Workshop on Many-body theory of inhomogeneous superfluids, Pisa, July 2007

1. Workshop on Many-body theory of inhomogeneous superfluids, Pisa, July 2007 Quantum Turbulence: Turbulence in a Superfluid Three introductory lectures W F Vinen School of Physics and Astronomy, University of Birmingham, UK

2. Introduction: turbulence in classical fluids • Turbulence (random chaotic motion) in a classical fluid has attracted great interest for many centuries. Leonardo da Vinci • Its existence depends on the non-linear term in the Navier-Stokes equation. • Hence a challenging problem in non-linear dynamics, often of practical importance in fields ranging from aerospace engineering to stellar evolution. • Truly interdisciplinary, involving applied mathematicians, physicists and engineers, different disciplines taking interestingly different approaches.

3. Introduction: quantum effects • Turbulence in a classical fluid is a purely classical phenomenon, perhaps of limited interest to many physicists?! • Are there circumstances where turbulence can be influenced by quantum effects? • The flow of a superfluid is strongly influenced by quantum effects. • Can we have turbulence in a superfluid, and if so how is it influenced by these quantum effects? • The answer is YES: we can have turbulence in a superfluid. Hence the subject of these lectures and those of Carlo Barenghi. • Turbulence always involves rotational motion. Only special, quantized, forms of rotational motion are possible in a superfluid. These special forms – quantized vortices - are forms of inhomogeneity (topological defects) in the superfluid. Hence the relevance to this Workshop.

4. History Some aspects of QT have a long history. First discussed as a theoretical possibility by Feynman in 1955. A special form of quantum turbulence had been discovered experimentally by Hall and Vinen in about 1953-54; this form exists in superfluid 4He in a heat current, and it has no classical analogue. There followed much study of this form of QT, but it was not until the 1990s that there was serious study of types of QT that do have classical analogues. This serious study revealed that there are both similarities and differences between CT and QT. Other forms of QT without classical analogues were discovered in the late 1990s, confirming that in some ways QT is a richer field than is CT. But study of QT has been seriously impeded by a shortage of really effective experimental techniques; a problem to which we shall return and which Carlo Barenghi will address. Most obviously we cannot visualize the flow! Also we have is no anemometer to measure local velocities. The search for new experimental techniques provides us with a serious challenge.

5. Plan of lectures • An introduction to the nature of superfluids and to the quantum restrictions that are relevant to QT. • An introduction to important ideas in classical turbulence. • A first discussion of the simple case of homogeneous QT in superfluid 4He. • How this discussion must be modified for superfluid 3He at high temperatures. • Special problems arising in QT at very low temperatures (no normal fluid). • A first look at the turbulent flow of a superfluid past an obstacle. • These last two areas of discussion will bring us into contact with much of the current research on QT and with many unsolved problems. • A brief look at counterflow turbulence in 4He as an example of a type of turbulence with no classical analogue.

6. References R P Feynman, Prog. Low Temp. Phys., 1, 17 (1955). (First theoretical reference to QT) W F Vinen & J J Niemela, J. Low Temp. Phys., 128, 167 (2002). (General review) A P Finne et al, Rep. Prog. Phys., 69, 3157 (2006). (Review of turbulence in 3He at high temperatures) W F Vinen, J.Low Temp. Phys., 145, 7 (2006). (General review) S Fisher & G Pickett, Physics World, 19, 22 (2006). (Popular review of 3He at very low temperatures) D I Bradley et al, Phys. Rev. Lett., 96, 035301 (2006). (Decay of QT in 3He at very low temperatures) W F Vinen & R J Donnelly, Physics Today, April 2007. (Popular review) Background R J Donnelly,Quantized vortices in helium II (CUP) U Frisch, Turbulence (CUP) P A Davidson, Turbulence (OUP)

7. An introduction to superfluidity I • Occurrence of superfluidity in liquid 4He below at 2.2K and in liquid 3He below about 2 mK. Superfluidity in Bose and Fermi condensed gases at temperatures below about 1 K. Extraterrestrial occurrence in neutron stars. 3He 4He • Focus on 4He and 3He-B

8. An introduction to superfluidity II • A simple superfluid (4He or 3He-B) differs from a classical fluid in three ways: • Two-fluid behaviour: a normal fluid component + superfluid component; normal fluid disappears at zero temperature. • Separate velocity fields . Frictionless flow of the superfluid component. • The superfluid velocity field is subject to severe quantum restrictions. • The existence of the superfluid component is associated with Bose or BCS condensation, as we explain in a moment. • The normal component is composed of thermal excitations: phonons and rotons in 4He; pair breaking fermions in 3He.

