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In [[vector analysis]], a mathematical discipline, a '''poloidal–toroidal decomposition''' of a three-dimensional [[solenoidal vector field]] '''F''' writes it as a sum of a [[toroidal and poloidal|poloidal vector field and a toroidal vector field]]:
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: <math>\mathbf{F} = \nabla \times \Psi \mathbf{r} + \nabla \times (\nabla \times \Phi \mathbf{r}), </math>
 
Thus, the vector field can be considered to be generated by a pair of [[scalar potential]]s &Psi; and &Phi;. This [[vector decomposition|decomposition]] is analogous to a [[Helmholtz decomposition]], and has been used in [[dynamo theory]].
 
== Poloidal and toroidal vector fields ==
 
A vector field <math> \mathbf{F} </math> is called toroidal if it can be written as <math> \mathbf{F} = \nabla \times \Psi \mathbf{r} </math> for some scalar field <math> \Psi </math>.{{sfn|Backus|1986|p=87}} Every toroidal field is solenoidal, because the divergence of the curl vanishes. A solenoidal vector field <math> \mathbf{F} </math> is toroidal if and only if it is tangential to spheres around the origin (<math> \mathbf{F} \cdot \mathbf{r} = 0 </math>).{{sfn|Backus|Parker|Constable|1996|p=178}}
 
A vector field <math> \mathbf{F} </math> is called poloidal if it is the curl of a toroidal field; in other words, if there is a scalar field <math> \Phi </math> such that <math> \mathbf{F} = \nabla \times \nabla \times \Phi \mathbf{r} </math>.{{sfn|Backus|1986|p=88}} Thus, the curl of a toroidal field is poloidal; reversibly, the curl of a poloidal field is toroidal.{{sfn|Backus|Parker|Constable|1996|p=178}} This leads to another characterization of poloidal vector fields: a solenoidal vector field is poloidal if and only if its curl is tangential to spheres around the origin.{{sfn|Backus|Parker|Constable|1996|p=179}}
 
== The decomposition ==
 
Every solenoidal vector field <math> \mathbf{F} </math> can be written as the sum of a toroidal and poloidal field. This decomposition is unique if it is required that the average of the scalar fields <math> \Psi </math> and <math> \Phi </math> vanishes on every sphere of radius <math> r </math>.{{sfn|Backus|1986|p=88}}
 
Poloidal–toroidal decompositions also exist in Cartesian coordinates, but a mean-field flow has to included in this case. For example, every solenoidal vector field can be written as
 
: <math>\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}}, </math>
 
where <math>\hat{\mathbf{x}}, \hat{\mathbf{y}}, \hat{\mathbf{z}}</math> denote the unit vectors in the coordinate directions.{{sfn|Jones|2008|p=62}}
 
==See also==
* [[Toroidal and poloidal]]
 
== Notes ==
{{reflist}}
 
== References ==
* [http://cdsads.u-strasbg.fr/abs/1984JCoPh..55..461G Numerical simulations of stellar convective dynamos. I &mdash; The model and method], Glatzmaier, G. A.; Journal of Computational Physics, vol. 55, Sept. 1984, pp.&nbsp;461&ndash;484.
* [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.&nbsp;622.
* [http://www.springerlink.com/content/h584m7h23v1k5428/ 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.&nbsp;291&ndash;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.&nbsp;247&ndash;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 in Geophysics | volume = 24 | year = 1986 | pages = 75–109 | doi = 10.1029/RG024i001p00075 }}.
* {{citation | first1 = George | last1 = Backus | first2 = Robert | last2 = Parker | first3 = Catherine | last3 = Constable | title = Foundations of Geomagnetism | publisher = Cambridge University Press | year = 1996 | isbn = 0-521-41006-1 }}.
* {{citation | first = Chris | last = Jones | title = Course 2: Dynamo Theory | booktitle = Dynamos: Lecture Notes of the Les Houches Summer School 2007 | isbn = 9780080548128 | publisher = Elsevier | doi = 10.1016/S0924-8099(08)80006-6 | year = 2008 }}.
 
{{DEFAULTSORT:Poloidal-toroidal decomposition}}
[[Category:Vector calculus]]

Revision as of 01:23, 28 February 2014

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