Toroidal and poloidal coordinates

A diagram depicting the poloidal ($\theta$) direction, represented by the red arrow, and the toroidal ($\zeta$ or $\phi$) direction, represented by the blue arrow.

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 iCartesian Coordinates by these transformation rules:

$x=(R_{0}+r\cos \theta )\sin \zeta \,$

$y=(R_{0}+r\cos \theta )\cos \zeta \,$

$z=r\sin \theta .\,$

See also

• Toroidal coordinates
• Zonal and poloidal
• Poloidal - Toroidal Decomposition
• Zonal flow (plasma)

References

• "Oxford English Dictionary Online". poloidal. Oxford University Press. Retrieved 2007-08-10.
• Elsasser, W. M. (1946). "Induction Effects in Terrestrial Magnetism, Part I. Theory". Phys. Rev. 69: 106--116. doi:10.1103/PhysRev.69.106. Retrieved 2007-08-10.