Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-26T09:12:39.497Z Has data issue: false hasContentIssue false
  • English
  • Français

Interacting vorticity waves as an instability mechanism for magnetohydrodynamic shear instabilities

Published online by Cambridge University Press:  12 February 2015

E. Heifetz ,
J. Mak ,
J. Nycander  and
O. M. Umurhan
E. Heifetz
Affiliation:
Department of Geophysics and Planetary Sciences, Tel Aviv University, Tel Aviv, 69978, Israel Department of Meteorology (MISU), Stockholm University, SE-106 91 Stockholm, Sweden
J. Mak
Affiliation:
Department of Geophysics and Planetary Sciences, Tel Aviv University, Tel Aviv, 69978, Israel
J. Nycander
Affiliation:
Department of Meteorology (MISU), Stockholm University, SE-106 91 Stockholm, Sweden
O. M. Umurhan
Affiliation:
NASA Ames Research, Space Sciences Division, Mail Stop N-245-3, Moffett Field, CA 94043, USA SETI Institute, 189 Bernardo Avenue, Suite 100, Mountain View, CA 94043, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The interacting vorticity wave formalism for shear flow instabilities is extended here to the magnetohydrodynamic (MHD) setting, to provide a mechanistic description for stabilising and destabilising shear instabilities by the presence of a background magnetic field. The interpretation relies on local vorticity anomalies inducing a non-local velocity field, resulting in action at a distance. It is shown here that the waves supported by the system are able to propagate vorticity via the Lorentz force, and waves may interact. The existence of instability then rests upon whether the choice of basic state allows for phase locking and constructive interference of the vorticity waves via mutual interaction. To substantiate this claim, we solve the instability problem of two representative basic states, one where a background magnetic field stabilises an unstable flow and the other where the field destabilises a stable flow, and perform relevant analyses to show how this mechanism operates in MHD.

