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

Notes

1. Subrahmanyan Chandrasekhar (1961). Hydrodynamic and hydromagnetic stability. International Series of Monographs on Physics. Oxford: Clarendon. See discussion on page 622.
2. Backus 1986, p. 87.
3. Backus 1986, p. 88.
4. Backus, Parker & Constable 1996, p. 178.
5. Backus, Parker & Constable 1996, p. 179.
6. Jones 2008, p. 17.

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, Bibcode:1986RvGeo..24...75B, doi:10.1029/RG024i001p00075.
• Backus, George; Parker, Robert; Constable, Catherine (1996), Foundations of Geomagnetism, Cambridge University Press, ISBN 0-521-41006-1.
• Jones, Chris (2008), Dynamo Theory (PDF).