Chandrasekhar–Kendall function

Chandrasekhar–Kendall functions are the eigenfunctions of the curl operator derived by Subrahmanyan Chandrasekhar and P. C. Kendall in 1957 while attempting to solve the force-free magnetic fields. The functions were independently derived by both, and the two decided to publish their findings in the same paper.

If the force-free magnetic field equation is written as

∇

×

H

=

λ

H

{\displaystyle \nabla \times \mathbf {H} =\lambda \mathbf {H} }

, where

H

{\displaystyle \mathbf {H} }

is the magnetic field and

λ

{\displaystyle \lambda }

is the force-free parameter, with the assumption of divergence free field,

∇

⋅

H

=

0

{\displaystyle \nabla \cdot \mathbf {H} =0}

, then the most general solution for the axisymmetric case is

H

=

1

λ

∇

×

(

∇

×

ψ

n

^

)

+

∇

×

ψ

n

^

{\displaystyle \mathbf {H} ={\frac {1}{\lambda }}\nabla \times (\nabla \times \psi \mathbf {\hat {n}} )+\nabla \times \psi \mathbf {\hat {n}} }

where

n

^

{\displaystyle \mathbf {\hat {n}} }

is a unit vector and the scalar function

ψ

{\displaystyle \psi }

satisfies the Helmholtz equation, i.e.,

∇

2

ψ

+

λ

2

ψ

=

0.

{\displaystyle \nabla ^{2}\psi +\lambda ^{2}\psi =0.}

The same equation also appears in Beltrami flows from fluid dynamics where, the vorticity vector is parallel to the velocity vector, i.e.,

∇

×

v

=

λ

v

{\displaystyle \nabla \times \mathbf {v} =\lambda \mathbf {v} }

.

Derivation

Taking curl of the equation

∇

×

H

=

λ

H

{\displaystyle \nabla \times \mathbf {H} =\lambda \mathbf {H} }

and using this same equation, we get

∇

×

(

∇

×

H

)

=

λ

2

H

{\displaystyle \nabla \times (\nabla \times \mathbf {H} )=\lambda ^{2}\mathbf {H} }

.

In the vector identity

∇

×

(

∇

×

H

)

=

∇

(

∇

⋅

H

)

−

∇

2

H

{\displaystyle \nabla \times \left(\nabla \times \mathbf {H} \right)=\nabla (\nabla \cdot \mathbf {H} )-\nabla ^{2}\mathbf {H} }

, we can set

∇

⋅

H

=

0

{\displaystyle \nabla \cdot \mathbf {H} =0}

since it is solenoidal, which leads to a vector Helmholtz equation,

∇

2

H

+

λ

2

H

=

0

{\displaystyle \nabla ^{2}\mathbf {H} +\lambda ^{2}\mathbf {H} =0}

.

Every solution of above equation is not the solution of original equation, but the converse is true. If

ψ

{\displaystyle \psi }

is a scalar function which satisfies the equation

∇

2

ψ

+

λ

2

ψ

=

0

{\displaystyle \nabla ^{2}\psi +\lambda ^{2}\psi =0}

, then the three linearly independent solutions of the vector Helmholtz equation are given by

L

=

∇

ψ

,

T

=

∇

×

ψ

n

^

,

S

=

1

λ

∇

×

T

{\displaystyle \mathbf {L} =\nabla \psi ,\quad \mathbf {T} =\nabla \times \psi \mathbf {\hat {n}} ,\quad \mathbf {S} ={\frac {1}{\lambda }}\nabla \times \mathbf {T} }

where

n

^

{\displaystyle \mathbf {\hat {n}} }

is a fixed unit vector. Since

∇

×

S

=

λ

T

{\displaystyle \nabla \times \mathbf {S} =\lambda \mathbf {T} }

, it can be found that

∇

×

(

S

+

T

)

=

λ

(

S

+

T

)

{\displaystyle \nabla \times (\mathbf {S} +\mathbf {T} )=\lambda (\mathbf {S} +\mathbf {T} )}

. But this is same as the original equation, therefore

H

=

S

+

T

{\displaystyle \mathbf {H} =\mathbf {S} +\mathbf {T} }

, where

S

{\displaystyle \mathbf {S} }

is the poloidal field and

T

{\displaystyle \mathbf {T} }

is the toroidal field.