The initial-value problem for the Kelvin-Helmholtz instabilities of high-velocity and magnetized shear layers
Author:
S. Roy Choudhury
Journal:
Quart. Appl. Math. 54 (1996), 637-662
MSC:
Primary 76E25; Secondary 35Q35, 76W05
DOI:
https://doi.org/10.1090/qam/1417229
MathSciNet review:
MR1417229
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Abstract: The general initial-value problem for the linear Kelvin-Helmholtz instability of arbitrarily compressible velocity shear layers is considered for both the unmagnetized and magnetized cases. 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. The instability is found to remain, within the linear theory, of the translationally convective or shear type. No onset of rotational or vortex motion, i.e., formation of “coherent structures” occurs.
H. Aref and E. D. Siggia, Evolution and breakdown of a vortex street in two dimensions, J. Fluid Mech. 109, 435–463 (1981)
H. Aref and G. Tryggvason, Dynamics of passive and active interfaces, Physica 2D, 59–70 (1984)
M. C. Begelman, R. D. Blandford, and M. J. Rees, Theory of extragalactic radio sources, Rev. Modern Phys. 56, 255–351 (1984)
C. M. Bender and S. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978
R. D. Blandford and J. E. Pringle, Kelvin-Helmholtz instability of relativistic beams, Monthly Notices Roy. Astron. Soc. 176, 443–454 (1976)
W. Blumen, Shear layer instability of an inviscid compressible fluid, J. Fluid Mech. 40, 769–781 (1970)
W. Blumen, P. G. Drazin, and D. F. Billings, Shear layer instability of an inviscid compressible fluid, Part 2, J. Fluid Mech. 71, 305–316 (1975)
J. C. Brandt and D. A. Mendis, The solar wind, in Solar System Plasma Physics, edited by C. F. Kennel, L. Lanzerotti, and E. N. Parker, Amsterdam, North-Holland, 1979
H. S. Bridge, J. W. Belcher, A. J. Lazarus, J. D. Sullivan, F. Bagenal, R. L. McNutt, Jr., K. W. Ogilvie, J. D. Scudder, and E. C. Sittler, Plasma observations near Jupiter: Initial results, Science 206, 972–976 (1979)
G. L. Brown and A. Roshko, On density effects and large structure in turbulent mixing layers, J. Fluid Mech. 64, 775–816 (1974)
K. M. Case, Stability of inviscid plane Couette flow, Phys. Fluids 3, 143–148; and Stability of an idealized atmosphere I. Discussion of results, Phys. Fluids 3, 149–157 (1960)
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover (originally published 1961, Oxford, Clarendon), New York, 1981
H. Dobrowolny and N. D’Angelo, Wave motion in type I comet tails, in Cosmic Plasma Physics (K. Schindler, ed.), Plenum, New York, 1972
A. I. Ershkovich, A. A. Nusmov, and A. A. Chernikov, Oscillations of type I comet tails, Planetary and Space Science 20, 1235–1243 (1972)
J. A. Fejer, Hydromagnetic stability at a fluid velocity discontinuity between compressible fluids, Phys. Fluids 7, 499–503 (1964)
R. A. Gerwin, Stability of the interface between two fluids in relative motion, Rev. Modern Phys, 40, 652–658 (1968)
E. L. Ince, Ordinary Differential Equations, Dover, New York, 1956
J. R. Jokipii and L. Davis, Long wavelength turbulence and the heating of the solar wind, Astrophysical J. 156, 1101–1106 (1969)
K. J. Kaup, Coherent structures in the planar magnetron, Phys. Fluids B2, 2253–2258 (1990)
N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics, McGraw-Hill, New York, 1973
L. D. Landau, The instability of moving superposed fluids, Akad. Nauk. S.S.S.R., Comptes Rendus (Doklady) 44, 139–144 (1944)
