Toroidal coordinates

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A toroidal co-ordinate system

Co-ordinate systems used in toroidal systems:

Eulerian

(X, Y, Z)

Cylindrical

(R, φ, Z), where

  • R2 = X2 + Y2, and
  • tan φ = Y/X.

Simple toroidal

(r, φ, θ), where

  • R = R0 + r cos θ, and
  • Z = r sin θ

(R0 corresponding to the axis of the torus)

Toroidal

(ζ, η, φ), where [1] [2] [3]

where Rp is the pole of the coordinate system. Surfaces of constant ζ are tori with major radii R = Rp/tanh ζ and minor radii r = Rp/sinh ζ. At R = Rp, ζ = ∞, while at infinity and at R = 0, ζ = 0. The coordinate η is a poloidal angle and runs from 0 to 2π. This system is orthogonal.

Magnetic

See Flux surface.

References

  1. Morse and Feshbach, Methods of theoretical physics, McGraw-Hill, New York, 1953 ISBN 007043316X
  2. Wikipedia:Toroidal_coordinates
  3. F. Alladio, F. Chrisanti, Analysis of MHD equilibria by toroidal multipolar expansions, Nucl. Fusion 26 (1986) 1143