Joe Vinen

School of Physics and Astronomy

University of Birmingham

Symposium on Universal Features of Turbulence: Warwick, December 2005

An Introduction to Quantum Turbulence

General introductory review: Vinen & Niemela: J Low Temp Physics 128, 167 (2002)

The aims of this presentation

Quantum turbulence is the name we give to turbulence in a superfluid, in which fluid motion is strongly influenced by quantum effects.

It is an old subject, first discussed about 50 years ago. But recently interest in it has grown strongly, and much of this symposium is concerned with it.

My aims are

• to describe a little of the relevant history;

• to provide a background and introduction to many of the papers on quantum turbulence that will be presented at this symposium.

• to emphasize links with problems in classical turbulence

History I

Two parallel and independent developments in the 1950s:

• Feynman’s suggestion that superfluids rotate through the presence of quantized vortex lines, and that these lines could allow a form of turbulence in a superfluid.

• Experiments (Hall and Vinen) showing that the Gorter-Mellink mutual friction accompanying thermal counterflow in superfluid 4He was due to turbulence (rotational motion) in the superfluid component.

These developments merged when Hall and I realized that vortex lines can give rise to mutual friction, due to scattering of the thermal excitations that constitute the normal fluid by vortex lines. This led directly to a quantitative theory of mutual friction in the uniformly rotating superfluid: a comparison of this theory with experiment provided the first evidence in favour of the existence of quantized vortex lines.

• These experiments were followed by the successful search for mutual friction in uniformly rotating 4He.

History II

Between 1955 and 1995 attention focussed on:

• The theory of thermal counterflow turbulence.

• Problems relating to the nucleation of quantized vortex lines.

• The structure and properties of vortices in superfluid 3He.

Thermal counterflow turbulence has no classical analogue. Strangely, there were no serious experiments on types of superfluid turbulence that do have classical analogues until the mid-1990s. These experiments opened up a whole new range of interesting questions, which have been responsible for much of the recent renaissance of interest in quantum turbulence.

At the same time experiments started to appear on turbulence in superfluid 3He-B, which again raised new and interesting questions. And of course experiments started on Bose-condensed cold atoms, stimulating still further interest.

I want to focus especially on these various new questions, which will be taken up in more detail by other speakers.

Simple superfluids I

Simple superfluids (4He; 3He-B; cold atoms) exhibit

• Two fluid behaviour: a viscous normal component + an “inviscid” superfluid component. Normal component disappears at lowest temps.

• Quantization of rotational motion in the superfluid component.

Quantization of rotational motion: , except on quantized vortex lines, each with one quantum of circulation

0svcurl

34 2mhmhds or rv

(Consequencies of Bose or BCS condensation.)

Viscosity of normal fluid: 4He very small; 3He-B very large. Turbulence in normal fluid? 4He: YES; 3He-B: NO.

( ~0.05 nm for 4He; ~80nm for 3He-B; larger for Bose gases).

round a core of radius equal to the coherence length

Calculating vortex motion

In absence of normal fluid an element of vortex generally moves with the local superfluid velocity, often calculated with the vortex filament model using either the full non-local Biot-Savart law

3

0

00 4 rr

rrrrv

ds

or in appropriate cases the local induction approximation (LIA)

rrrrr

rv

140 ln)()(loc

s

In the presence of normal fluid we must add the force of mutual friction

Lnd vvf ~ (+ a transverse component)

This force causes the vortex to move relative to the local superfluid velocity in accord with to the classical Magnus effect.

A useful dimensionless parameter is . sIf (4He) motion of vortex is only weakly perturbed by mutual friction.. 1If (3He-B at high temperatures) motion is strongly perturbed. 1

The Non-Linear Schrodinger Equation and vortex reconnections

Our description of vortex motion has been essentially classical. It will fail on length scales where quantum effects become important, notably on scales of order . The only available quantum theory is that leading to the NLSE:

In the context of quantum turbulence the most important effect not described by the classical theory is the vortex reconnection, first described in terms of the NLSE by Koplik & Levine.

