Quantum TurbulenceAndrew W. Baggaley   Carlo F. Barenghi   Yuri Sergeev              28th November, 2011                  ...
Outline   Introduction       Quantum Fluids       Quantised vortices       Vortex reconnections       Kelvin wave cascade ...
Classical picture   Turbulence in a normal (classical) fluid,                                              3 / 57
Classical vortices     Velocity v,     vorticity ω =    × v.     In a classical fluid,     vortices are     unconstrained, ...
Superfluidity      Below a critical temperature ( 2.2K) 4 He becomes superfluid,      a component of the fluid can flow withou...
Quantised vortices     Gross-Pitaevskii Equation (weakly     interacting gas)            ∂Ψ     2                         ...
Quantum Turbulence      Quantum turbulence (QT) is a tangle of these quantised      vortices.      Hence QT is more simple...
Quantum vortex reconnections      If vortices become close (core size) they can reconnect.      Directly observed (Paolett...
Kelvin waves  Quantised vortices  reconnect, creating  pronounced cusps.  Also the interaction of  vortices can lead to  p...
10 / 57
Energy spectrum      At scale larger than intervortex spacing - classical      (Kolmogorov) regime. Experimentally (Maurer...
Vortex filament model   Helium experiments: average distance between lines ( ≈ 10−1 to   10−4 cm) is much bigger than core-...
Tree approximation      Biot-Savart law scales as N 2 .      Use tree methods (Barnes and Hut [1983]) used in      astroph...
Outline   Introduction       Quantum Fluids       Quantised vortices       Vortex reconnections       Kelvin wave cascade ...
Grenoble experiment                      15 / 57
A suprising result?       Intensity of quantum turbulence is characterized by the vortex       line density L (vortex leng...
Finite temperature effects      0 < T < Tλ - two fluid system,      normal (viscous) fluid and superfluid Helium.      mutual ...
Modelling the normal fluid      We wish to avoid DNS of turbulent normal fluid,      we use Kinematic Simulations (KS) model...
4/3                k1|vn |,   Re =              ≈ 180                kM                                   19 / 57
Numerical results                                    Frequency spectrum Vortex line density L =   Λ/D3                    ...
An explanation      Roche et al. argued randomly oriented vortex lines have some      of the statistical properties of pas...
Frequency spectrum of passive line      Line density grows and saturates as before but at larger values.      Understandab...
Outline   Introduction       Quantum Fluids       Quantised vortices       Vortex reconnections       Kelvin wave cascade ...
Velocity statistics     Ordinary viscous flows     Homogeneous isotropic turbulence     Turbulent velocity v(r, t)     PDFs...
Classical velocity statistics           Experiment:                Theory:           Noullez & al         Vincent & Menegu...
Classical velocity statistics       Velocity v at point r is determined by vorticity ω =    × v via       Biot-Savart law:...
Maryland experiment         Measurement of velocity PDF in superfluid 4 He                                                 ...
Maryland experiment    Turbulent superfluid 4 He,    solid hydrogen tracers radius    ≈ 10−4 cm.    Paoletti et al found   ...
Velocity statistics in quantum turbulence      Calculation of velocity PDF in various vortex configurations:               ...
Velocity statistics in quantum turbulence                                            30 / 57
Velocity statistics in quantum turbulence                       Vortex filament method   Kolmogorov turbulence in 4 He (AWB...
Grenoble experiment                      32 / 57
Grenoble experiment                             10−3                      p(v)                             10−4           ...
Why power-law PDFs rather than classical Gaussian ?       Paoletti, Lathrop & Sreenivasan:       vortex reconnections are ...
From power law to Gaussian             Solution of the Maryland-Grenoble puzzle:    = average vortex spacing   ∆ = size of...
Conclusion        = average intervortex spacing      At scales >> quantum turbulence is similar to ordinary      turbulenc...
