Quantum Turbulence in Helium Superfluids

1. Quantum Turbulence in Helium Superfluids L. Skrbek Joint Low Temperature Laboratory, Institute of Physics ASCR and Charles University V Holešovičkách 2, 180 00 Prague 8,Czech Republic Thanks to collaboration and discussions with: A.V. Gordeev Prague T. Chagovets W.F. Vinen University of Birmingham R.J. Donnelly University of Oregon S.R. Stalp J.J. Niemela Trieste K.R. Sreenivasan P. Skyba Košice M. Krusius - LTL HUT (ROTA), Helsinki V.B. Eltsov, A.P. Finne J. Hosio N.B. Kopnin, G.E. Volovik P.V.E. McClintock, G. Pickett - Lancaster University D.I. Bradley, D. Charalambous, D.O. Clubb, S.N. Fisher, A.M. Guénault, P.C. Hendry, H. A. Nichol M. Tsubota, R. Hanninen, A. Mitani Osaka City University C. Barenghi Newcastle and others University of Warwick, December 8th, 2005

2. Outline • An experimentalist’s attempt of presenting a unified perhaps (over)simplified picture of quantum turbulence in He II and superfluid 3He-B based on current theoretical understanding and available experiments • Recent results on experimental investigation of He II (counterflow and pure superflow – generated) turbulence

3. 4He 3He Solid He He I P (kPa) Superfluid He II Criticalpoint gas T (K)

4. T (p) (cm2/s) / air 20 C 0,15 0,122 water 20 C 1,004x10-2 14,4 Normal 3He above Tc ~ 1, olive oil Normal fluid of 3HeB around 0.6 Tc ~ 0.2, air Helium I 2,25 K (VP) 1,96x10-4 3,25x10-5 Helium II 1,8 K (VP) 9,01x10-5 X He-gas 5,5 K (2,8 bar) 3,21x10-4 1,41x108 Classical turbulence – measures of its intensity Reynolds number For isothermal flows Rayleigh number for thermally driven flows in a gravitational field • He II and 3He B – so far the only two media where quantum turbulence has been systematically studied under controlled laboratory conditions • Cryogenic He Gas, He I and He II are probably the best known working fluids for the controlled, laboratory high Re and Ra turbulence experiments

5. Heat transfer efficiency in cryogenic turbulent convection convection cryostat Niemela, Skrbek, Seenivasan, Donnelly TurbulentConvection at Very High Rayleigh Numbers, Nature 404 (2000) 837 1 m

6. Systems (simplified) to understand turbulence in 4He 3He (B) normal liquid He I Classical Navier-Stokes fluid of extremely low kinematic viscosity T normal liquid 3He Classical Navier-Stokes fluid of kinematic viscosity comparable with that of air 1 Superfluid transition at Tc He II – a “mixture” of two fluids superfluid 3He B normal fluid of extremely normal fluid of of kinematic viscosity low kinematic viscositycomparable with that of air + + Inviscid superfluid Inviscid superfluid Circulation is quantized Circulation is quantized T 0 limit Pure superfluidPure superfluid 3 0 2

7. Step 1 – classical turbulence in a viscous fluid Grid turbulence- visualisation Smoke Ink or dye Kalliroscope flakes Hydrogen bubbles Baker’s pH technique Hot wire anemometry Laser Doppler velocimetry Particle image velocimetry … Classical turbulence - Richardson cascade

8. 3d energy spectrum of the homogeneous isotropic turbulence Energy containing eddies Inertial range Spectral energy density (log scale) Dissipation range Viscosity is unimportant ….. …….. up to here k Energy containing length scale Dissipative (Kolmogorov) length scale

9. Spectral decay model for HIT Skrbek, Stalp: On the decay of homegeneous isotropic turbulence, Physics of Fluids12, No 8, (2000) 1997 Diff. equation for HIT Classical turbulence The model has been recently verified by computer simulations by Touil, Bertoglio and Shao (J. Turbulence 3 (2002) 049) Direct and large eddy simulations, as well as the eddy-damped quasi-normal Markovian closure

