A study by computer simulation is reported of the behavior of a quantized vortex line at a very low temperature when there is continuous excitation of low-frequency Kelvin waves. There is no dissipation except by phonon radiation at a very high frequency. It is shown that nonlinear coupling leads to a net flow of energy to higher wave numbers and to the development of a simple spectrum of Kelvin waves that is insensitive to the strength and frequency of the exciting drive. The results are likely to be relevant to the decay of turbulence in superfluid 4He at very low temperatures.

A zero-temperature superfluid is arguably the simplest system in which to study complex fluid dynamics, such as turbulence. We describe computer simulations of such turbulence and compare the results directly with recent experiments in superfluid 3 He-B. We are able to follow the entire process of the production, evolution, and decay of quantum turbulence. We find striking agreement between simulation and experiment and gain insights into the mechanisms involved.

We consider superfluid helium inside a container which rotates at constant angular velocity and investigate numerically the stability of the array of quantized vortices in the presence of an imposed axial counterflow. This problem was studied experimentally by Swanson et al., who reported evidence of instabilities at increasing axial flow but were not able to explain their nature. We find that Kelvin waves on individual vortices become unstable and grow in amplitude, until the amplitude of the waves becomes large enough that vortex reconnections take place and the vortex array is destabilized. The eventual nonlinear saturation of the instability consists of a turbulent tangle of quantized vortices which is strongly polarized. The computed results compare well with the experiments. Finally we suggest a theoretical explanation for the second instability which was observed at higher values of the axial flow.

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...

We study numerically vortex dynamics in superfluid 3 He-B by solving the full Biot-Savart equations inside a rotating cylinder. The initial vortex configuration seems to have an essential role whether the growth process starts or not. The growth process is, at least at the early stages of simulations, mostly governed by the reconnections with cylinder boundary. In order to see a large increase in vortex density one should go below 0.5T c in temperature, somewhat lower than what is observed in the experiments.