JFM classification

Instability: Instability MHD and Electrohydrodynamics: Magnetic fluids MHD and Electrohydrodynamics: MHD and Electrohydrodynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 767 , 25 March 2015 , pp. 199 - 225
Check for updates
Copyright
© 2015 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baines, P. G. & Mitsudera, H. 1994 On the mechanism of shear instabilities. J. Fluid Mech. 276, 327–342.Google Scholar
Balmforth, N. J., Roy, A. & Caulfield, C. P. 2012 Dynamics of vorticity defects in stratified shear flow. J. Fluid Mech. 694, 292–331.Google Scholar
Biskamp, D. 2003 Magnetohydrodynamic Turbulence. Cambridge University Press.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, 335–345.CrossRefGoogle Scholar
Cairns, R. A. 1979 The role of negative energy waves in some instabilities of parallel flows. J. Fluid Mech. 92, 1–14.Google Scholar
Carpenter, J. R., Balmforth, N. J. & Lawrence, G. A. 2010 Identifying unstable modes in stratified shear layers. Phys. Fluids 22, 054104.Google 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, 061001.Google Scholar
Caulfield, C. P. 1994 Multiple linear instability of layered stratified shear flow. J. Fluid Mech. 258, 255–285.Google Scholar
Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Chen, X. L. & Morrison, P. J. 1991 A sufficient condition for the ideal instability of shear flow with parallel magnetic field. Phys. Fluids B 3, 863–865.CrossRefGoogle Scholar
Cho, J. Y.-K. 2008 Atmospheric dynamics of tidally synchronized extrasolar planets. Phil. Trans. R. Soc. Lond. A 366, 4477–4488.Google ScholarPubMed
Constantinou, N. C. & Ioannou, P. J. 2011 Optimal excitation of two-dimensional Holmboe instabilities. Phys. Fluids 23, 074102.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability, 2nd edn. Cambridge University Press.Google Scholar
Gilman, P. A. & Cally, P. S. 2007 Global MHD instabilities of the tachocline. In The Solar Tachocline (ed. Hughes, D. W., Rosner, R. & Weiss, N. O.). Cambridge University Press.Google Scholar
Gilman, P. A. & Fox, P. A. 1997 Joint instability of latitudinal differential rotation and toroidal magnetic fields below the solar convection zone. Astrophys. J. 484, 439–454.Google Scholar
Goldston, R. J. & Rutherford, P. H. 1995 Introduction to Plasma Physics. Taylor & Francis.CrossRefGoogle Scholar
Guha, A. & Lawrence, G. A. 2013 A wave interaction approach to studying non-modal homogeneous and stratified shear instabilities. J. Fluid Mech. 755, 336–364.CrossRefGoogle 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, 2615–2630.Google Scholar
Heifetz, E., Bishop, C. H., Hoskins, B. J. & Methven, J. 2004a The counter-propagating Rossby-wave perspective on baroclinic instability I: Mathematical basis. Q. J. R. Meteorol. Soc. 130, 211–231.Google Scholar
Heifetz, E., Methven, J., Hoskins, B. J. & Bishop, C. H. 2004b The counter-propagating Rossby-wave perspective on baroclinic instability II: application to the Charney model. Q. J. R. Meteorol. Soc. 130, 233–258.CrossRefGoogle Scholar
van Heijst, G. J. F. & Clercx, H. J. H. 2009 Laboratory modeling of geophysical vortices. Annu. Rev. Fluid Mech. 41, 143–164.Google Scholar
Holmboe, J. 1962 On the behaviour of symmetric waves in stratified shear layers. Geophys. Publ. 24, 67–113.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, 877–946.Google Scholar
Hughes, D. W. & Tobias, S. M. 2001 On the instability of magnetohydrodynamic shear flows. Proc. R. Soc. Lond. A 457, 1365–1384.Google Scholar
Kent, A. 1966 Instability of laminar flow of a perfect magnetofluid. Phys. Fluids 9, 1286–1289.Google Scholar
Kent, A. 1968 Stability of laminar magnetofluid flow along a parallel magnetic field. J. Plasma Phys. 2, 543–556.CrossRefGoogle Scholar
Lecoanet, D., Zweibel, E. G., Townsend, R. H. D. & Huang, Y.-M. 2010 Violation of Richardson’s criterion via introduction of a magnetic field. Astrophys. J. 712, 1116–1128.CrossRefGoogle Scholar
Lott, F. 2003 Large-scale flow response to short gravity waves breaking in a rotating shear flow. J. Atmos. Sci. 1704, 1691–4862.Google Scholar
Lundquist, S. 1951 On the stability of magneto-hydrostatic fields. Phys. Rev. 83, 307–311.CrossRefGoogle Scholar
Mak, J.2013 Shear instabilities in shallow-water magnetohydrodynamics. PhD thesis, University of Leeds.Google Scholar
Methven, J., Heifetz, E., Hoskins, B. J. & Bishop, C. H. 2005 The counter-propagating Rossby-wave perspective on baroclinic instability III: primitive-equation disturbances on the sphere. Q. J. R. Meteorol. Soc. 131, 1393–1424.CrossRefGoogle Scholar
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10, 496–508.CrossRefGoogle 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
Ray, T. P. & Ershkovich, A. I. 1983 Kelvin–Helmholtz instabilities in a sheared compressible plasma. Mon. Not. R. Astron. Soc. 204, 821–831.Google Scholar
Ruderman, M. S. & Belov, N. A. 2010 Stability of MHD shear flows: application to space physics. J. Phys.: Conf. Ser. 216, 012016.Google Scholar
Stern, M. E. 1963 Joint instability of hydromagnetic fields which are separately stable. Phys. Fluids 6, 636–642.Google Scholar
Tatsuno, T. & Dorland, W. 2006 Magneto-flow instability in symmetric field profiles. Phys. Plasmas 13, 092107.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