Kelvin-Helmholtz instabilities of high-velocity magnetized shear layers with generalized polytrope laws
Authors:
Kevin G. Brown and S. Roy Choudhury
Journal:
Quart. Appl. Math. 58 (2000), 401-423
MSC:
Primary 76E25; Secondary 35Q35, 76E20, 76N99, 76W05
DOI:
https://doi.org/10.1090/qam/1770646
MathSciNet review:
MR1770646
Full-text PDF Free Access
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Abstract: The linear stability of zero and finite width, arbitrarily compressible, and magnetized velocity shear layers with isotropic or anisotropic pressure is investigated. Such flows, modeled by the magnetohydrodynamic equations with generalized polytrope laws for the pressure parallel and perpendicular to the magnetic field, are relevant in various astrophysical, geophysical and space plasma configurations. The conditions for instability of shear layers of zero width are derived. For layers of finite width, shooting numerical schemes are employed to satisfy the Sommerfeld radiation conditions of outgoing, spatially damping modes in a frame comoving with the plasma flow. The resulting eigenvalues for the angular frequency and linear growth rate are mapped out for different regions of the wave number/Mach number space. For polytrope indices corresponding to the double adiabatic and magnetohydrodynamic equations, the results reduce to those obtained earlier using these models.
A. K. Sen, Stability of the magnetosphere boundary, Planetary and Space Science 13, 131–141 (1965)
J. F. Mckenzie, Hydromaqnetic oscillations of the geomagnetic tail and plasma sheet, J. Geophysical Res. 75, 5331–5339 (1970)
D. J. Southwood, Some features of the field line resonances in the magnetosphere, Planetary and Space Science 22, 483–491 (1974)
L. Chen and A. Hasegawa, A theory of long-period magnetic pulsations, J. Geophysical Res. 79, 1024–1032 (1974)
F. L. Scarf, W. S. Kurth, D. A. Gurnett, H. S. Bridge, and J. D. Sullivan, Jupiter tail phenomena upstream from Saturn, Nature 292, 585–586 (1981)
H. Dobrowolny and N. D’Angelo, Wave motion in type I comet tails, in Cosmic Plasma Physics (K. Schindler, ed.), Plenum Press, New York, 1972
A. I. Ershkovich, A. A. Nusnov, and A. A. Chernikov, Oscillations of type I comet tails, Planetary and Space Science 20, 1235–1243 (1972); and, Nonlinear waves in type I comet tails 21, 663–673 (1973)
B. D. Turland and P. A. G. Scheuer, Instabilities of Kelvin-Helmholtz type for relativistic streaming, Monthly Notices Roy. Astron. Soc. 176, 421–441 (1976)
R. D. Blandford and J. E. Pringle, Kelvin-Helmholtz instability of relativistic beams, Monthly Notices Roy. Astron. Soc. 176, 443–454 (1976)
- S. Chandrasekhar, Hydrodynamic and hydromagnetic stability, The International Series of Monographs on Physics, Clarendon Press, Oxford, 1961. MR 0128226
R. A. Gerwin, Stability of the interface between two fluids in relative motion, Rev. Modern Phys. 40, 652–658 (1968)
T. P. Ray and A. I. Ershkovich, Kelvin-Helmholtz instabilities of magnetized shear layers, Monthly Notices Roy. Astron. Soc. 204, 821–826 (1983)
A. Miura, Anomalous transport by magnetohydrodynamic Kelvin-Helmholtz instabilities in the solar wind-magnetosphere interaction, J. Geophysical Res. 89, 801–818 (1984)
S. Roy Choudhury and R. V. Lovelace, On the Kelvin-Helmholtz instabilities of supersonic shear layers, Astrophysical J. 283, 331–342 (1984); and, On the Kelvin-Helmholtz instabilities of high-velocity magnetized shear layers 302, 188–199 (1986); A. Miura and P. L. Pritchett, Nonlinear stability analysis of the MHD Kelvin-Helmholtz instability in a compressible plasma, J. Geophysical Res. 87, 7431–7444 (1982)
