Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T14:26:31.030Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear saturation of resistive modified tearing modes in a toroidal plasma. Part 1. Simon saturation theory in quasi-linear approximation

Published online by Cambridge University Press:  13 March 2009

J. B. Ehrman
J. B. Ehrman*
Affiliation:
Department of Applied Mathematics, University of Western Ontario, London, Canada
Get access Rights & Permissions [Opens in a new window]

Abstract

The nonlinear saturation of a resistive modified tearing mode in a toroidal plasma is calculated by Simon saturation theory for a weakly linearly unstable (over-stable) regime. The linearly unstable mode is a (1, 1) mode, where the first 1 indicates n, the toroidal mode number about an axisymmetric equilibrium and the second 1 indicates the multiple of the non-zero frequency at which the n = 1 mode oscillates when linearly marginally stable. The saturation amplitude is found to be proportional to (Δ – Δc)½ where Δ, a measure of the driving energy of the instability, is proportional to the difference between the logarithmic derivatives of the radial (i.e. perpendicular to the magnetic surface) perturbation magnetic field in the simplest case, and Δc measures toroidal stabilization due to average magnetic line curvature. To obtain the Simon saturation condition, one must go to third order in the small parameter (Δ –Δc)½. Starting with (1, 1), (1, – 1),( – 1, – 1), and( –1, 1 ) modes in first order in (Δ – Δc)½, the modes (0, 0), (0, 2), (2, 2), (2, 0) are obtained in second order, and these are driven with first-order modes to a third-order (1,1) mode which yields a saturation condition. In the quasi-linear approximation, only the near-zero frequency modes (0, 0) and (2, 0) are considered in the second order. In the present paper, only the axisymmetric perturbation (0,0) is used in second order. This gives a relation between nonlinear saturation amplitude and frequency shift, but does not determine either uniquely because of the undetermined parameter γ in the perturbation solutions. This parameter is determined exactly by the requirement of finiteness of the solutions when the (2, 0) non-axisymmetric near-zero frequency perturbation is taken into account in second order. However, magnetic island width at saturation can still be estimated taking only the (0,0) second order mode because this width depends so insensitively on γ, namely as γ–¼. Under this restriction to (0, 0) in second order, the frequency shift is found to be negative. Its absolute value is proportional to Δ – Δc, while the magnetic island width is proportional to (Δ – Δc)¼ and has a scale determined by t, the thickness of the resistive layer, proportional to η⅓, the cube root of the electrical resistivity.

Type
Research Article
Information
Journal of Plasma Physics , Volume 30 , Issue 3 , December 1983 , pp. 423 - 461
Copyright
Copyright © Cambridge University Press 1983

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

Bailey, W. N. 1935 Generalized Hypergeometric Functions. Cambridge University Press.Google Scholar
Bateman, G. 1978 MHD Instabilities. MIT Press.Google Scholar
Coppi, B., Greene, J. M. & Johnson, J. L. 1966 Nucl. Fusion, 6, 101.CrossRefGoogle Scholar
Erdélyi, A. 1953 Higher Transcendental Functions, vol. 2. McGraw-Hill.Google Scholar
Furth, H. P., Killeen, J. & Rosenbluth, M. N. 1963 Phys. Fluid, 6, 459.CrossRefGoogle Scholar
Glasser, A. H., Greene, J. M. & Johnson, J. L. 1975 Phys. Fluids, 18, 875.CrossRefGoogle Scholar
Luke, Y. L. 1962 Integrals of Bessel Functions. McGraw-Hill.Google Scholar
Luke, Y. L. 1969 The Special Functions and Their Approximations, vol. 1. Academic.Google Scholar
McLachlan, N. W. 1934 Bessel Functions for Engineers. Oxford University Press.Google Scholar
Mercier, C. 1962 Nucl. Fusion Suppl. Pt. 2, 801.Google Scholar
Rutherford, P. H. 1973 Phys. Fluids, 16, 1903.CrossRefGoogle Scholar
Simon, A. 1968 Phys. Fluids, 11, 1181.CrossRefGoogle Scholar
Simon, A. & Rosenbluth, M. N. 1976 Phys. Fluids, 19, 1567.CrossRefGoogle Scholar
Sykes, A. & Wesson, J. A. 1980 Proceedings of 8th International Conference on Plasma Physics and Controlled Fusion, Brussels, vol. 1, p. 237. IAEA.Google Scholar
White, R. B., Monticello, D. A., Rosenbluth, M. N. & Waddell, B. V. 1977 Phys. Fluids, 20, 800.CrossRefGoogle Scholar