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On the analytic solution of the helical equilibrium equation in the MHD approximation

  • M. L. Woolley (a1)
Abstract

The second-order elliptic partial differential equation, which describes a class of static ideally conducting magnetohydrodynamic equilibria with helical symmetry, is solved analytically. When the equilibrium is contained within an infinitely long conducting cylinder, the appropriate Dirichiet boundary-value problem may be solved in general in terms of hypergeometric functions. For a countably infinite set of particular cases, these functions are polynomials in the radial co-ordinate; and a solution may be obtained in a closed form. Necessary conditions are given for the existence of the equilibrium, which is described by the simplest of these functions. It is found that the Dirichlet boundary-value problem is not well-posed for these equiilbria; and additional information (equivalent to locating a stationary value of the hydrodynamic pressure) must be provided, in order that the solution be unique.

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References
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Abramowitz M. & Stegun I. A. (ed.) 1965 Handbook of Mathematical Functions. Dover.
Burkill J. C. 1956 Theory of Ordinary Differential Equations. Oliver and Boyd.
Courant R. & Hilbert D. 1965 Methods of Mathematical Physics. Interscience.
Jeffreys H. J. & Jeffreys B. S. 1962 Methods of Mathematical Physics. Cambridge University Press.
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Journal of Plasma Physics
  • ISSN: 0022-3778
  • EISSN: 1469-7807
  • URL: /core/journals/journal-of-plasma-physics
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