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Poloidal–toroidal decomposition

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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.



For a three-dimensional vector field F with zero divergence

F = 0 ,

this F can be expressed as the sum of a toroidal field T and poloidal vector field P

F = T + 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 ) )

and the poloidal field is derived from another scalar field Φ(r, θ, φ), as a twice-iterated curl,

P = × ( × ( r Φ ( r ) ) ) .

This decomposition is symmetric in that the curl of a toroidal field is poloidal, and the curl of a poloidal field is toroidal.


A toroidal vector field is tangential to spheres around the origin,

r T = 0

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

r ( × 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 ^ ,

where x ^ , y ^ , z ^ denote the unit vectors in the coordinate directions.


Poloidal–toroidal decomposition Wikipedia

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