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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.

Definition

Further information: Vector operator

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

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} }}} denote the unit vectors in the coordinate directions.

See also

Notes

  1. Subrahmanyan Chandrasekhar (1961). Hydrodynamic and hydromagnetic stability. International Series of Monographs on Physics. Oxford: Clarendon. See discussion on page 622.
  2. Backus 1986, p. 87.
  3. ^ Backus 1986, p. 88.
  4. ^ Backus, Parker & Constable 1996, p. 178.
  5. Backus, Parker & Constable 1996, p. 179.
  6. Jones 2008, p. 17.

References

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