Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



The initial-value problem for the 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. 60 (2002), 657-673
MSC: Primary 76E25; Secondary 76E20, 76X05
DOI: https://doi.org/10.1090/qam/1939005
MathSciNet review: MR1939005
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Abstract: The general initial-value problem for the linear Kelvin-Helmholtz instability of arbitrarily compressible magnetized anisotropic velocity shear layers is considered. The time evolution of the physical quantities characterizing the layer is treated using Laplace transform techniques. Singularity analysis of the resulting equations using Fuchs-Frobenius theory yields the large-time asymptotic solutions. Since all the singular points turned out to be real, the instability is found to remain, within the linear theory, of the translationally convective shear type. No onset of rotational or vortex motion, i.e., formation of ``coherent structures'' occurs because there are no imaginary singularities.

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  • 1. Abraham-Shrauner, B., Small amplitude hydromagnetic waves for a plasma with a generalized polytrope law, Plasma Phys. 15, 375-385 (1973)
  • 2. Aref, H. and Siggia, E. D., Evolution and breakdown of a vortex street in two dimensions, Journal of Fluid Mechanics 109, 435-463 (1981)
  • 3. Aref, H. and Tryggvason, G., Dynamics of passive and active interfaces, Physica 2D, 59-70 (1984)
  • 4. Begelman, M. C., Blandford, R. D., and Rees, M. J., Theory of extragalactic radio sources, Review of Modern Physics 56, 255-351 (1984)
  • 5. Carl M. Bender and Steven A. Orszag, Advanced mathematical methods for scientists and engineers, McGraw-Hill Book Co., New York, 1978. International Series in Pure and Applied Mathematics. MR 538168
  • 6. Blandford, R. D. and Pringle, J. E., Kelvin-Helmholtz instability of relativistic beams, Monthly Notices Roy. Astron. Soc. 176, 443-454 (1976)
  • 7. Blumen, W., Shear layer instability of an inviscid compressible fluid, Journal of Fluid Mechanics 40, 769-781 (1970)
  • 8. Blumen, W., Drazin, P. G., and Billings, D. F., Shear layer instability of an inviscid compressible fluid, Part 2, Journal of Fluid Mechanics 71, 305-316 (1975)
  • 9. Brandt, J. C. and Mendis, D. A. (1979), The solar wind in ``Solar System Plasma Physics'' (C. F. Kennel, L. Lanzerotti, and E. N. Parker, Eds.), Amsterdam: North Holland
  • 10. Bridge, H. S., Belcher, J. W., Lazarus, A. J., Sullivan, J. D., Bagenal, F., McNutt, Jr., R. L., Oglivie, K. W., Scudder, J. D., and Sittler, E. C., Plasma observations near Jupiter: Initial results, Science 206, 972-976 (1979)
  • 11. Brown, G. L. and Roshko, A., On density effects and large structure in turbulent mixing layers, Journal of Fluid Mechanics 64, 775-816 (1974)
  • 12. 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
  • 13. Kevin G. Brown and S. Roy Choudhury, An analytical study of the Kelvin-Helmholtz instabilities of compressible, magnetized tangential velocity discontinuities with generalized polytrope laws, Quart. Appl. Math. 60 (2002), no. 4, 601–630. MR 1938343, https://doi.org/10.1090/qam/1938343
  • 14. K. M. Case, Stability of inviscid plane Couette flow, Phys. Fluids 3 (1960), 143–148. MR 0128230, https://doi.org/10.1063/1.1706010
  • 15. S. Chandrasekhar, Hydrodynamic and hydromagnetic stability, The International Series of Monographs on Physics, Clarendon Press, Oxford, 1961. MR 0128226
  • 16. Dobrowolny, H. and D'Angelo, N., Wave motion in type I comet tails, in Cosmic Plasma Physics (I. Schindler, ed.), Plenum, New York, 1972
