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 ${\displaystyle \nabla \times \mathbf {H} =\lambda \mathbf {H} }$, where ${\displaystyle \mathbf {H} }$ is the magnetic field and ${\displaystyle \lambda }$ is the force-free parameter, with the assumption of divergence free field, ${\displaystyle \nabla \cdot \mathbf {H} =0}$, then the most general solution for the 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 Beltrami flows from fluid dynamics where, the vorticity vector is parallel to the velocity vector, i.e., ${\displaystyle \nabla \times \mathbf {v} =\lambda \mathbf {v} }$.