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

:$\nabla \cdot \mathbf = 0,$

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

:$\mathbf = \mathbf + \mathbf$

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,

:$\mathbf = \nabla \times (\mathbf \Psi(\mathbf))$

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

:$\mathbf = \nabla \times (\nabla \times (\mathbf \Phi (\mathbf)))\,.$

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,

:$\mathbf \cdot \mathbf = 0$

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

:$\mathbf \cdot (\nabla \times \mathbf) = 0.$

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

: $\mathbf(x,y,z) = \nabla \times g(x,y,z) \hat + \nabla \times (\nabla \times h(x,y,z) \hat) + b_x(z) \hat + b_y(z)\hat,$

where $\hat, \hat, \hat$ denote the unit vectors in the coordinate directions.

See also

* Toroidal and poloidal
* Chandrasekhar–Kendall function

Notes

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

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G. D. McBain
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{{DEFAULTSORT:Poloidal-toroidal decomposition
Vector calculus