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

Further information: Vector operator

For a three-dimensional vector field F with zero divergence

[math]\displaystyle{ \nabla \cdot \mathbf{F} = 0, }[/math]

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

[math]\displaystyle{ \mathbf{F} = \mathbf{T} + \mathbf{P} }[/math]

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,

[math]\displaystyle{ \mathbf{T} = \nabla \times (\mathbf{r} \Psi(\mathbf{r})) }[/math]

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

[math]\displaystyle{ \mathbf{P} = \nabla \times (\nabla \times (\mathbf{r} \Phi (\mathbf{r})))\,. }[/math]

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,

[math]\displaystyle{ \mathbf{r} \cdot \mathbf{T} = 0 }[/math]

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

[math]\displaystyle{ \mathbf{r} \cdot (\nabla \times \mathbf{P}) = 0. }[/math]

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

[math]\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}}, }[/math]

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

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)
• Backus, George (1986), "Poloidal and toroidal fields in geomagnetic field modeling", Reviews of Geophysics 24: 75–109, doi:10.1029/RG024i001p00075, Bibcode: 1986RvGeo..24...75B .
• Backus, George; Parker, Robert; Constable, Catherine (1996), Foundations of Geomagnetism, Cambridge University Press, ISBN 0-521-41006-1 .
• Jones, Chris, Dynamo Theory, http://www1.maths.leeds.ac.uk/~cajones/LesHouches/chapter.pdf .