Toroidal and poloidal

The earliest use of these terms cited by the Oxford English Dictionary (OED) is by Walter M. Elsasser (1946) in the context of the generation of the Earth's magnetic field by currents in the core, with "toroidal" being parallel to lines of latitude and "poloidal" being in the direction of the magnetic field (i.e. towards the poles).

The OED also records the later usage of these terms in the context of toroidally confined plasmas, as encountered in magnetic confinement fusion. In the plasma context, the toroidal direction is the long way around the torus, the corresponding coordinate being denoted by z in the slab approximation or \zeta or \phi in magnetic coordinates; the poloidal direction is the short way around the torus, the corresponding coordinate being denoted by y in the slab approximation or \theta in magnetic coordinates. (The third direction, normal to the magnetic surfaces, is often called the "radial direction", denoted by x in the slab approximation and variously \psi, \chi, r, \rho, or s in magnetic coordinates.)

Toroidal and Poloidal Coordinates

As a simple example, consider an axisymmetric system with circular, concentric magnetic surfaces (a crude approximation to the magnetic field geometry in an early Tokamak) and denote the toroidal angle by \zeta, and the poloidal angle by \theta. Then the Toroidal/Poloidal coordinate system relates to standard Cartesian Coordinates by these transformation rules:

x = (R_0%2Br \cos \theta) \sin\zeta \,
y = (R_0%2Br \cos \theta) \cos\zeta \,
z = r \sin \theta. \,

See also

References