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


 * $$ \nabla \cdot \mathbf{F} = 0, $$

this F can be expressed as the sum of a toroidal field T and poloidal vector field P


 * $$\mathbf{F} = \mathbf{T} + \mathbf{P} $$

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,


 * $$ \mathbf{T} = \nabla \times (\mathbf{r} \Psi(\mathbf{r})) $$

and the poloidal field is derived from another scalar field Φ(r, θ, φ), as a twice-iterated curl,


 * $$ \mathbf{P} = \nabla \times (\nabla \times (\mathbf{r} \Phi (\mathbf{r})))\,.$$

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.

Geometry
A toroidal vector field is tangential to spheres around the origin,


 * $$ \mathbf{r} \cdot \mathbf{T} = 0 $$

while the curl of a poloidal field is tangential to those spheres


 * $$ \mathbf{r} \cdot (\nabla \times \mathbf{P}) = 0. $$

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.

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


 * $$\mathbf{F}(x,y,z) = \nabla \times g(x,y,z) \hat{\mathbf{z}} + \nabla \times (\nabla \times h(x,y,z) \hat{\mathbf{z}}) + b_x(z) \hat{\mathbf{x}} + b_y(z)\hat{\mathbf{y}}, $$

where $$\hat{\mathbf{x}}, \hat{\mathbf{y}}, \hat{\mathbf{z}}$$ denote the unit vectors in the coordinate directions.