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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 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 COSLAB, Smolenice, September 2, 2005

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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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**for thermally driven flows in a gravitational field**

Measures of turbulence intensity Reynolds number For isothermal flows Rayleigh number for thermally driven flows in a gravitational field Ra Re Sun 1021 1013 Ocean 1020 109 Atmosphere 1017 Navy (ship) Aerospace (aircraft) T (p) (cm2/s) / air 20 C 0,15 0,122 water 1,004x10-2 14,4 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) 1,96x10-4 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 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 Cryogenic He gas T (K) P (kPa)**

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

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**Quantum mechanical description of He II**

Macroscopic wave function Circulation- multiply connected region Circulation –singly 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 4He - 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) Very low kinematic viscosity of the normal fluid (above appr. 1 K) Two-fluid model is also applicable to superfluid 3He phases (talk of Matti Krusius)

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**Systems (simplified) to understand turbulence in**

4He 3He T normal liquid He I Classical Navier-Stokes fluid of extremely low kinematic viscosity 1 normal liquid 3He Classical Navier-Stokes fluid of kinematic viscosity comparable with that of air 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 viscosity comparable with that of air Inviscid superfluid Inviscid superfluid Circulation is quantized Circulation is quantized T limit Pure superfluid Pure superfluid 3 2

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**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

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**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

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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 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 Superfluid turbulence

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**For decaying vorticity, theree subsequent regimes 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:

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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**

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

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**Numerical evidence 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. △: 2 < k < ○: 2 < k < □: 2 < k < 16 E(k)〜 k -n(t) E(k) There are three works to date which studied the energy spectrum of a vortex tangle without mutual friction. One is the contribution of the Paris group, and the other two are ours. 5/3

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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. In order to overcome this difficulty, we have studied the turbulence of the Gross-Pitaevskii model in the wave number k-space by introducing the dissipative mechanism. First I will talk about the decaying turbulence, which is reported in this paper. Next, I will discuss steady turbulence of our very recent work. By introducing,..

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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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**Experimental evidence in 3He B – Lancaster (courtesy of S. Fisher)**

Classical spectral decay model – universal decay

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**Step 3 – turbulence within the two-fluid model**

Viscous dissipation Phonon emission by Kelvin waves Richardson cascade 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 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

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**He II Could the entire energy spectrum be experimentally observed?**

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 [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

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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

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**Classical flow - Navier-Stokes equation Re depends on the geometry of flow in question**

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

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**Classical or Kolmogorov regime quantum cutoff**

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

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**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

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**He II - experiment Inertial range Spectral energy density (log scale)**

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

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**NS HeII 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**

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

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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) 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

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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….) Richardson cascade 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 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

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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) 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

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

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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 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

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**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

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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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**L 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 %

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**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) for t > tsat the decay of vortex line density (vorticity) both in grid-generated and counterflow-generated He II turbulence displays the classical exponent of -3/2

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**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) S10 The late decay -consistent with Kolmogorov – type turbulence S6

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**Turbulence in He II generated by pure superflow**

(there is no net normal fluid flow through the channel) He II bath level Sintered silver superleaks The vortex line density decays exponentially 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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