The Energy and Helicity of Knotted Magnetic Flux Tubes
Magnetic relaxation of a magnetic field embedded in a perfectly
conducting incompressible fluid to minimum energy magnetostatic
equilibrium states is considered. It is supposed that the magnetic field
is confined 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 with respect to this system. The helicity of the field is
then determined; this vanishes for a field that is either purely toroidal
or purely poloidal. The magnetic energy functional is calculated under
the simplifying assumptions that the tube is axially uniform and of
circular cross-section. The case of a tube with helical axis is first
considered, and new results concerning kink mode instability and
associated bifurcations are obtained. The case of flux tubes in the form
of torus knots is then considered and the `ground-state' energy function
m'(h) (where h is an internal twist parameter) is
obtained; as expected, m'(h), which is a topological
invariant of the knot, increases with increasing knot complexity. The
function m'(h) provides an upper bound on the
corresponding function m(h) that applies when the above
constraints on tube structure are removed. The technique is applicable
to any knot admitting a parametric representation, on condition that
points of vanishing curvature are excluded.
This paper is published in
Chui, A.Y.K. & Moffatt, H.K. 1995 (fluxtube.ps.gz, 203K, 21 pages)
The energy and helicity of knotted magnetic flux tubes. Proc. R. Soc. Lond. A451, pp. 609-629.