Outline   Introduction       Quantum Fluids       Quantised vortices       Vortex reconnections       Kelvin wave cascade ...
Classical uniform and isotropic turbulence                           ∞         1       v2    E=              dV =       E(...
Decay of vortex line density in T → 0 limit   Two regimes have been observed:   1) Quasiclassical (Kolmogorov),   L ∼ t−3/...
Decay regimes                   Assumptions: Energy is distributed over scales (hence t−3/2 law: [Quasiclassical   k−5/3 s...
Ion injection experiment (Walmsley & Golov, PRL 2008)   Negative ions (electron bubbles) generate vortex rings (Winiecki &...
Ion injection experiment – decay regimes Ultraquantum, L ∼ t−1 (relatively        Quasiclassical, L ∼ t−3/2 (prolonged sho...
Numerical calculation      Biot-Savart calculation (Tree), periodic box D = 0.03 cm.      Beam of vortex rings, R = 6 × 10...
Vortex line density, L vs time     Small ion injection rate:   Large ion injection rate:       Ultraquantum decay         ...
Local curvature of the vortex lines Probability distribution function of the local curvature, C (cm−1 ) solid line – ultra...
Time-dependent spectra: formation and decay         Quasiclassical            Ultraquantum         solid: t = 0.1 s       ...
Generation of large scale motion   Energy input at k = k∗ (2π/k∗ ≈ 2πR, where R is the ring’s   radius)                   ...
General scenario: ultraquantum decay     Energy is concentrated at small scales     (k > k∗ ), “Kelvulence”     Kelvin wav...
General scenario: quasiclassical decay     Full time-dependent spectrum, consisting     of the Kolmogorov and ultraquantum...
Mechanism of formation of the large scale motion                  Isotropic initial conditions                   Anisotrop...
Mechanism of formation of the large scale motion     PDF(R) of loop sizes, R (cm)                                         ...
Outline   Introduction       Quantum Fluids       Quantised vortices       Vortex reconnections       Kelvin wave cascade ...
Bundling of filaments =⇒ Coherent structures.                                               53 / 57
Bundling of filaments =⇒ Coherent structures.                                               54 / 57
Relationship between vortex length and energy   Energy is constant throughout the run =⇒ Energy per unit length   is not c...
Open Questions      Bottleneck at crossover length scale?      Kelvin wave cascade in both strong and weak regimes.      T...
Thank you for listening                          57 / 57
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Exeter_2011

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A talk I gave at the University of Exeter in November 2011. Recent work in quantum turbulence is discussed with some future perspectives.

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Exeter_2011

  1. 1. Quantum TurbulenceAndrew W. Baggaley Carlo F. Barenghi Yuri Sergeev 28th November, 2011 1 / 57
  2. 2. Outline Introduction Quantum Fluids Quantised vortices Vortex reconnections Kelvin wave cascade Fluctuations in vortex density A suprising result? Numerical simulations Velocity statistics Classical picture Deviation from Gaussianity A U-turn? Decay of quantum turbulence Experimental results Ultra-quantum and quasiclassical regimes An explanation? Current/Future Work 2 / 57
  3. 3. Classical picture Turbulence in a normal (classical) fluid, 3 / 57
  4. 4. Classical vortices Velocity v, vorticity ω = × v. In a classical fluid, vortices are unconstrained, they can be big or small weak or strong. In quantum fluids · · · 4 / 57