9. Bose condensation and superfluidity • Bose or BCS condensation: the accumulation of a macroscopic fraction of the particles in a single quantum state:  coherent particle field (r,t), the condensate wave function. The “particles” are single atoms in 4He; atom pairs in 3He. (Rigorous definition of  is based on the single or two particle density matrices). • Write ; then is the velocity with which the condensed particles are moving. Note that • Particles in an annulus: integrate v round the circuit  where n is an integer if  is to be single-valued. • The circulation K cannot change unless all particles in the condensate change quantum state simultaneously. So the circulation is metastable. In other respects the system can be in equilibrium with the walls of the channel. We have described a metastable supercurrent.Superfluid velocity • We have described a system with broken symmetry (cf a ferromagnet).

10. The superfluid density • Strictly we should identify a supercurrent: where is some effective density. • Superfluidity will exist only if . • This condition places requirements on the form of the spectrum of thermal excitations: phonons only at low energy (4He); energy gap in 3He. Ideal Bose gas is not a superfluid!

11. Quantized vortex lines • We showed that and the hydrodynamiccirculation (m = mass of helium atom for 4He; mass of Cooper pair for 3He) • We can have a finite  in a simply-connected volume if there is a line singularity along which • This topological defect is a quantized vortex line. These lines usually have n=1. The region, radius 0, over which s is depressed is called the vortex core. For 4He: 0~0.05 nm; for 3He-B: 0~80 nm. • An array of these lines allows a macroscopic volume of a superfluid to rotate, with, eg, a containing vessel. On a scale >> vortex separation the velocity field looks the same as classical solid-body rotation.

12. Quantized vortex lines and quantum turbulence • Turbulence involves irregular and chaotic rotational motion. The only form of rotational motion in a superfluid is a quantized vortex line. Therefore quantum turbulence must involve some random array, or tangle, of quantized vortex lines. • To understand how such a tangle evolves we need to know how individual vortices move. • Brief description here; more details from Barenghi

13. The dynamics of quantized vortex lines • For the most part the dynamical behaviour is classical: the same as that of a very thin cylinder of radius  (therefore negligible mass) round which there is a circulation . • If the cylinder moves relative to the surrounding superfluid a Magnus force (lift force) acts on it. If the cylinder has no mass there can be no relative velocity, and the vortex must move with the local superfluid velocity. • This is true only if there is no external force on the vortex. Such an external force can arise if the normal fluid is moving relative to the vortex: the normal fluid excitations are scattered by the vortex  a force of the form S=unit vector along line This is called the force of mutual friction longitudinal transverse • Then the vortex moves according to the Magnus formula • The dimensionless parameter determines how seriously vortex motion is perturbed by mutual friction.For 4He,  <<1; for 3He,  ~1.

14. Calculating vortex motion • Much of our understanding of quantum turbulence comes from computer simulations. • The classical view in the previous slide leads to the vortex filament model. • The local superfluid velocity is often due mostly to the vortices themselves, and it is calculated by using the Biot-Savart law where the integration is along all vortices, including neighbouring sections of the same vortex.These neighbouring sections require special care (to avoid divergences) and give rise to the local induction (LI) contribution • Sometimes this LI contribution (due to curvature of the lines) is dominant, and the LI approximation (LIA) can be used (equivalent to neglect of long-range interactions).

