Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-29T13:10:59.308Z Has data issue: false hasContentIssue false
  • English
  • Français

Parametric instabilities of circularly polarized small-amplitude Alfvén waves in Hall plasmas

Published online by Cambridge University Press:  01 February 2008

MICHAEL S. RUDERMAN  and
PHILIPPE CAILLOL
MICHAEL S. RUDERMAN
Affiliation:
Department of Applied Mathematics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, UK ([email protected])
PHILIPPE CAILLOL
Affiliation:
Department of Applied Mathematics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, UK ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the stability of circularly polarized Alfvén waves (pump waves) in Hall plasmas. First we re-derive the dispersion equation governing the pump wave stability without making an ad hoc assumption about the dependences of perturbations on time and the spatial variable. Then we study the stability of pump waves with small non-dimensional amplitude a (a ≪ 1) analytically, restricting our analysis to b < 1, where b is the ratio of the sound and Alfvén speed. Our main results are the following. The stability properties of right-hand polarized waves are qualitatively the same as in ideal MHD. For any values of b and the dispersion parameter τ they are subject to decay instability that occurs for wave numbers from a band with width of order a. The instability increment is also of order a. The left-hand polarized waves can be subject, in general, to three different types of instabilities. The first type is the modulational instability. It only occurs when b is smaller than a limiting value that depends on τ. Only perturbations with wave numbers smaller than a limiting value of order a are unstable. The instability increment is proportional to a2. The second type is the decay instability. It has the same properties as in the case of right-hand polarized waves; however, it occurs only when b < 1 τ. The third type is the beat instability. It occurs for any values of b and τ, and only perturbations with the wave numbers from a narrow band with the width of order a2 are unstable. The increment of this instability is proportional to a2, except for τ close to τc when it is proportional to a, where τc is a function of b.

