Eigenfunctions of the curl in annular cylindrical and rectangular geometry

Abstract

A method for determining the eigenfunctions of the equation ∇⃗×B⃗=λB⃗ for finite cylindrical geometry with normal boundary condition B⃗∙n̂=0 and nonaxisymmetric modes ∼eimθ,m≠0 was given by Morse [J. Math. Phys. 46, 113511 (2005)] This method uses series solution of a Chandrasekhar-Kendall [Astrophys. J. 126, 157 (1957)] function. Here this method is generalized to include annular (coaxial) cylindrical geometry of finite length. The method is also applied to the case of rectangular duct geometry with periodic z-dependence. For slender aspect ratios in the annular cylinder, the eigenfunctions and eigenfields correspond to the rectangular duct. The eigenfunctions for finite-length rectangular boxes can also be obtained from the rectangular duct case. The robust convergence properties (an∼1∕n4) and simple eigenvalue equations from Morse are retained in these other geometries.