S. Roy Choudhury, Kelvin-Helmholtz instabilities of supersonic, magnetized shear layers, J. Plasma Phys. 35, 375–392 (1986)
C. Uberoi, On the Kelvin-Helmholtz instabilities of structured plasma layers in the magnetosphere, Planetary and Space Science 34, 1223–1227 (1986)
M. Fujimota and T. Terasawa, Ion inertia effect on the Kelvin-Helmholtz instability, J. Geophysical Res. 96, 15725–15734 (1991)
A. C. Sharma and K. M. Shrivastava, Magnetospheric plasma waves, Astrophys. Space Sci. 200, 107–115 (1993)
- S. K. Malik and M. Singh, Chaos in Kelvin-Helmholtz instability in magnetic fluids, Phys. Fluids A 4 (1992), no. 12, 2915–2922. MR 1192761, DOI https://doi.org/10.1063/1.858518
S. Roy Choudhury and V. L. Patel, Kelvin-Helmholtz instabilities of high-velocity, magnetized anisotropic shear layers, Phys. Fluids 28, 3292–3301 (1985)
S. Duhau, F. Gratton, and J. Gratton, Hydromagnetic oscillations of a tangential discontinuity in the CGL approximation, Phys. Fluids 13, 1503–1509 (1970)
S. Duhau, F. Gratton, and J. Gratton, Radiation of hydromagnetic waves from a tangential velocity discontinuity, Phys. Fluids 14, 2067–2071 (1971)
S. Duhau and J. Gratton, Effect of compressibility on the stability of a vortex sheet in an ideal magnetofluid, Phys. Fluids 16, 150–152 (1972)
R. Rajaram, G. L. Kalra, and J. N. Tandon, Discontinuities and the magnetospheric phenomena, J. Atmos. Terr. Phys. 40, 991–1000 (1978)
- R. Rajaram, G. L. Kalra, and J. N. Tandon, Stability of a collisionless contact discontinuity, Astrophys. and Space Sci. 67 (1980), no. 1, 137–145. MR 562535, DOI https://doi.org/10.1007/BF00651873
S. P. Talwar, Hydromagnetic stability of the magnetospheric boundary, J. Geophysical Res. 69, 2707–2713 (1964)
S. P. Talwar, Kelvin-Helmholtz instability in an anisotropic plasma, Phys. Fluids 8, 1295–1299 (1965)
Zu-Yin Pu, Kelvin-Helmholtz instability in collisionless space plasmas, Phys. Fluids B 1, 440–447 (1989)
B. A. Shrauner, Small amplitude hydromagnetic waves for a plasma with a generalized polytrope law, Plasma Phys. 15, 375–385 (1973)
- S. Roy Choudhury, An analytical study of the Kelvin-Helmholtz instabilities of compressible, magnetized, anisotropic, and isotropic tangential velocity discontinuities, Phys. Fluids 29 (1986), no. 5, 1509–1519. MR 839380, DOI https://doi.org/10.1063/1.865669
S. Roy Choudhury, Global asymptotic analysis of the Kelvin-Helmholtz instability of supersonic shear layers, Canadian J. Physics 68, 334–342 (1990)
A. K. Sen, Stability of the magnetosphere boundary, Planetary and Space Science 13, 131–141 (1965)
J. F. Mckenzie, Hydromaqnetic oscillations of the geomagnetic tail and plasma sheet, J. Geophysical Res. 75, 5331–5339 (1970)
D. J. Southwood, Some features of the field line resonances in the magnetosphere, Planetary and Space Science 22, 483–491 (1974)
L. Chen and A. Hasegawa, A theory of long-period magnetic pulsations, J. Geophysical Res. 79, 1024–1032 (1974)
F. L. Scarf, W. S. Kurth, D. A. Gurnett, H. S. Bridge, and J. D. Sullivan, Jupiter tail phenomena upstream from Saturn, Nature 292, 585–586 (1981)
H. Dobrowolny and N. D’Angelo, Wave motion in type I comet tails, in Cosmic Plasma Physics (K. Schindler, ed.), Plenum Press, New York, 1972
A. I. Ershkovich, A. A. Nusnov, and A. A. Chernikov, Oscillations of type I comet tails, Planetary and Space Science 20, 1235–1243 (1972); and, Nonlinear waves in type I comet tails 21, 663–673 (1973)
B. D. Turland and P. A. G. Scheuer, Instabilities of Kelvin-Helmholtz type for relativistic streaming, Monthly Notices Roy. Astron. Soc. 176, 421–441 (1976)