Type
Papers
Information
Journal of Plasma Physics , Volume 74 , Issue 1 , February 2008 , pp. 119 - 138
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barnes, A. and Hollweg, J. V. 1974 Large-amplitude hydromagnetic waves. J. Geophys. Res. 79, 2302–2318.CrossRefGoogle Scholar
Brodin, G. and Lundberg, J. 1990 On the stability of a finite-amplitude electromagnetic-wave in an astrophysic-plasma. Phys. Scripta 42, 343–346.CrossRefGoogle Scholar
Brodin, G. and Stenflo, L. 1988 Parametric instabilities of finite amplitude Alfvén waves. Phys. Scripta 37, 89–92.CrossRefGoogle Scholar
Brodin, G. and Stenflo, L. 1990 Coupling coefficients for ion-cyclotron Alfvén waves. Contr. Plasma Phys. 30, 413–419.CrossRefGoogle Scholar
Cramer, N. F., Hertzberg, M. P. and Vladimirov, S. V. 2003 Parametric instabilities in magnetized bi-ion and dusty plasmas. Pramana-J. Phys. 61, 1171–1177.CrossRefGoogle Scholar
Del Zanna, L. and Velli, M. 2002 Coronal heating through Alfvén waves. Adv. Space. Res. 30, 471–480.CrossRefGoogle Scholar
Del Zanna, L., Velli, M. and Londrillo, P. 2001 Parametric decay of circularly polarized Alfvén waves: Multidimensional simulations in periodic and open domains. Astron. Astrophys. 367, 705–718.CrossRefGoogle Scholar
Derby, N. F. Jr., 1978 Modulational instability of finite-amplitude circularly polarized Alfvén waves. Astrophys. J. 224, 1013–1016.CrossRefGoogle Scholar
Forslund, D. W., Kindel, J. M. and Lindman, E. L. 1972 Parametric excitation of electromagnetic waves. Phys. Rev. Lett. 29, 249–252.CrossRefGoogle Scholar
Galeev, A. A. and Oraevskii, V. N. 1963 The stability of Alfvén waves. Sov. Phys. Dokl. 7, 988–993.Google Scholar
Ghosh, S. and Goldstein, M. L. 1994 Nonlinear evolution of a large-amplitude circularly-polarized Alfvén-wave—low-beta. J. Geophys. Res. 99, 13351–13362.CrossRefGoogle Scholar
Ghosh, S., Vinas, A. F. and Goldstein, M. L. 1993 Parametric-instabilities of a large-amplitude circularly-polarized Alfvén-wave—linear growth in 2-dimensional geometries. J. Geophys. Res. 98, 15561–15570.CrossRefGoogle Scholar
Ghosh, S., Vinas, A. F. and Goldstein, M. L. 1994 Nonlinear evolution of a large-amplitude circularly-polarized Alfvén-wave—high-beta. J. Geophys. Res. 99, 19289–19300.CrossRefGoogle Scholar
Goldstein, M. L. 1978 An instability of finite-amplitude circularly polarized Alfvén waves. Astrophys. J. 219, 700–704.CrossRefGoogle Scholar
Hertzberg, M. P., Cramer, N. F. and Vladimirov, S. V. 2003 Parametric instabilities of Alfvén waves in a dusty plasma. Phys. Plasmas 10, 3160–3167.CrossRefGoogle Scholar
Hertzberg, M. P., Cramer, N. F. and Vladimirov, S. V. 2004a Parametric instabilities in magnetized multicomponent plasmas. Phys. Rev. E 69, 056402.Google ScholarPubMed
Hertzberg, M. P., Cramer, N. F. and Vladimirov, S. V. 2004b Modulational and decay instabilities of Alfvén waves in a multicomponent plasma. J. Geophys. Res. 109, A02103.CrossRefGoogle Scholar
Hollweg, J. V., Esser, R. and Jayanti, V. 1993 Modulational and decay instabilities of Alfvén waves: effect of streaming He++. J. Geophys. Res. 98, 3491–3500.CrossRefGoogle Scholar
Inhester, B. 1990 A drift-kinetic treatment of the parametric decay of large-amplitude Alfvén waves. J. Geophys. Res. 95, 10525–10539.CrossRefGoogle Scholar
Jayanti, V. and Hollweg, J. V. 1993a On the dispersion relation for parametric instabilities of parallel-propagating Alfvén waves. J. Geophys. Res. 98, 13247–13252.CrossRefGoogle Scholar
Jayanti, V. and Hollweg, J. V. 1993b Parametric instabilities of parallel-propagating Alfvén waves: some analytical results. J. Geophys. Res. 98, 19049–19063.CrossRefGoogle Scholar
Jayanti, V. and Hollweg, J. V. 1994 Growth-rate of new parametric-instabilities occuring in a plasma with streaming He++. J. Geophys. Res. 99, 23449–23459.CrossRefGoogle Scholar
Khanna, M. and Rajaram, R. 1982 Evolution of nonlinear Alfvén waves propagating along the magnetic field in a collisionless plasma. J. Plasma Phys. 28, 459–468.CrossRefGoogle Scholar
Lashmore-Davies, C. N. and Stenflo, L. 1979 MHD stability of a helical magnetic-field of arbitrary amplitude. Plasma Phys. Control. Fusion 21, 735–740.Google Scholar
Ling, K. M. and Abraham-Shrauner, B. 1979 Parametric decay instability in high-speed solar wind streams. J. Geophys. Res. 84, 6713–6716.CrossRefGoogle Scholar
Longtin, M. and Sonnerup, B. U. Ö. 1986 Modulational instability of circularly polarized Alfvén waves. J. Geophys. Res. 91, –6824.CrossRefGoogle Scholar
Matsukiyo, S. and Hada, T. 2003 Parametric instabilities of circularly polarized Alfvén waves in a relativistic electron-positron plasma. Phys. Rev. E 67, 046406.Google Scholar
Mjølhus, E. and Wyller, J. 1988 Nonlinear Alfvén waves in a finite-beta plasma. J. Plasma Phys. 40, 299–318.CrossRefGoogle Scholar
Ovenden, C. R., Shah, H. A. and Schwartz, S. J. 2004 Alfvén Solitons in the solar wind. J. Geophys. Res. 88, 6095–6101.CrossRefGoogle Scholar
Ruderman, M. S. and Simpson, D. 2004a The stability of parallel-propagating circularly polarized Alfvén waves revisited. J. Plasma Phys. 70, 143–153.CrossRefGoogle Scholar
Ruderman, M. S. and Simpson, D. 2004b Absolute and convective instabilities of parallel-propagating circularly polarized Alfvén waves: decay instability. Phys. Plasmas 11, 4178–4187.CrossRefGoogle Scholar
Ruderman, M. S. and Simpson, D. 2005 Absolute and convective instabilities of parallel-propagating circularly polarized Alfvén waves. Space Sci. Rev. 121, 287–297.CrossRefGoogle Scholar
Sagdeev, R. Z. and Galeev, A. A. 1969 Non-linear Plasma Theory. New York: W. A. Benjamin.Google Scholar
Sakai, J. I. and Sonnerup, B. U. Ö. 1983 Modulational instability of finite-amplitude dispersive Alfvén waves. J. Geophys. Res. 88, 9069–9079.CrossRefGoogle Scholar
Shukla, P. K. and Stenflo, L. 1985 Nonlinear propagation of electromagnetic ion-cyclotron Alfvén waves. Phys. Fluids 28, 1576–1578.CrossRefGoogle Scholar
Simpson, D. and Ruderman, M. S. 2005 Absolute and convective instabilities of parallel-propagating circularly polarized Alfvén waves: beat instability. Phys. Plasma 12, 062103.CrossRefGoogle Scholar
Simpson, D., Ruderman, M. S. and Erdélyi, R. 2006 Absolute and convective instabilities of parallel-propagating circularly polarized Alfvén waves: numerical results. Astron. Astrophys., 452, 641–646.CrossRefGoogle Scholar
Spangler, S. R. 1989 Kinetic effects of Alfvén wave non-linearity. I. Ponderomotive density fluctuations. Phys. Fluids 8, 1738–1746.CrossRefGoogle Scholar
Spangler, S. R. 1990 Kinetic effects of Alfvén wave non-linearity. II. The modified nonlinear-wave equation. Phys. Fluids B 2, 407–418.CrossRefGoogle Scholar
Spangler, S. R. 1997 Nonlinear evolution of MHD waves at the Earth's bow shock: opinions on the confrontations between theory, simulations, and measurements. Nonlinear waves and Chaos in Space Plasmas (ed. Hada, T. and Matsumoto, H.). Tokyo: TERRAPUB, pp. 171–223.Google Scholar
Stenflo, L. 1976 Influence of a circularly polarized electromagnetic-wave on a magnetized plasma. Phys. Scripta 14, 320–323.CrossRefGoogle Scholar
Vinas, A. F and Goldstein, M. L. 1991 Parametric instabilities of circularly polarized, large amplitude, dispersive Alfvén waves—excitation of obliquely-propagating daughter and side-band waves. J. Plasma. Phys. 46, 129–152.CrossRefGoogle Scholar
Wong, H. K. and Goldstein, M. L. 1986 Parametric instabilities of circularly polarized Alfvén waves. J. Geophys. Res. 91, 5617–5628.CrossRefGoogle Scholar