More about simulations of quantum turbulence

Transcription

1 More about simulations of quantum turbulence Discussion session on Monday, May 2, 2011 Chair: Workshop on Classical and Quantum Turbulence, Abu Dhabi, 2-5 May 2011

2 Outline 1 Speeding up the simulations Barenghi: Vortex tree code 2 Vortex reconnections Nemirovskii: Variety of the reconnection procedures 3 Energy spectrum & identification of the Kelvin waves Tsubota: Energy spectrum for the Lancaster 3 He-B exp. Identification of the Kelvin waves Brachet: The k 3 spectrum observed by Yeppez et al. 4 Truncated HVBK model Roche: HVBK model truncated at the inter-vortex length 5 Numerical verification of the Kelvin spectrum Repetition of Kozik s arxiv: calculation Harmonically driven vortex

3 Outline 1 Speeding up the simulations Barenghi: Vortex tree code 2 Vortex reconnections Nemirovskii: Variety of the reconnection procedures 3 Energy spectrum & identification of the Kelvin waves Tsubota: Energy spectrum for the Lancaster 3 He-B exp. Identification of the Kelvin waves Brachet: The k 3 spectrum observed by Yeppez et al. 4 Truncated HVBK model Roche: HVBK model truncated at the inter-vortex length 5 Numerical verification of the Kelvin spectrum Repetition of Kozik s arxiv: calculation Harmonically driven vortex

4 A.W. Baggaley and C.F. Barenghi VORTEX FILAMENT METHOD Exploit core radius a 0 << l intervortex distance Points s j (j = 1, N) Biot-Savart law: v s (s j ) = κ (s j r) 4π L s j r 3 dr CPU time N 2 (very slow) Schwarz (1988) used LIA: v(s j ) βs j s j but LIA can make wrong predictions (Ricca, Samuels & Barenghi, JFM 1999) and does not capture vortex-vortex interaction and self-organization of flow What to do?

5 A.W. Baggaley and C.F. Barenghi N-BODY SIMULATIONS IN ASTROPHYSICS Tree algorithms: CPU time N log N (Barnes & Hut 1984) Millennium Run (2005) Virgo Consortium, Garching 10 billion particles

6 A.W. Baggaley and C.F. Barenghi VORTEX TREE CODE CPU time vs N for various θ vortex tangle with N = 400, 000

7 For numerical stability and to resolve the fastest Kelvin waves t ( ξ) 2 1/N 2 Therefore, in order to simulate a given time window t = [0, T ], the required cpu operations scales as N 4, normal Biot-Savart N 3 log N, with the tree code (scale separation)

8 Outline 1 Speeding up the simulations Barenghi: Vortex tree code 2 Vortex reconnections Nemirovskii: Variety of the reconnection procedures 3 Energy spectrum & identification of the Kelvin waves Tsubota: Energy spectrum for the Lancaster 3 He-B exp. Identification of the Kelvin waves Brachet: The k 3 spectrum observed by Yeppez et al. 4 Truncated HVBK model Roche: HVBK model truncated at the inter-vortex length 5 Numerical verification of the Kelvin spectrum Repetition of Kozik s arxiv: calculation Harmonically driven vortex

9 Reconnection ansatz The problem of single reconnection is more or less tractable (at least numerically) in the both classic and quantum fluids. However, because of huge amount of reconnections in calculations of the dense vortex tangles (up to many thousands) the treatment of each reconnection event is impossible. Schwartz proposed to use the so called reconnection ansatz which was to change the detailed process by the instantaneous reconnection whenever the vortex lines approach at a critical distance. Instantaneous reconnection is a considerable simplification of the general problem because it "separates" the various scales of motion, eliminating the huge range of small scales needed to describe the collapse of the vortex filaments. But the very delicate question arises, namely how to define this critical distance? We will review very shortly the variety of different approaches.

10 Schwarz s reconnection ansatz

11 Samuels and Barenghi Reconnection ansatz

12 Tsubota s ansatz In large series of numerical simulations of superfluid turbulence made by M. Tsubota with coauthors it were also discussed some of the shortcomings of the Schwarz s criterion. Tsubota proposed to fulfill reconnection within the space resolution \Delta \xi (along line). Authors argued that this procedure was best for the filament reconnection under the full Biot-Savart calculation. Of course, a question arises - how does this artificial selection of critical distance influence on the reliability of the results? To clarify this question authors made the spacial investigation how the space resolution affects their results, and concluded that this influence is not too large. Authors explained the insensitivity of their results by that the reconnection process is robust.

