Chandrasekhar–Kendall function

Chandrasekhar–Kendall functions are the axisymmetric eigenfunctions of the curl operator, derived by Subrahmanyan Chandrasekhar and P.C. Kendall in 1957,^[1][2] in attempting to solve the force-free magnetic fields. The results were independently derived by both, but were agreed to publish the paper together.

If the force-free magnetic field equation is written as ${\displaystyle \nabla \times \mathbf {H} =\lambda \mathbf {H} }$ with the assumption of divergence free field (${\displaystyle \nabla \cdot \mathbf {H} =0}$), then the most general solution for axisymmetric case is

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

where ${\displaystyle \mathbf {\hat {n}} }$ is a unit vector and the scalar function ${\displaystyle \psi }$ satisfies the Helmholtz equation, i.e.,

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

The same equation also appears in fluid dynamics in Beltrami flows where, vorticity vector is parallel to the velocity vector, i.e., ${\displaystyle \nabla \times \mathbf {v} =\lambda \mathbf {v} }$.

DerivationEdit

Taking curl of the equation ${\displaystyle \nabla \times \mathbf {H} =\lambda \mathbf {H} }$ and using this same equation, we get

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

In the vector identity ${\displaystyle \nabla \times \left(\nabla \times \mathbf {H} \right)=\nabla (\nabla \cdot \mathbf {H} )-\nabla ^{2}\mathbf {H} }$, we can set ${\displaystyle \nabla \cdot \mathbf {H} =0}$ since it is solenoidal, which leads to a vector Helmholtz equation,

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

Every solution of above equation is not the solution of original equation, but the converse is true. If ${\displaystyle \psi }$ is a scalar function which satisfies the equation ${\displaystyle \nabla ^{2}\psi +\lambda ^{2}\psi =0}$, then the three linearly independent solutions of the vector Helmholtz equation are given by

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

where ${\displaystyle \mathbf {\hat {n}} }$ is a fixed unit vector. Since ${\displaystyle \nabla \times \mathbf {S} =\lambda \mathbf {T} }$, it can be found that ${\displaystyle \nabla \times (\mathbf {S} +\mathbf {T} )=\lambda (\mathbf {S} +\mathbf {T} )}$. But this is same as the original equation, therefore ${\displaystyle \mathbf {H} =\mathbf {S} +\mathbf {T} }$, where ${\displaystyle \mathbf {S} }$ is the poloidal field and ${\displaystyle \mathbf {T} }$ is the toroidal field. Thus, substituting ${\displaystyle \mathbf {T} }$ in ${\displaystyle \mathbf {S} }$, we get the most general solution as

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

Cylindrical polar coordinatesEdit

Taking the unit vector in the ${\displaystyle z}$ direction, i.e., ${\displaystyle \mathbf {\hat {n}} =\mathbf {e} _{z}}$, with a periodicity ${\displaystyle L}$ in the ${\displaystyle z}$ direction with vanishing boundary conditions at ${\displaystyle r=a}$, the solution is given by^[3][4]

${\displaystyle \psi =J_{m}(\mu _{j}r)e^{im\theta +ikz},\quad \lambda =\pm (\mu _{j}^{2}+k^{2})^{1/2}}$

where  is the Bessel function, , the integers  and  is determined by the boundary condition  The eigenvalues for  has to be dealt separately. Since here , we can think of  direction to be toroidal and  direction to be poloidal, consistent with the convention. 

This article uses material from the Wikipedia article  Metasyntactic variable, which is released under the  Creative Commons Attribution-ShareAlike 3.0 Unported License.