# Poloidal–toroidal decomposition

In vector calculus, a topic in pure and applied mathematics, a **poloidal–toroidal decomposition** is a restricted form of the Helmholtz decomposition. It is often used in the spherical coordinates analysis of solenoidal vector fields, for example, magnetic fields and incompressible fluids.[1]

## Definition

For a three-dimensional vector field **F** with zero divergence

this **F** can be expressed as the sum of a toroidal field **T** and poloidal vector field **P**

where **r** is a radial vector in spherical coordinates (*r*, *θ*, *φ*). The toroidal field is obtained from a scalar field, Ψ(*r*, *θ*, *φ*),[2] as the following curl,

and the poloidal field is derived from another scalar field Φ(*r*, *θ*, *φ*),[3] as a twice-iterated curl,

This decomposition is symmetric in that the curl of a toroidal field is poloidal, and the curl of a poloidal field is toroidal, known as Chandrasekhar-Kendall function.[4]

## Geometry

A toroidal vector field is tangential to spheres around the origin,[4]

while the curl of a poloidal field is tangential to those spheres

- .[5]

The poloidal–toroidal decomposition is unique if it is required that the average of the scalar fields Ψ and Φ vanishes on every sphere of radius *r*.[3]

## Cartesian decomposition

A poloidal–toroidal decomposition also exists in Cartesian coordinates, but a mean-field flow has to be included in this case. For example, every solenoidal vector field can be written as

where denote the unit vectors in the coordinate directions.[6]

## Notes

- Subrahmanyan Chandrasekhar (1961).
*Hydrodynamic and hydromagnetic stability*. International Series of Monographs on Physics. Oxford: Clarendon. See discussion on page 622. - Backus 1986, p. 87.
- Backus 1986, p. 88.
- Backus, Parker & Constable 1996, p. 178.
- Backus, Parker & Constable 1996, p. 179.
- Jones 2008, p. 17.

## References

*Hydrodynamic and hydromagnetic stability*, Chandrasekhar, Subrahmanyan; International Series of Monographs on Physics, Oxford: Clarendon, 1961, p. 622.- Decomposition of solenoidal fields into poloidal fields, toroidal fields and the mean flow. Applications to the boussinesq-equations, Schmitt, B. J. and von Wahl, W; in
*The Navier-Stokes Equations II — Theory and Numerical Methods*, pp. 291–305; Lecture Notes in Mathematics, Springer Berlin/ Heidelberg, Vol. 1530/ 1992. - Anelastic Magnetohydrodynamic Equations for Modeling Solar and Stellar Convection Zones, Lantz, S. R. and Fan, Y.; The Astrophysical Journal Supplement Series, Volume 121, Issue 1, Mar. 1999, pp. 247–264.
- Plane poloidal-toroidal decomposition of doubly periodic vector fields: Part 1. Fields with divergence and Part 2. Stokes equations. G. D. McBain. ANZIAM J. 47 (2005)
- Backus, George (1986), "Poloidal and toroidal fields in geomagnetic field modeling",
*Reviews of Geophysics*,**24**: 75–109, Bibcode:1986RvGeo..24...75B, doi:10.1029/RG024i001p00075. - Backus, George; Parker, Robert; Constable, Catherine (1996),
*Foundations of Geomagnetism*, Cambridge University Press, ISBN 0-521-41006-1. - Jones, Chris,
*Dynamo Theory*(PDF).