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]

See also

• Toroidal and poloidal

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. 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.