The spectrum of the curl operator on spherically symmetric domains

This paper presents a mathematically complete derivation of the minimum-ene rgy divergence-free vector fields of fixed helicity, defined on and tangent to the boundary of solid balls and spherical shells. These fields satisfy the equation del xV=lambda V, where lambda is the eigenvalue of curl having smallest nonzero absolute value among such fields. It is shown that on the ball the energy minimizers are the axially symmetric spheromak fields foun d by Woltjer and Chandrasekhar-Kendall, and on spherical shells they are sp heromak-like fields. The geometry and topology of these minimum-energy fiel ds, as well as of some higher-energy eigenfields, are illustrated. (C) 2000 American Institute of Physics. [S1070-664X(00)04005-2].