15. The Non-Linear Schrodinger Equation + vortex reconnections • Our description of vortex motion so far has been essentially classical. This fails on length scales comparable with 0. Then we must use a proper quantum description of the dynamics of the condensate. This is the NLSE, to be discussed by Barenghi. BUT valid only for weakly interacting Bose gases. Quantum analogue of Euler and continuity equations for superfluid component. • In the context of quantum turbulence the most important effect not described by the classical theory in the vortex reconnection • Reconnections can be included in the vortex filament model, but • inclusion is artificial • real reconnections are dissipative, which can be important Adams et al) • Therefore NLSE is often used in connection with quantum turbulence, in spite of its shortcomings for real helium.

16. Vortex nucleation: is superflow really metastable? • Circulation round the annulus can be reduced by moving a vortex line across it. • If the supercurrent round the annulus is not to decay, thus movement of the vortex must be forbidden.. • Or nucleation of the vortex must be difficult. • Attraction of vortex to its image in a wall  energy barrier opposing nucleation. • Often this barrier is too large to be overcome thermally or by tunnelling, especially in 4He. • So how are vortex lines generated? Often by growth of existing (remanent) lines, pinned in metastable equilibrium (extrinsic nucleation). • Intrinsic nucleation much easier in 3He.

17. Quasi-classical quantum turbulence? • 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.? • First we must introduce some basic ideas about classical turbulence.

18. Classical Turbulence • Based on the Navier-Stokes equation. Dissipative term Non-linear inertial term • In turbulent flows the non-linear (inertial) term of this equation becomes important relative to the dissipative term. • The ratio non-linear inertial term is the Reynolds number, Re. viscous term At high Re dissipation can be neglected. • Two extreme aspects: • initial instability leading to a transition from laminar flow to turbulent flow at sufficiently high Re. Focus on this initially • behaviour of turbulence when fully developed

19. The classical Richardson cascade • In a typical homogeneous turbulent flow at high Re, energy is injected into large scale motion. Non-linear coupling transfers this energy to smaller scales. If Re for the smaller scales remains large, this transfer takes place with conservation of energy (an inertial regime). Eventually turbulent energy reaches scales where Re ~1, and then it is dissipated by viscosity.  • Role of vortex stretching • Interactions are local in k-space. Hence acascade, taking energy in a step-wise manner to smaller scales • In the steady state, how is the turbulent energy distributed over the different length scales?

20. The Kolmogorov energy spectrum • The non-linear transfer process gives an eddy a lifetime (the turnover time): (Assumes that interactions local in k-space) • Therefore rate of energy flow down inertial cascade per unit mass  = rate of energy loss by eddies of size r ; i.e.  In steady state . Therefore Equivalent to the Kolmogorov energy spectrum: where is the energy in wave number range

21. Words of warning! • The real world is more complicated than is suggested by the previous slide. • Intermittency: we assumed that the smaller eddies fill space (related to self-similarity of the random velocity field) Introducethe nth order structure functions, defined by is related to : But Kolmogorov arguments can be used to show that It is observed that for these relations fail for n 3, most seriously for n >>3. Due, we believe, to intermittency. • Two-dimensional turbulence and inverse cascades: Energy flows to larger scales: an inverse cascade! Negative eddy viscosity? A direct cascade can also exist, with , but it carries enstrophy  , not energy. Both E and  are conserved in an inertial regime in 2D.

22. Quasi-classical quantum turbulence? • So we return to our question: is fully-developed quantum turbulence in any way similar to classical turbulence? • Turn to experiments.

23. Quasi-classical quantum turbulence: experimental evidence • Two classic experiments, both on 4He above 1K: • Observation of the spectrum of 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, using second sound (Stalp, Skrbek & Donnelly). • In neither case do we visualize the flow; a serious disadvantage!

