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

For a three-dimensional vector field F with zero divergence

this F can be expressed as the sum of a toroidal field T and poloidal vector field 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),[2] as the following curl,

and the poloidal field is derived from another scalar field Φ(r, θ, φ),[3] as a twice-iterated curl,

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]

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

.[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

where denote the unit vectors in the coordinate directions.[6]

See also

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. 1 2 Backus 1986, p. 88.
  4. 1 2 Backus, Parker & Constable 1996, p. 178.
  5. ↑ Backus, Parker & Constable 1996, p. 179.
  6. ↑ Jones 2008, p. 62.

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 in 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 .
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