The spectrum of the curl operator on spherically symmetric domains

Abstract

This paper presents a mathematically complete derivation of the minimum-energy divergence-free vector elds of xed helicity, dened on and tangent to the boundary of solid balls and spherical shells. These elds satisfy the equationrV =V ,w here is the eigenvalue of curl having smallest non-zero absolute value among such elds. It is shown that on the ball the energy-minimizers are the axially symmetric spheromak elds found by Woltjer and Chandrasekhar-Kendall, and on spherical shells they are spheromak-like elds. The geometry and topology of these minimum-energy elds, as well as of some higher-energy eigenelds, is illustrated.