Reconnections can be included in the vortex filament model, but

• describes the static and dynamic behaviour of the condensate, • but applies quantitatively only to the weakly-interacting Bose gas.

• inclusion is artificial• real reconnections are dissipative, which can be important (Barenghi, Adams et al)

Therefore NLSE is often used in connections with quantum turbulence, in spite of its shortcomings for real helium.

Thermal counterflow turbulence

4He above 1K.

Experiments indicated homogeneous turbulence in the superfluid component, maintained by the relative motion of the two fluids. No classical analogue.

A understanding was provided by the pioneering simulations of Schwarz, based on the vortex filament model and the LIA. He showed that self-sustaining tangles of lines could arise from the mutual friction, provided that one allows for reconnections (artificially introduced).

Schwarz provided us with a quantitative theory, but some problems remain: • Is the normal fluid turbulent (Melotte & Barenghi)?

• Artificial introduction of reconnections (correct criteria?).• Is the LIA adequate (high values of )?

Vortex nucleation

An energy barrier opposes the creation of vortices, except at the highest velocities.

Of course such a barrier is crucial to the existence of superfluidity!

But at typical velocities and temperatures at which turbulence is observed the barrier is often too large to be overcome thermally or by tunnelling, especially in 4He .

Therefore in these cases the nucleation must be extrinsic: i.e. dependent on remanent vortices.

Until recently, study of extrinsic nucleation was hampered by ignorance of, and lack of control over, the configuration of the nucleating vortex. However, recent experiments in Helsinki have shown that with 3He-B there can be better control, and this work will be described in later papers.

Both nucleation and subsequent propagation of turbulence have been studied in 3He-B in a rotating vessel (analogous to classical spin-up experiments).

Intrinsic vortex nucleation

Intrinsic nucleation can be observed:

• with some care in 3He-B

• with much care in 4He (small makes it difficult to remove remanent vortices)

An early demonstration with 4He involved vortex nucleation by a small sphere in the form of a negative-ion bubble (McClintock et al)

• an energy barrier of about 3 K was observed at a bubble velocity ~ 40 ms-1

• at higher temperatures there was thermal activation

• at lower temperatures there was quantum tunnelling

The 3K barrier was correctly predicted by a modified vortex filament model (Muirhead, Vinen & Donnelly).

Modelling of this type of nucleation with the NLSE (Roberts & Berloff; Frisch, Pomeau & Rica ; Huepe & Brachet; Winiecki, McCann & Adams), can be very instructive, but the NLSE may not provide a good enough model for real helium (is there an energy barrier?).

Also seen in later work on flow through small apertures by Varoquaux et al

Quasi-classical quantum turbulence I

It is strange that for many years the only form of quantum turbulence to be studied seriously was that produced by thermal counterflow in 4He, which does not have a classical analogue.

An obvious question:

• What happens if you replace the classical liquid in a typical example of classical turbulence by a superfluid?

• For example in flow through a grid, which classically produces the much-studied case of homogeneous isotropic turbulence.

• Do we get analogues of Richardson cascades; Kolmogorov energy spectra; etc.?

Quasi-classical quantum turbulence II

Even now there are only two detailed experiments, both on 4He above 1K:

• Observation of the spectrum pressure fluctuations in turbulence produced by counter-rotating discs (Maurer & Tabeling).

• Observation of the decay of vortex-line density in the wake of a steadily moving grid (Stalp, Skrbek & Donnelly).

The pressure fluctuations are observed with a pressure transducer with size ~ 0.5mm. They show that on scales 0.5 mm there is a Kolmogorov spectrum,

The moving grid experiments are more difficult to interpret, but are consistent with:

3532 kCkE

vorticity square mean effective 22L

indistinguishable from that above the superfluid transition

• a similar Kolmogorov spectrum on scales >> mean vortex spacing

• dissipation, on a scale ~ , given by the quasi-classical expressionL=vortex line density

Why quasi-classical behaviour? (Vinen 2000)

Start by thinking about the probable outcome of a grid-flow experiment at a very low temperature (no normal fluid).

