Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-30T11:15:38.816Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized Case—van Kampen modes in a multidimensional non-uniform plasma with application to gyroresonance heating

Published online by Cambridge University Press:  13 March 2009

E. R. Tracy ,
A. J. Brizard  and
A. N. Kaufman
E. R. Tracy
Affiliation:
Physics Department, College of William and Mary Williamsburg, Virginia 23185, U.S.A.
A. J. Brizard
Affiliation:
Lawrence Berkeley National Laboratory, Berkeley, California 94720, U.S.A.
A. N. Kaufman
Affiliation:
Lawrence Berkeley National Laboratory, Berkeley, California 94720, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

The generalization of the Case—van Kampen analysis to a multidimensional non-uniform plasma is presented. Application of this analysis is made to minority-ion gyrolesonant heating in an axisymmetric tokamak. In previous work the Case—van Kampen analysis, in conjunction with the Bateman—Kruskal algorithm, was used in a one-dimensional slab model to compute the collective wave spin-off (to the minority-ion Bernstein wave) and the gyroballistic continuum for minority gyroresonant absorption. The generalization to many gyroresonant dimensions and non-trivial geometries requires several important new developments: In tokamak geometry particles can be trapped, an effect that is absent in the slab model. Also, the ray propagation dynamics for both the flee gyroballistic waves and the collective minority-ion Bernstein wave is far more complicated than in the slab model. In particular, a resonance zone is identified wherein the gyroballistic waves interact strongly and cannot be treated as free. We use the Weyl calculus to construct a local form of the self-consistent gyroballistic equation within the resonance zone. This reduced equation is simplified via a metaplectic transformation (a generalization of the Fourier tiansformation). After this simplification, the equation is shown to be of Case—van Kampen type with weak non-uniformities; hence there are no true Case—van Kampen eigenfunctions. Using the Bateman—Kruskal approach, a local Case—van Kampen basis can be constructed and the initial-value problem solved. The self-consistent interactions of the gyroballistic continuum lead to a collective wave, the minority-ion Bernstein wave. The Bernstein wave is extracted by the spectral deformation approach of Crawford and Hislop. The relevance of this work to the theory of collective phenomena in nonlinear oscillator ensembles is briefly discussed.

Type
Research Article
Information
Journal of Plasma Physics , Volume 55 , Issue 3 , June 1996 , pp. 449 - 486
Copyright
Copyright © Cambridge University Press 1996

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

REFERENCES

Arnold, V. I. 1989 Mathematical Methods of Classical Mechanics, 2nd edn.Springer-Verlag, New York.Google Scholar
Baldwin, D. E. 1964 Phys. Fluids 7, 892.Google Scholar
Bateman, G. & Kruskal, M. D. 1972 Phys. Fluids 15, 277.CrossRefGoogle Scholar
Brizard, A. J. 1994 Phys. Plasmas 1, 2460.CrossRefGoogle Scholar
Brizard, A. J. & Kaufman, A. N. 1996 Phys. Plasmas 3, 64.Google Scholar
Cairns, R. A. 1991 Radiofrequency Heating of Plasmas. Adam Hilger, Bristol.Google Scholar
Case, K. M. 1959 Ann. Phys. (NY) 7, 349.CrossRefGoogle Scholar
Cook, D. R. 1993 Wave conversion in phase space and plasma gyroresonance. PhD thesis, University of California at Berkeley.Google Scholar
Cook, D. R., Kaufman, A. N., Tracy, E. R. & Flå, T. 1993 a Phys. Lett. 175A, 326.CrossRefGoogle Scholar
Cook, D. R., Kaufman, A. N., Brizard, A. J., Ye, H. & Tracy, E. R. 1993 b Phys. Lett. 178A, 413.CrossRefGoogle Scholar
Crawford, J. D. & Hislop, P. D. 1989 Ann. Phys. (NY) 189, 265.CrossRefGoogle Scholar
Fuchs, V. & Bers, A. 1988 Phys. Fluids 31, 3702.CrossRefGoogle Scholar
Hislop, P. D. & Crawford, J. D. 1989 J. Math. Phys. 30, 2819.Google Scholar
Hörmander, L. 1979 Commun. Pure Appl. Maths 32, 359.CrossRefGoogle Scholar
Imre, K. 1967 Phys. Fluids 10, 2226.CrossRefGoogle Scholar
Kaufman, A. N., Brizard, A. J., Cook, D. R., Tracy, E. R. & Ye, H. 1991 RF Power in Plasmas. AIP Conf. Proc., Vol. 244, p. 205.CrossRefGoogle Scholar
Kaufman, A. N., Brizard, A. J. & Tracy, E. R. 1995 RF Power in Plasmas. AIP Conf. Proc., Vol. 355, p. 239.Google Scholar
Larsson, J. 1991 Phys. Rev. Lett. 66, 1466.CrossRefGoogle Scholar
Littlejohn, R. G. 1983 J. Plasma Phys. 29, 111.CrossRefGoogle Scholar
Littlejohn, R. G. 1986 Phys. Rep. 138, 193.Google Scholar
Mcdonald, S. W. 1983 Wave dynamics of regular and chaotic rays. PhD thesis, University of California at Berkeley.Google Scholar
Mcdonald, S. W. & Kaufman, A. N. 1985 Phys. Rev. A32, 1708.Google Scholar
Muldrew, D. B. & Hagg, E. L. 1966 Can. J. Phys. 44, 925.CrossRefGoogle Scholar
Sedláček, Z. 1972 Czech. J. Phys. B22, 439.Google Scholar
Springer, G. 1981 Introduction to Riemann Surfaces. Chelsea, New York.Google Scholar
Strogatz, S. H., Mirollo, R. E. & Matthews, P. C. 1992 Phys. Rev. Lett. 68, 2730.Google Scholar
Tracy, E. R. & Kaufman, A. N. 1993 Phys. Rev. E48, 2196.Google Scholar
Van, Kampen N. G. 1955 Physica 21, 949.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley, New York.Google Scholar
Ye, H. & Kaufman, A. N. 1988 Phys. Rev. Lett. 60, 1642.CrossRefGoogle Scholar
Ye, H. & Kaufman, A. N. 1992 Phys. Fluids B4, 1735.CrossRefGoogle Scholar