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
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 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.,
Taking curl of the equation
= λ
and using this same equation, we get In the vector identity
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.
ψ
is a scalar function which satisfies the equation
ψ +
ψ = 0
, then the three linearly independent solutions of the vector Helmholtz equation are given by where
is a fixed unit vector.
But this is same as the original equation, therefore
is the poloidal field and
is the toroidal field.
, we get the most general solution as Taking the unit vector in the
direction, i.e.,
direction with vanishing boundary conditions at
, the solution is given by[3][4] where
is the Bessel function,
is determined by the boundary condition
has to be dealt separately.
direction to be toroidal and
direction to be poloidal, consistent with the convention.