24. The Maurer-Tabeling experiment (4He: T > 1K) • Observation of Pressure fluctuations in a swirling flow generated by two counter-rotating blades (1998)  turbulent energy spectrum in frequency space. (a) 2.3 K (b) 2.08K (c) 1.4 K No change as we go through the superfluid phase transition! But pressure sensor sees only the large scale motion (> 1mm). This large scale motion seems to be the same above and below the superfluid transition! • Thus there is indeed a Richardson cascade and a single Kolmogorov energy spectrum, as in a classical fluid. • Also classical deviations from Kolmogorov (higher order structure functions – intermittency)

25. Title Add a large constant bias velocity, U Pressure fluctuations are then related to the energy spectrum

26. Moving grid experiment (Stalp, Skrbek & Donnelly) (4He; T>1) • Observation of the decay in vortex line density behind a moving grid with second sound. (Second sound is a form of wave motion in a two-fluid system in which the two fluids move in anti-phase with each other. It is attenuated by mutual friction; a measurement of this attenuation  vortex density (length per unit volume) L.) • More difficult to interpret than Maurer-Tabeling, but results are consistent with • a single Kolmogorov spectrum on scales >> mean vortex spacing  • dissipation, on a scale ~ , given by the expression where ’ is a constant with dimensions of kinematic viscosity. • This expression for the dissipation is similar to that known to apply to classical homogeneous turbulence, if ’ = .

27. Why quasi-classical behaviour? Zero temperature. • Start by thinking about the probable outcome of a grid-flow experiment at a very low temperature (no normal fluid). • No really definitive experiments at these temperatures, although it is known that turbulence can be created by a grid and does decay. • Therefore, make reasonable guesses. • On small length scales (<~) the turbulence must be very different from any classical type. • But what happens on large scales (>>, containing many vortices)? The vortex lines can then be arranged to mimic any classical turbulent flow pattern (eg uniform rotation). Is the time evolution also classical? Probably YES. The vortex equations of motion are classical. And systems containing many quanta tend to behave classically • So we guess that on scales >> , in the absence of normal fluid, there could be a Richardson cascade and Kolmogorov energy spectrum. This is provided that there is dissipation on a small scale. We discuss the origin of this dissipation later.

28. Why quasi-classical behaviour? T0 • 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: 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)

29. Review of experimental and computational evidence • Summarize the evidence for quasi-classical behaviour. • Evidence, already noted, that quasi-classical behaviour can be seen in 4He at high temperatures. • Experimental evidence for quasi-classical behaviour at very low temperatures is incomplete (see later). Such evidence is necessary because of suggestion that quasi-classical behaviour in the superfluid at a finite temperature is forced by the normal fluid. • There is evidence from the spin-up experiments that 3He-B does behave at high temperatures in the way suggested (laminar for  > 1; turbulent for  < 1), but no experiments yet on homogeneous turbulence in 3He-B. • Computational evidence for behaviour at T = 0. Eg: Kobayashi & Tsubota, based on NLSE.

30. Dissipation in quantum turbulence and the structure of quantum turbulence at small length scales • If we are to have quasi-classical behaviour, we must have dissipation at small length scales (high wave number). • In a classical fluid turbulent energy is dissipated by viscosity. • A superfluid may have no viscosity, especially if there is no normal fluid! • It is therefore fundamental that we understand the origin of dissipation in quantum turbulence, especially at very low temperatures. • Closely associated with this dissipation is the important question of the structure of quantum turbulence at small (<  ) length scales. • This is an area of much current research.

31. Dissipation in quantum turbulence in 4He at high temps • In principle this is straightforward, because there is plenty of normal fluid • there is viscous dissipation in the normal fluid; • there is dissipation in the superfluid due to mutual friction if there is relative motion between the two fluids; i.e. if the two velocity fields are not the same. In 4He this relative motion exists only on length scales . But this is sufficient to provide high-k dissipation. Indeed it is possible to predict the effective kinematic viscosity ’ at temperatures above 1K. Effective viscosity is similar in magnitude to that of normal fluid.

32. Dissipation in quantum turbulence at very low temperatures I • No normal fluid; no viscous dissipation; no mutual friction. What other mechanisms can there be? • Focus on 4He. 3He is slightly different, and we consider it later. • Up to a point the answer to our question is obvious. Vortex motion can radiate sound. So there is phonon radiation. But typical frequencies associated with vortex motion on a scale  (  /22) are too small to produce significant radiation. • We need energy flow to smaller length scales. There is a natural tendency for turbulent energy to flow to smaller length scales. But how does this happen? And what is the structure of the turbulence on these very small length scales?

