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.

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. The toroidal field is obtained from a scalar field, Ψ, as the following curl,
and the poloidal field is derived from another scalar field 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.

Geometry

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.

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.