Abu Dhabi, May 2011

Transitions to quantum turbulence

W F Vinen

School of Physics and Astronomy, University of Birmingham

Introduction

In principle we can study transitions in• steady pipe flow, including steady thermal counterflow in 4He (in

practice very little except in thermal counterflow and related flows)

• steady flow past an obstacle (in practice only limited study, except for steady flow through a grid).

• oscillatory flow past an obstacle (many studies, involving spheres, cylinders (wires), grids and tuning forks - experimentally easy)

• spin up and spin down (spin-down in 4He – Golov; spin-up and down in 3He - Eltsov)

• oscillatory pipe flow (only in early work before anything was really known about quantum turbulence)

• transitions in a Bose-condensed gas: Tsubota and discussion session on Thursday

Introduction

In principle we can study transitions in• steady pipe flow, including thermal counterflow in 4He (in practice

very little except for thermal counterflow and related flows)

• steady flow past an obstacle (in practice only limited study, except for steady flow through a grid)

• oscillatory flow past an obstacle (many studies, involving spheres, cylinders (wires), grids and tuning forks - experimentally easy)

• oscillatory pipe flow (only in early work before anything was really known about quantum turbulence)

I could try to describe the results of many experiments on these transitions, and then try to draw conclusions. But the effect would be confusing, partly because of the large volume of material, and partly because many of the experimental data are confusingly incomplete. So I shall try to talk about general principles, as I understand them, appealing occasionally to experiment, and then try to raise what I see as important questions for possible discussion.

The nature of the transitions

The transitions are from states in which there is potential flow of the superfluid, with perhaps a few pinned remanent vortices, to fully turbulent superfluid, containing a more or less dense array of vortex lines in irregular motion.

These transitions involve a succession of two processes:

At a finite temperature the normal fluid must be involved

• the nucleation of free vortex line

• the growth of this line into a turbulent tangle (may involve >1 step).

• In 4He the normal fluid has a very small viscosity, so it too can become turbulent

• In 3He-B the normal fluid is very viscous and cannot become turbulent

• Mutual friction accompanying the vortex lines tends to couple motion in the two fluids (4He)

or damp turbulent motion in the superfluid (3He)

or have a role in the generation of turbulence (thermal counterflow)

• Often I shall talk about very low temperatures - little normal fluid.

Connection with classical turbulent transitions?

No obvious connection, because in the classical case the instability is from a state of viscous flow (eg in a viscous boundary layer) in contrast to the state of potential flow in the superfluid with slip at a solid boundary (albeit with a few remanent vortices)

However, we shall discuss the possibility that the growth of vortex line, following nucleation, may have classical features.

Intrinsic nucleation (a)

Suppose that we start from a state where there are no vortices.

Nucleation is impeded by a potential barrier: eg at a solid boundary (otherwise no superfluidity!)

This barrier can be overcome at a high velocity by• thermal activation• (macroscopic) quantum tunnelling

In 4He the barrier can be overcome at a reasonable rate (by thermal activation) only at temperatures very close to the superfluid transition, except at velocities exceeding ~10 m s-1 (experiments by Reppy et al very close to the transition).

At very high velocities in 4He nucleation has been observed by both thermal activation and quantum tunnelling (experiments by McClintock et al)

• A linear instability when vortex core touches boundary at the maximum in E.

Intrinsic nucleation (b)

In 3He-B intrinsic nucleation can be observed at a solid boundary at much smaller velocities ~ mm s-1 (owing to much larger vortex core): experiments by Ruutu et al (JLTP 106, 93, 1997) showing a dependence on surface roughness.

Comparison with Landau critical velocities (for generation of excitations – rotons (4He) or pair breaking (3He):

• in 4He the roton critical velocity is comparable with the intrinsic nucleation velocity at low temperatures - but unimportant here because quantum turbulence in 4He is always studied at much smaller velocities

• in 3He-B the pair breaking velocity can often be comparable with the vortex nucleation velocity, leading to difficulties in interpretation of experimental results

Note that intrinsic nucleation in 4He involves a situation where the laminar state (potential flow) is linearly stable. Return later to the significance of this fact in relation to turbulent lifetimes.

Extrinsic nucleation in 4He

Vortex nucleation is almost always observed to take place in 4He at velocities of order cm s-1.

Nucleation must then be extrinsic and based on pinned remanent vortices, left over in metastable equilibrium from previous experiments or from the Kibble-Zurek process when the helium was cooled through its phase transition.

Experimental evidence (Hashimoto et al. PRB 76, 020504, 2007)

Closed cell except for small (0.1 mm) pinhole

Extrinsic nucleation in 4He: steady flow

Evolution of pinned remanent vortex under influence of superflow (or relative motion of two fluids)?

Continued pinning at low velocities: otherwise no superfluidity

Critical velocity for channel with some characteristic dimension D

DUSC

~ = quantum of circulation

Above some critical velocity there must be some process, involving surface and/or volume reconnections, that leads to the multiplication of vortices.

