THE ENERGY AND HELICITY OF KNOTTED MAGNETIC-FLUX TUBES

Magnetic relaxation of a magnetic field embedded in a perfectly conduc ting incompressible fluid to minimum energy magnetostatic equilibrium states is considered. It is supposed that the magnetic field is confin ed to a single flux tube which may be knotted. A local non-orthogonal coordinate system, zero-framed with respect to the knot, is introduced , and the field is decomposed into toroidal and poloidal ingredients w ith respect to this system. The helicity of the field is then determin ed; this vanishes for a field that is either purely toroidal or purely poloidal. The magnetic energy functional is calculated under the simp lifying assumptions that the tube is axially uniform and of circular c ross-section. The case of a tube with helical axis is first considered , and new results concerning kink mode instability and associated bifu rcations are obtained. The case of flux tubes in the form of torus kno ts is then considered and the 'ground-state' energy function (m) over bar(h) (where h is an internal twist parameter) is obtained; as expect ed, (m) over bar(h), which is a topological invariant of the knot, inc reases with increasing knot complexity. The function (m) over bar(h) p rovides an upper bound on the corresponding function m(h) that applies when the above constraints on tube structure are removed. The techniq ue is applicable to any knot admitting a parametric representation, on condition that points of vanishing curvature are excluded.