  5. 5. Superfluidity Below a critical temperature ( 2.2K) 4 He becomes superfluid, a component of the fluid can flow without viscosity. What is the nature of turbulence in the superfluid (quantum turbulence)? How is energy distributed between scales? Here we focus on superfluid 4 He, QT also possible in 3 He and Bose-Einstein condensates. 5 / 57
  6. 6. Quantised vortices Gross-Pitaevskii Equation (weakly interacting gas) ∂Ψ 2 2 i =− Ψ + gΨ |Ψ |2 − µΨ. ∂t 2m √ Macroscopic wave function Ψ = neiφ Velocity v = ( /m) φ (irrotational) Vorticity constrained to filaments of fixed circulation Γ v · dr = Γ C Vortex core size dictated by atom size, hence thin! (∼ 10−8 cm) 6 / 57
  7. 7. Quantum Turbulence Quantum turbulence (QT) is a tangle of these quantised vortices. Hence QT is more simple than classical turbulence as vortices are well defined elements. But turbulence requires an energy sink, what is the dissipation mechanism? 7 / 57
  8. 8. Quantum vortex reconnections If vortices become close (core size) they can reconnect. Directly observed (Paoletti, Lathrop & Sreenivasan, [2008]). Numerically modelled used GPE (Koplik and Levine, PRL, [1993]), No violation of Kelvin’s circulation theorem (inside vortex core, ρ → 0) 8 / 57
  9. 9. Kelvin waves Quantised vortices reconnect, creating pronounced cusps. Also the interaction of vortices can lead to perturbations along the vortices, these propagate as Kelvin waves. Nonlinear (3 wave) interactions lead to the creation of smaller scales. Simulations taken from Kivotedes et al. [2003] At high k energy dissipated as phonon (sound) emission. 9 / 57
  10. 10. 10 / 57
  11. 11. Energy spectrum At scale larger than intervortex spacing - classical (Kolmogorov) regime. Experimentally (Maurer and Tabeling [1998]) and numerically (Nore et al. [1997], Araki et al. [2002]) veryfied. At small (< ) scales quantum regime - Kelvin wave cascade. At crossover scale arguments for, L’vov et al. [2007] (above), and against (Kozik & Svistunov [2008]) bottleneck . 11 / 57
  12. 12. Vortex filament model Helium experiments: average distance between lines ( ≈ 10−1 to 10−4 cm) is much bigger than core-size 10−8 cm ∴ Model vortex lines as reconnecting space curves s(ξ, t) Biot-Savart law: ds Γ (s − r) × dr =− dt 4π |s − r|3 LIA: ds ≈ βs × s dt N = number of discretization points Biot-Savart is slow: CPU ∼ N 2 Tree algorithm is faster: CPU ∼ N log N - (AWB & Barenghi, JLTP [2012]) 12 / 57
  13. 13. Tree approximation Biot-Savart law scales as N 2 . Use tree methods (Barnes and Hut [1983]) used in astrophysical simulations - N log N scaling. 13 / 57
  14. 14. Outline Introduction Quantum Fluids Quantised vortices Vortex reconnections Kelvin wave cascade Fluctuations in vortex density A suprising result? Numerical simulations Velocity statistics Classical picture Deviation from Gaussianity A U-turn? Decay of quantum turbulence Experimental results Ultra-quantum and quasiclassical regimes An explanation? Current/Future Work 14 / 57
  15. 15. Grenoble experiment 15 / 57
  16. 16. A suprising result? Intensity of quantum turbulence is characterized by the vortex line density L (vortex length per unit volume). Roche et al. [2007] measured the fluctuations of L in turbulent 4 He. They observed that the frequency spectrum scales as f −5/3 , Interpret L as a measure of the rms superfluid vorticity (ωs = Γ L). Contradiction with the classical scaling of vorticity expected from the Kolmogorov energy spectrum. Can numerical simulations shed light on the problem? 16 / 57