On small length scales (<~) the turbulence must be very different from any classical type.

So we can argue that on scales >> there could be a Richardson cascade and Kolmogorov energy spectrum. This is provided that, as seems to be the case, there is dissipation on a small scale. We return to the origin of this dissipation later.

There are no detailed experiments at these temperatures, although it is known that turbulence can be created by a grid and does decay.

All this could apply equally to 4He and 3He-B.

But on large scales (>>, containing many vortices) the vortex lines can be arranged, with local polarization, to mimic classical turbulent flow, including, probably, the time-evolution of this flow.

Why quasi-classical behaviour? II

Now raise the temperature, to produce some normal fluid.

We must now distinguish between 4He and 3He-B.

• In 4He the normal fluid has a very small viscosity. Therefore it too becomes turbulent in the wake of the grid, with a Richardson cascade and Kolmogorov energy spectrum. Thus the flow in each fluid is likely to display Kolmogorov spectra. But the two fluids are coupled by mutual friction. The two velocity fields become locked together, and we get a single velocity field with a single Kolmogorov spectrum, as observed.

• In 3He-B the normal fluid is too viscous to become turbulent. Therefore its effect is the damp the turbulence in the superfluid, through the effect of mutual friction. The result can be predicted (Vinen 2005; Lvov et al 2005): it turns out that a small mutual friction ( << 1) damps only the largest quasi-classical

eddies;

a large mutual friction ( 1) will kill the turbulence in the superfluid.

(-1 acts as a kind of Reynolds number)

Experimental and computational evidence?

BUT, no detailed experimental evidence yet for quasi-classical behaviour at very low temperatures.

There is evidence from the spin-up experiments that 3He-B does behave at high temperatures in the way suggested (importance of the parameter ), but no experiments yet on homogeneous turbulence in 3He-B .

Evidence, already noted, that quasi-classical behaviour can be seen in 4He at high temperatures.

Computational evidence for behaviour at T = 0. Eg: Kobayashi & Tsubota, based on NLSE.

Dissipation in quantum turbulence

When there is normal fluid this is easy:

• there is viscous dissipation in the normal fluid;

• there is dissipation in the superfluid due to mutual friction. In 4He this occurs only on length scales , where the two velocity fields cannot match, but this is sufficient to provide high-k dissipation required for the Kolmogorov spectrum. Indeed it is possible to predict the effective kinematic viscosity at temperatures above 1K.

22L

Dissipation in quantum turbulence at very low temperatures

No normal fluid; no viscous dissipation; no mutual friction. What other mechanisms can there be?

We need energy flow to smaller length scales. Look at a simulation: the evolution of a tangle of vortex lines at very low temperatures.

Tsubota et al

The kinks involve smaller length scales and are produced by large numbers of reconnections.

Vortex motion can radiate sound. But typical frequencies associated with this motion on a scale are too small to produce significant radiation.

Dissipation associated with reconnections I

Two sources of dissipation:

• phonon emission during reconnections.

• phonon emission from high-frequency Kelvin waves produced by reconnections.

It turns out that phonon emission during reconnections is likely to be very important in cases where the vortex spacing is not much more than the vortex core size . This is the case in Bose gases modelled by the NLSE: simulations by Nore & Brachet; and by Kobayashi & Tsubota.

Dissipation associated with reconnections II

But in helium (especially 4He) dissipation during reconnections is relatively unimportant owing to the small size of the vortex core.

In that case we note that repeated reconnections lead to the continual generation of Kelvin waves on each length of vortex (cf plucking of a string).

• Some of these Kelvin waves have a very high frequency and can generate phonons very efficiently;

• Others have a lower frequency, but non-linear interactions can lead to transfer of energy (in a cascade?) to the required high frequencies (Svistunov; Vinen; Vinen, Tsubota & Mitani; Kozik & Svistunov: numerical work and weak turbulence theory).

