Quantum turbulence

Quantum turbulence is the name given to the turbulent flow – the chaotic motion of a fluid at high flow rates – of quantum fluids, such as superfluids which have been cooled to temperatures close to absolute zero.

Introduction

A simulated vortex tangle representing quantum turbulence in a cubic volume and showing the quantized vortices

The turbulence of classical fluids is an everyday phenomenon, which can be readily observed in the flow of a stream or river. When turning on a water tap, one notices that at first the water flows out in a regular fashion (called laminar flow), but if the tap is turned up to higher flow rates, the flow becomes decorated with irregular bulges, unpredictably splitting into multiple strands as it spatters out in an ever-changing torrent, known as turbulent flow. Turbulent flow comprises randomly sized regions of circulating fluid called eddies and vortices, which can be ordered, giving rise to large-scale motions such as tornados or whirlpools, but in general are completely irregular.

Under normally experienced conditions, all fluids have a resistance to flow, called viscosity, which governs the switch from laminar to turbulent flow, and causes the turbulence to decay (for example, after a cup of coffee is stirred it will eventually return to rest). A superfluid is a fluid which has no viscosity, or resistance to flow, meaning that flow around a closed loop will last forever. These strange fluids exist only at temperatures close to absolute zero, being in effect a more ordered and separate fluid state, arising due to the macroscopic influence of quantum mechanics brought about by the low temperatures involved.

Despite having no viscosity, turbulence is possible in a superfluid. This was first suggested theoretically by Richard Feynman in 1955,[1] and was soon found experimentally. Since the flow of a superfluid is an inherently quantum phenomenon (see macroscopic quantum phenomena and superfluid helium-4), turbulence in superfluids is often given the name quantum turbulence to reflect the key role played by quantum mechanics. A recent overview of quantum turbulence is given by Skrbek.[2]

In these so-called "superfluids", the vortices have a fixed size, and are identical. This is another startling property of superfluids, being very different from the random vortices in a classical fluid, and arises out of the quantum physics whose effects become observable on a larger scale at low temperatures. Quantum turbulence, then, is a tangle of these quantized vortices, making it a pure form of turbulence which is much simpler to model than classical turbulence, in which myriad possible interactions of the eddies quickly make the problem too complex to be able to predict what will happen.

Turbulence in classical fluid is often modelled simply using virtual vortex filaments, around which there is a certain circulation of the fluid, to get a grasp on what is happening in the fluid. In quantum turbulence, these vortex lines are real – they can be observed, and have a very definite circulation – and moreover they provide the whole of the physics of the situation.

The two-fluid model

Helium II is usefully regarded theoretically as a mixture of normal fluid and superfluid, having a total density equal to the sum of the densities of the two components. The normal part behaves like any other liquid, and the superfluid part flows without resistance. The proportions of the two components change continuously from all normal fluid at the transition temperature (2.172 K) to all superfluid at zero temperature. More details can be found in the articles on superfluid helium-4 and macroscopic quantum phenomena.

In turbulence, the normal fluid behaves as a classical fluid, and has a classically turbulent velocity field when a superfluid experiences turbulence. In the superfluid component, however, vorticity is restricted to the quantized vortex lines, and there is no viscous dissipation. In turbulence, the vortex lines arrange themselves in an irregular fashion, and this is described as a "vortex tangle". This vortex tangle mediates an interaction between the superfluid and the normal component known as mutual friction.

Experimental developments

Superfluidity is only observed "naturally" in two liquids: helium-4 and the rarer isotope, helium-3. Quantum turbulence was first discovered in pure ⁴He in a counterflow (where the normal and superfluid components are made to flow in opposite directions) generated by a steady heat current. See superfluid helium-4. Since the two-fluid model, and therefore counterflow itself, is unique to superfluids, this counterflow turbulence is not observed classically; the first observations of turbulence with direct classical counterparts has come much more recently through the investigation of pressure fluctuations in rotational flow and grid turbulence.

In ³He-⁴He mixtures, like in dilution refrigerators, quantum turbulence can be created far below 1 K if the velocities exceed certain critical values.[3] For velocities above the critical velocity there is a dissipative interaction between the superfluid component and the ³He which is called mutual friction.

Second sound

Second sound is a wave in which the densities of the superfluid and normal components oscillate out of phase with each other. Much of our knowledge about the turbulence in superfluids comes from the measurement of the attenuation of second sound, which gives a measure of the density of vortex lines in the superfluid.

Theoretical developments

The idea that a form of turbulence might be possible in a superfluid via the quantized vortex lines was first suggested by Richard Feynman. Since then, the theoretical understanding of quantum turbulence has posed many challenges, some similar to those of classical fluid mechanics, but also new phenomena peculiar to superfluids and not encountered elsewhere. Some of the theoretical work in this field is quite speculative, and there are a number of areas of divergence between theoretical speculations and what has been obtained experimentally.

Computer simulations play a particularly important role in the development of a theoretical understanding of quantum turbulence.[4][5] They have allowed theoretical results to be checked, and simulations of vortex dynamics to be developed.

Numerical simulations of vortex tangles, basis for vortex reconnections,[6] connections between bundles recently investigated.

References

R.P. Feynman (1955). "Application of quantum mechanics to liquid helium". II. Progress in Low Temperature Physics. 1. Amsterdam: North-Holland Publishing Company.
L. Skrbek (2011). "Quantum turbulence". Journal of Physics: Conference Series. 318 (1): 012004. Bibcode:2011JPhCS.318a2004S. doi:10.1088/1742-6596/318/1/012004.
J.C.H. Zeegers; R.G.K.M. Aarts; A.T.A.M. de Waele & H.M. Gijsman (1992). "Critical velocities in ³He-⁴He mixtures below 100 mK". Physical Review B. 45 (21): 12442–12456. Bibcode:1992PhRvB..4512442Z. doi:10.1103/PhysRevB.45.12442.
K.W. Schartz (1983). "Critical Velocity for a Self-Sustaining Vortex Tangle in Superfluid Helium". Physical Review Letters. 50 (5): 364. Bibcode:1983PhRvL..50..364S. doi:10.1103/PhysRevLett.50.364.
R.G.K.M. Aarts & A.T.A.M. de Waele (1994). "Numerical investigation of the flow properties of He II". Physical Review B. 50 (14): 10069–10079. Bibcode:1994PhRvB..5010069A. doi:10.1103/PhysRevB.50.10069.
A.T.A.M. de Waele & R.G.K.M. Aarts (1994). "Route to vortex reconnection". Physical Review Letters. 72 (4): 482–485. Bibcode:1994PhRvL..72..482D. doi:10.1103/PhysRevLett.72.482. PMID 10056444.