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] For a three-dimensional vector field F with zero divergence this
is a radial vector in spherical coordinates
[4] A toroidal vector field is tangential to spheres around the origin,[4] 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.[3] 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.