Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-28T21:08:05.607Z Has data issue: false hasContentIssue false
  • English
  • Français

Variational formulation for weakly nonlinear perturbations of ideal magnetohydrodynamics

Published online by Cambridge University Press:  26 January 2011

M. HIROTA
M. HIROTA*
Affiliation:
Japan Atomic Energy Agency, Naka, Ibaraki, 311-0193, Japan ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

A new equation of motion that governs weakly nonlinear phenomena in ideal magnetohydrodynamics (MHDs) is derived as a natural extension of the well-known linearized equation of motion for the displacement field. This derivation is made possible by expanding the MHD Lagrangian explicitly up to third order with respect to the displacement of plasma, which necessitates an efficient use of the Lie series expansion. The resultant equation of motion (i.e. the Euler–Lagrange equation) includes a new quadratic force term which is responsible for various mode–mode coupling due to the MHD nonlinearity. The third-order potential energy serves to quantify the coupling coefficient among resonant three modes and its cubic symmetry proves the Manley–Rowe relations. In contrast to earlier works, the coupling coefficient is expressed only by the displacement vector field, which is already familiar in the linear MHD theory, and both the fixed and free boundary cases are treated systematically.

Type
Papers
Information
Journal of Plasma Physics , Volume 77 , Issue 5 , October 2011 , pp. 589 - 615
Copyright
Copyright © Cambridge University Press 2011

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

[1]Bernstein, I. B., Frieman, E. A., Kruskal, M. D. and Kulsrud, R. M. 1958 Proc. Roy. Soc. Lond. A 244, 17.Google Scholar
[2]Newcomb, W. A. 1962 Nucl. Fusion Suppl. Part 2, 451.Google Scholar
[3]Galeev, A. A. and Oraevski, V. N. 1963 Sov. Phys. Dokl. 7, 988.Google Scholar
[4]Sagdeev, R. Z. and Galeev, A. A. 1969 Nonlinear Plasma Theory, New York: Benjamin.Google Scholar
[5]Brodin, G. and Stenflo, L. 1988 J. Plasma Phys. 39, 277.CrossRefGoogle Scholar
[6]Webb, G. M., Zakharian, A. R., Brio, M. and Zank, G. P. 2000 J. Plamsa Phys. 63, 393.Google Scholar
[7]Larsson, J. 1996 J. Plasma Phys. 55, 261.CrossRefGoogle Scholar
[8]Larsson, J. 2000 J. Plasma Phys. 64, 155.CrossRefGoogle Scholar
[9]Axelsson, P. 2000 Nonlinearity 13, 19.CrossRefGoogle Scholar
[10]Morrison, P. J. and Greene, J. M. 1980 Phys. Rev. Lett. 45, 790. (ERRATA) 1982 Phys. Rev. Lett. 48, 569.CrossRefGoogle Scholar
[11]Frieman, E. and Rotenberg, M. 1960 Rev. Mod. Phys. 32, 898.CrossRefGoogle Scholar
[12]Dewar, R. L. 1970 Phys. Fluids 13, 2710.CrossRefGoogle Scholar
[13]Brizard, A. J. and Kaufman, A. N. 1995 Phys. Rev. Lett. 74, 4567.CrossRefGoogle Scholar
[14]Whitham, G. B. 1967 Proc. Roy. Soc. A 299, 6.Google Scholar
[15]Larsson, J. 2003 J. Plasma Phys. 69, 211.CrossRefGoogle Scholar
[16]Sattinger, D. H. and Weaver, O. L. 1986 Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics, New York: Springer-Verlag.CrossRefGoogle Scholar
[17]Coppi, B., Rosenbluth, M. N. and Sudan, R. N. 1969 Ann. Phys. 55, 207.CrossRefGoogle Scholar
[18]Hirota, M. and Fukumoto, Y. 2008 J. Math. Phys. 49, 083101.CrossRefGoogle Scholar
[19]Hirota, M. and Fukumoto, Y. 2008 Phys. Plasmas 15, 122101.CrossRefGoogle Scholar
[20]Andrews, D. G. and Mcintyre, M. E. 1978 J. Fluid Mech. 89, 609.CrossRefGoogle Scholar
[21]Hirota, M. and Tokuda, S. 2010 Phys. Plasmas 17, 082109.CrossRefGoogle Scholar
[22]Fukumoto, Y. and Hirota, M. 2008 Phys. Scr. T132, 014041.CrossRefGoogle Scholar