13 Kivotides Procedure

14 Kondaurova reconnection procedure In series of works made by Kondaurova with co-authors the new, different reconnection algorithm was used. The authors selected out all the line elements, they are the pairs of neighboring on the lines, and followed their fate. In the case, when the set of equations describing motion and collision of line elements has a solution, authors fulfilled the reconnection at this time step. The procedure of reconnection, described above do not require any criterion, lines "themselves" find the moment of collision.

15 Comparison of the Vortex Line Density

16 Summary We have described the various ways, which are used by various authors to implement the reconnection between lines. The authors of various works using different procedures of reconnection assert that the chosen variants of the reconnection procedure did not lead to significantly different results, in other words, it is robust and it is not too sensitive to the concrete realization. We think it is not correct assertion, and our comparison confirms this conclusion. The question how the reconnection procedure affects the final results, although frequently touched, has not been studied thoroughly. I think the necessity of such study is obvious, especially in cooperation with theoretical investigation.

17 Outline 1 Speeding up the simulations Barenghi: Vortex tree code 2 Vortex reconnections Nemirovskii: Variety of the reconnection procedures 3 Energy spectrum & identification of the Kelvin waves Tsubota: Energy spectrum for the Lancaster 3 He-B exp. Identification of the Kelvin waves Brachet: The k 3 spectrum observed by Yeppez et al. 4 Truncated HVBK model Roche: HVBK model truncated at the inter-vortex length 5 Numerical verification of the Kelvin spectrum Repetition of Kozik s arxiv: calculation Harmonically driven vortex

18 Energy spectrum for the Lancaster 3 He-B experiments S. Yamamoto(Osaka), M. Tsubota, W. F. Vinen(Birmingham) We calculated the energy spectrum through the vortex filament model. Generation of turbulence by a flapping grid at very low temperatures PRL 96, (2006); PRB 81,180512(2010)

19 t = 0.10 sec -5/3-2/3-1 t = 1.00 sec t = 2.50 sec Most of energy is in the quantum range of wave numbers.

20 Decay of the line density -1 This decay of t -1 is consistent with the energy spectrum in the previous slide.

21 Outline 1 Speeding up the simulations Barenghi: Vortex tree code 2 Vortex reconnections Nemirovskii: Variety of the reconnection procedures 3 Energy spectrum & identification of the Kelvin waves Tsubota: Energy spectrum for the Lancaster 3 He-B exp. Identification of the Kelvin waves Brachet: The k 3 spectrum observed by Yeppez et al. 4 Truncated HVBK model Roche: HVBK model truncated at the inter-vortex length 5 Numerical verification of the Kelvin spectrum Repetition of Kozik s arxiv: calculation Harmonically driven vortex

22 Kelvin Waves (KWs) Simple helical waves: x(z) = w k cos (kz), y(z) = w k sin (kz) Left-handed (k < 0) Combination Right handed (k > 0) For KWs on a straight vortex with periodic b.c.: x(z) + iy(z) = P k w k exp(ikz) = P m wm exp(i2πmz/l).

23 Smoothed Vortex Baggaley and Barenghi, PRB 83, (2011): Smooth the vortex by fitting a spline through every N 15 points. Take Fourier transformation for the deviation a(ξ) = s(ξ) s smooth (ξ).

24 Smoothed Vortex Baggaley and Barenghi, PRB 83, (2011): Smooth the vortex by fitting a spline through every N 15 points. Take Fourier transformation for the deviation a(ξ) = s(ξ) s smooth (ξ). Simple test with spectrum (A(k) = n k ): n k = Ak 2 for kl/2π = n k = 0 for kl/2π > 100 IMPROVEMENTS NEEDED!

