Internal structure of localized quantized vortex tangles

TL;DR

This study numerically analyzes the internal structure of localized quantum turbulence in superfluid helium, revealing self-similarity through fractal dimension and vortex length distribution, with findings on anisotropic and isotropic vortex tangles.

Contribution

It introduces a detailed numerical investigation of the fractal and size distribution properties of localized vortex tangles in superfluid helium, highlighting self-similarity and anisotropy effects.

Findings

Fractal dimension saturates around 1.8 with increasing vortex line density.

Anisotropic vortex tangles exhibit a power-law vortex length distribution.

Isotropic tangles do not show a power-law distribution.

Abstract

In this study, we numerically investigate the internal structure of localized quantum turbulence in superfluid $^4$He at zero temperature with the expectation of self-similarity in the real space. In our previous study, we collected the statistics of vortex rings emitted from a localized vortex tangle. As a result, the power law between the minimum size of detectable vortex rings and the emission frequency is obtained, which suggests that the vortex tangle has self-similarity in the real space [Nakagawa $et$ $al.$, Phys. Rev. B $\boldsymbol{101}$, 184515 (2020)]. In this work, we study the fractal dimension and vortex length distribution of localized vortex tangles, which can show their self-similar structure. We generate statistically steady and localized vortex tangles by injecting vortex rings with a fixed size. We used two types of injection methods that produce anisotropic or isotropic tangles. The injected vortex rings develop into a localized vortex tangle consisting of vortex rings of various sizes through reconnections (fusions and splitting of vortices). The fractal dimension is an increasing function of the vortex line density and becomes saturated to a value of approximately 1.8, as the density increases sufficiently. The behavior of the fractal dimension was independent of the anisotropy of the vortex tangles. The vortex length distribution indicates the number of vortex rings of each size that are distributed in a tangle. The distribution of the anisotropic vortex tangle shows the power law in the range above the injected vortex size, although the distribution of the isotropic vortex does not.