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Experiments on homogeneous quantum turbulence in 4 He: what do we really know, and what can we hope to do in the future? Andrei Golov Thanks: Paul Walmsley, Sasha Levchenko, Joe Vinen, Henry Hall, Peter Tompsett, Dmitry Zmeev, Fatemeh Pakpour 1.Important points and questions 2.High-temperature limit: counter-flow and co-flow turbulence 3.Low-temperature limit: one-component turbulence, Manchester results 4.Conclusions and Outlook for T=0 regime Abu Dhabi Workshop, 3 May 2011

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Introduction Two interacting fluid components: -normal (fluid of excitations): hydrodynamics at T > 0.7K; -superfluid (ordered condensate): inviscid & irrotational; In superfluid, vorticity is concentrated along lines - with preserved circulation; location of these lines is the only degree of freedom; Interaction between the two components is via vortex lines. Large-scale approximations exist, e.g. HVBK equations

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Some tasks: Find conditions when Quantum Turbulence (QT) resembles Classical Turbulence, if any Observe behaviours specific to either two-fluid system or quantized superfluid vorticity Investigate the dynamics of different types of tangles of quantized vortices, especially the dissipation mechanisms and rates Find if QT helps generalize the science of turbulence: - QT has its own keys to understanding of CT (?); - QT is another example alongside hydrodynamic, wave and MHD turbulence (?)

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Tangles of quantized vortices in 4 He d dissipation k l = L -1/2 Classical Quantum 0.03 – 3 mm45 mm ~ 3 nm From simulations by Tsubota, Araki, Nemirovskii (2000) T = 1.6 K T = 0 Microscopic dynamics of each vortex filament is well-understood since ~1860 (Helmholtz). It is the consequences of their interactions and especially reconnections – that are non-trivial. An important observable – length of vortex line per unit volume (vortex density) L. However, without specifying correlations in polarization of lines, this is insufficient. The dynamics of tangles is determined by energy, impulse, angular momentum – which strongly depend on line correlations over various length scales. mean inter-vortex distance vortex bundles, etc. Kelvin waves

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Two-tier picture of velocity field Sometimes it is possible to separate energy into: “quantum”, E q ~ ρ ln( Ɩ /a 0 ) L, and “classical”,

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Types of vortex tangles Uncorrelated (Vinen) tangle of vortex loops (v c ~ 0, E c << E q ) : Free decay: L(t) = B ’ -1 t -1, where B = ln(l/a 0 )/4 =1.2, if dE/dt = - ’( L) 2 Correlated tangles (e.g. eddies of various size as in HIT of Kolmogorov type). When E c >> E q, free decay L(t) = (3C) 3/2 -1 k 1 -1 -1/2 t -3/2 where C ≈ 1.5 and k 1 ≈ 2 /d, if size of energy-containing eddy is constant in time, its energy lifetime dE c /dt = d(u 2 /2)/dt = - Cu 3 d -1, dE/dt = - ’( L) 2. k EkEk l -1 k EkEk d -1

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Purely classical 2-fluid description for high T (no Kelvin cascade = no bottleneck) L’vov, Nazarenko, Skrbek (2006): (See also Vinen, Barenghi et al., Volovik... Roche, Barenghi, Leveque (2009))

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One-fluid quasi-classical turbulence at T=0 L’vov, Nazarenko, Rudenko, 2007-2008 (bottleneck, pile-up of vorticity at mesosclaes ~ l) Kozik and Svistunov, 2007-2008 (reconnections, fractalization, build-up of vorticity at mesoscales ~ l) I.e. at T = 0, it is expected to have excess L at scales ~ l. Unlike classical techniques (which are usually sensitive to velocity at large scales), For QT, the convenient observable is L (vorticity ~ ( L) 2 ).

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From Kolmogorov to Kelvin-wave cascade (Kozik & Svistunov, 2007) crossover to QT reconnections of vortex bundles reconnections between neighbors in the bundle self – reconnections (vortex ring generation) purely non-linear cascade of Kelvin waves (no reconnections) length scale phonon radiation

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Simulations (T=0): -5/3 spectrum Gross-Pitaevskii (non-linear Schrodinger): Nore, Abid and Brachet (1997) Kobayashi and Tsubota (2005) C = 0.32-0.55 Machida et al. (2008) Filament model (Biot-Savart): Araki, Tsubota, Nemirovskii (2002) C = 0.7 But: in classical turbulence, C = 1.5 Eq. for coarse-grained superfluid velocity v :

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Simulation: Kelvin wave cascade (T=0) Barenghi, Tsubota, Vinen, Kozik&Svistunov Most recent: Baggaley & Barenghi (2011): As yet, no satisfactory simulations of both cascades at once

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Experiments with two-component flows (high T): Counter-flow turbulence Pure supeflow Co-flow turbulence

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Steady-state counterflow: rich structure Experiments: Baehr, Courts, Tough 1982, 1985, 1988, 1989 In square channels, two different states (T1 and T2) depending on heat flux; T2 is well-described by theory

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Sampling normal velocity in counterflow Melotte & Barenghi (1998): normal velocity profile can be either parabolic (laminar, T-1) at low flux or flat (turbulent, T-2) at high flux Probing by injected ions (Awschalom, Milliken, Schwarz, 1984): v n is spatially-homogeneous across channel Probing by molecules He 2 * (Guo et al., 2008-2010): Hint at parabolic profile at low flux