25 Kivotides et al, PRL 86, 3080 (2001): Calculation of the energy spectrum E(k) is difficult due to 1/r velocity profile. (E kin = R E(k)dk)

26 Kivotides et al, PRL 86, 3080 (2001): Calculation of the energy spectrum E(k) is difficult due to 1/r velocity profile. (E kin = R E(k)dk) One should always verify that E(k) C/k when k k res. Most of the the KWs (wiggles) are additionally due to bad numerics /(2kR) log 10 (E/ρ s κ 2 R 2 ) (kr) 2 / log 10 (kr)

27 Energy spectrum of straight vortex with KWs KWs on a straight vortex with periodic b.c. (period L 0 = 1 mm): 10 1 x(z) + iy(z) = X k w k exp(ikz) 10 2 with w k k η = k 17/10 (KS). Length converges only for η > 3/2. Blue: Vortex with KWs. Magenta: Straight vortex, length L 0. Red: Straight vortex, length L KWs 1.43 mm. At high-k the slope is -1! E(k)/ρ s κ 2 L kl 0 /2π

28 Outline 1 Speeding up the simulations Barenghi: Vortex tree code 2 Vortex reconnections Nemirovskii: Variety of the reconnection procedures 3 Energy spectrum & identification of the Kelvin waves Tsubota: Energy spectrum for the Lancaster 3 He-B exp. Identification of the Kelvin waves Brachet: The k 3 spectrum observed by Yeppez et al. 4 Truncated HVBK model Roche: HVBK model truncated at the inter-vortex length 5 Numerical verification of the Kelvin spectrum Repetition of Kozik s arxiv: calculation Harmonically driven vortex

29 Giorgio Krstulovic and Marc Brachet PRL 105, (2010) Comment on Superfluid Turbulence from Quantum Kelvin Wave to Classical Kolmogorov Cascades by Yepez et al. They «observed» high-k «Kelvin wave» spectrum of k 3 Power law is an artifact stemming from the definition of the kinetic energy spectra and is not directly related to a Kelvin wave cascade jeudi 28 avril 2011

30 !"!"!" '" &" #"!" % -, ffiffiffi!" t c ¼ ic=ð 2 Þðc jc j 2 c þ 2 r 2 c Þ;!" p pffiffiffi c ¼ ffiffiffi expði ffiffiffi E kin ¼ 1=2ð vj Þ 2 % %!" pffiffi 2c Þ,!!" #!"!!" # $ $ ffiffi pffiffiffiffiffiffiffiffiffi 1Þ 2 % pffiffiffiffiffiffiffiffiffi, verifies ðrþ r as r! 0 and ðrþ ¼ 1 þ Oðr 2 Þ pffiffiffi for r!1. Thus vj has a small r singular behavior of the type r 0 $ $ %!#* and behaves as r 1 $ ) ) ) ily values of ξk c. at large r. In general, for a ( function scaling as gðrþr s the (2D) Fourier transform is ^gðkþk s 2 and the associated spectrum scales as k 2s 3 %. Thus E kin ðkþ scales as k 3 ( for k k zone. 1 and as k 1 for b) Kinetic k k [3]. ), Following the above discussion,!" the k 3 power law /, 0 $12 %!"!& #" #& FIG. 6: (Color online) a) Temporal evolution of energies (as in Fig4.a) for ξk max =1.48, 2.97 and 6.01 (resolution 64 2, and respectively). Yellow stars are the kinetic energy reconstructed from fit data using Eq.(63). energy spectrum at t =17.4 for ξk max =1.48, 2.97 and 6.01; the dashed black line indicates k 2 power-law scaling. c) Temporal evolution of kinetic energy spectrum; the solid red lines correspond to fits using Eq.(62) and the dashes black line indicate k 3 power-law scaling. d) Temporal evolution of effective self-truncation wavenumber k c (Eq.(62)) at different resolutions. enough? Note that this problem is related to the classical Fermi-Pasta-Ulam-Tsingu problem [42]. To try to answer this question within the Taylor-Green jeudi 28 avril 2011 Artifact in TYG initial data in: arxiv: v2. $ tation dk c /dt v.s. k c has been used on Fig Fig.7.c. With this representation, a self-similar e k c (t) t η corresponds to a line of slope χ =(η Figure 7.b, shows transient self-similar evolutio all terminate by a vertical asymptote, correspo logarithmic growth (η = 0). This self-truncati place for small values of k c /k max strongly sugges 1 0 $12 the self-truncation happens in0 3 a40regime 12( indepe cut-off. Finally, Fig.7.c suggests 0 $12 that, %!#* dependin tial conditions, self-truncation can take place at 1 0 %!#* $ %!#* 3 12( As the dynamics of modes at 0 wave-numbers $12 lar 1 k c is weakly nonlinear, it should0 $12 be amenable to a tion in terms of wave turbulence theory; 12( this co haps explain the slowdown of the 0 51( thermalizatio 0 The new regime indicates (6( that total the tion is delayed when increasing the amount of d % (controlled by ξk max ) but is preceded by a par malization (quasi-equilibrium up to k c ) within a We now turn to estimations of order of magni evant to physical BEC. At low-temperature, th known [3] to give an accurate description of the (!" dynamics of physical BEC at scales larger tha teratomic separation. At finite temperature th gives a good approximation$ of Bose-Einstein co (BEC) only for the phonon modes with high oc!" number, see [3, 13]. At very low temperature t a limited % range of low-wavenumber density wav # equipartition.!"!!" # $ This limited range has consequences on temperature thermodynamics of BEC that ca tained by the following considerations. The e

