Abstract
We examine the magnetohydrodynamic (MHD) stability of a magnetic loop, taking into account field line tying at its foot points. We use the ideal MHD energy equation to derive a stability equation for a specific class of perturbations.
We found that for a loop with large aspect ratio (≳10) the field line tying effect is negligible to the m = 1 kink mode but important to the localized modes. The stability criterion for high m localized modes is derived and compared with the Suydam criterion. The result shows that for the perturbation of the class studied, there are two effects of field line tying; one is a field line bending effect which is always stabilizing and the other is a shear effect which is stabilizing or destabilizing depending on the sign of the gradient of potential magnetic field. The net effect of field line tying is determined by the sum of these two effects.
The result of this work is contrary to the result of Hood and Priest, in which they found that the field line tying effect is significant to the m = 1 mode. We believe that the contradiction comes from their incomplete minimization of the energy equation.
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References
An, C. H. and Bateman, G.: 1980, Oak Ridge National Laboratory Rep. ORNL/TM-7074.
Bateman, G.: 1978, MHD Instabilities, M.I.T. Press, Cambridge, Mass.
Bateman, G. and An, C. H.: 1977, not published.
Bernstein, I. B., Frieman, E. A., Kruskal, M. D., and Kulsrud, R. M.: 1958, Proc. Roy. Soc. A244, 17.
Cheng, C. C.: 1977, Astrophys. J. 213, 558.
Chuideri, C., Giachett, R., and Van Hoven, G.: 1977, Solar Phys. 54, 107.
Foukal, P. V.: 1975, Solar Phys. 43, 327.
Foukal, P. V.: 1976, Astrophys. J. 210, 575.
Furth, H. P., Killeen, J., and Rosenbluth, M. N.: 1963, Phys. Fluids 6, 459.
Giachett, R., Van Hoven, G., and Chiuderi, C.: 1977, Solar Phys. 55, 371.
Gibson, E. G.: 1977, Solar Phys. 53, 123.
Hood, A. W. and Priest, E. R.: 1979, Solar Phys. 64, 303.
Kadomtsev, B. B.: 1966, in M. A. Leontovich (ed.), Review of Plasma Physics 2, 153, Consultants Bureau, N.Y.
Low, B. C.: 1973a, Astrophys. J. 181, 917.
Low, B. C.: 1973b, Astrophys. J. 184, 917.
Newcomb, W. A.: 1960, Ann. Phys. 10, 232.
Raadu, M. A.: 1972, Solar Phys. 22, 425.
Robinson, D. C. and King, R. E.: 1969 Proc. 3rd Int. Conf. Plasma Physics and Controlled Nuclear Fusion Research, Novosibirsk, USSR 1968, IAEA, Vienna, Austria.
Rosner, R., Tucker, W. H., and Vaiana, G. S.: 1978, Astrophys. J. 220, 643.
Rust, D. M.: 1973, Sacramento Peak Obs. Contr. No. 209.
Shafranov, V. D.: 1970, Sov. Phys. - Tech. Phys. 15, 175.
Solovl'ev, L. S.: 1971, Soviet Atomic Energy 30, 14.
Somov, B. V. and Syrovatskii, S. I.: 1976, Sov. Phys. Usp. 19, No. 10, 813.
Spicer, D. S.: 1977, Solar Phys. 53, 305.
Suydam, B. R.: 1958, Proc. Second Internat. Conf. Peaceful Uses of Atomic Energy, (Geneva: United Nation) 31, 157.
Švestka, Z.: 1976, Solar Flares, D. Reidel Publ. Co., Dordrecht, Holland, p. 28.
Taylor, J. B.: 1974, in Proceedings of the Fifth International Conference on Plasma Physics and Controlled Nuclear Fusion Research, Tokyo, Japan, (International Atomic Energy Agency, Vienna), (1975), Vol. I, p. 161.
Van Hoven, G., Chiuderi, C., and Giachetti, R.: 1977, Astrophys. J. 213, 869.
Vorpahl, J. A., Gibson, E. G., Landecker, P. B., McKenzie, D. L., and Underwood, J. H.: 1975, Solar Phys. 45, 199.
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An, CH. The effects on the MHD stability of field line tying to the end faces of a cylindrical magnetic loop. Sol Phys 75, 19–34 (1982). https://doi.org/10.1007/BF00153457
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DOI: https://doi.org/10.1007/BF00153457