An analytical study of the Kelvin-Helmholtz instabilities of compressible, magnetized tangential velocity discontinuities with generalized polytrope laws

Authors:
Kevin G. Brown and S. Roy Choudhury

Journal:
Quart. Appl. Math. **60** (2002), 601-630

MSC:
Primary 76E25; Secondary 76E20, 76X05

DOI:
https://doi.org/10.1090/qam/1938343

MathSciNet review:
MR1938343

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The linear Kelvin-Helmholtz instability of tangential velocity discontinuities in high velocity magnetized plasmas with isotropic or anisotropic pressure is investigated. A new analytical technique applied to the magnetohydrodynamic equations with generalized polytrope laws (for the pressure parallel and perpendicular to the magnetic field) yields the complete structure of the unstable, standing waves in the (inverse plasma beta, Mach number) plane for modes at arbitrary angles to the flow and the magnetic field. The stable regions in the (inverse plasma beta, propagation angle) plane are mapped out via a level curve analysis, thus clarifying the stabilizing effects of both the magnetic field and the compressibility. For polytrope indices corresponding to the double adiabatic and magnetohydrodynamic equations, the results reduce to those obtained earlier using these models. Detailed numerical results are presented for other cases not considered earlier, including the cases of isothermal and mixed waves. Also, for modes propagating along or opposite to the magnetic field direction and at general angles to the flow, a criterion is derived for the absence of standing wave instability--in the isotropic MHD case, this condition corresponds to (plasma beta) .

**[1]**A. K. Sen,*Stability of the magnetosphere boundary*, Planetary and Space Science**13**, 131-141 (1965)**[2]**J. F. Mckenzie,*Hydromagnetic oscillations of the geomagnetic tail and plasma sheet*, J. Geophysical Res.**75**, 5331-5339 (1970)**[3]**D. J. Southwood,*Some features of the field line resonances in the magnetosphere*, Planetary and Space Science**22**, 483-491 (1974)**[4]**L. Chen and A. Hasegawa,*A theory of long-period magnetic pulsations*, J. Geophysical Res.**79**, 1024-1032 (1974)**[5]**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)**[6]**H. Dobrowolny and N. D'Angelo,*Wave motion in type I comet tails*, in*Cosmic Plasma Physics*(K. Schindler, ed.), Plenum, New York, 1972**[7]**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)**[8]**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)**[9]**R. D. Blandford and J. E. Pringle,*Kelvin-Helmholtz instability of relativistic beams*, Monthly Notices Roy. Astron. Soc.**176**, 443-454 (1976)**[10]**S. Chandrasekhar,*Hydrodynamic and hydromagnetic stability*, The International Series of Monographs on Physics, Clarendon Press, Oxford, 1961. MR**0128226****[11]**R. A. Gerwin,*Stability of the interface between two fluids in relative motion*, Rev. Modern Phys.**40**, 652-658 (1968)**[12]**T. P. Ray and A. I. Ershkovich,*Kelvin-Helmholtz instabilities of magnetized shear layers*, Monthly Notices Roy. Astron. Soc.**204**, 821-826 (1983)**[13]**A. Miura,*Anomalous transport by magnetohydrodynamic Kelvin-Helmholtz instabilities in the solar wind-magnetosphere interaction*, J. Geophysical Res.**89**, 801-818 (1984)**[14]**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)**[15]**S. Roy Choudhury,*Kelvin-Helmholtz instabilities of supersonic, magnetized shear layers*, J. Plasma Phys.**35**, 375-392 (1986)**[16]**C. Uberoi,*On the Kelvin-Helmholtz instabilities of structured plasma layers in the magnetosphere*, Planetary and Space Science**34**, 1223-1227 (1986)**[17]**M. Fujimota and T. Terasawa,*Ion inertia effect on the Kelvin-Helmholtz instability*, J. Geophysical Res.**96**, 15725-15734 (1991)**[18]**A. C. Sharma and K. M. Shrivastava,*Magnetospheric plasma waves*, Astrophys. Space Sci.**200**, 107-115 (1993)**[19]**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**, https://doi.org/10.1063/1.858518**[20]**S. Roy Choudhury and V. L. Patel,*Kelvin-Helmholtz instabilities of high-velocity, magnetized anisotropic shear layers*, Phys. Fluids**28**, 3292-3301 (1985)**[21]**S. Duhau, F. Gratton, and J. Gratton,*Hydromagnetic oscillations of a tangential discontinuity in the CGL approximation*, Phys. Fluids**13**, 1503-1509 (1970)**[22]**S. Duhau, F. Gratton, and J. Gratton,*Radiation of hydromagnetic waves from a tangential velocity discontinuity*, Phys. Fluids**14**, 2067-2071 (1971)**[23]**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)**[24]**R. Rajaram, G. L. Kalra, and J. N. Tandon,*Discontinuities and the magnetospheric phenomena*, J. Atm. Terr. Phys.**40**, 991-1000 (1978)**[25]**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**, https://doi.org/10.1007/BF00651873**[26]**S. P. Talwar,*Hydromagnetic stability of the magnetospheric boundary*, J. Geophysical Res.**69**, 2707-2713 (1964)**[27]**S. P. Talwar,*Kelvin-Helmholtz instability in an anisotropic plasma*, Phys. Fluids**8**, 1295-1299 (1965)**[28]**Zu-Yin Pu,*Kelvin-Helmholtz instability in collisionless space plasmas*, Phys. Fluids B**1**, 440-447 (1989)**[29]**B. A. Shrauner,*Small amplitude hydromagnetic waves for a plasma with a generalized polytrope law*, Plasma Phys.**15**, 375-385 (1973)**[30]**Giulio Mattei,*Jeans’s gravitational instability of an anisotropic plasma with generalized polytropic equations of state*, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)**64**(1978), no. 2, 170–176 (Italian, with English summary). MR**551644****[31]**D. Summers,*Polytropic solutions to the problem of spherically symmetric flow of an ideal gas*, Canad. J. Phys.**58**(1980), no. 8, 1085–1092 (English, with French summary). MR**586788**, https://doi.org/10.1139/p80-148**[32]**M. Dobróka,*Coupled waves in stratified plasmas, treated by generalized polytropic equation of state*, Plasma Phys.**24**, 1401-1410 (1982)**[33]**Kevin G. Brown and S. Roy Choudhury,*Kelvin-Helmholtz instabilities of high-velocity magnetized shear layers with generalized polytrope laws*, Quart. Appl. Math.**58**(2000), no. 3, 401–423. MR**1770646**, https://doi.org/10.1090/qam/1770646**[34]**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**, https://doi.org/10.1063/1.865669**[35]**H. P. Furth,*Prevalent instability of nonthermal plasma*, Phys. Fluids**6**, 48-57 (1963)**[36]**W. B. Thompson,*An introduction to plasma physics*, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto, Calif.-London, 1962. MR**0153257****[37]**Much of the content of Section 3C was motivated by the very constructive and focused comments of an anonymous referee, which led to significant improvement in the presentation of all the subsequent results.**[38]**Zu-Yin Pu and M. G. Kivelson,*Kelvin-Helmholtz instability at the magnetopause: Solution for compressible plasmas*, J. Geophysical Res.**88**, 841-852 (1983)

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
76E25,
76E20,
76X05

Retrieve articles in all journals with MSC: 76E25, 76E20, 76X05

Additional Information

DOI:
https://doi.org/10.1090/qam/1938343

Article copyright:
© Copyright 2002
American Mathematical Society