  17. 17. Finite temperature effects 0 < T < Tλ - two fluid system, normal (viscous) fluid and superfluid Helium. mutual friction (scattering of quasi-particles) means we must modify equations of motion. At high T , ρn > ρs : no back reaction from superfluid on normal fluid, ds = vs + αs × (vn − vs ) − α s × s × (vn − vs ) , dt Γ (s − r) vs = − × dr. 4π L |s − r|3 17 / 57
  18. 18. Modelling the normal fluid We wish to avoid DNS of turbulent normal fluid, we use Kinematic Simulations (KS) model, Fung et al. [1992] M vn (s, t) = (Am × km cos φm + Bm × km sin φm ) , m=1 φm = km · s + ωm t, where km and ωm = 3 km E(km ) are wavevectors and frequencies. Easily enforce energy spectrum of vn to reduce to −5/3 E(km ) ∝ km , simple to impose periodic boundary conditions. 18 / 57
  19. 19. 4/3 k1|vn |, Re = ≈ 180 kM 19 / 57
  20. 20. Numerical results Frequency spectrum Vortex line density L = Λ/D3 Energy Spectrum Quantised vortex dynamo: rapid initial growth in L, saturation due to increasing dissipation (reconnections). (AWB & Barenghi PRB [2011]) 20 / 57
  21. 21. An explanation Roche et al. argued randomly oriented vortex lines have some of the statistical properties of passive scalars. These randomly oriented vortices contribute to vortex line density, and so second sound attenuation (temperature waves), which is experimental method to detect quantised vortices. We test with by modelling passive lines, ds/dt = vKS . Must continue to reconnect lines or line length (density) will never saturate. 21 / 57
  22. 22. Frequency spectrum of passive line Line density grows and saturates as before but at larger values. Understandable, consider mutual friction term: αs × (vn − vs ). Normal fluid cannot contribute to stretching of vortices. Frequency spectrum of passive line seems to agree with f −5/3 , Result also similar if ds/dt = αs × (vKS − vs ) 22 / 57
  23. 23. Outline Introduction Quantum Fluids Quantised vortices Vortex reconnections Kelvin wave cascade Fluctuations in vortex density A suprising result? Numerical simulations Velocity statistics Classical picture Deviation from Gaussianity A U-turn? Decay of quantum turbulence Experimental results Ultra-quantum and quasiclassical regimes An explanation? Current/Future Work 23 / 57
  24. 24. Velocity statistics Ordinary viscous flows Homogeneous isotropic turbulence Turbulent velocity v(r, t) PDFs of components of v are Gaussian Flow past a grid Wind speed 24 / 57
  25. 25. Classical velocity statistics Experiment: Theory: Noullez & al Vincent & Meneguzzi (JFM 1997) (JFM 1991) 25 / 57
  26. 26. Classical velocity statistics Velocity v at point r is determined by vorticity ω = × v via Biot-Savart law: 1 ω(r , t) × (r − r ) v(r, t) = dr , 4π |r − r |3 If r is surrounded by many randomly oriented eddies, Gaussianity results from Central Limit Theorem 26 / 57
  27. 27. Maryland experiment Measurement of velocity PDF in superfluid 4 He 27 / 57
  28. 28. Maryland experiment Turbulent superfluid 4 He, solid hydrogen tracers radius ≈ 10−4 cm. Paoletti et al found −3 PDF(vx ) ∼ vx unlike ordinary turbulence. They say vortex reconnections are responsible for tails. 28 / 57
  29. 29. Velocity statistics in quantum turbulence Calculation of velocity PDF in various vortex configurations: Always power-laws (even without vortex reconnections) 29 / 57
  30. 30. Velocity statistics in quantum turbulence 30 / 57