T. N. Larosa and R. L. Moore, A mechanism for bulk energization in the impulsive phase of solar flares, Astrophysical J. 418, 912–918 (1993)
I. Lerche, Validity of the hydromagnetic approach in discussing instability of the magnetospheric boundary, J. Geophysical Res. 71, 2365–2371 (1966)
A. Michalke, On the inviscid instability of the hyperbolic-tangent velocity profile, J. Fluid Mech. 19, 543–556 (1964)
J. W. Miles, On wind over water, J. Acoustical Soc. America 29, 226–230 (1957)
J. W. Miles, On the disturbed motion of a plane vortex sheet, J. Fluid Mech. 4, 538–552 (1958)
A. Miura and P. L. Pritchett, Nonlocal stability analysis of the MHD Kelvin-Helmholtz instability in a compressible plasma, J. Geophysical Res. 87, 7431–7444 (1982)
A. Miura, Anomalous transport by magnetohydrodynamic Kelvin-Helmholtz instabilities in the solar wind-magnetosphere interaction, J. Geophysical Res. 89, 801–818 (1984)
M. Nepveu, Cylindrical jets, Astronomy and Astrophysics 84, 14–21 (1980)
N. F. Ness, M. H. Acuna, R. P. Lepping, J. E. P. Connerney, K. W. Behannon, and L. F. Burlaga, Magnetic field studies by Voyager I, Science 212, 211–217 (1981)
M. L. Norman, L. Smarr, K. H. A. Winkler, and M. D. Smith, Instabilities of cylindrical jets, Astronomy and Astrophysics 113, 285–351 (1982)
E. N. Parker, Interplanetary Dynamical Processes, Interscience, New York, 1963
P. L. Pritchett and F. V. Coroniti, The collisionless macroscopic Kelvin-Helmholtz instability 1. Transverse electrostatic mode, J. Geophysical Res. 89, 168–178 (1984)
Z. Y. Pu and M. G. Kivelson, Kelvin-Helmholtz instability at the magnetopause: Solution for compressible plasmas, J. Geophysical Res. 88, 841–852; and, Energy flux into the magnetosphere, J. Geophysical Res. 88, 853–861 (1983)
T. P. Ray, The effects of a simple shear layer on the growth of Kelvin-Helmholtz instabilities, Monthly Notices Roy. Astron. Soc. 198, 617–625 (1982)
T. P. Ray and A. I. Ershkovich, Kelvin-Helmholtz instabilities of magnetized shear layers, Monthly Notices Roy. Astron. Soc. 204, 821–826 (1983)
S. Roy Choudhury and R. V. E. Lovelace, On the Kelvin-Helmholtz instabilities of supersonic shear layers, Astrophysical J. 283, 331–342 (1984)
S. Roy Choudhury and R. V. E. Lovelace, On the Kelvin-Helmholtz instabilities of high-velocity magnetized shear layers, Astrophysical J. 302, 188–199 (1986)
S. Roy Choudhury, Kelvin-Helmholtz instabilities of supersonic, magnetized shear layers, J. Plasma Phys. 35, 375–392 (1986)
S. Roy Choudhury and V. L. Patel, Kelvin-Helmholtz instabilities of high-velocity, magnetized anisotropic shear layers, Phys. Fluids 28, 3292–3301 (1985)
S. Roy Choudhury, Global asymptotic analysis of the Kelvin-Helmholtz instability of supersonic shear layers, Canadian J. Phys. 68, 334–342 (1990)
A. K. Sen, Stability of the magnetospheric boundary, Planetary and Space Science 13, 131–141 (1965)
A. K. Sen, Effect of compressibility on Kelvin-Helmholtz instability in a plasma, Phys. Fluids 7, 1293–1298 (1964)
D. J. Southwood, The hydromagnetic stability of the magnetospheric boundary, Planetary and Space Science 16, 587–605 (1968)
D. J. Southwood, Some features of field-line resonances in the magnetosphere, Planetary and Space Science 22, 483–491 (1974)
P. A. Sturrock and R. E. Hartle, Two-fluid model of the solar wind, Physical Review Lett. 16, 628–631 (1966)
T. Tajima and J. N. Leboeuf, Kelvin-Helmholtz instability in supersonic and super-Alfvenic fluids, Phys. Fluids 23, 884–888 (1980)
