Abstract
Superfluid turbulence consists of a very complex, apparently disordered, tangle of quantized vortex filaments. Until now it has been usual to characterize the vortex tangle mainly in terms of its density (length of vortex line per unit volume). The vortex line density is related to energy, so it has a simple physical interpretation; moreover, it is directly measured in the experiments and is easily computed in the numerical simulations. Unfortunately, the vortex line density does not describe the intrinsic disorder, coiling and linking which occurs within the turbulent vortex tangle due to the combined action of the Biot–Savart law and vortex reconnections. Using ideas borrowed from modern geometry and knot theory, firstly we introduce new measures to describe the geometrical and topological complexity of superfluid turbulence. Secondly, we test these measures on a model problem--the growth of a patch of quantized vorticity--and compare the rate of growth of complexity against the rate of growth of energy and length. Finally, we determine how vortex reconnections depend on the vortex line density.
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