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Free decay of counterflow turbulence Barenghi and Skrbek (2007): Free decay of T2-state, L~ t -3/2, implies formation of classical eddies of size of channel width

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Vortex vizualization and velocity analysis Paoletti, Fisher, Sreenivasan, Lathrop (2008): non-classical velocity statistics in counterflow Bewley, Paoletti1, Sreenivasan, Lathrop (2008) Barenghi (2010), Adachi&Tsubota(2011): reconnections not necessary to account for this velocity statistics turbulence in water (Gaussian over 5 decades)

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Nearly pure superflow turbulence (high T) Experiment: Ashton, Opatowsky, Tough (1981) Normal component locked in experimental volume, superfluid flows through During pumping, QT with flat profile in stationary normal component: perfect agreement with theory (Schwarz, Tsubota)

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Chagovets and Skrbek (2008): observed a new state! State A: agreement with simulations for flat v n = 0, State B: perhaps NC starts moving? Nearly pure superflow turbulence (high T)

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Mechanically generated turbulence (high T) Maurer and Tabelung, Europhys. Lett., 43 (1998) 29

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Roche et al., EPL, 77 (2007) 66002. Roche, Barenghi (2007) Mechanically generated turbulence (high T)

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Followed by: Skrbek, Niemela, Donnelly, PRL 2000; Skrbek and Stalp, Phys. Fluids 2000; Stalp, Niemela, Vinen, Donnelly, Phys Fluids 2002; Niemela, Sreenivasan, and Donnelly, JLTP 2005. Model for the free decay of quasi-classical homogeneous turbulence with saturated energy-containing length (scale of largest eddies limited by the container size d): L(t) ~ t -3/2 Mechanically generated turbulence (high T

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One-component turbulence: (T < 0.7K) Ultra-quantum regime: preparation by a limited number of colliding vortex rings Quasi-classical regime. Generation: spin-down with different boundary conditions, torsional oscillations, ion pump Switching between these regimes

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4.5 cm Experimental Cell We can inject rings from the side We can also inject rings from the bottom We can create an array of vortices by rotating the cryostat The experiment is a cube with sides of length 4.5 cm containing 4 He (P = 0.1 bar).

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Trapped negative ions When inside helium at T < 0.7K, electrons (in bubbles of R ~ 19Å) nucleate vortex rings Charged vortex rings can be manipulated and detected. Charged vortex rings of suitable radius used as detectors of L: Force on a charged vortex tangle can be used to engage liquid into motion

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Free decay of ultra-quantum turbulence (little large-scale flow) V = 0.1 L(t) = 1.2 ’ -1 t -1 Simulations of non-structured tangles: Tsubota, Araki, Nemirovskii (2000): n ~ 0.06 k (frequent reconnections) Leadbeater, Samuels, Barenghi, Adams (2003): n ~ 0.001 k (no reconnections)

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Means of generating large-scale flow 1. Change of angular velocity of container (e.g. impulsive spin-down from to rest or AC modulation of ) 2. Dragging liquid by current of ions (injected impulse ~ I×∆t ) I ×∆ t

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Free decay of quasi-classical turbulence (dominant large-scale flow) t -3/2

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Free decay of quasi-classical turbulence (E c > E q ) k EkEk l -1 d -1 L(t) = (3C) 3/2 -1 k 1 -1 -1/2 t -3/2 where C ≈ 1.5 and k 1 ≈ 2 /d. Puzzle 1 (Vinen): if < (2 -1/2 ( /3C) 3/2 dL -1/2 < 0.14 , then at late-time E q > E c, hence, L ~ t -3/2 should not hold !

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Different means of generation – puzzles at low T Puzzle 2: at T < 0.7 K, spin-down and ion jet result in quite different values of ’ Quasi-classical turbulences of different spectra, both decaying as L ~ t -3/2 ? L(t) = (3C) 3/2 (d /2 ) ’ -1/2 t -3/2

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Inject ions for 2 – 500 s, then wait time t and probe L at right angle. Transient states of ion-jet generated turbulence (free decay) Initially, the cross-section of backflow (tangle) grows Eventually, the largest scale of flow is saturated by container size II Inject ions for 30 s in E = 10 – 100 V/cm, then probe L at right angle.

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Conclusions (T = 0) 1: Quasiclassical tangles decay as L ~ t -3/2. This is consistent with a developed cascade truncated at cell size, E ~ t -2. At T < 0.5K ( < 10 -5 ), two different states with effective kinematic viscosity = 0.003 and = 0.05 observed. A better characterization of the structure of the tangle and comparison of classical and quantum energy is required. Ultraquantum tangles decay as L ~ t -1. This is consistent with Vinen’s equation, E ~ t -1, and the effective kinematic viscosity = 0.1 .. However, more research is required to understand ion- generated turbulence at low temperatures and turbulences that decay as L ~ t -1.

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Outlook (T=0): beyond classical range Probing individual vortices – say, Kelvin wave cascade on rectilinear vortex lines Space- and time- resolved measurements of velocity/pressure fluctuations in quantum regime (~1mm and ~10s in large containers) Tagging/visualizing vortices with small particles (He 2 *)

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