33. Small scale structures: a first look • Look at some simulations (Tsubota et al). T = 1.6K T = 0K • At the limited resolution of these simulations the small-scale motion takes the form of kinks on the vortex lines, produced by reconnections.

34. What do reconnections do? • As we have seen, they produce kinks on the vortex lines. • More importantly, they act like the plucking of a string  waves on the vortices. • Hence we must understand the nature of waves on a vortex: Kelvin waves. • Hence the next two slides.

35. An introduction to Kelvin waves • Take a rectilinear vortex (along z)and apply a small z-dependent local transverse displacement (, ) parallel to x-y plane. z • The displacement will propagate as aKelvin wave.   • If you bend a vortex it will move perpendicular to the plane in which it is bent, at a speed proportional to the curvature (compare with vortex ring or smoke ring) x y • Solutions of the form • Circularly polarized modes (sense of rotation opposite to ), with dispersion relation

36. Kelvin waves (cond) • Another view: vortices have a tension equal to the energy per unit length . (Integrate kinetic energy) • Tension in vortex bent in x-z plane gives a force in x-direction equal to • Vortex moves according to Magnus effect. Yields same dispersion relation. • Observation of Kelvin waves in a superfluid (Hall 1958) • Kelvin waves are damped by mutual friction, unless the temperature is very low; otherwise damped only by phonon emission, but only at high frequencies.

37. Repeated reconnections in a vortex tangle • Each vortex line in the tangle is subject to repeated reconnection, leading to the continuous generation of Kelvin waves. • The Kelvin-wave amplitude continuously increases, so that non-linear effects become important. • What happens? • We can get a good picture by doing a simulation.

38. Some simulations • Take a rectilinear vortex and apply oscillating transverse force to drive Kelvin waves at a particular wavenumber (frequency). We apply damping at a high frequency to mimic effect of sound radiation damping. Use full Biot-Savart. k = amplitude of k Fourier component Steady state

39. Kelvin-wave turbulence and the Kelvin-wave cascade • Note that we generated an random chaotic wave motion on the vortex, with strong coupling between different Fourier components. This is just like turbulence. It is an example of wave turbulence. • We fed energy into this turbulent field at a low wave number (large length scale); the energy flowed to a high wave number, where it was dissipated. So we have a kind of cascade, analogous to the classical Richardson cascade: the Kelvin-wave cascade.

40. The Kelvin wave cascade • The subject of much research during the past few years. • Important questions: Is it really a cascade? What is the energy spectrum associated with this cascade (the analogue of the Kolmogorov energy spectrum)? • The spectrum can be obtained from • Dimensional analysis (in part) • Simulations • Weak turbulence theory (see later: confirms local interactions  cascade) • The result: • Dimensional analysis Or with • Simulations and theory i.e.

41. Small scale structures and dissipation in quantum turbulence at very low temperatures I • We can now start to understand these structures and how they lead to dissipation. • Tentatively we see that, on scales less than the vortex spacing , the vortex lines carry Kelvin waves with wave numbers extending from ~ 1/ to much larger values at which waves are attenuated by phonon radiation. • These waves are generated largely by vortex reconnections. • The reconnections themselves are dissipative; each reconnection radiates phonons and/or other thermal excitations. • The energy loss per reconnection is of order 3 x energy per unit length of vortex core; i.e. of order . • If the core parameter is very small, as in 4He, this source of energy loss is expected to be relatively unimportant. It is likely to be more important in 3He or in a Bose-condensed gas.

42. Small scale structures and dissipation in quantum turbulence at very low temperatures II • So we start to build up the following tentative picture of homogeneous quantum turbulence at a very low temperature. (CM excitations refer to 3He: see later) Scale  • We are left with three questions: • At what wave number is the Kelvin wave cascade cut off by phonon radiation? • Exactly what happens at the cross-over from Richardson to Kelvin? And value of ’? • Inverse Kelvin wave cascades?