Extrinsic nucleation in 4He: oscillating flow

Behaviour of an oscillating sphere, with remanent vortex shown.

Oscillation at frequency generates Kelvin waves with half wavelength ~(/)1/2.

A reconnection at the surface of the sphere produces a vortex loop of radius r ~ (/)1/2.

This loop will expand under the influence of the velocity U of the sphere if U > ~ /r.

Similar surface reconnections might occur in steady flow past a sphere, leading to the generation of attached vortex loops with size ~ radius, R, of the sphere,

21~SCU Therefore critical velocity

RUSC ~

Change from one regime to the other at 2R ~

Some experimental evidence in favour of these formulae; reality is more complicated

A comment on remanent vortices

We have assumed that the density of remanent vortices is small.

There have been suggestions that this may not be true, in the sense that a solid surface may be covered in a thin but dense layer of vortices. It is not clear how such a layer could exist in metastable equilibrium, and the evidence for it (anomalously high effective mass for oscillating structures) is not fully reproducible and therefore questionable.

Vortex nucleation in 3He-B

As already explained, the study of vortex nucleation in 3He-B is made complicated by the fact that intrinsic nucleation tends to occur at a velocity close to the pair breaking velocity. Furthermore extrinsic nucleation can also occur at a similar velocity. In many experiments with oscillating structures in 3He-B it is not clear what nucleation process is actually occurring. In recent work by Yano’s group (Nago et al, PRB 82, 224511 (2010), significant disentangling of this situation for the case of vortex generation by a vibrating wire has been achieved by careful study of frequency and pressure dependences. (The extrinsic critical velocity is proportional to 1/2 and independent of pressure, while the depairing and intrinsic critical velocities are independent of frequency and increase with increasing pressure.)

This is not to say that experiments on vibrating structures in 3He-B are not valuable, and we shall return later to interesting results with a vibrating grid in 3He-B.

Transitions to fully-developed turbulence: thermal counterflow

Experiments homogeneous turbulence in the superfluid component, maintained by relative motion of the two fluids, with gradual build up.

A understanding was provided by the pioneering simulations of Schwarz, based on the vortex filament model. He showed that self-sustaining tangles of lines could probably arise from the mutual friction, provided that one allows for reconnections. Confirmed by more recent simulations by Tsubota.

First observed in detail in thermal counterflow in 4He above 1K: no classical analogue

Complications? Turbulence in the normal fluid? An understanding of this doubly turbulent system presents us with a major challenge.

So far we have considered only the initial breakdown of laminar (potential) flow; as the velocity increases we get a gradual transition to more fully-developed turbulence

Self-sustaining turbulence is possible only above a critical velocity, dependent on width of channel. (Return later to precise meaning of self-sustaining)

Transitions to fully-developed turbulence: flow past an obstacle in 4He

Look at two examples involving 4He.

In the wake of a moving grid, we get homogeneous isotropic turbulence, very similar to classical case

Behaviour of the drag on the prongs of a tuning fork

AUCF D2

2

1 Definition of drag coefficient

Transitions to fully-developed turbulence: quasi-classical at high velocities?

Strong hints here that at high velocities there is a tendency towards quasi-classical (Q-C) behaviour.

We are fairly certain that this is right in the case of flow through a grid because there is evidence for a Kolmogorov spectrum. Can similar things happen in other cases?

For example, for flow past a sphere.

Increasing Reynolds number

• Perhaps a similar sequence in a superfluid?

• Or perhaps a direct transition to a pattern similar to that at the highest Reynolds number?

Quasi-classical behaviour at high velocities?

Q-C behaviour expected only if the vortex density in the neighbourhood of the structure is such that << all characteristic scales of the quasi-classical flow. (This is how we understand the appearance of a Kolmogorov spectrum in quantum grid turbulence.) How does this high density build up in order that Q-C behaviour can appear?

Appeal to the results of simulations

Smooth oscillating sphere 200Hz, T = 0, R = 100 m, U = 150 mm s-1, in a tube,

with one remanent vortex (Hanninen, Tsubota, Vinen, PRB 75, 064502, 2007)

When the velocity of the sphere exceeds a critical value the vortex density near the sphere grows very rapidly (in an avalanche?).

The vortex avalanche

So simulations for an oscillating sphere in 4He do suggest that a high vortex density can be set up above a critical velocity by surface reconnections (generated by high amplitude Kelvin waves) and vortex stretching. This leads to a number of questions.

• Is there a corresponding process that can operate in a steady flow? Not known

• Does the process that seems to operate in an oscillating flow facilitate the generation of Q-C flow patterns? In the case of an oscillating sphere, what is the classical flow pattern? pend_movie9.avi (Donnelly)

• Do we understand this process? An attempt has been made to describe it analytically by Hanninen & Schoepe (JLTP 153, 189, 2008). Based on an equation due to Kopnin, but the validity of this application of the equation is questionable. Can we do better? Not known.

The situation in 3He? Experimental evidence for the case of an vibrating grid. A different process, but again leading initially to a dense random tangle of vortex lines. (Subsequent evolution to Kolmogorov spectrum raises interesting questions.)

