Chandrasekhar–Kendall function explained
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.[1] [2] 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
, where
is the magnetic field and
is the force-free parameter, with the assumption of divergence free field,
, then the most general solution for the axisymmetric case is
H=
\nabla x (\nabla x \psi\hatn)+\nabla x \psi\hatn
where
is a unit vector and the scalar function
satisfies the
Helmholtz equation, i.e.,
The same equation also appears in
Beltrami flows from fluid dynamics where, the
vorticity vector is parallel to the velocity vector, i.e.,
.
Derivation
Taking curl of the equation
and using this same equation, we get
\nabla x (\nabla x H)=λ2H
.
In the vector identity
\nabla x \left(\nabla x H\right)=\nabla(\nabla ⋅ H)-\nabla2H
, we can set
since it is solenoidal, which leads to a vector
Helmholtz equation,
.
Every solution of above equation is not the solution of original equation, but the converse is true. If
is a scalar function which satisfies the equation
, then the three linearly independent solutions of the vector
Helmholtz equation are given by
L=\nabla\psi, T=\nabla x \psi\hatn, S=
\nabla x T
where
is a fixed unit vector. Since
, it can be found that
. But this is same as the original equation, therefore
, where
is the poloidal field and
is the toroidal field. Thus, substituting
in
, we get the most general solution as
H=
\nabla x (\nabla x \psi\hatn)+\nabla x \psi\hatn.
Cylindrical polar coordinates
Taking the unit vector in the
direction, i.e.,
, with a periodicity
in the
direction with vanishing boundary conditions at
, the solution is given by
[3] [4]
where
is the Bessel function,
k=\pm2\pin/L, n=0,1,2,\ldots
, the integers
and
is determined by the boundary condition
ak\mujJm'(\muja)+mλJm(\muja)=0.
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.
See also
Notes and References
- Chandrasekhar. Subrahmanyan. 1956. On force-free magnetic fields. Proceedings of the National Academy of Sciences. 42. 1. 1–5. 10.1073/pnas.42.1.1. 16589804. 534220. 0027-8424. free. en.
- Chandrasekhar. Subrahmanyan. Kendall. P. C.. September 1957. On Force-Free Magnetic Fields. The Astrophysical Journal. en. 126. 1. 1–5 . 10.1086/146413. 16589804. 0004-637X. 1957ApJ...126..457C. 534220.
- Montgomery. David. Turner. Leaf. Vahala. George. 1978. Three-dimensional magnetohydrodynamic turbulence in cylindrical geometry. Physics of Fluids. en. 21. 5. 757–764. 10.1063/1.862295.
- Yoshida. Z.. 1991-07-01. Discrete Eigenstates of Plasmas Described by the Chandrasekhar–Kendall Functions. Progress of Theoretical Physics. en. 86. 1. 45–55. 10.1143/ptp/86.1.45. 0033-068X. free.
