Vortex length, vortex energy and fractal dimension of superfluid turbulence at very low temperature

TL;DR

This paper models the fractal dimension of superfluid vortex tangles at very low temperatures by assuming a self-similar Kelvin wave structure, linking energy distribution to fractal geometry.

Contribution

It introduces a model relating vortex energy scaling to fractal dimension, independent of specific assumptions, and connects these to Kelvin wave cascade properties.

Findings

Fractal dimension exceeds unity when smaller scales contribute less energy per unit length.

The model relates energy scaling to fractal geometry of vortex tangles.

Provides a relation between Kelvin wave cascade exponents and vortex fractal dimension.

Abstract

By assuming a self-similar structure for Kelvin waves along vortex loops with successive smaller scale features, we model the fractal dimension of a superfluid vortex tangle in the zero temperature limit. Our model assumes that at each step the total energy of the vortices is conserved, but the total length can change. We obtain a relation between the fractal dimension and the exponent describing how the vortex energy per unit length changes with the length scale. This relation does not depend on the specific model, and shows that if smaller length scales make a decreasing relative contribution to the energy per unit length of vortex lines, the fractal dimension will be higher than unity. Finally, for the sake of more concrete illustration, we relate the fractal dimension of the tangle to the scaling exponents of amplitude and wavelength of a cascade of Kelvin waves.