Abstract
Incontrovertible evidence is presented that the force-free magnetic fields exhibit strong stochastic behavior. Arnold's solution is given with the associated first integral of energy. A subset of the solution is shown to be non-ergodic whereas the full solution is shown to be ergodic. The first integral of energy is applied to the study of these fields to prove that the equilibrium points of such magnetic configurations are saddle points. Finally, the potential function of the first integral of energy is shown to be a member of the Helmholtz family of solutions. Numerical results corroborate the theoretical conclusions and demonstrate the robustness of the energy integral, which remains constant for arbitrarily long computing times.
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References
Arnold, V. I.: 1965, Comptes Rend. Acad. Sci., Paris 261, 17.
Bartle, R. G: 1976, The Elements of Real Analysis, John Wiley & Sons, New York, p. 379.
Birkhoff, G.: 1966, Dynamical Systems, American Mathematical Society, Vol. 106.
Chandrasekhar, S.: 1956a, Proc. Natl. Acad. Sci. 42, 1.
Chandrasekhar, S.: 1956b, Astrophys. J. 126, 232.
Chandrasekhar, C. and Kendall, P. C.: 1957, Astrophys. J. 126, 457.
Chandrasekhar, S. and Woltjer, L.: 1958, Proc. Natl. Acad. Sci. 44, 285.
Chintsin, A. J.: 1964, Mathematische Grundlagen der Statistischen Mechanik, Bibliographisches Institut, Mannheim.
Démoulin, P.: 1999, J. Atmospheric Solar-Terrest. Phys. 61, 101.
Dombre, T., Frisch, U., Greene, J. M., Hénon, M., Mehr A., and Soward, A. M.: 1986, J. Fluid Mech. 167, 353. After submitting our paper we discovered this paper with results similar to Equation (33).
Ferrar, W. L.: 1957, Algebra, Oxford University Press, Oxford, p. 148.
Golub, L. and Pasachoff, J. M.: 1997, The Solar Corona, Cambridge University Press, Cambridge.
Hansen, W. W.: 1935, Phys. Rev. 47, 139.
Hénon, M.: 1966, Comptes Rend. Acad. Sci., Paris 262, 312.
Jahnke, E. and Emde, F.: 1945, Tables of Functions, Dover Publications, New York, p. 168.
Landau, L. D. and Lifshitz, E. M.: 1993, Fluid Mechanics, Pergamon Press, Oxford, p. 13.
Lüst, R. and Schlüter, A.: 1954, Z. Astrophys. 34, 263.
Poletto, G., Vaiana, G. S., Zombeck, M. V., Krieger, A. S., and Timothy, A. F.: 1975, Solar Phys. 44, 83.
Priest, E. R.: 1984, Solar Magnetohydrodynamics, D. Reidel Publ. Co., Dordrecht, Holland.
Taylor, J. B.: 1986, Rev. Mod. Phys. 58, 741.
Woltjer, L.: 1958, Proc. Natl. Acad. Sci. 44, 489, 833.
Zirker, J. B., Martin, S. F., Harvey, K., and Gaizauskas, V.: 1997, Solar Phys. 175, 27.
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Evangelidis, E., Vaughan, L. & Botha, G. The structure of force-free magnetic fields. Solar Physics 193, 17–32 (2000). https://doi.org/10.1023/A:1005236824271
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DOI: https://doi.org/10.1023/A:1005236824271