• This transfer process involves a form of wave turbulence (again a link with classical fluid mechanics), which will be discussed rather fully in later papers, along with the form of energy spectra associated with this process and questions about direct and inverse cascades.

The overall picture?

So perhaps we have the following picture of the evolution of turbulence in superfluid 4He at a very low temperature. Energy flows to smaller and smaller length scales:

phonons

• First in a classical Richardson cascade

• Followed by a Kelvin-wave cascade

• With final dissipation by radiation of phonons

• The length scale (= vortex spacing) at which we change from Richardson to Kelvin-wave cascades adjusts itself automatically to achieve the correct dissipation.

3He-B may be similar except that energy can be lost from the Kelvin waves into quasi-particle bound states in the cores of the vortices (Caroli-Matricon states), which do not exist in 4He. This occurs at a frequency much smaller than that required for phonon radiation.

Oscillating wires and grids at very low temperatures

Recent experiments on both 4He and 3He-B (Lancaster; Osaka).

Turbulence produced is inhomogeneous.

Too early to draw conclusions?

No systematic classical results with which to compare.

Comments and conclusions

We have focussed on cases of homogeneous turbulence, because it seems best to try to understand these cases first.

Much of our discussion has been speculative, although it has thrown up many interesting theoretical questions. We have also ignored potentially interesting details, such as deviations from Kolmogorov scaling and the existence of analogues of coherent structures in classical turbulence.

There is still a serious shortage of experimental data, especially at very low temperatures, and the data we do have are based on techniques that do not provide the kind of detailed information (about eg velocity fields) available to those studying classical fluid mechanics. Simulations provide some kind of “experimental data”. But are they reliable and can they extend over the wide ranges of length scale that seem to be important in quantum turbulence?

Major problems and challenges face us in the development of new techniques relating to very low temperatures and to the acquisition of more sophisticated data. Papers by Carlo Barenghi, Gary Ihas, and the Lancaster 3He Group will address some of these questions.

Finally I have emphasized relationships between quantum turbulence and classical turbulence (including wave turbulence). Other links will be emphasized later in the symposium.

Many friends, colleagues and organizations.

• Grant support from EPSRC

Cryogenic Turbulence Laboratory, University of Oregon (NSF Grant D MR-9529609);

Newton Institute for Mathematical Sciences, Cambridge;

The Royal Society.

Tsunehiko ArakiCarlo Barenghi,Demetris Charalambous Russell Donnelly, Marie Farge,Shaun Fisher.Andrei GolovHenry HallDemos Kivotides,Matti Krusius,Akira MitaniPeter McClintock,Joe Niemela,Alastair Rae,David Samuels,Ladik Skrbek,Edouard Sonin,Steve Stalp, Boris Svistunov,Makoto Tsubota, Grisha Volovik.

Acknowledgements

Thank you

Comments and conclusions

We have focussed on cases of homogeneous turbulence, because it seems best to try to understand these cases first.

Much of our discussion has been speculative, although it has thrown up many interesting theoretical questions. We have also ignored potentially interesting details, such as deviations from Kolmogorov scaling and the existence of analogues of coherent structures in classical turbulence.

There is still a serious shortage of experimental data, especially at very low temperatures, and the data we do have are based on techniques that do not provide the kind of detailed information (about eg velocity fields) available to those studying classical fluid mechanics. Simulations provide some kind of “experimental data”. But are they reliable and can they extend over the wide ranges of length scale that seem to be important in quantum turbulence?

Major problems and challenges face us in the development of new techniques relating to very low temperatures and to the acquisition of more sophisticated data. Papers by Carlo Barenghi, Gary Ihas, and the Lancaster 3He Group will address some of these questions.

Finally I have emphasized relationships between quantum turbulence and classical turbulence (including wave turbulence). Other links will be emphasized later in the symposium.