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
\nabla ⋅ F=0,
this F can be expressed as the sum of a toroidal field T and poloidal vector field P
F=T+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,
T=\nabla x (r\Psi(r))
and the poloidal field is derived from another scalar field Φ(r, θ, φ), as a twice-iterated curl,
P=\nabla x (\nabla x (r\Phi(r))).
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,
r ⋅ T=0
while the curl of a poloidal field is tangential to those spheres
r ⋅ (\nabla x P)=0.
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
F(x,y,z)=\nabla x g(x,y,z)\hat{z
where
\hat{x