[Table of Contents]

Plasma and Fusion Research

Volume 8, 1401057 (2013)

Regular Articles

Canonical Hamiltonian Model of Reduced MHD and its Comparison with the Two-Dimensional Euler System
Yuta KANEKO and Zensho YOSHIDA
Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa, Chiba 277-8561, Japan
(Received 14 November 2012 / Accepted 1 March 2013 / Published 22 May 2013)

Abstract

While expressing the ideal fluid/plasma equations in terms of Eulerian variables, we encounter a non-canonical Hamiltonian structure. In other words, Poisson operators determining symplectic geometry have nontrivial kernels that foliate phase spaces. There are several different recipes for “canonicalizing” such Hamiltonian formalisms by either reducing or extending phase spaces. Clebsch parametrization is a well-known method for reducing phase spaces. Here we introduce a new scheme that generalizes the Clebsch parametrization. Using the new set of variables, we delineate a fundamental difference between the reduced magnetohydrodynamic equations and the two-dimensional Euler equations.

Keywords

reduced MHD, Euler equation, Hamiltonian formalism, Clebsch parametrization, canonicalization

DOI: 10.1585/pfr.8.1401057

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References

  • [1] B.B. Kadomtsev and O.P. Pogutse, Sov. Phys. JETP 38, 283 (1974).
  • [2] M.N. Rosenbluth, D.A. Monticello, H.R. Strauss and R.B. White, Phys. Fluids 19, 1987 (1976).
  • [3] H.R. Strauss, Phys. Fluids 19, 134 (1976).
  • [4] H.R. Strauss, Phys. Fluids 20, 1354 (1977).
  • [5] P.J. Morrison and J.M. Greene, Phys. Rev. Lett. 45, 790 (1980).
  • [6] P.J. Morrison and R.D. Hazeltine, Phys. Fluids 27, 886 (1984).
  • [7] A. Clebsch, J. Reine Angew. Math. 56, 1 (1859).
  • [8] D.P. Stern, Am. J. Phys. 38, 494 (1970).
  • [9] P.J. Morrison, Rev. Mod. Phys. 70, 467 (1998).
  • [10] Z. Yoshida, J. Math. Phys. 50, 113101 (2009).
  • [11] Z. Yoshida and S.M. Mahajan, Plasma Phys. Control. Fusion 54, 014003 (2012).
  • [12] Z. Yoshida, Nonlinear Science–The Challenge of Complex Systems (Springer, Heidelberg, 2010) p. 102.
  • [13] Y. Fukumoto, Topologica 1, 003 (2008).
  • [14] V.S. Imshennik and S.I. Syrovatskii, Sov. Phys. JETP 25, 656 (1967).
  • [15] D. Biskamp, Magnetic Reconnection in Plasmas (Cambridge University Press, Cambridge, 2000) p. 47.

This paper may be cited as follows:

Yuta KANEKO and Zensho YOSHIDA, Plasma Fusion Res. 8, 1401057 (2013).