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Shear flow instabilities in shallow-water magnetohydrodynamics

Published online by Cambridge University Press:  14 January 2016

J. Mak ,
S. D. Griffiths  and
D. W. Hughes
J. Mak
Affiliation:
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
S. D. Griffiths
Affiliation:
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
D. W. Hughes
Affiliation:
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
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Abstract

Within the framework of shallow-water magnetohydrodynamics, we investigate the linear instability of horizontal shear flows, influenced by an aligned magnetic field and stratification. Various classical instability results, such as Høiland’s growth-rate bound and Howard’s semi-circle theorem, are extended to this shallow-water system for quite general flow and field profiles. In the limit of long-wavelength disturbances, a generalisation of the asymptotic analysis of Drazin & Howard (J. Fluid Mech., vol. 14, 1962, pp. 257–283) is performed, establishing that flows can be distinguished as either shear layers or jets. These possess contrasting instabilities, which are shown to be analogous to those of certain piecewise-constant velocity profiles (the vortex sheet and the rectangular jet). In both cases it is found that the magnetic field and stratification (as measured by the Froude number) are generally each stabilising, but weak instabilities can be found at arbitrarily large Froude number. With this distinction between shear layers and jets in mind, the results are extended numerically to finite wavenumber for two particular flows: the hyperbolic-tangent shear layer and the Bickley jet. For the shear layer, the instability mechanism is interpreted in terms of counter-propagating Rossby waves, thereby allowing an explication of the stabilising effects of the magnetic field and stratification. For the jet, the competition between even and odd modes is discussed, together with the existence at large Froude number of multiple modes of instability.

JFM classification

Instability: Instability Geophysical and Geological Flows: Shallow water flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 788 , 10 February 2016 , pp. 767 - 796
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© 2016 Cambridge University Press 

