Hydrodynamic flow in classical and quantum fluids can be either laminar or turbulent. Vorticity in turbulent flow is often modelled with vortex filaments. While this represents an idealization in classical fluids, vortices are topologically stable quantized objects in superfluids. Superfluid turbulence is therefore thought to be important for the understanding of turbulence more generally. The fermionic 3He superfluids are attractive systems to study because their characteristics vary widely over the experimentally accessible temperature regime. Here we report nuclear magnetic resonance measurements and numerical simulations indicating the existence of sharp transition to turbulence in the B phase of superfluid 3He. Above 0.60T(c) (where T(c) is the transition temperature for superfluidity) the hydrodynamics are regular, while below this temperature we see turbulent behaviour. The transition is insensitive to the fluid velocity, in striking contrast to current textbook knowledge of turbulence. Rather, it is controlled by an intrinsic parameter of the superfluid: the mutual friction between the normal and superfluid components of the flow, which causes damping of the vortex motion.

The energy spectrum of the superfluid turbulence without the normal fluid is studied numerically under the vortex filament model. Time evolution of the Taylor-Green vortex is calculated under the full nonlocal Biot-Savart law. It is shown that for k < 2π/l, the energy spectrum is very similar to the Kolmogorov's -5/3 law which is the most important statistical property of the conventional turbulence, where k is the wave number of the Fourier component of the velocity field and l the average intervortex spacing. The vortex length distribution becomes to obey a scaling property reflecting the self-similarity of the tangle. 67.40.Vs, 67.40.Bz Particular attention has been focused recently on the similarity between superfluid turbulence and conventional turbulence [1][2][3]. Early work of the superfluid turbulence has been concerned with the counterflow where the normal fluid and superfluid flow oppositely [4], having no classical analog. However Stalp et al. studied recently the superfluid turbulence produced by the towed grid, thus finding the similarity between the superfluid turbulence and the conventional turbulence above 1.4 K [2]. They observed indirectly the Kolmogorov law which is one of the most important statistical properties of the conventional turbulence. This is understood by the idea that the superfluid and the normal fluid are likely to be coupled together by the mutual friction between them, and to behave like a conventional fluid [1,5]. Since the normal fluid is negligible at mK temperatures, an important question arises: even free from the normal fluid, is the superfluid turbulence still similar to the conventional turbulence or not?As the physical model to describe the vortex dynamics in superfluid He at very low temperatures, two types of models are well-known: the Gross-Pitaevskii (GP) equation which describes the motion of a weakly interacting Bose condensate, and the vortex filament model governed by the incompressible Euler dynamics. The former reduces to the Euler vortex filament model when variations of the wave function over scales of the order of the superfluid healing length are neglected. The GP equation includes such complicated compressible effects as the radiation of sound from the vortex lines [6,7], the vortexsound interactions, etc. In order to consider the intrinsic property of superfluid turbulence in a simpler situation, we study the energy spectrum of the 3D velocity field induced by the vortex tangle in the absence of the normal fluid under the vortex filament model.The energy spectrum of the vortices in superfluid was numerically calculated by other authors. Nore et al. studied the energy spectrum of the decaying superfluid turbulence by using the GP equation, and finding the transient spectrum for small k has the Kolmogorov law [3]. However at late stage some complicated compressible effects become dominant. On the other hand, an advantage of the vortex filament model compared with the GP equation is the followings. First this model enables us to calculate the energy s...

A recent experiment has shown that a tangle of quantized vortices in superfluid 4 He decayed even at mK temperatures where the normal fluid was negligible and no mutual friction worked. Motivated by this experiment, this work studies numerically the dynamics of the vortex tangle without the mutual friction, thus showing that a self-similar cascade process, whereby large vortex loops break up to smaller ones, proceeds in the vortex tangle and is closely related with its free decay. This cascade process which may be covered with the mutual friction at higher temperatures is just the one at zero temperature Feynman proposed long ago. The full Biot-Savart calculation is made for dilute vortices, while the localized induction approximation is used for a dense tangle. The former finds the elementary scenario: the reconnection of the vortices excites vortex waves along them and makes them kinked, which could be suppressed if the mutual friction worked. The kinked parts reconnect with the vortex they belong to, dividing into small loops. The latter simulation under the localized induction approximation shows that such cascade process actually proceeds self-similarly in a dense tangle and continues to make small vortices. Considering that the vortices of the interatomic size no longer keep the picture of vortex, the cascade process leads to the decay of the vortex line density. The presence of the cascade process is supported also by investigating the classification of the reconnection type and the size distribution of vortices. The decay of the vortex line density is consistent with the solution of the Vinen's equation which was originally derived on the basis of the idea of homogeneous turbulence with the cascade process. The cascade process revealed by this work is an intrinsic process in the superfluid system free from the normal fluid. The obtained result is compared with the recent Vinen's theory which discusses the Kelvin wave cascade with sound radiation.67.40.Vs, 67.40.Bz

The spatial diffusion of an inhomogeneous vortex tangle is studied numerically with the vortex filament model. A localized initial tangle is prepared by applying a counterflow, and the tangle is allowed to diffuse freely after the counterflow is turned off. Comparison with the solution of a generalization of the Vinen equation that takes diffusion into account leads to a very small diffusion constant, as expected from simple theoretical considerations. The relevance of this result to recent experiments on the generation and decay of superfluid turbulence at very low temperatures is discussed.

Almost all studies of vortex states in helium II have been concerned with either ordered vortex arrays or disordered vortex tangles. This work numerically studies what happens in the presence of both rotation (which induces order) and thermal counterflow (which induces disorder). We find a new statistically steady state in which the vortex tangle is polarized along the rotational axis. Our results are used to interpret an instability that was discovered experimentally by Swanson et al. [Phys. Rev. Lett. 50, 190 (1983)]] and the vortex state beyond the instability that has been unexplained until now.