Chandrasekhar–Kendall function

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

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

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Taking curl of the equation   and using this same equation, we get

 .

In the vector identity  , 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

 

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

 

Cylindrical polar coordinates

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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,  , the integers   and   is determined by the boundary condition   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

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References

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  1. ^ Chandrasekhar, Subrahmanyan (1956). "On force-free magnetic fields". Proceedings of the National Academy of Sciences. 42 (1): 1–5. doi:10.1073/pnas.42.1.1. ISSN 0027-8424. PMC 534220. PMID 16589804.
  2. ^ Chandrasekhar, Subrahmanyan; Kendall, P. C. (September 1957). "On Force-Free Magnetic Fields". The Astrophysical Journal. 126 (1): 1–5. Bibcode:1957ApJ...126..457C. doi:10.1086/146413. ISSN 0004-637X. PMC 534220. PMID 16589804.
  3. ^ Montgomery, David; Turner, Leaf; Vahala, George (1978). "Three-dimensional magnetohydrodynamic turbulence in cylindrical geometry". Physics of Fluids. 21 (5): 757–764. doi:10.1063/1.862295.
  4. ^ Yoshida, Z. (1991-07-01). "Discrete Eigenstates of Plasmas Described by the Chandrasekhar–Kendall Functions". Progress of Theoretical Physics. 86 (1): 45–55. doi:10.1143/ptp/86.1.45. ISSN 0033-068X.