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References

Balmforth, N. J. 1999 Shear instability in shallow water. J. Fluid Mech. 387, 97–127.Google Scholar
Bazdenkov, S. V. & Pogutse, O. P. 1983 Supersonic stabilization of a tangential shear in a thin atmosphere. J. Expl Theor. Phys. Lett. 37, 375–377.Google Scholar
Blumen, W., Drazin, P. G. & Billings, D. F. 1975 Shear layer instability of an inviscid compressible fluid. Part 2. J. Fluid Mech. 71, 305–316.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.Google Scholar
Cally, P. S. 2003 Three-dimensional magneto-shear instabilities in the solar tachocline. Mon. Not. R. Astron. Soc. 339, 957–972.Google Scholar
Carpenter, J. R., Tedford, E. W., Heifetz, E. & Lawrence, G. A. 2012 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
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.Google Scholar
Collings, I. L. & Grimshaw, R. H. J. 1980 The effect of topography on the stability of a barotropic coastal current. Dyn. Atmos. Oceans 5, 83–106.CrossRefGoogle Scholar
De Sterck, H. 2001 Hyperbolic theory of the ‘shallow water’ magnetohydrodynamics equations. Phys. Plasmas 8, 3293–3304.Google Scholar
Dellar, P. J. 2002 Hamiltonian and symmetric hyperbolic structures of shallow water magnetohydrodynamics. Phys. Plasmas 9, 1130–1136.Google Scholar
Dikpati, M. & Gilman, P. A. 2001 Analysis of hydrodynamic stability of solar tachocline latitudinal differential rotation using a shallow-water model. Astrophys. J. 551, 536–564.CrossRefGoogle Scholar
Dikpati, M., Gilman, P. A. & Rempel, M. 2003 Stability analysis of tachocline latitudinal differential rotation and coexisting toroidal band using a shallow-water model. Astrophys. J. 596, 680–697.Google Scholar
Drazin, P. G. & Howard, L. N. 1962 The instability to long waves of unbounded parallel inviscid flow. J. Fluid Mech. 14, 257–283.Google Scholar
Drazin, P. G. & Howard, L. N. 1966 Hydrodynamic stability of parallel flow of inviscid fluid. Adv. Appl. Mech. 9, 1–89.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability, 2nd edn. Cambridge University Press.Google Scholar
Gedzelman, S. D. 1973 Hydromagnetic stability of parallel flow of an ideal heterogeneous fluid. J. Fluid Mech. 58, 777–794.Google Scholar
Gill, A. E. 1965 Instabilities of ‘top-hat’ jets and wakes in compressible fluids. Phys. Fluids 8, 1428–1430.Google Scholar
Gill, A. E. 1982 Atmosphere-Ocean Dynamics. Academic.Google Scholar
Gill, A. E. & Drazin, P. G. 1965 Note on instability of compressible jets and wakes to long-wave disturbances. J. Fluid Mech. 22, 415.Google Scholar
Gilman, P. A. 1967 Stability of baroclinic flows in a zonal magnetic field: part I. J. Atmos. Sci. 24, 101–118.Google Scholar
Gilman, P. A. 2000 Magnetohydrodynamic ‘shallow water’ equations for the solar tachocline. Astrophys. J. 544, L79–L82.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. & Dikpati, M. 2002 Analysis of instability of latitudinal differential rotation and toroidal field in the solar tachocline using a magnetohydrodynamic shallow-water model. I. Instability for broad toroidal field profiles. Astrophys. J. 576, 1031–1047.Google Scholar
Gough, D. O. 2007 An introduction to the solar tachocline. In The Solar Tachocline (ed. Hughes, D. W., Rosner, R. & Weiss, N. O.), Cambridge University Press.Google Scholar
Harnik, N. & Heifetz, E. 2007 Relating overreflection and wave geometry to the counter-propagating Rossby wave perspective: toward a deeper mechanistic understanding of shear instability. J. Atmos. Sci. 64, 2238–2261.Google Scholar
Hayashi, Y.-Y. & Young, W. R. 1987 Stable and unstable shear modes of rotating parallel flows in shallow water. J. Fluid Mech. 184, 477–504.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.Google Scholar
Heng, K. & Spitkovsky, A. 2009 Magnetohydrodynamic shallow water waves: linear analysis. Astrophys. J. 703, 1819–1831.Google Scholar
Høiland, E. 1953 On two-dimensional perturbation of linear flow. Geophys. Publ. 18, 333–342.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
Howard, L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509–512.Google Scholar
Howard, L. N. 1963 Neutral curves and stability boundaries in stratified flow. J. Fluid Mech. 16, 333–342.Google Scholar
Hughes, D. W., Rosner, R. & Weiss, N. O. 2007 The Solar Tachocline. Cambridge University Press.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
Lipps, F. B. 1962 The barotropic stability of the mean winds in the atmosphere. J. Fluid Mech. 12, 397–407.Google Scholar
Mak, J.2013 Shear instabilities in shallow-water magnetohydrodynamics. PhD thesis, University of Leeds.Google Scholar
Michael, D. H. 1955 Stability of a combined current and vortex sheet in a perfectly conducting fluid. Proc. Camb. Phil. Soc. 51, 528–532.Google Scholar
Michalke, A. 1964 On the inviscid instability of the hyperbolic-tangent velocity profile. J. Fluid Mech. 19, 543–556.Google Scholar
Miles, J. W. 1958 On the disturbed motion of a vortex sheet. J. Fluid Mech. 4, 538–552.Google Scholar
Miura, A. & Pritchett, P. L. 1982 Nonlocal stability analysis of the MHD Kelvin–Helmholtz instability in a compressible plasma. J. Geophys. Res. 87, 7431–7444.Google Scholar
Pedlosky, J. 1964 The stability of currents in the atmosphere and the ocean: part I. J. Atmos. Sci. 21, 201–219.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, L. 1878 On the instability of jets. Proc. Lond. Math. Soc. 10, 4–12.Google Scholar
Ripa, P. 1983 General stability conditions for zonal flows in a one-layer model on the ${\it\beta}$ -plane or the sphere. J. Fluid Mech. 126, 463–489.Google Scholar
Satomura, T. 1981 An investigation of shear instability in a shallow water. J. Met. Soc. Japan 59, 148–170.Google Scholar
Schecter, D. A., Boyd, J. F. & Gilman, P. A. 2001 ‘Shallow-water’ magnetohydrodynamic waves in the solar tachocline. Astrophys. J. 551, L185–L188.Google Scholar
Shivamoggi, B. K. & Debnath, L. 1987 Stability of magnetohydrodynamic stratified shear flows. Acta Mechanica 68, 33–42.Google Scholar
Stern, M. E. 1963 Joint instability of hydromagnetic fields which are separately stable. Phys. Fluids 6, 636–642.Google Scholar
Sutherland, B. R. & Peltier, W. R. 1992 The stability of stratified jets. Geophys. Astrophys. Fluid Dyn. 66, 101–131.Google Scholar
Takehiro, S. I. & Hayashi, Y. Y. 1992 Over-reflection and shear instability in a shallow-water model. J. Fluid Mech. 236, 259–279.Google Scholar
Vallis, G. K. 2006 Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press.Google Scholar
Zaqarashvili, T. V., Oliver, R., Ballester, J. L. & Shergelashvili, B. M. 2007 Rossby waves in ‘shallow water’ magnetohydrodynamics. Astron. Astrophys. 470, 815–820.Google Scholar