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

$\mathrm{\nabla }\cdot 𝐅=0,$

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 $(r,\theta ,\varphi )$. The toroidal field is obtained from a scalar field,$\mathrm{\Psi }(r,\theta ,\varphi )$,Template:Sfn as the following curl,

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

and the poloidal field is derived from another scalar field $\mathrm{\Phi }(r,\theta ,\varphi )$,Template:Sfn as a twice-iterated curl,

$𝐏=\mathrm{\nabla }\times (\mathrm{\nabla }\times (𝐫\mathrm{\Phi }(𝐫))).$

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.Template:Sfn

Geometry

A toroidal vector field is tangential to spheres around the origin,Template:Sfn

$𝐫\cdot 𝐓=0$

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

$𝐫\cdot (\mathrm{\nabla }\times 𝐏)=0.$Template:Sfn

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.Template:Sfn

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

$𝐅(x,y,z)=\mathrm{\nabla }\times g(x,y,z)\widehat{𝐳}+\mathrm{\nabla }\times (\mathrm{\nabla }\times h(x,y,z)\widehat{𝐳})+b_x(z)\widehat{𝐱}+b_y(z)\widehat{𝐲},$

where $\widehat{𝐱},\widehat{𝐲},\widehat{𝐳}$ denote the unit vectors in the coordinate directions.Template:Sfn

See also

• Toroidal and poloidal
• Chandrasekhar–Kendall function

References

• Hydrodynamic and hydromagnetic stability, Chandrasekhar, Subrahmanyan; International Series of Monographs on Physics, Oxford: Clarendon, 1961, p. 622.
• Decomposition of solenoidal fields into poloidal fields, toroidal fields and the mean flow. Applications to the boussinesq-equations, Schmitt, B. J. and von Wahl, W; in The Navier–Stokes Equations II — Theory and Numerical Methods, pp. 291–305; Lecture Notes in Mathematics, Springer Berlin/ Heidelberg, Vol. 1530/ 1992.
• Anelastic Magnetohydrodynamic Equations for Modeling Solar and Stellar Convection Zones, Lantz, S. R. and Fan, Y.; The Astrophysical Journal Supplement Series, Volume 121, Issue 1, Mar. 1999, pp. 247–264.
• Plane poloidal-toroidal decomposition of doubly periodic vector fields: Part 1. Fields with divergence and Part 2. Stokes equations. G. D. McBain. ANZIAM J. 47 (2005)
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