  • 17. Duhau, S., Gratton, F., and Gratton, J., Hydromagnetic oscillations of a tangential discontinuity in the CGL approximation, Phys. Fluids 13, 1503-1509 (1970)
  • 18. Duhau, S., Gratton, F., and Gratton, J., Radiation of hydromagnetic waves from a tangential velocity discontinuity, Phys. Fluids 14, 2067-2071 (1971)
  • 19. Duhau, S. and Gratton, J., Effect of compressibility on the stability of a vortex sheet in an ideal magnetofluid, Phys. Fluids 16, 150-152 (1972)
  • 20. Ershkovich, A. I., Nusnov, A. A., and Chernikov, A. A., Oscillations of type I comet tails, Planetary and Space Science 20, 1235-1243 (1972)
  • 21. Ershkovich, A. I. and Chernikov, A. A., Nonlinear waves in type I comet tails, Planetary and Space Science 21, 663-673 (1973)
  • 22. J. A. Fejer, Hydromagnetic stability at a fluid velocity discontinuity between compressible fluids, Phys. Fluids 7 (1964), 499–503. MR 0163553, https://doi.org/10.1063/1.1711229
  • 23. Gerwin, R. A., Stability of the interface between two fluids in relative motion, Rev. Modern Phys. 40, 652-658 (1968)
  • 24. E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. MR 0010757
  • 25. Jokipii, J. R. and Davis, L., Long wavelength turbulence and the heating of the solar wind, Astrophys. Journal 156, 1101-1106 (1969)
  • 26. D. J. Kaup, Coherent structures in the planar magnetron, Phys. Fluids B 2 (1990), no. 9, 2253–2258. MR 1068288, https://doi.org/10.1063/1.859407
  • 27. Krall, N. A. and Trivelpiece, A. W. (1973), ``Principles of Plasma Physics.'' New York: McGraw-Hill
  • 28. Landau, L. D., The instability of moving superposed fluids, Akad. Nauk SSSR, Comptes Rendus (Doklady) 44, 139-144 (1944)
  • 29. L. D. Landau and E. M. Lifshitz, Course of theoretical physics. Vol. 8, Pergamon International Library of Science, Technology, Engineering and Social Studies, Pergamon Press, Oxford, 1984. Electrodynamics of continuous media; Translated from the second Russian edition by J. B. Sykes, J. S. Bell and M. J. Kearsley; Second Russian edition revised by Lifshits and L. P. Pitaevskiĭ. MR 766230
  • 30. Larosa, T. N. and Moore, R. L., A mechanism for bulk energization in the impulsive phase of solar flares, Astrophysical Journal 418, 912-918 (1993)
  • 31. Lerche, I., Validity of the hydromagnetic approach in discussing instability of the magnetospheric boundary, Journal of Geophys. Res. 71, 2365-2371 (1966)
  • 32. A. Michalke, On the inviscid instability of the hyperbolic-tangent velocity profile, J. Fluid Mech. 19 (1964), 543–556. MR 0184516, https://doi.org/10.1017/S0022112064000908
  • 33. Miles, J. W., On wind over water, Journal Acoust. Soc. Amer. 29, 226-230 (1957)
  • 34. John W. Miles, On the disturbed motion of a plane vortex sheet, J. Fluid Mech. 4 (1958), 538–552. MR 0097930, https://doi.org/10.1017/S0022112058000653
  • 35. Miura, A. and Pritchett, P. L., Nonlocal stability analysis of the MHD Kelvin-Helmholtz instability in a compressible plasma, J. Geophysical Res. 87, 7431-7444 (1982)
  • 36. Miura, A., Anomalous transport by magnetohydrodynamic Kelvin-Helmholtz instabilities in the solar wind-magnetosphere interaction, J. Geophysical Res. 89, 801-818 (1984)
  • 37. Nepveu, M., Cylindrical jets, Astronom. and Astrophys. 84, 14-21 (1980)
  • 38. Ness, N. F., Acuna, M. H., Lepping, R. P., Connerney, J. E. P., Behannon, K. W., and Burlaga, L. F., Magnetic field studies by Voyager I, Science 212, 211-217 (1981)
  • 39. Norman, M. L., Smarr, L., Winkler, K. H. A., and Smith, M. D., Instabilities of cylindrical jets, Astronomy and Astrophysics 113, 285-351 (1982)
