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.
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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