R. D. Blandford and J. E. Pringle, Kelvin-Helmholtz instability of relativistic beams, Monthly Notices Roy. Astron. Soc. 176, 443–454 (1976)
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover (originally published 1961, Oxford, Clarendon), New York, 1981; A. Syrovatskii, The Helmholtz instability, Soviet Physics Uspekhi 62, 247–253 (1957); T. G. Northrop, Helmholtz instability of a plasma, Physical Review (Second series) 103, 1150–1154 (1956)
R. A. Gerwin, Stability of the interface between two fluids in relative motion, Rev. Modern Phys. 40, 652–658 (1968)
T. P. Ray and A. I. Ershkovich, Kelvin-Helmholtz instabilities of magnetized shear layers, Monthly Notices Roy. Astron. Soc. 204, 821–826 (1983)
A. Miura, Anomalous transport by magnetohydrodynamic Kelvin-Helmholtz instabilities in the solar wind-magnetosphere interaction, J. Geophysical Res. 89, 801–818 (1984)
S. Roy Choudhury and R. V. Lovelace, On the Kelvin-Helmholtz instabilities of supersonic shear layers, Astrophysical J. 283, 331–342 (1984); and, On the Kelvin-Helmholtz instabilities of high-velocity magnetized shear layers 302, 188–199 (1986); A. Miura and P. L. Pritchett, Nonlinear stability analysis of the MHD Kelvin-Helmholtz instability in a compressible plasma, J. Geophysical Res. 87, 7431–7444 (1982)
S. Roy Choudhury, Kelvin-Helmholtz instabilities of supersonic, magnetized shear layers, J. Plasma Phys. 35, 375–392 (1986)
C. Uberoi, On the Kelvin-Helmholtz instabilities of structured plasma layers in the magnetosphere, Planetary and Space Science 34, 1223–1227 (1986)
M. Fujimota and T. Terasawa, Ion inertia effect on the Kelvin-Helmholtz instability, J. Geophysical Res. 96, 15725–15734 (1991)
A. C. Sharma and K. M. Shrivastava, Magnetospheric plasma waves, Astrophys. Space Sci. 200, 107–115 (1993)
S. K. Malik and M. Singh, Chaos in Kelvin-Helmholtz instability in magnetic fluids, Phys. Fluids A 4, 2915–2922 (1992)
S. Roy Choudhury and V. L. Patel, Kelvin-Helmholtz instabilities of high-velocity, magnetized anisotropic shear layers, Phys. Fluids 28, 3292–3301 (1985)
S. Duhau, F. Gratton, and J. Gratton, Hydromagnetic oscillations of a tangential discontinuity in the CGL approximation, Phys. Fluids 13, 1503–1509 (1970)
S. Duhau, F. Gratton, and J. Gratton, Radiation of hydromagnetic waves from a tangential velocity discontinuity, Phys. Fluids 14, 2067–2071 (1971)
S. Duhau and J. Gratton, Effect of compressibility on the stability of a vortex sheet in an ideal magnetofluid, Phys. Fluids 16, 150–152 (1972)
R. Rajaram, G. L. Kalra, and J. N. Tandon, Discontinuities and the magnetospheric phenomena, J. Atmos. Terr. Phys. 40, 991–1000 (1978)
R. Rajaram, G. L. Kalra, and J. N. Tandon, Discontinuities in the magnetosphere, Astrophys. Space Sci. 67, 137–150 (1980)
S. P. Talwar, Hydromagnetic stability of the magnetospheric boundary, J. Geophysical Res. 69, 2707–2713 (1964)
S. P. Talwar, Kelvin-Helmholtz instability in an anisotropic plasma, Phys. Fluids 8, 1295–1299 (1965)
Zu-Yin Pu, Kelvin-Helmholtz instability in collisionless space plasmas, Phys. Fluids B 1, 440–447 (1989)
B. A. Shrauner, Small amplitude hydromagnetic waves for a plasma with a generalized polytrope law, Plasma Phys. 15, 375–385 (1973)
S. Roy Choudhury, An analytical study of the Kelvin-Helmholtz instabilities of compressible, magnetized anisotropic tangential velocity discontinuities, Phys. Fluids 29, 1509–1519 (1986)
S. Roy Choudhury, Global asymptotic analysis of the Kelvin-Helmholtz instability of supersonic shear layers, Canadian J. Physics 68, 334–342 (1990)
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