The vortex avalanche: does it lead to Q-C behaviour?

Is there any indication of Q-C behaviour in the simulations of Hanninen et al?

• Apparently NO! The vortices seem to be unpolarized (random tangle). There is no polarization required to produce something like a macroscopic vortex ring.

• The instability that leads the production of the macroscopic vortex rings by the oscillating sphere arises in the viscous penetration depth (Taylor-Görtler instability), which has no obvious analogue in the quantum case

• Nevertheless, a random tangle of vortex lines ought to behave to some extent like a classical fluid, with an eddy kinematic viscosity of order (characteristic velocity times characteristic length), at least on length scales >> . Something is missing.

• In fact the unpolarized tangle probably produces too small a drag.

Q-C behaviour: the right boundary conditions

• But the classical fluid has a no-slip condition at a solid boundary. No analogous condition holds in the simulations of Hanninen et al, since the sphere is smooth, with no vortex pinning.

• To mimic classical flow the vortex tangle must have not only an effective viscosity, but also the right boundary conditions.

• Probably vortex pinning would mimic the no-slip condition.

• So we need to repeat the Hanninen et al simulations with vortices pinned at the surface of the sphere. Quite hard, and not yet done.

• A question: precisely what density of vortices is required to get Q-C behaviour? It could be that must be much less than the thickness of a boundary layer or a viscous penetration depth. Hard to achieve?

More than one critical velocity?

Two critical velocities? (a) random tangle; (b) Q-C transition?

Vibrating wires at 10 mK (Bradley et al, JLTP 154, 97, 2009)

Vibrating tuning fork at 10 mK (Deepak et al, to be published)

Lack of reproducibility?

Vibrating wire at 1.5K (Bradley

et al, JLTP 162, 375, 2011) Various vibrating forks at very low temperatures (Lancaster)

We see two types of behaviour:

• A cusp in CD followed by a sharp rise towards a value close to unity.

• No cusp in CD, and a levelling off at a value significantly less than unity.

Suggestion by Blazkova et al (PRB 79, 054522, 2009) that a cusp corresponds to transition to quasi-classical turbulent flow pattern; perhaps this transition does not always occur because density of vortices is insufficient.

Hysteresis and switching

Vibrating wire (Bradley et al, JLTP 138, 493, 2005)

Oscillating sphere (Niemetz and Schoepe, JLTP 135, 447, 2004)

Switching has been studied in great detail by Schoepe et al. It appears that over a range of driving forces neither the turbulent state nor the laminar state is stable; each has a finite lifetime, after which it switches rapidly (collapses) to the other regime.

Switching from laminar (potential) flow to turbulent flow must be associated with remanent vortices. What about switching the other way?

Quantum turbulence: is it self-sustaining?

Thermal counterflow turbulence is believed to be “self-sustaining”, although it is likely that the corresponding instability is convective, not absolute (Schwarz: PRL 64,1130 (1990). But needs to be checked with modern computations (no LIA)) Observation of switching behaviour suggests that quantum turbulence produced by flow past an obstacle need not be self-sustaining, at least at low velocities.

Simulations are not yet able to answer this question. Experiments are difficult because it is difficult to remove all nucleating vortices!

Our question can be put in a different way. Is any particular form of quantum turbulence “self sustaining”, in the sense that it is maintained even if the initial source of nucleating vortices is removed?

But we recall that Yano et al do know how to start with no nucleating vortices. Have they been able to answer our question for their particular wires?

Quantum turbulence not self-sustaining for vibrating structures

The blue wire is free from remanent vortices. Set this wire vibrating. Use the red wire to bombard the blue wire with vortex rings, Turbulence develops around the blue wire, and drag increases, Switch off beam of rings from red wire. Observe what happens to the drag on the blue wire (Yano et al, PRB 81, 220507 (2010)

Strong suggestion that lifetime of turbulent state is always finite (in absence of perturbations), although it becomes very large at high drives. Generally true, for all structures?

Compare with classical systems exhibiting collapse of turbulent state into linearly stable laminar flow (eg pipe flow - linearly stable, both absolutely and

convectively).

Concluding remarks

The transition to quantum turbulence is fundamentally different from its classical analogue because it involves a transition from potential flow rather than viscous laminar flow.

The transition must involve initially the generation of quantized vortex line; this process can be either intrinsic or extrinsic.

However, as soon as the vortex line density becomes large enough the vortices might arrange themselves so that the flow patterns can mimic those in the transition to classical turbulence. There is some evidence in favour of this view. But the view must remain largely speculative until more powerful experiments become possible (especially visualization). If the view is correct the transition to fully developed turbulence may be a three-stage process: nucleation of vortex lines; evolution into a dense random tangle; evolution into quasi-classical turbulent patterns.

The role of simulations is still limited. They are unable to handle large enough number of vortices or realistic boundary conditions.

The observation of the spontaneous collapse of quantum turbulence offers another interesting classical-quantum comparison.

Thank you