Toroidal coordinates: Difference between revisions

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[[File:Toroidal coordinates.png|400px|thumb|right|A toroidal co-ordinate system]]
[[File:Toroidal coordinates.png|400px|thumb|right|A toroidal coordinate system]]


Co-ordinate systems used in toroidal systems:
Coordinate systems used in toroidal systems:


== Cartesian ==
== Cartesian ==

Revision as of 18:42, 13 September 2009

A toroidal coordinate system

Coordinate systems used in toroidal systems:

Cartesian

(X, Y, Z) [1]

Cylindrical

(R, φ, Z), where [2]

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

φ is called the toroidal angle (and not the cylindrical angle, at least not in the context of magnetic confinement).

Simple toroidal

(r, φ, θ), where

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

R0, corresponding to the torus axis, is called the major radius and r the minor radius. θ is called the poloidal angle.

Toroidal

(ζ, η, φ), where [3] [4] [5]

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. Wikipedia:Cartesian coordinate system
  2. Wikipedia:Cylindrical coordinate system
  3. Morse and Feshbach, Methods of theoretical physics, McGraw-Hill, New York, 1953 ISBN 007043316X
  4. Wikipedia:Toroidal coordinates
  5. F. Alladio, F. Chrisanti, Analysis of MHD equilibria by toroidal multipolar expansions, Nucl. Fusion 26 (1986) 1143