31 Outline 1 Speeding up the simulations Barenghi: Vortex tree code 2 Vortex reconnections Nemirovskii: Variety of the reconnection procedures 3 Energy spectrum & identification of the Kelvin waves Tsubota: Energy spectrum for the Lancaster 3 He-B exp. Identification of the Kelvin waves Brachet: The k 3 spectrum observed by Yeppez et al. 4 Truncated HVBK model Roche: HVBK model truncated at the inter-vortex length 5 Numerical verification of the Kelvin spectrum Repetition of Kozik s arxiv: calculation Harmonically driven vortex

32 J. Salort, P.-E. Roche (Institut Néel, Grenoble) & E. Lévêque (ENS-Lyon) EPL 94, (2011) The HVBK model ρ s Dv s Dt = p s F + f ext ρ n Dv n Dt = p n + F + µ 2 v n Coarse-grained vortex tangle Pros: - simulation of high Re possible - accounts for mutual coupling in a consistent fashion Cons: - energy cascade continuously beyond the intervortex cut-off scale A Truncated HVBK model Phase space is restricted to scales larger than intervortex spacing Intervortex spacing is estimated self-consistently from the superfluid vorticity field intervortex 1 V LD with V LD ω s 2 1/2 κ

33 HIGH TEMP (ρs /ρn = 0.1, T K) High Temperature!2 10 k!5/3!4 E(k) 10 Velocity!6 10!8 10 k ~ inertial range Vorticity ω s P(!s ) k!5/3!2 10! Consistent with He-II experiments Similar to Navier-Stokes fluids Provides some validation of the model k 10 Inter-vortex spacing

34 Further validation : Inter-vortex spacing Simulations versus Experiments Re µ/# !1 Experiments (T K) " / L 10!2 10!3!3/4 0.5 Re! ~ 3!3/4 Reµ/# Ijsselstein et al. Holmes & Van Sciver Walstrom unpublished (Roche et al.) 10! Re! Present simulations ( T 1.44 K, ρ s =10.ρ n )

35 Velocity spectra versus Temperature 10!2 k!5/3 VERY LOW (ρ s /ρ n = 40, T 1.15 K) LOW TEMP (ρ s /ρ n = 10, T 1.44 K) INTERMED. (ρ s /ρ n = 1, T 1.96 K) HIGH TEMP (ρ s /ρ n =0.1, T K) E(k) 10!4 10!6 10!8 k k 2 intervortex spacing Consistent with experiments (Maurer et al., Salort et al.) Equipartition of energy : inertial cascade < meso-scales < quantum scales

36 Superfluid Vorticity spectra versus Temperature k!5/3 ~ inertial range VERY LOW (ρ s /ρ n = 40, T 1.15 K) LOW TEMP (ρ s /ρ n = 10, T 1.44 K) INTERMED. (ρ s /ρ n = 1, T 1.96 K) HIGH TEMP (ρ s /ρ n =0.1, T K) P (! s ) 10!2 10! k intervortex spacing Consistent with experiments (miniature second sound probe) At low (finite) temperature : «vorticity» concentrates at low scales

37 Outline 1 Speeding up the simulations Barenghi: Vortex tree code 2 Vortex reconnections Nemirovskii: Variety of the reconnection procedures 3 Energy spectrum & identification of the Kelvin waves Tsubota: Energy spectrum for the Lancaster 3 He-B exp. Identification of the Kelvin waves Brachet: The k 3 spectrum observed by Yeppez et al. 4 Truncated HVBK model Roche: HVBK model truncated at the inter-vortex length 5 Numerical verification of the Kelvin spectrum Repetition of Kozik s arxiv: calculation Harmonically driven vortex

