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.{{cite book |title=Hydrodynamic and hydromagnetic stability |url=http://cdsads.u-strasbg.fr/abs/1961hhs..book.....C |author=Subrahmanyan Chandrasekhar |series=International Series of Monographs on Physics |publisher=Oxford: Clarendon |year=1961 |at=See discussion on page 622}}

Definition

For a three-dimensional vector field F with zero divergence

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

this $\mathbf{F}$ can be expressed as the sum of a toroidal field $\mathbf{T}$ and poloidal vector field $\mathbf{P}$

: $\mathbf{F} = \mathbf{T} + \mathbf{P}$

where $\mathbf{r}$ is a radial vector in spherical coordinates $(r,\theta,\phi)$ . The toroidal field is obtained from a scalar field, $\Psi(r,\theta,\phi)$ ,{{sfn|Backus|1986|p=87}} as the following curl,

: $\mathbf{T} = \nabla \times (\mathbf{r} \Psi(\mathbf{r}))$

and the poloidal field is derived from another scalar field $\Phi(r,\theta, \phi)$ ,{{sfn|Backus|1986|p=88}} as a twice-iterated curl,

: $\mathbf{P} = \nabla \times (\nabla \times (\mathbf{r} \Phi (\mathbf{r})))\,.$

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.{{sfn|Backus|Parker|Constable|1996|p=178}}

Geometry

A toroidal vector field is tangential to spheres around the origin,{{sfn|Backus|Parker|Constable|1996|p=178}}

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

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

: $\mathbf{r} \cdot (\nabla \times \mathbf{P}) = 0.$ {{sfn|Backus|Parker|Constable|1996|p=179}}

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.{{sfn|Backus|1986|p=88}}

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.{{sfn|Jones|2008|p=17}}

Notes

References

[http://cdsads.u-strasbg.fr/abs/1961hhs..book.....C Hydrodynamic and hydromagnetic stability], Chandrasekhar, Subrahmanyan; International Series of Monographs on Physics, Oxford: Clarendon, 1961, p. 622.
[https://doi.org/10.1007%2FBFb0090349 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.
[http://cdsads.u-strasbg.fr/abs/1999ApJS..121..247L 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: [http://www.austms.org.au/Publ/Jamsb/V47P1/2148.html Part 1. Fields with divergence] and [http://www.austms.org.au/Publ/Jamsb/V47P1/2203.html Part 2. Stokes equations]. G. D. McBain. [http://www.austms.org.au/Publ/ANZIAM/index.shtml ANZIAM J.] [http://www.austms.org.au/Publ/ANZIAM/V47P1/contents.html 47 (2005)]
{{citation | first = George | last = Backus | title = Poloidal and toroidal fields in geomagnetic field modeling | journal = Reviews of Geophysics | volume = 24 | year = 1986 | pages = 75–109 | doi = 10.1029/RG024i001p00075 | bibcode = 1986RvGeo..24...75B }}.
{{citation | first1 = George | last1 = Backus | first2 = Robert | last2 = Parker | first3 = Catherine | last3 = Constable |authorlink3=Catherine Constable | title = Foundations of Geomagnetism | publisher = Cambridge University Press | year = 1996 | isbn = 0-521-41006-1 }}.
{{citation | first1 = Chris | last1 = Jones | title = Dynamo Theory | url = http://www1.maths.leeds.ac.uk/~cajones/LesHouches/chapter.pdf |year=2008}}.

Category:Vector calculus