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. 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 ∇ × H = λ H {\displaystyle \nabla \times \mathbf {H} =\lambda \mathbf {H} } , where H {\displaystyle \mathbf {H} } is the magnetic field and λ {\displaystyle \lambda } is the force-free parameter, with the assumption of divergence free field, ∇ ⋅ H = 0 {\displaystyle \nabla \cdot \mathbf {H} =0} , then the most general solution for the axisymmetric case is
H = 1 λ ∇ × ( ∇ × ψ n ^ ) + ∇ × ψ n ^ {\displaystyle \mathbf {H} ={\frac {1}{\lambda }}\nabla \times (\nabla \times \psi \mathbf {\hat {n}} )+\nabla \times \psi \mathbf {\hat {n}} }
where n ^ {\displaystyle \mathbf {\hat {n}} } is a unit vector and the scalar function ψ {\displaystyle \psi } satisfies the Helmholtz equation, i.e.,
∇ 2 ψ + λ 2 ψ = 0. {\displaystyle \nabla ^{2}\psi +\lambda ^{2}\psi =0.}
The same equation also appears in Beltrami flows from fluid dynamics where, the vorticity vector is parallel to the velocity vector, i.e., ∇ × v = λ v {\displaystyle \nabla \times \mathbf {v} =\lambda \mathbf {v} } .
Derivation Taking curl of the equation ∇ × H = λ H {\displaystyle \nabla \times \mathbf {H} =\lambda \mathbf {H} } and using this same equation, we get
∇ × ( ∇ × H ) = λ 2 H {\displaystyle \nabla \times (\nabla \times \mathbf {H} )=\lambda ^{2}\mathbf {H} } . In the vector identity ∇ × ( ∇ × H ) = ∇ ( ∇ ⋅ H ) − ∇ 2 H {\displaystyle \nabla \times \left(\nabla \times \mathbf {H} \right)=\nabla (\nabla \cdot \mathbf {H} )-\nabla ^{2}\mathbf {H} } , we can set ∇ ⋅ H = 0 {\displaystyle \nabla \cdot \mathbf {H} =0} since it is solenoidal, which leads to a vector Helmholtz equation,
∇ 2 H + λ 2 H = 0 {\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 ψ {\displaystyle \psi } is a scalar function which satisfies the equation
∇ 2 ψ + λ 2 ψ = 0 {\displaystyle \nabla ^{2}\psi +\lambda ^{2}\psi =0} , then the three linearly independent solutions of the vector Helmholtz equation are given by
L = ∇ ψ , T = ∇ × ψ n ^ , S = 1 λ ∇ × T {\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 n ^ {\displaystyle \mathbf {\hat {n}} } is a fixed unit vector. Since ∇ × S = λ T {\displaystyle \nabla \times \mathbf {S} =\lambda \mathbf {T} } , it can be found that ∇ × ( S + T ) = λ ( S + T ) {\displaystyle \nabla \times (\mathbf {S} +\mathbf {T} )=\lambda (\mathbf {S} +\mathbf {T} )} . But this is same as the original equation, therefore H = S + T {\displaystyle \mathbf {H} =\mathbf {S} +\mathbf {T} } , where S {\displaystyle \mathbf {S} } is the poloidal field and T {\displaystyle \mathbf {T} } is the toroidal field. Thus, substituting T {\displaystyle \mathbf {T} } in S {\displaystyle \mathbf {S} } , we get the most general solution as
H = 1 λ ∇ × ( ∇ × ψ n ^ ) + ∇ × ψ n ^ . {\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 z {\displaystyle z} direction, i.e., n ^ = e z {\displaystyle \mathbf {\hat {n}} =\mathbf {e} _{z}} , with a periodicity L {\displaystyle L} in the z {\displaystyle z} direction with vanishing boundary conditions at r = a {\displaystyle r=a} , the solution is given by
ψ = J m ( μ j r ) e i m θ + i k z , λ = ± ( μ j 2 + k 2 ) 1 / 2 {\displaystyle \psi =J_{m}(\mu _{j}r)e^{im\theta +ikz},\quad \lambda =\pm (\mu _{j}^{2}+k^{2})^{1/2}}
where J m {\displaystyle J_{m}} is the Bessel function, k = ± 2 π n / L , n = 0 , 1 , 2 , … {\displaystyle k=\pm 2\pi n/L,\ n=0,1,2,\ldots } , the integers m = 0 , ± 1 , ± 2 , … {\displaystyle m=0,\pm 1,\pm 2,\ldots } and μ j {\displaystyle \mu _{j}} is determined by the boundary condition a k μ j J m ′ ( μ j a ) + m λ J m ( μ j a ) = 0. {\displaystyle ak\mu _{j}J_{m}'(\mu _{j}a)+m\lambda J_{m}(\mu _{j}a)=0.} The eigenvalues for m = n = 0 {\displaystyle m=n=0} has to be dealt separately. Since here n ^ = e z {\displaystyle \mathbf {\hat {n}} =\mathbf {e} _{z}} , we can think of z {\displaystyle z} direction to be toroidal and θ {\displaystyle \theta } direction to be poloidal, consistent with the convention.
See also Poloidal–toroidal decomposition Woltjer's theorem
References