Chandrasekhar–Kendall function

Chandrasekhar–Kendall functions are the <a href="/facts/Eigenfunction/62EJS0XB">eigenfunctions</a> of the <a href="/facts/Curl_(mathematics)/GuS8dc7o">curl</a> operator derived by <a href="/facts/Subrahmanyan_Chandrasekhar/LQID778M">Subrahmanyan Chandrasekhar</a> and P. C. Kendall in 1957 while attempting to solve the <a href="/facts/Force-free_magnetic_field/4Ye5ENeB">force-free magnetic fields</a>. 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 <a href="/facts/Helmholtz_equation/SWfaXrTB">Helmholtz equation</a>, i.e.,

∇
          
            2
          
        
        ψ
        +
        
          λ
          
            2
          
        
        ψ
        =
        0.
      
    
    {\displaystyle \nabla ^{2}\psi +\lambda ^{2}\psi =0.}

The same equation also appears in <a href="/facts/Beltrami_flow/c93elPk8">Beltrami flows</a> from fluid dynamics where, the <a href="/facts/Vorticity/2oQ6eT4d">vorticity</a> vector is parallel to the velocity vector, i.e., 
 
 
 
 ∇
 ×
 
 v
 
 =
 λ
 
 v
 
 
 
 {\displaystyle \nabla \times \mathbf {v} =\lambda \mathbf {v} }
 
.

Chandrasekhar–Kendall function open-in-new

Chandrasekhar–Kendall function