Abstract
Formal stability, i.e., positive energy linear stability, and nonlinear stability are defined. The generic way in which a system with infinite degrees of freedom may be nonlinearly unstable in spite of formal stability is illustrated. The “energy-Casimir” method of proof of nonlinear stability is applied to “relaxed states” of anisotropic magnetohydrodynamics (MHD). Such states are consistent with ergodic magnetic field lines and are formally stable under mild restrictions. Specifically, there is no free energy to drive interchange modes. Nonlinear stability is guaranteed under somewhat more restrictive conditions than is formal stability.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
D. Holm, J. Marsden, T. Ratiu, and A. Weinstein, Phys. Rep. 123, 1 (1985).
V. I. Arnold, Am. Math. Soc. Trans. 19, 267 (1969).
See, for example, R. Littlejohn, All-Conference Proceedings No. 88, 47 (1981).
J. Finn and T. Antonsen, Jr., Phys. Fluids 26, 3540 (1983).
P. Morrison and J. Greene, Phys. Rev. Lett. 45, 790 (1980).
G. Sun and J. Finn, to be submitted to Phys. Fluids.
C. Grebogi, E. Ott, J. Yorke, these proceedings, Fig. Ia.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1987 Springer-Verlag
About this paper
Cite this paper
Finn, J.M., Sun, GZ. (1987). Nonlinear stability in anisotropic magnetohydrodynamics. In: Kim, Y.S., Zachary, W.W. (eds) The Physics of Phase Space Nonlinear Dynamics and Chaos Geometric Quantization, and Wigner Function. Lecture Notes in Physics, vol 278. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-17894-5_320
Download citation
DOI: https://doi.org/10.1007/3-540-17894-5_320
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-17894-1
Online ISBN: 978-3-540-47901-7
eBook Packages: Springer Book Archive