Self-organization of vortex-length distribution in quantum turbulence: An approach based on the Barabási-Albert model

Abstract

The energy spectrum of decaying quantum turbulence at T = 0 obeys Kolmogorov's law. In addition to this, recent studies revealed that the vortex-length distribution ͑VLD͒, meaning the size distribution of the vortices, in decaying Kolmogorov quantum turbulence also obeys a power law. This power-law VLD suggests that the decaying turbulence has scale-free structure in real space. Unfortunately, however, there has been no practical study that answers the following important question: why can quantum turbulence acquire a scale-free VLD?We propose here a model to study the origin of the power law of the VLD from a generic point of view. The nature of quantized vortices allows one to describe the decay of quantum turbulence with a simple model that is similar to the Barabási-Albert model, which explains the scale-invariance structure of large networks. We show here that such a model can reproduce the power law of the VLD well.It has long been known that classical fluid turbulence has a scale-free ͑scale-invariant͒ property in its structure. Theoretical and experimental studies have shown that classical turbulence has scale invariance in its energy spectrum E͑k͒ ϵ 4k 2 ͉ṽ͑k͉͒ 2 , where ṽ͑k͒ is the Fourier transform of the fluid velocity v͑r͒. 1 This energy spectrum obeys the Kolmogorov power law E͑k͒ ϰ k −5/3 , which means that classical turbulence has a scaling property in wave number space. There has also been an exponential growth of interest in the scalefree structure of quantum turbulence in recent years. Here, quantum turbulence is an irregular tangle of quantized vortex lines. 2 Several years ago, Stalp et al. studied the quantum turbulence produced by the towed grid above 1.4 K, where the normal fluid and superfluid coexist, and observed indirectly the Kolmogorov power-law energy spectrum E͑k͒ ϰ k −5/3 . 3 Motivated by this pioneering experiment, some numerical simulations have revealed that the energy spectrum of decaying quantum turbulence obeys a Kolmogorov power law even at T = 0, where the normal fluid does not exist. 4,5 These authors calculated the energy spectrum of the velocity field directly and found that it obeys the Kolmogorov power law E͑k͒ ϰ k −5/3 as in classical turbulence. The Kolmogorov law is a scaling property in wave number space, but should be closely related to the self-similarity of the turbulent velocity field in real space. Thus, we ask the following question: how can the scale invariance in wave number space be translated into scale invariance in real space? A possible answer was proposed by Araki et al. 4 Unlike eddies in classical turbulence, the length of each quantum vortex in quantum turbulence is definitely measurable. Hence, they calculated the vortex-length distribution ͑VLD͒ n͑l͒ in decaying Kolmogorov quantum turbulence, where n͑l͒⌬l represents the number of vortices in Kolmogorov quantum turbulence of length from l to l + ⌬l. They found that the VLD obeys a scaling property n͑l͒ ϰ l −␣ with ␣ = 1.34. Kobayashi also found a scaling property of n͑l͒ with ␣ Ϸ 1.5. 6 Scale-fr...

Quantum hydrodynamics

“…If self-similar Richardson cascade appears, it is expected to yield some self-similar power-law in the vortex length distribution. There are still a few numerical works [83,118,119]. It is impossible to observe the distribution in superfluid helium, but possible in atomic BECs.…”

“…Then the system is called dynamically unstable. The dynamic instability occurs when g 2 12 > g 11 g 22 for the dispersion (119). The instability means that homogeneous condensates are unstable; if g 12 > √ g 11 g 22 , the two components will undergo phase separations due to the strong inter-component interaction, while if −g 12 > √ g 11 g 22 , the condensates are unstable to the formation of a denser droplet containing both components.…”

Influence of the medium's dimensionality on defect-mediated turbulence

“…It would be interesting to see if DDDT can be observed in an experimental system such as the BZ reaction, where simple filament motions have been measured in 3D using various tomographic techniques [46^-8]. The question whether DDDT can be linked to certain instabilities or breakup mechanisms remains a challenge for the future but might have further important applications in diverse settings such as optical systems [49], superfluid turbulence [50]-including Bose-Einstein condensates [51]and cosmology [52] as well as the morphogenesis of slime mold [53], Copyright of Physical Review E: Statistical, Nonlinear & Soft Matter Physics is the property of American Physical Society and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.…”

Fractality of Light’s Darkness

“…The interpretation of the vortices in the model of Ref. 16 was as the configuration of cosmic strings in the early universe; however, cosmic strings, as with other quantised vortices in physics such as those in superfluids, evolve according to nonlinear dynamics, which affects the overall fractal dimension [27] and loop length distribution [28]. We emphasize that the optical fields considered here are both linear and monochromatic, so there are no energetic considerations, and the statistics depend only on the probability distribution of the superposed random waves.…”