Poloidal–toroidal decomposition

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), 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.

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 x ^ , y ^ , z ^ denote the unit vectors in the coordinate directions.