10. For decaying vorticity, several subsequent decaying regimes are predicted: Simplest case – assume that the energy- containing scale is saturated - D Note – on first approximation, intermittency does not change the power of the decay:

11. Step 2 – superfluid turbulence in a pure superfluid (T=0) Quantum length scale

12. 3d energy spectrum of the superfluid homogeneous isotropic turbulence Energy containing eddies Inertial range, no dissipation Spectral energy density (log scale) Dissipation range due to phonon irradiation Hypothesis: The form of the superfluid energy spectrum at T=0 is of the form as that of classical turbulence Is there any, at least partial evidence ??? k Energy containing length scale Quantum dissipative length scale

13. Numerical evidence T. Araki, M.Tsubota and S.K.Nemirovskii, Phys.Rev.Lett.89, 145301(2002): Energy Spectrum of Superfluid Turbulence with No Normal-Fluid Component The energy spectrum containing an inertial range of a Kolmogorov form was obtained under the vortex filament formulation. C. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9, 2644(1997) By solving the Gross-Pitaevskiiequation, they studied the energy spectrum of a Taylor-Green flow. The spectrum shows the -5/3 power on the way of the decay, but the acoustic emission makes the situation complicated. M. Kobayashi and M. Tsubota, PRL94, 65302 (2005) obtained the Kolmogorov spectrum more clearly, by introducing the small- scale dissipation

14. Experimental evidence in He II, Lancaster, P. McClintock’s group Davis et al, Physica B 280, 43 (2000) It is clear that He II turbulence decays (on time scale of seconds)at very low temperature Due to complexity of this experiment, quantitative comparison with the decay model is difficult

15. Experimental evidence in 3He B, Lancaster, G. Pickett’s group

16. 3He-B

17. Experimental evidence in 3He B – Lancaster (courtesy of S. Fisher) Classical spectral decay model – universal decay

18. Step 3 – turbulence within the two-fluid model Viscous dissipation Phonon emission by Kelvin waves Energy spectrum is steeper here, due to action of the dissipative mutual friction, not any more strong enough to lock the N and S eddies together The turbulent energy partly leaves the cascade, being transferred into heat, which makes the spectrum steeper Richardson cascade Up to here the normal and superfluid eddies are locked together by the mutual friction force. They evolve together and there is negligibly small energy dissipation – hence Kolmogorov scaling Mutual friction beyond some critical k becomes negligibly small again and Kolmogorov scaling is recovered Phonon emission by high k Kelvin waves D – size of our laboratory box

19. Could the entire energy spectrum be experimentally observed? He II Typical cooling power does not exceed about 1 [W] into 1 [liter] of liquid In fluid dynamics, all energies are considered per unit mass, i.e. E [m2/s2] Asssuming HIT, we use Main distance between vortices Kolmogorov dissipation length Using a typical-size laboratory cells or flow channels (of order 1 cm) we can, in principle, observe the entire energy spectrum of He II turbulence. Note that during the decay the energy containing length scale, as well as the relevant dissipation scales grow. We can therefore use decaying turbulence to study them more closely

20. 3He B Typical cooling power does not exceed about 1 [nW] into 1 [cm3] of liquid Asssuming HIT, we get Main distance between vortices Kolmogorov dissipation length Using a typical-size laboratory cells or flow channels (of order 1 cm) we cannot observe the entire energy spectrum of He II turbulence, the normal fluid will always stay stationary. We need a theory for superfluid turbulence in a stationary normal fluid

21. Classical flow - Navier-Stokes equation Re depends on the geometry of flow in question Dissipation Superfluid flow – by averaging the Euler equation over distance >> intervortex spacing plays a role of inverse Reynolds number and does NOT depend on the geometry