43. Phonon radiation by Kelvin waves • Can be calculated either • Classically or • Quantum mechanically • The dominant mechanism is quadrupole radiation, or equivalently two Kelvon (Kelvin-wave quanta) processes. • These processes lead to a cut-off on the Kelvin-wave cascade at Typically ()

44. The cross-over from Richardson to Kelvin • A crucial question is whether the Kelvin-wave cascade can carry energy away at a rate to match the flux in the Richardson cascade.. • The energy flux per unit mass in the Richardson cascade can be written i.e. at the crossover at r =  • The energy flux per unit mass in the Kelvin wave cascade is • The cross-over is at . At this wave vector the maximum value of k is presumably given by putting i.e. maximum flux in the Kelvin-wave cascade is • No problem if A ~ 1.

45. The cross-over from Richardson to Kelvin: is there a bottleneck? • Very recently it was shown (L’vov, Nazarenko & Rudenko) from weak turbulence theory that So • Therefore a serious bottleneck?

46. Experimental evidence: the effective kinematic viscosity for homogeneous turbulence at T = 0 • If there is a serious bottleneck then we must expect ’ << . • Preliminary experimental evidence is confusing • In 3He-B, ’ ~ 0.3. Pickett et al, Lancaster (see later) (decay of vorticity in 4He when a containing bucket ceases to rotate) • In 4He,’ ~ 0.005. Golov et al, Manchester • Comment on the possibility of infinite Reynolds numbers! Owing to a numerical accident, the kinematic viscosity of the normal phase of 4He is roughly 0.2 . So the effective viscosity relevant to turbulence in a superfluid 4He is not very different from that in the normal phase. So superfluidity (zero viscosity) does not lead to infinite Reynolds numbers!

47. The absence of a bottleneck? • Discussion by Kozik & Svistunov. Complicated, but in essence perhaps as follows. • We have made the unrealistic assumption that there is a sudden transition from the classical Richardson cascade to the Kelvin-wave cascade at precisely . • In reality there will be complicated transition region, in which it is not clear what happens. • If the Kelvin-wave cascade starts at, say, , and if we suppose that at this wave number the Kelvin wave amplitude can be as large as , then there need be no bottleneck. • If the average wave number is, say, , and if we continue to assume that the maximum amplitude of the Kelvin waves at this wave number is , then the problem disappears.

48. The speculative nature of our ideas on dissipation at very low temperatures • There are as yet few relevant experiments! Those we have give conflicting results.

49. Behaviour of 3He-B at very low temperatures • Up to now we have suggested that at low temperatures the two isotopes are very similar (if the difference in ’ is real we do not understand it). • But there ought to be small differences, because structure of vortex core is different. • Core is much larger than in 4He, giving greater energy loss during reconnections, but, more importantly, the core contains low-lying thermal excitations (Caroli-Matricon, in context of type II superconductors). • Spacing of these energy levels • They can absorb energy strongly from Kelvin waves of frequency ~ 10 kHz, so we expect that there is no need to rely on phonon radiation. • No relevant experimental evidence.

50. The Kelvin-wave cascade reconsidered: inverse cascades • The Kelvin-wave cascade can be treated analytically by “weak turbulence theory”, in which the non-linear interactions are treated as scattering events. For Kelvin waves the lowest relevant scattering process is a 6-Kelvon process.Locality in k-space  cascade. • This process conserves the number of Kelvons. • Now the existence of the Kolmogorov energy spectrum in 3D turbulence depends on existence of an inertial regime in which energy is conserved. • We see that in the weak turbulence theory of the Kelvin wave cascade there are two conserved quantities: the energy; and the Kelvon number (or wave action). • It has been suggested that there are then two simple forms of cascade: • finite energy flux; zero Kelvon flux: ~ (forward) Nazarenko • finite Kelvon flux; zero energy flux: ~ (inverse) • A real cascade will then involve a “superposition” of these two cascades, in proportions depending on the boundary conditions; increased complication.