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 $\nabla \times \mathbf {H} =\lambda \mathbf {H}$ with the assumption of divergence free field ( $\nabla \cdot \mathbf {H} =0$ ), then the most general solution for axisymmetric case is

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

where $\mathbf {\hat {n}}$ 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 fluid dynamics in Beltrami flows where, vorticity vector is parallel to the velocity vector, i.e., $\nabla \times \mathbf {v} =\lambda \mathbf {v}$ .

Derivation

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

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

.

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

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

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

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

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

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

References

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

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[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.

Chandrasekhar-Kendall function

Derivation

Cylindrical polar coordinates

See also

References