22. energy in energy in Dissipation due to mutual friction at large scales viscous dissipation Continuous approach is justified if circulation at the smallest scale >>circulation quantum Classical or Kolmogorov regime quantum cutoff L’vov, Nazarenko, Volovik: They describe cascade in the simplest possible manner, using the differential form of the energy transfer term in the energy budget equation Ek is the one-dimensional density of the turbulent kinetic energy in the k-space If then the Komogorov scaling is recovered

23. Vinen considers the effect of mutual friction, which can be written in the form Considering the effect of mutual friction at different length scales Vinen obtains (for all length scales) a characteristic decay time time The Kolmogorov spectrum will survive if this time exceeds the corresponding eddy turn-over time. This is satisfied only for eddies smaller than Bigger eddies cannot exist – they become fast dissipated by mutual friction The flow of energy toward larger wave numbers can be described as a diffusion of energy in k space. It can be shown that this diffusion must obey a nonlinear diffusion equation, which can be written, for the case of homogeneous, isotropic turbulence, in the form

24. He II - experiment Energy containing eddies Inertial range Spectral energy density (log scale) dissipation Maurer, Tabeling: Europhysics Lett. 43 (1998) 29 Flow between counterrotating discs U=80cm/s; Re=2x106 a-2.3K; b-2.08K; c-1.4 K k Energy containing length scale Dissipative (Kolmogorov) length scale

25. Decay of the grid generated turbulence in He II – main features Character of the decay does not change with temprature (1,1 K < T < 2,1 K), although the normal to superfluid density ratio does by more than order of magnitude There are four different regimes of the decay of vortex line density (vorticity – if defined as ) in the finite channel, characterized by: Stalp, Skrbek, Donnelly: Decay of Grid Turbulence in a Finite Channel,Phys. Rev. Lett. 82 (1999) 4831 Skrbek, Niemela,Donnelly: Four Regimes of Decaying Grid Turbulence in a Finite Channel, Phys. Rev. Lett. 85 (2000) 2973

26. Energy spectrum – He II grid turbulence Skrbek, Niemela, Sreenivasan: The Energy Spectrum of Grid – Generated Turbulence, Phys. Rev. E 64 (2001) 067301 Quantum length scale Is the classically generated He II coflow turbulence the same as classical? Yes – on large length scales, exceeding the quantum length No – on length scales comparable or smaller

27. Conclusions (1) • It seems that co-flow quantum turbulence in He II and 3He B can be, at least approximately, understood, based on the suggested form of the energy spectrum • Depending on the energy decay rate and size of the sample, various parts of the spectrum can be probed • Does this form of the energy spectrum apply to other systems? • (BEC, cosmic strings….) Energy spectrum is steeper here, due to action of the dissipative mutual friction, not any more strong enough to lock the N and S eddies together The turbulent energy partly leaves the cascade, being transferred into heat, which makes the spectrum steeper Richardson cascade Up to here the normal and superfluid eddies are locked together by the mutual friction force. They evolve together and there is negligibly small energy dissipation – hence Kolmogorov scaling Mutual friction beyond some critical k becomes negligibly small again and Kolmogorov scaling is recovered Phonon emission by high k Kelvin waves D – size of our laboratory box

28. Does this model also work for counterflow (pure superflow) turbulence in He II above 1K? No Why ? Due to counterflow, large normal and superfluid eddies cannot be locked together, so the dissipationless inertial range cannot exist (Steady state) counterflow turbulence = thermally driven turbulence, similarly as thermal convection Counterflow turbulence phenomenology (Vinen 1957) Vortex ring T 0 2b Finite T In counterflow, though, if rings with expand Dimensional analysis and analogy with classical fluid dynamics leads to the Vinen equation: Reproduced by Schwarz (1988) – computer simulations Local induction approximation Importance of reconnections production decay