38 Kozik and Svistunov, PRL 94, (2005); arxiv: (2010):

39 Recalculation of arxiv: log 10 ( w k ) KS imitation sign(k)log 10 ( k ) log 10 ( w k ) 10 Vortex filament model (L =1 mm) sign(k)log 10 ( k ) Kelvin amplitudes at t = 0 (red), 10τ res (green), and 1500τ res (blue). The initial spectrum is LN spectrum with w k k 11/6.

40 Recalculation of arxiv: KS imitation Vortex filament model (L =1 mm) k 17/5 n k k 17/5 n k k k Time averaged and k-smoothed spectrum n k = w k 2 + w k 2. The initial t = 0 (red) spectrum is LN spectrum with n k k 11/3. Time averages are for t/τ res = (green), t/τ res = (magenta), t/τ res = (cyan) and t/τ res = (blue).

41 Recalculation #2 of arxiv: KS imitation Vortex filament model (L =1 mm) log 10 ( w k ) sign(k)log 10 ( k ) log 10 ( w k ) sign(k)log 10 ( k ) Kelvin amplitudes at t = 0 (red), 10τ res (green), and 1500τ res (blue). The initial spectrum is LN spectrum with w k k 11/6 (but only modes 30).

42 Recalculation #2 of arxiv: KS imitation Vortex filament model (L =1 mm) k 17/5 n k k 17/5 n k k k Time averaged and k-smoothed spectrum n k = w k 2 + w k 2. The initial t = 0 (red) spectrum is LN spectrum with n k k 11/3 (but only modes 30). Time averages are for t/τ res = (green), t/τ res = (magenta), t/τ res = (cyan) and t/τ res = (blue).

43 Outline 1 Speeding up the simulations Barenghi: Vortex tree code 2 Vortex reconnections Nemirovskii: Variety of the reconnection procedures 3 Energy spectrum & identification of the Kelvin waves Tsubota: Energy spectrum for the Lancaster 3 He-B exp. Identification of the Kelvin waves Brachet: The k 3 spectrum observed by Yeppez et al. 4 Truncated HVBK model Roche: HVBK model truncated at the inter-vortex length 5 Numerical verification of the Kelvin spectrum Repetition of Kozik s arxiv: calculation Harmonically driven vortex

44 Shaking a vortex (for details see arxiv: ) Use the VFM with full Biot-Savart. Vortex of length L along z-axis. Drive the endpoints, z = 0, L: x = x 0 sin(ωt), y = 0 ω κk2 4π [ln(2/ka 0) γ], k = k drive = 2πm drive /L Assume periodic b.c. Include m.f. and assume v n = 0. Vortex is a sum of Kelvin waves: x(z) + iy(z) = m w m exp(i2πmz/l). Seek for a steady spectrum w m m η. drive amplitude x 0 = mm α = 0.1 w /x m w m /x t (s) t (s) w /x m α = α = t (s) α = t (s) α = t (s) 5 x α = t (s)

45 Kelvin spectrum x 0 = mm Resolutions: 256, 1024, and 4096 Red: α = 0.1 Blue: α = 0.01 Magenta: α = w m = w m Fits w m m η : α = 0.1: η = 1.88 α = 0.01: η = 1.86 For small x 0: w m x 0. w m /x mode, m

46 Power dissipated due to mutual friction (v n = 0) Z P mf = Z v L f mf dξ = αρ sκ LIA: Converges only when η > 5/2. For a wide range of scales: P mf = αcξ β cutoff, For η = 17/10 (KS): β For η = 11/6 (LN): β ŝ v s 2 dξ P mf (N KWs )/P mf (1) If P mf is fixed by the drive and assuming a (sharp) cutoff at ξ cutoff. Then ξ cutoff,2 ξ cutoff,1 = where 1/β 1. α2 α 1 «1/β, N KWs Figure: P mf as a function of the largest Kelvin mode N KWs for different spectra w k k η. From top to bottom η = 1.7, 1.833, 2.0, and 2.5.

47 Other Topics Let s thank all the speakers!

48 Other Topics Let s thank all the speakers! Other topics?