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.
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), as the following curl,
and the poloidal field is derived from another scalar field Φ(r, θ, φ), 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.
A toroidal vector field is tangential to spheres around the origin,
while the curl of a poloidal field is tangential to those spheres
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.
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.
- ^ Subrahmanyan Chandrasekhar (1961). Hydrodynamic and hydromagnetic stability. International Series of Monographs on Physics. Oxford: Clarendon. See discussion on page 622.
- ^ Backus 1986, p. 87.
- ^ a b Backus 1986, p. 88.
- ^ a b Backus, Parker & Constable 1996, p. 178.
- ^ Backus, Parker & Constable 1996, p. 179.
- ^ Jones 2008, p. 17. sfn error: no target: CITEREFJones2008 (help)
- 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).