Tsunehiko Araki

An intrinsic velocity-independent criterion for superfluid turbulence

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 0.60Tc (where Tc 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.Hydrodynamic flow in both classical and quantum fluids can be either laminar or turbulent. To describe the latter, vortices in turbulent flow are modelled with stable vortex filaments. While this is an idealization in classical fluids, vortices are real topologically stable quantized objects in superfluids. Thus superfluid turbulence is thought to hold the key to new understanding on turbulence in general. The fermion superfluid 3He offers further possibilities owing to a large variation in its hydrodynamic characteristics over the experimentally accessible temperatures. While studying the hydrodynamics of the B phase of superfluid 3He, we discovered a sharp transition at 0.60Tc between two regimes, with regular behaviour at high-temperatures and turbulence at low-temperatures. Unlike in classical fluids, this transition is insensitive to velocity and occurs at a temperature where the dissipative vortex damping drops below a critical limit. This discovery resolves the conflict between existing high- and low-temperature measurements in 3He-B: At high temperatures in rotating flow a vortex loop injected into superflow has been observed to expand monotonically to a single rectilinear vortex line, while at very low temperatures a tangled network of quantized vortex lines can be generated in a quiescent bath with a vibrating wire. The solution of this conflict reveals a new intrinsic criterion for the existence of superfluid turbulence.

Energy spectrum of superfluid turbulence with no normal-fluid component

The energy of 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<2pi/l the energy spectrum is very similar to the Kolmogorovs -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 is the average intervortex spacing. The vortex length distribution converges to a scaling property reflecting the self-similarity of the tangle.

Dynamics of vortex tangle without mutual friction in superfluid 4 He

A recent experiment has shown that a tangle of quantized vortices in superfluid

Rotating superfluid turbulence.

{}^{4}\mathrm{He}

Instability of vortex array and transitions to turbulence in rotating helium II

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 Vinens 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 Vinens theory which discusses the Kelvin wave cascade with sound radiation.

Diffusion of an inhomogeneous vortex tangle

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.

Energy spectrum of the random velocity field induced by a Gaussian vortex tangle in HeII

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. We find that 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. We suggest a theoretical explanation for the second instability which was observed at higher values of the axial flow and conclude by making an analogy between the alignment of vortices in the presence of rotation and the alignment of dipole moments in the presence of an applied magnetic field.

Vortex tangle polarized by rotation

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.

Transient growth of Kelvin waves on quantized vortices

Using the Gaussian model of the vortex tangle (VT) arising in the turbulent superfluid Hell, we calculate the energy spectrum E(k) of the 3D random velocity field induced by that VT. If the VT is assumed to be a purely fractal object with Haussdorf dimension HD, the E(k) is a power-like function E(k)∝k-2+HD. A more realistic VT in HeII is a semi-fractal object, behaving as smooth line for small separations Δξ≪R (ξ is the label coordinate, R is mean curvature) and having a random walk structure for large Δξ with HD=2. For that case calculations give a spectrum E(k) that is k-independent for k smaller than 1/R (but larger than the inverse size of the system) and that scales as k−1 for larger k. The latter reflects the fact, that for small, scales a vortex filament behaves as a smooth line. Our results agree with recent numerical simulations.

Cascade process of vortex tangle dynamics in superfluid 4He without mutual friction

Almost all studies of vortex states in superfluid 4He have been concerned with either ordered vortex arrays driven by rotation or disordered vortex tangles driven, for example, by thermal counterfiow. In this work we study numerically what happens to vortices in the presence of both effects. We find that a rotating vortex array becomes unstable, exciting Kelvin waves when it is subject to a counterfiow which is parallel to the rotation axis and which is sufficiently large. After the initial growth of the instability, the vortices enter a new, statistically steady, turbulent state, in which the vortex tangle is polarized along the rotational axis. We determine the polarization of the tangle as a function of the rotation frequency and the counterfiow velocity.