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.^[1]

Definition

Further information: Vector operator

For a three-dimensional vector field F with zero divergence

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

this ${\displaystyle \mathbf {F} }$ can be expressed as the sum of a toroidal field ${\displaystyle \mathbf {T} }$ and poloidal vector field ${\displaystyle \mathbf {P} }$

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

where ${\displaystyle \mathbf {r} }$ is a radial vector in spherical coordinates ${\displaystyle (r,\theta ,\phi )}$. The toroidal field is obtained from a scalar field,${\displaystyle \Psi (r,\theta ,\phi )}$,^[2] as the following curl,

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

and the poloidal field is derived from another scalar field ${\displaystyle \Phi (r,\theta ,\phi )}$,^[3] as a twice-iterated curl,

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

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.^[4]

Geometry

A toroidal vector field is tangential to spheres around the origin,^[4]

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

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

${\displaystyle \mathbf {r} \cdot (\nabla \times \mathbf {P} )=0.}$^[5]

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]

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

${\displaystyle \mathbf {F} (x,y,z)=\nabla \times g(x,y,z){\hat {\mathbf {z} }}+\nabla \times (\nabla \times h(x,y,z){\hat {\mathbf {z} }})+b_{x}(z){\hat {\mathbf {x} }}+b_{y}(z){\hat {\mathbf {y} }},}$

where ${\displaystyle {\hat {\mathbf {x} }},{\hat {\mathbf {y} }},{\hat {\mathbf {z} }}}$ denote the unit vectors in the coordinate directions.^[6]

See also

• Toroidal and poloidal
• Chandrasekhar–Kendall function