  31. 31. Velocity statistics in quantum turbulence Vortex filament method Kolmogorov turbulence in 4 He (AWB & Barenghi, PRB [2011]) Heat flow turbulence in 4 He (Adachi & Tsubota, PRB [2011]) 31 / 57
  32. 32. Grenoble experiment 32 / 57
  33. 33. Grenoble experiment 10−3 p(v) 10−4 10−5 −4 −2 0 2 4 v− v σ Superfluid wind tunnel Kolmogorov energy spectrum Gaussian velocity statistics 33 / 57
  34. 34. Why power-law PDFs rather than classical Gaussian ? Paoletti, Lathrop & Sreenivasan: vortex reconnections are responsible for power-law. Additional interpretation based on Min & Leonard (1996): velocity statistics of singular and non-singular vortices are qualitatively different. For N singular vortices: −3 N = 1: straight vortex: P DF (vj ) = vj N > 1: provided that the velocity contribution of each vortex can be considered an independent random variable, the PDF converges to Gaussian but extremely slowly, e.g. N ∼ 106 vortices has still significant tails (Weiss, & al, PoF [1998]) 34 / 57
  35. 35. From power law to Gaussian Solution of the Maryland-Grenoble puzzle: = average vortex spacing ∆ = size of region over which the velocity is averaged ∆=2 ∆= ∆ = /6 • Grenoble: nozzle a = 0.06 cm a >> ≈ 0.5 to 2 × 10−3 cm • Maryland: tracer a = 10−4 cm tracer a << ≈ 10−3 to 10−2 cm (AWB & Barenghi, PRE [2011]) 35 / 57
  36. 36. Conclusion = average intervortex spacing At scales >> quantum turbulence is similar to ordinary turbulence (same Kolmogorov energy spectrum, same Gaussian velocity statistics) At scales << the quantum nature of vortices causes differences (Kelvin energy spectrum, non-Gaussian velocity statistics) 36 / 57
  37. 37. Outline Introduction Quantum Fluids Quantised vortices Vortex reconnections Kelvin wave cascade Fluctuations in vortex density A suprising result? Numerical simulations Velocity statistics Classical picture Deviation from Gaussianity A U-turn? Decay of quantum turbulence Experimental results Ultra-quantum and quasiclassical regimes An explanation? Current/Future Work 37 / 57
  38. 38. Classical uniform and isotropic turbulence ∞ 1 v2 E= dV = E(k) dk V 2 V 0 Kolmogorov spectrum: E(k) = C 2/3 k−5/3 Decay of classical turbulence Energy (per unit mass) is concentrated at scale D with velocity v: E ∼ v 2 /2 Characteristic timescale (lifetime of largest eddies): τ = D/v Dissipation rate: dE E v3 E 3/2 =− ∼ ∼ ∼ dt τ D D Decay: D D E∼ 2 ∼ 3 t t 38 / 57
  39. 39. Decay of vortex line density in T → 0 limit Two regimes have been observed: 1) Quasiclassical (Kolmogorov), L ∼ t−3/2 spindown (Walmsley, Golov, et al., PRL [2008]) injected ions which form rings (Walmsley & Golov, PRL [2008]) oscillating grid in 3 He-B (Bradley et al., PRL [2006] and Nature Phys. [2008]) 2) Ultraquantum (Vinen), L ∼ t−1 injected ions (Walmsley & Golov PRL [2008]) oscillating grid in 3 He-B (Bradley et al., PRL [2006] and Nature Phys. [2008]) 39 / 57
  40. 40. Decay regimes Assumptions: Energy is distributed over scales (hence t−3/2 law: [Quasiclassical k−5/3 spectrum) ≈ ν ω 2 (in analogy with classical (“Kolmogorov”) turbulence) ω ≈ κL (this implies existence of coherent turbulence] structures!) =⇒ L ∼ t−3/2 The only length-scale is the intervortex distance, = L−1/2 =⇒ the only velocity scale v = κ/(2π ) t−1 law: dE v3 [Ultraquantum =⇒ =− ∼ = ν (κL)2 dt (“Vinen”) turbulence] dL ν κ2 2 If E = cL, then ∼− L dt c =⇒ L ∼ t−1 40 / 57
  41. 41. Ion injection experiment (Walmsley & Golov, PRL 2008) Negative ions (electron bubbles) generate vortex rings (Winiecki & Adams, EPL [2000]) 41 / 57
