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., ${\displaystyle \nabla \times \mathbf {v} =\lambda \mathbf {v} }$.

Derivation

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 coordinates

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 ${\displaystyle J_{m}}$ is the Bessel function, ${\displaystyle k=\pm 2\pi n/L,\ n=0,1,2,...}$, the integers ${\displaystyle m=0,\pm 1,\pm 2,...}$ and ${\displaystyle \mu _{j}}$ is determined by the boundary condition ${\displaystyle ak\mu _{j}J_{m}'(\mu _{j}a)+m\lambda J_{m}(\mu _{j}a)=0.}$ The eigenvalues for ${\displaystyle m=n=0}$ has to be dealt separately. Since here ${\displaystyle \mathbf {\hat {n}} =\mathbf {e} _{z}}$, we can think of ${\displaystyle z}$ direction to be toroidal and ${\displaystyle \theta }$ direction to be poloidal, consistent with the convention.

See also

• Poloidal–toroidal decomposition
• Woltjer's theorem

References

1. Chandrasekhar, S. (1956). On force-free magnetic fields. Proceedings of the National Academy of Sciences, 42(1), 1-5.
2. Chandrasekhar, S., & Kendall, P. C. (1957). On force-free magnetic fields. The Astrophysical Journal, 126, 457.
3. Montgomery, D., Turner, L., & Vahala, G. (1978). Three‐dimensional magnetohydrodynamic turbulence in cylindrical geometry. The Physics of Fluids, 21(5), 757-764.
4. Yoshida, Z. (1991). Discrete eigenstates of plasmas described by the Chandrasekhar-Kendall functions. Progress of theoretical physics, 86(1), 45-55.