  • 40. Parker, E. N. (1963), ``Interplanetary Dynamical Processes.'' New York: Interscience
  • 41. Pritchett, P. L. and Coroniti, F. V., The collisionless macroscopic Kelvin-Helmholtz instability I. Transverse electrostatic mode, Journal Geophys. Res. 89, 168-178 (1984)
  • 42. Pu, Z. Y., Kelvin-Helmholtz instability in collisionless space plasmas, Phys. Fluids B 1, 440-447 (1989)
  • 43. Pu, Z. Y. and Kivelson, M. G., Kelvin-Helmholtz instability at the magnetopause: Solution for compressible plasmas, Journal Geophys. Res. 88, 841-852; and, Energy flux into the magnetosphere, Journal Geophys. Res. 88, 853-861 (1983)
  • 44. Rajaram, R., Kalra, G. L., and Tandon, J. N., Discontinuities and the magnetospheric phenomena, J. Atm. Terr. Phys. 40, 991-1000 (1978)
  • 45. 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
  • 46. Ray, T. P., The effects of a simple shear layer on the growth of Kelvin-Helmholtz instabilities, Monthly Notices Roy. Astronom. Soc. 198, 617-625 (1982)
  • 47. Ray, T. P. and Erschkovich, A. I., Kelvin-Helmholtz instabilities of magnetized shear layers, Monthly Notices Roy. Astron. Soc. 204, 821-826 (1983)
  • 48. Roy Choudhury, S. and Lovelace, R. V. E., On the Kelvin-Helmholtz instabilities of supersonic shear layers, Astrophysical J. 283, 331-342 (1984)
  • 49. Roy Choudhury, S. and Lovelace, R. V. E., On the Kelvin-Helmholtz instabilities of high-velocity magnetized shear layers, Astrophysical J. 302, 188-199 (1986)
  • 50. Roy Choudhury, S., Kelvin-Helmholtz instabilities of supersonic, magnetized shear layers, J. Plasma Phys. 35, 375-392 (1986)
  • 51. Roy Choudhury, S. and Patel, V. L., Kelvin-Helmholtz instabilities of high-velocity, magnetized anisotropic shear layers, Phys. Fluids 28, 3292-3301 (1985)
  • 52. Roy Choudhury, S., Global asymptotic analysis of the Kelvin-Helmholtz instability of supersonic shear layers, Canadian J. Physics 68, 334-342 (1990)
  • 53. Sen, A. K., Stability of the magnetosphere boundary, Planetary and Space Science 13, 131-141 (1965)
  • 54. Amiya K. Sen, Effect of compressibiliy on Kelvin-Helmholtz instability in a plasma, Phys. Fluids 7 (1964), 1293–1298. MR 0165823, https://doi.org/10.1063/1.1711374
  • 55. Southwood, D. J., The hydromagnetic stability of the magnetospheric boundary, Planetary and Space Science 16, 587-605 (1968)
  • 56. Southwood, D. J., Some features of the field line resonances in the magnetosphere, Planetary and Space Science 22, 483-491 (1974)
  • 57. Sturrock, P. A. and Hartle, R. E., Two-fluid mode of the solar wind, Physical Review Lett. 16, 628-631 (1966)
  • 58. Syrovatskii, A., The Helmholtz instability, Soviet Physics Uspekhi 62, 247-253 (1957)
  • 59. John W. Miles, On Kelvin-Helmholtz instability, Phys. Fluids 23 (1980), no. 9, 1915–1916. MR 585970, https://doi.org/10.1063/1.863218
  • 60. Talwar, S. P., Hydromagnetic stability of the magnetospheric boundary, J. Geophysical Res. 69, 2707-2713 (1964)
  • 61. Talwar, S. P., Kelvin-Helmholtz instability in an anisotropic plasma, Phys. Fluids 8, 1295-1299 (1965)
  • 62. Turland, B. D. and Scheuer, P. A. G., Instabilities of Kelvin-Helmholtz type for relativistic streaming, Monthly Notices Roy. Astron. Soc. 176, 421-441 (1976)
  • 63. Winant, C. D. and Browand, F. K., Vortex pairing: The mechanism of turbulent mixing-layer growth, Journal Fluid Mech. 63, 237-255 (1974)

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DOI: https://doi.org/10.1090/qam/1939005
Article copyright: © Copyright 2002 American Mathematical Society

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