S. P. Talwar, Kelvin-Helmholtz instability in an anisotropic plasma, Phys. Fluids 8, 1295–1299 (1965)
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)
C. D. Winant and F. K. Browand, Vortex pairing: The mechanism of turbulent mixing-layer growth, J. Fluid Mech. 63, 237–255 (1974)
H. Aref and E. D. Siggia, Evolution and breakdown of a vortex street in two dimensions, J. Fluid Mech. 109, 435–463 (1981)
H. Aref and G. Tryggvason, Dynamics of passive and active interfaces, Physica 2D, 59–70 (1984)
M. C. Begelman, R. D. Blandford, and M. J. Rees, Theory of extragalactic radio sources, Rev. Modern Phys. 56, 255–351 (1984)
C. M. Bender and S. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978
R. D. Blandford and J. E. Pringle, Kelvin-Helmholtz instability of relativistic beams, Monthly Notices Roy. Astron. Soc. 176, 443–454 (1976)
W. Blumen, Shear layer instability of an inviscid compressible fluid, J. Fluid Mech. 40, 769–781 (1970)
W. Blumen, P. G. Drazin, and D. F. Billings, Shear layer instability of an inviscid compressible fluid, Part 2, J. Fluid Mech. 71, 305–316 (1975)
J. C. Brandt and D. A. Mendis, The solar wind, in Solar System Plasma Physics, edited by C. F. Kennel, L. Lanzerotti, and E. N. Parker, Amsterdam, North-Holland, 1979
H. S. Bridge, J. W. Belcher, A. J. Lazarus, J. D. Sullivan, F. Bagenal, R. L. McNutt, Jr., K. W. Ogilvie, J. D. Scudder, and E. C. Sittler, Plasma observations near Jupiter: Initial results, Science 206, 972–976 (1979)
G. L. Brown and A. Roshko, On density effects and large structure in turbulent mixing layers, J. Fluid Mech. 64, 775–816 (1974)
K. M. Case, Stability of inviscid plane Couette flow, Phys. Fluids 3, 143–148; and Stability of an idealized atmosphere I. Discussion of results, Phys. Fluids 3, 149–157 (1960)
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover (originally published 1961, Oxford, Clarendon), New York, 1981
H. Dobrowolny and N. D’Angelo, Wave motion in type I comet tails, in Cosmic Plasma Physics (K. Schindler, ed.), Plenum, New York, 1972
A. I. Ershkovich, A. A. Nusmov, and A. A. Chernikov, Oscillations of type I comet tails, Planetary and Space Science 20, 1235–1243 (1972)
J. A. Fejer, Hydromagnetic stability at a fluid velocity discontinuity between compressible fluids, Phys. Fluids 7, 499–503 (1964)
R. A. Gerwin, Stability of the interface between two fluids in relative motion, Rev. Modern Phys, 40, 652–658 (1968)
E. L. Ince, Ordinary Differential Equations, Dover, New York, 1956
J. R. Jokipii and L. Davis, Long wavelength turbulence and the heating of the solar wind, Astrophysical J. 156, 1101–1106 (1969)
K. J. Kaup, Coherent structures in the planar magnetron, Phys. Fluids B2, 2253–2258 (1990)
N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics, McGraw-Hill, New York, 1973
L. D. Landau, The instability of moving superposed fluids, Akad. Nauk. S.S.S.R., Comptes Rendus (Doklady) 44, 139–144 (1944)
T. N. Larosa and R. L. Moore, A mechanism for bulk energization in the impulsive phase of solar flares, Astrophysical J. 418, 912–918 (1993)
I. Lerche, Validity of the hydromagnetic approach in discussing instability of the magnetospheric boundary, J. Geophysical Res. 71, 2365–2371 (1966)
A. Michalke, On the inviscid instability of the hyperbolic-tangent velocity profile, J. Fluid Mech. 19, 543–556 (1964)
