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 .[ 1]
Definition
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 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 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
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
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} }}}
See also
Notes
References
Hydrodynamic and hydromagnetic stability 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) 
