Poloidal–toroidal decomposition

In vector calculus, a poloidal–toroidal decomposition is a restricted form of the Helmholtz decomposition that is often used in the spherical-coordinates analysis of solenoidal vector fields, for example, magnetic fields and incompressible fluids.^[1] For a three-dimensional F, such that

\nabla \cdot \mathbf {F} =0,

can be expressed as the sum of a toroidal and poloidal vector fields:

\mathbf {F} =\mathbf {T} +\mathbf {P} =\nabla \times \Psi \mathbf {r} +\nabla \times (\nabla \times \Phi \mathbf {r} ),

where $\mathbf {r}$ is a radial vector in spherical coordinates $(r,\theta ,\phi )$ , and where ${\mathbf {T}}$ is a toroidal field

\mathbf {T} =\nabla \times \Psi \mathbf {r}

for scalar field $\Psi (r,\theta ,\phi )$ ,^[2] and where ${\mathbf {P}}$ is a poloidal field

\mathbf {P} =\nabla \times \nabla \times \Phi \mathbf {r}

for scalar field $\Phi (r,\theta ,\phi )$ .^[3] This decomposition is symmetric in that the curl of a toroidal field is poloidal, and the curl of a poloidal field is toroidal.^[4] A toroidal vector field is tangential to spheres around the origin

\mathbf {r} \cdot \mathbf {T} =0

,^[4]

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

\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 $\Psi$ and $\Phi$ 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

\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 ${\hat {\mathbf {x} }},{\hat {\mathbf {y} }},{\hat {\mathbf {z} }}$ denote the unit vectors in the coordinate directions.^[6]

Notes

↑ Subrahmanyan Chandrasekhar (1961). Hydrodynamic and hydromagnetic stability. International Series of Monographs on Physics. Oxford: Clarendon. See discussion on page 622.
↑ Backus 1986, p. 87.
1 2 Backus 1986, p. 88.
1 2 Backus, Parker & Constable 1996, p. 178.
↑ Backus, Parker & Constable 1996, p. 179.
↑ 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, 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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Poloidal–toroidal decomposition

Cartesian decomposition

See also

Notes

References