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Quantum turbulence - an overview L. Skrbek Joint Low Temperature Laboratory, Institute of Physics ASCR and Charles University V Holešovičkách 2, Prague 8, Czech Republic COSLAB, Smolenice, September 2, 2005 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 Osaka City University and others 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

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Turbulence - outstanding problem of classical physics that remains unsolved to this day Leonardo Da Vinci This talk - An experimentalist’s attempt of presenting a unified, perhaps (over)simplified, picture of quantum turbulence, based on current theoretical understanding and experiments in He II and superfluid 3He-B

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Characteristic length scales in turbulence Quantized vortex in He II

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Measures of turbulence intensity Reynolds number For isothermal flows Rayleigh number for thermally driven flows in a gravitational field RaRe Sun Ocean Atmosphere Navy (ship)10 9 Aerospace (aircraft) T (p) (cm 2 /s) // air20 C0,150,122 water20 C1,004x ,4 Normal 3He above Tc ~ 1, olive oil Normal fluid of 3He B around 0.6 Tc ~ 0.2, air Helium I 2,25 K (VP) 1,96x ,25x10 -5 Helium II1,8 K (VP)9,01x10 -5 X He-gas 5,5 K (2,8 bar) 3,21x ,41x10 8 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 He II and 3He B – so far the only two media where quantum turbulence has been studied under controlled laboratory conditions

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Phase diagram of 3He Normal liquid 3He Classical Navier-Stokes fluid

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Phase diagram of 4He T (K) P (kPa) He I – normal liquid Classical Navier-Stokes Solid He Cryogenic He gas Superfluid He II Critical point

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Quantum mechanical description of He II Macroscopic wave function Circulation –singly connected region Circulation- multiply connected region Quantized vortices in He II Rotating bucket of He II -thanks to the existence of rectilinear vortex lines He II mimics solid body rotation vorticity

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Hydrodynamics of superfuid 4 He - He II Two-fluid model (Landau) He II flow is well described only in a limit of low velocities Thermal counterflow Second sound prediction – ( entropy or temerature waves) Two-fluid model is also applicable to superfluid 3 He phases (talk of Matti Krusius) Very low kinematic viscosity of the normal fluid (above appr. 1 K)

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

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

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

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Spectral decay model for HIT Skrbek, Stalp: On the decay of homegeneous isotropic turbulence, Physics of Fluids 12, No 8, (2000) 1997 Diff. equation for HIT Classical turbulence Superfluid 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

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

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Step 2 – superfluid turbulence in a pure superfluid (T=0)

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3d energy spectrum of the superfluid homogeneous isotropic turbulence k Spectral energy density (log scale) Energy containing eddies Energy containing length scale Quantum dissipative length scale Dissipation range due to phonon irradiation Inertial range, no dissipation 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 ???

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Numerical evidence △ : 2 < k < 12 ○ : 2 < k < 14 □ : 2 < k < 16 E(k) 5/3 E(k) 〜 k -n(t) T. Araki, M.Tsubota and S.K.Nemirovskii, Phys.Rev.Lett.89, (2002): Energy Spectrum of Superfluid Turbulence with No Normal-Fluid Component The energy spectrum of a Taylor-Green vortex was obtained under the vortex filament formulation. The absolute value with the energy dissipation rate was consistent with the Kolmogorov law. C. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9, 2644(1997) By solving the Gross-Pitaevskii equation, 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.

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Gross-Pitaevskii model by introducing the small-scale dissipation M. Kobayashi and M. Tsubota, PRL94, (2005) obtained the Kolmogorov spectrum more clearly.

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Experimental evidence in He II 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

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

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Step 3 – turbulence within the two-fluid model 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 Mutual friction beyond some critical k becomes negligibly small again and Kolmogorov scaling is recovered Phonon emission by high k Kelvin waves 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 D – size of our laboratory box Viscous dissipation Phonon emission by Kelvin waves

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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 ment per unit volume, i.e. E [m 2 /s 2 ] 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

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3 He B Typical cooling power does not exceed about 1[nW] into 1 [cm 3 ] 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

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plays a role of inverse Reynolds number and does NOT depend on the geometry Superfluid flow –by averaging the Euler equation over distance >> the intervortex spacing Classical flow - Navier-Stokes equation Re depends on the geometry of flow in question assuming the normal fluid is at rest : After rescaling the time (Volovik) Dissipation is distinctly different, depends on the large-scale velocity U: Dissipation:

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energy in viscous dissipation energy in Dissipation due to mutual friction at large scales 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 E k is the one-dimensional density of the turbulent kinetic energy in the k-space If then the Komogorov scaling is recovered

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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 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

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He II - experiment k Spectral energy density (log scale) Energy containing eddies Energy containing length scale Dissipative (Kolmogorov) length scale dissipation Inertial range Maurer, Tabeling: Europhysics Lett. 43 (1998) 29 Flow between counterrotating discs U=80cm/s; Re=2x10 6 a-2.3K; b-2.08K; c-1.4 K

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Local pressure fluctuations are measured using a small total-head pressure tube local pressure upstream flow After removing the mean parts NS HeII

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Grid turbulence in He II - detection scheme used in the Oregon decay experiments Detection method: Second sound attenuation Second sound is generated and detected by oscillating gold-plated porous membranes Quantity detected: quantized vortex line density, L (its projection to a plane perpendicular to the direction of the second sound propagation, averaged over channel crossection )

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Decay of the grid generated turbulence in He II – main features There are four different regimes of the decay of vortex line density (vorticity – if defined as ) in the finite channel, characterized by: 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 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

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Energy spectrum – He II grid turbulence Skrbek, Niemela, Sreenivasan: The Energy Spectrum of Grid – Generated Turbulence, Phys. Rev. E 64 (2001) 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 Quantum length scale

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Conclusions (1) 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 Mutual friction beyond some critical k becomes negligibly small again and Kolmogorov scaling is recovered Phonon emission by high k Kelvin waves 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 D – size of our laboratory box 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….)

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Does this model also work for counterflow (pure superflow) turbulence in He II ? 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) 2b Vortex ring T 0 Finite T In counterflow, though, if rings with expand Dimensional analysis and analogy with classical fluid dynamics leads to the Vinen equation: production decay Reproduced by Schwarz (1988) – computer simulations Local induction approximation Importance of reconnections

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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 superfluid normal fluid Vinen equation predicts the decay of vortex line density of the form Steady-state data

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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 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 Note: without knowing anything of quantized vortices, this simple model predicts that the extra attenuation decays linearly with time!

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Estimating the effective volume (i.e., V +half of the counterflow channel volume) to be 7.5 cm 3 Thermally driven decaying turbulence Isothermal decaying turbulence

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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 ?

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Depolarization of the vortex tangle? Wang, Swanson, Donnelly, PRB 36 (1987) 5236 vortex tangle generated in the steady state counterflow turbulence is polarized (i)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 L “randomization” - transverse second sound would indicate a net growth by a factor up to i.e., up to about 23 %

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

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After saturation of the energy-containing length scale – universal decay law S10 S6 Dependence on the channel size experimentally confirmed for the first time (even for classical turbulence) The late decay -consistent with Kolmogorov – type turbulence

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

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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 Do quantum fluids hold the key to unlocking the underlying physics of fluid turbulence?

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