  42. 42. Ion injection experiment – decay regimes Ultraquantum, L ∼ t−1 (relatively Quasiclassical, L ∼ t−3/2 (prolonged short injection time) Energy contained ion injection) Kolmogorov spectrum at at small scales, k 1/ wavenumbers k 1/ 42 / 57
  43. 43. Numerical calculation Biot-Savart calculation (Tree), periodic box D = 0.03 cm. Beam of vortex rings, R = 6 × 10−4 cm, confined within π/10 angle Vortex rings interact, reconnect, and form a tangle, 43 / 57
  44. 44. Vortex line density, L vs time Small ion injection rate: Large ion injection rate: Ultraquantum decay Quasiclassical decay 44 / 57
  45. 45. Local curvature of the vortex lines Probability distribution function of the local curvature, C (cm−1 ) solid line – ultraquantum dashed line – quasiclassical 45 / 57
  46. 46. Time-dependent spectra: formation and decay Quasiclassical Ultraquantum solid: t = 0.1 s solid: t = 0.07 s dot-dashed: t = 1.1 s dashed: t = 0.12 s dashed: t = 3.4 s dot-dashed: t = 0.6 s 46 / 57
  47. 47. Generation of large scale motion Energy input at k = k∗ (2π/k∗ ≈ 2πR, where R is the ring’s radius) ∞ E= Ek dk = EL + ES 0 k∗ ∞ EL = Ek dk – transferred to large scales EL = Ek dk – 0 k∗ transferred to small scales Quasiclassical Ultraquantum ES ES 1 (< 0.13 at all times) 1 (at least > 1) EL EL 47 / 57
  48. 48. General scenario: ultraquantum decay Energy is concentrated at small scales (k > k∗ ), “Kelvulence” Kelvin wave cascade, phonon emission at a very small scale (k = kc ) =⇒ decay E ∼ t−1 and L ∼ t−1 consistent with the rate of phonon emission dEtot 2 ∼ Etot dt (Vinen & Niemela, JLTP [2002]) Precise form of the KW spectrum is not important 48 / 57
  49. 49. General scenario: quasiclassical decay Full time-dependent spectrum, consisting of the Kolmogorov and ultraquantum parts. Most of the energy is contained at large scales, decay is determined by the quasiclassical (“Kolmogorov”) part of the spectrum (from the viewpoint of the Kolmogorov part, sink = KW + phonon emission) dE dL ∼ t−2 , ∼ t−3/2 dt dt The precise form of the KW spectrum as well as the possible bottleneck are not important for quasiclassical decay. 49 / 57
  50. 50. Mechanism of formation of the large scale motion Isotropic initial conditions Anisotropic initial ‘beam’ 50 / 57
  51. 51. Mechanism of formation of the large scale motion PDF(R) of loop sizes, R (cm) Reconnection of vortex rings Bottom: head-on reconnection Dashed: initial PDF Top: reconnection of rings travelling Resulting PDFs: in the same direction Grey: isotropic injection of rings =⇒ formation of large loops Black: anisotropic beam Anisotropy of the beam is important! 51 / 57
  52. 52. Outline Introduction Quantum Fluids Quantised vortices Vortex reconnections Kelvin wave cascade Fluctuations in vortex density A suprising result? Numerical simulations Velocity statistics Classical picture Deviation from Gaussianity A U-turn? Decay of quantum turbulence Experimental results Ultra-quantum and quasiclassical regimes An explanation? Current/Future Work 52 / 57
  53. 53. Bundling of filaments =⇒ Coherent structures. 53 / 57
  54. 54. Bundling of filaments =⇒ Coherent structures. 54 / 57
  55. 55. Relationship between vortex length and energy Energy is constant throughout the run =⇒ Energy per unit length is not constant. 55 / 57
  56. 56. Open Questions Bottleneck at crossover length scale? Kelvin wave cascade in both strong and weak regimes. Topology of structures in QT, define a knot ‘spectrum’ ? Tracer particles for flow visualisation. 56 / 57
  57. 57. Thank you for listening 57 / 57

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