Poloidal–toroidal decomposition: Difference between revisions
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In [[vector calculus]], a '''poloidal–toroidal decomposition''' is a restricted form of the [[Helmholtz decomposition]] that is often used in the analysis, for example, of [[magnetic fields]] and [[incompressible flow|incompressible fluids]].<ref>{{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|page |
In [[vector calculus]], a '''poloidal–toroidal decomposition''' is a restricted form of the [[Helmholtz decomposition]] that is often used in the analysis, for example, of [[magnetic fields]] and [[incompressible flow|incompressible fluids]].<ref>{{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|see discussion on page 622}}</ref> For of a three-dimensional [[solenoidal vector field]] '''F''', such that |
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:<math> \nabla \cdot \mathbf{F} = 0, </math> |
:<math> \nabla \cdot \mathbf{F} = 0, </math> |
Revision as of 14:07, 13 June 2015
In vector calculus, a poloidal–toroidal decomposition is a restricted form of the Helmholtz decomposition that is often used in the analysis, for example, of magnetic fields and incompressible fluids.[1] For of a three-dimensional solenoidal vector field F, such that
can be expressed as the sum of a toroidal and poloidal vector fields:
where is a radial vector in spherical coordinates, and where is a toroidal field
for scalar field ,[2] and where is a poloidal field
for scalar field .[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
- ,[4]
while the curl of a poloidal field is tangential to spheres
- .[5]
The decomposition
Every solenoidal vector field 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 and vanishes on every sphere of radius .[3]
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
where denote the unit vectors in the coordinate directions.[6]
See also
Notes
- ^ Subrahmanyan Chandrasekhar (1961). Hydrodynamic and hydromagnetic stability. International Series of Monographs on Physics. Oxford: Clarendon.
{{cite book}}
: Text "see discussion on page 622" ignored (help) - ^ Backus 1986, p. 87.
- ^ a b Backus 1986, p. 88.
- ^ a b 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.