29. Counterflow channel V S

30. Steady state data When heater is switched off, the following amount of heat must flow away through the channel (neglecting the channel volume) Assuming quasi-equilibrium, for decaying temperature gradient we get Note: without knowing anything of quantized vortices, this simple model predicts that the extra attenuation decays linearly with time! We can use this simple model to estimate the time over which the temperature difference disappears • It is easy to take into account • the volume of the channel itself, • assuming linear T gradient along it • (i.e., add half of its volume) During this time, although the heater is switched off, the turbulence is NOT isothermal, but is driven by the (decaying) temperature gradient

31. Estimating the effective volume (i.e., V +half of the counterflow channel volume) to be 7.5 cm3 Isothermal decaying turbulence Thermally driven decaying turbulence

32. Experiments on counterflow turbulence in superfluid He II Joint Low Temperature Laboratory, Institute of Physics and Charles University, Prague Detection method: Second sound attenuation Second sound is generated and detected by oscillating gold-plated porous nuclepore membranes Steady-state data superfluid normal fluid Vinen equation predicts the decay of vortex line density of the form

33. Can we treat our data on decaying counterflow turbulence as isothermal ? • Yes(all our second sound data within experimentally accessible time domain) • Direct measurement of the temperature difference across the channel • Simple model for decaying temperature gradient when the heater is switched off • (Gordeev, Chagovets, Soukup, Skrbek, JLTP 138 (2005) 554) A net increase of the vortex line density during the decay ?

34. Depolarization of the vortex tangle? Wang, Swanson, Donnelly, PRB 36 (1987) 5236 vortex tangle generated in the steady state counterflow turbulence is polarized L • The tangle is fully polarized • vortices lying randomly in planes perpendicular counterflow velocity. • Second sound is effectively attenuated by their projection to the plane perpendicular to the second sound wave, i.e., by • (ii) The tangle is random in 3D • Again, the second sound is effectively attenuated by the projection of • these vortices to the plane perpendicular to the second sound wave, this time by “randomization” - transverse second sound would indicate a net growth by a factor up to i.e., up to about 23 %

35. Decay of counterflow turbulence in He II above 1K Stalp, Skrbek, Donnelly: Phys. Rev. Lett. 82 (1999) 4831 Skrbek, Gordeev, Soukup:Phys. Rev. E (2003) fort > tsatthe decay of vortex line density (vorticity) both in grid-generated and counterflow-generated He II turbulence displays the classical exponent of -3/2

36. After saturation of the energy-containing length scale – universal decay law Dependence on the channel size experimentally confirmed for the first time (even for classical turbulence) Gordeev, Chagovets, Soukup, Skrbek: JLow Temp. Phys.138 (2005) 554 S10 The late decay -consistent with Kolmogorov – type turbulence S6

37. Turbulence generated by pure superflow He II bath level Sintered silver superleaks He II heater

38. Turbulence in He II generated by pure superflow (there is no net normal fluid flow through the channel) The vortex line density decays exponentially Superfluid flow profile is flat – the channel width is no more a relevant parameter

39. Conclusions (2) • Developed turbulence in He superfluids displays features consistent with two different turbulent states • Kolmogorov state (classical-like regime) • Vinen state (quantum regime) • Transitions between these states most likely occur and have been observed (T1 to TII transition as classified by Tough, decay of thermal counterflow) • There are experimental data for He II and 3He-B, consistent with the existence of the flow phase diagram predicted by Volovik Volovik:JETP Letters 78 (2003) 553 Skrbek: JETP Letters 80 (2004) 474 Do quantum fluids hold the key to unlocking the underlying physics of fluid turbulence?

40. Typical decay data – pure superflow, T=1.72 K, 6x6 mm channel No heating Decay heating on Overshoot

41. Stationary data – pure superflow thermal counterflow of He II through superleaks T=1.72 K

42. Shaft to the cryostat flange and linear driving motor Two counterrotating discs with six blades each Second sound transmitter/receiver