Chandrasekhar–Kendall Function

Chandrasekhar–Kendall functions are the axisymmetric 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 $\nabla\times\mathbf=\lambda\mathbf$, where $\mathbf$ is the magnetic field and $\lambda$ is the force-free parameter, with the assumption of divergence free field, $\nabla\cdot\mathbf=0$, then the most general solution for the axisymmetric case is
:$\mathbf = \frac\nabla\times(\nabla\times\psi\mathbf) + \nabla \times \psi \mathbf$
where $\mathbf$ is a unit vector and the scalar function $\psi$ satisfies the Helmholtz equation, i.e.,
:$\nabla^2\psi + \lambda^2\psi=0.$
The same equation also appears in Beltrami flows from fluid dynamics where, the vorticity vector is parallel to the velocity vector, i.e., $\nabla\times\mathbf=\lambda\mathbf$.

Derivation

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

:$\nabla\times(\nabla\times\mathbf) = \lambda^2\mathbf$.

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

:$\nabla^2\mathbf+\lambda^2\mathbf=0$.

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

:$\mathbf = \nabla\psi,\quad \mathbf = \nabla\times\psi\mathbf, \quad \mathbf = \frac\nabla\times\mathbf$

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

:$\mathbf = \frac\nabla\times(\nabla\times\psi\mathbf) + \nabla \times \psi \mathbf.$

Cylindrical polar coordinates

Taking the unit vector in the $z$ direction, i.e., $\mathbf=\mathbf_z$, with a periodicity $L$ in the $z$ direction with vanishing boundary conditions at $r=a$, the solution is given by

:$\psi = J_m(\mu_jr)e^, \quad \lambda =\pm(\mu_j^2+k^2)^$

where $J_m$ is the Bessel function, $k=\pm 2\pi n/L, \ n = 0,1,2,\ldots$, the integers $m =0,\pm 1,\pm 2,\ldots$ and $\mu_j$ is determined by the boundary condition $a k\mu_j J_m'(\mu_j a)+m \lambda J_m(\mu_j a) =0.$ The eigenvalues for $m=n=0$ has to be dealt separately.
Since here $\mathbf=\mathbf_z$, we can think of $z$ direction to be toroidal and $\theta$ direction to be poloidal, consistent with the convention.

See also

*Poloidal–toroidal decomposition
*Woltjer's theorem

References

{{DEFAULTSORT:Chandrasekhar-Kendall function
Astrophysics
Plasma physics