Toroidal coordinates

A toroidal co-ordinate system

Co-ordinate systems used in toroidal systems:

Cartesian

(X, Y, Z)

Cylindrical

(R, φ, Z), where

• 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

$R=R_{p}{\frac {\sinh \zeta }{\cosh \zeta -\cos \eta }}$

$Z=R_{p}{\frac {\sin \eta }{\cosh \zeta -\cos \eta }}$

where R_p is the pole of the coordinate system. Surfaces of constant ζ are tori with major radii R = R_p/tanh ζ and minor radii r = R_p/sinh ζ. At R = R_p, 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.