density jump the perturbation can be decomposed into two coherent gravity waves propagating horizontally in opposite directions. We show, in detail, how the instability results from a phase-locking action-at-a-distance interaction between the four waves (two at each jump) but can as well be reasonably approximated by the interaction between only the two counter-propagating waves (one at each jump). From this perspective the nature of the instability mechanism is similar to that of the barotropic and baroclinic ones. Next we add a small ambient stratification to examine how the critical-level dynamics alters our conclusions. We find that a strong vorticity anomaly is generated at the critical level because of the persistent vertical velocity induction by the interfacial waves at the jumps. This critical-level anomaly acts in turn at a distance to dampen the interfacial waves. When the ambient stratification is increased so that the Richardson number exceeds the value of a quarter, this destructive interaction overwhelms the constructive interaction between the interfacial waves, and consequently the flow becomes stable. This effect is manifested when considering the different action-at-a-distance contributions to the energy flux divergence at the critical level. The interfacial-wave interaction is found to contribute towards divergence, that is, towards instability, whereas the critical-level–interfacial-wave interaction contributes towards an energy flux convergence, that is, towards stability.

Copyright
Corresponding author
Email address for correspondence: eyalh@post.tau.ac.il
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.



F. P. Bretherton 1966 Baroclinic instability, the short wave cutoff in terms of potential vorticity. Q. J. R. Meteorol. Soc. 92, 335345.

J. G. Charney & M. E. Stern 1962 On the stability of internal baroclinic jets in a rotating atmosphere. J. Atmos. Sci. 19, 159172.

H. C. Davies & C. H. Bishop 1994 Eady edge waves and rapid development. J. Atmos. Sci. 51, 19301946.

E. T. Eady 1949 Long waves and cyclone waves. Tellus 1, 3352.

S. Goldstein 1931 On the stability of superposed streams of fluids of different densities. Proc. R. Soc. Lond. A 132, 524548.

N. Harnik & E. Heifetz 2007 Relating over-reflection and wave geometry to the counter propagating Rossby wave perspective: toward a deeper mechanistic understanding of shear instability. J. Atmos. Sci. 64, 22382261.

N. Harnik , E. Heifetz , O. M. Umurhan & F. Lott 2008 A buoyancy–vorticity wave interaction approach to stratified shear flow. J. Atmos. Sci. 65, 26152630.


E. Heifetz , C. H. Bishop , B. J. Hoskins & P. Alpert 1999 Counter-propagating Rossby waves in barotropic Rayleigh model of shear instability. Q. J. R. Meteorol. Soc. 125, 28352853.

E. Heifetz , C. H. Bishop , B. J. Hoskins & J. Metheven 2004 The counter-propagating Rossby-wave perspective on baroclinic instability. Part I. Mathematical basis. Q. J. R. Meteorol. Soc. 130, 211232.

E. Heifetz , N. Harnik & T. Tamarin 2009 A Hamiltonian counter-propagating Rossby wave perspective of pseudoenergy in the context of shear instability. Q. J. R. Meteorol. Soc. 135, 21612167.

E. Heifetz & J. Methven 2005 Relating optimal growth to counterpropagating Rossby waves in shear instability. Phys. Fluids 17, 064107.

B. J. Hoskins , M. E. Mcintyre & A. W. Robertson 1985 On the use and significance of isentropic potential vorticity maps. Q. J. R. Meteorol. Soc. 111, 877946.





R. S. Lindzen & A. J. Rosenthal 1981 A WKB asymptotic analysis of baroclinic instability. J. Atmos. Sci. 38, 619629.

R. S. Lindzen & K. K. Tung 1978 Wave overreflection and shear instability. J. Atmos. Sci. 35, 16261632.

J. Metheven , B. J. Hoskins , E. Heifetz & C. H. Bishop 2005 The counter-propagating Rossby-wave perspective on baroclinic instability. Part IV. Nonlinear life cycles. Q. J. R. Meteorol. Soc. 131, 14251440.



T. G. Shepherd 1990 Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics. Adv. Geophys. 32, 287352.

G. I. Taylor 1931 Effect of variation in density on the stability of superposed streams of fluid. Proc. R. Soc. Lond. A 132, 499523.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords: