Lower and upper bounds on the energy of braided magnetic fields

Abstract

Let ℬ be a class of topologically equivalent regular magnetic fields occupying a cylindrical domain of axis Oz and produced by ideal MHD deformations of the uniform field , and let ℬff ⊂ ℬ be constituted of the fields of ℬ which are force-free. We derive topological lower and upper bounds on the energy of the fields in ℬ and in ℬff, respectively. We also establish new formulae for the field line helicity and the relative magnetic helicity of the fields in ℬ, and give a sufficient criterion of ideal MHD linear stability for the fields in ℬff.