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A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
( R e d i r e c t e d f r o m P o l o i d a l f i e l d )
For a three-dimensional vector field F with zero divergence
∇
⋅
F
=
0
,
{\displaystyle \nabla \cdot \mathbf {F} =0,}
this F can be expressed as the sum of a toroidal field T and poloidal vector field P
F
=
T
+
P
{\displaystyle \mathbf {F} =\mathbf {T} +\mathbf {P} }
where r is a radial vector in spherical coordinates (r , θ , φ ). The toroidal field is obtained from a scalar field , Ψ (r , θ , φ ), as the following curl ,
T
=
∇
×
(
r
Ψ
(
r
)
)
{\displaystyle \mathbf {T} =\nabla \times (\mathbf {r} \Psi (\mathbf {r} ))}
and the poloidal field is derived from another scalar field Φ(r , θ , φ ), as a twice-iterated curl,
P
=
∇
×
(
∇
×
(
r
Φ
(
r
)
)
)
.
{\displaystyle \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 .
Geometry [ edit ]
A toroidal vector field is tangential to spheres around the origin,
r
⋅
T
=
0
{\displaystyle \mathbf {r} \cdot \mathbf {T} =0}
while the curl of a poloidal field is tangential to those spheres
r
⋅
(
∇
×
P
)
=
0.
{\displaystyle \mathbf {r} \cdot (\nabla \times \mathbf {P} )=0.}
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 .
Cartesian decomposition [ edit ]
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
F
(
x
,
y
,
z
)
=
∇
×
g
(
x
,
y
,
z
)
z
^
+
∇
×
(
∇
×
h
(
x
,
y
,
z
)
z
^
)
+
b
x
(
z
)
x
^
+
b
y
(
z
)
y
^
,
{\displaystyle \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
x
^
,
y
^
,
z
^
{\displaystyle {\hat {\mathbf {x} }},{\hat {\mathbf {y} }},{\hat {\mathbf {z} }}}
denote the unit vectors in the coordinate directions.
See also [ edit ]
References [ edit ]
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 of Geophysics , 24 : 75–109, Bibcode :1986RvGeo..24...75B , doi :10.1029/RG024i001p00075 .
Backus, George; Parker, Robert; Constable, Catherine (1996), Foundations of Geomagnetism , Cambridge University Press, ISBN 0-521-41006-1 .
Jones, Chris (2008), Dynamo Theory (PDF) .
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Poloidal–toroidal_decomposition&oldid=1222227863 "
C a t e g o r y :
● V e c t o r c a l c u l u s
● T h i s p a g e w a s l a s t e d i t e d o n 4 M a y 2 0 2 4 , a t 1 8 : 0 7 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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