J. W. Miles, On wind over water, J. Acoustical Soc. America 29, 226–230 (1957)
J. W. Miles, On the disturbed motion of a plane vortex sheet, J. Fluid Mech. 4, 538–552 (1958)
A. Miura and P. L. Pritchett, Nonlocal stability analysis of the MHD Kelvin-Helmholtz instability in a compressible plasma, J. Geophysical Res. 87, 7431–7444 (1982)
A. Miura, Anomalous transport by magnetohydrodynamic Kelvin-Helmholtz instabilities in the solar wind-magnetosphere interaction, J. Geophysical Res. 89, 801–818 (1984)
M. Nepveu, Cylindrical jets, Astronomy and Astrophysics 84, 14–21 (1980)
N. F. Ness, M. H. Acuna, R. P. Lepping, J. E. P. Connerney, K. W. Behannon, and L. F. Burlaga, Magnetic field studies by Voyager I, Science 212, 211–217 (1981)
M. L. Norman, L. Smarr, K. H. A. Winkler, and M. D. Smith, Instabilities of cylindrical jets, Astronomy and Astrophysics 113, 285–351 (1982)
E. N. Parker, Interplanetary Dynamical Processes, Interscience, New York, 1963
P. L. Pritchett and F. V. Coroniti, The collisionless macroscopic Kelvin-Helmholtz instability 1. Transverse electrostatic mode, J. Geophysical Res. 89, 168–178 (1984)
Z. Y. Pu and M. G. Kivelson, Kelvin-Helmholtz instability at the magnetopause: Solution for compressible plasmas, J. Geophysical Res. 88, 841–852; and, Energy flux into the magnetosphere, J. Geophysical Res. 88, 853–861 (1983)
T. P. Ray, The effects of a simple shear layer on the growth of Kelvin-Helmholtz instabilities, Monthly Notices Roy. Astron. Soc. 198, 617–625 (1982)
T. P. Ray and A. I. Ershkovich, Kelvin-Helmholtz instabilities of magnetized shear layers, Monthly Notices Roy. Astron. Soc. 204, 821–826 (1983)
S. Roy Choudhury and R. V. E. Lovelace, On the Kelvin-Helmholtz instabilities of supersonic shear layers, Astrophysical J. 283, 331–342 (1984)
S. Roy Choudhury and R. V. E. Lovelace, On the Kelvin-Helmholtz instabilities of high-velocity magnetized shear layers, Astrophysical J. 302, 188–199 (1986)
S. Roy Choudhury, Kelvin-Helmholtz instabilities of supersonic, magnetized shear layers, J. Plasma Phys. 35, 375–392 (1986)
S. Roy Choudhury and V. L. Patel, Kelvin-Helmholtz instabilities of high-velocity, magnetized anisotropic shear layers, Phys. Fluids 28, 3292–3301 (1985)
S. Roy Choudhury, Global asymptotic analysis of the Kelvin-Helmholtz instability of supersonic shear layers, Canadian J. Phys. 68, 334–342 (1990)
A. K. Sen, Stability of the magnetospheric boundary, Planetary and Space Science 13, 131–141 (1965)
A. K. Sen, Effect of compressibility on Kelvin-Helmholtz instability in a plasma, Phys. Fluids 7, 1293–1298 (1964)
D. J. Southwood, The hydromagnetic stability of the magnetospheric boundary, Planetary and Space Science 16, 587–605 (1968)
D. J. Southwood, Some features of field-line resonances in the magnetosphere, Planetary and Space Science 22, 483–491 (1974)
P. A. Sturrock and R. E. Hartle, Two-fluid model of the solar wind, Physical Review Lett. 16, 628–631 (1966)
T. Tajima and J. N. Leboeuf, Kelvin-Helmholtz instability in supersonic and super-Alfvenic fluids, Phys. Fluids 23, 884–888 (1980)
S. P. Talwar, Kelvin-Helmholtz instability in an anisotropic plasma, Phys. Fluids 8, 1295–1299 (1965)
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)
C. D. Winant and F. K. Browand, Vortex pairing: The mechanism of turbulent mixing-layer growth, J. Fluid Mech. 63, 237–255 (1974)
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© Copyright 1996
American Mathematical Society
