Lagrangian intermittency in drift-wave turbulence Wouter Bos LMFA, Ecole Centrale de Lyon, Turbulence & Stability Team Acknowledgments Benjamin Kadoch, Kai Schneider, Laurent Chevillard, Julian Scott, Robert Rubinstein 1
2 Intermittency in plasma turbulence (Hidalgo et al. C.R. Phys. 2006) Experimentalists measure strongly intermittent statistics in the tokamak edge. It is important to understand this kind of intermittency in the context of design and control of fusion devices.
3 Overview Intermittency, different types Lagrangian dynamics in drift wave turbulence
4 How to measure Intermittency Flatness or Kurtosis : F = u4 u2 2. Measures the relative fluctuations of fluctuations. Example from Frisch (CUP 1995). Compare the signals : The intermittent signal is on during a fraction γ of time. Variance of signal b is γu2, the 4th order moment is γu4 1 γu4 = Fa Fb = γ (γu2 )2 Intermittency, drift-wave turbulence (1) Ecole de Physique des Houches 2011
5 PDFs and Flatness 0.5 0.45 Gaussian Laplace 0.4 0.35 0.3 P(x) 0.25 0.2 0.15 0.1 0.05 0-4 -2 0 2 4 x/σ(x) Gaussian distribution : F = 3, Laplace distribution : F = 6
6 Intermittency in turbulent flows, different types L η Kinetic Energy E(K) ǫ - Large-scale intermittency - Inertial range intermittency - Dissipation range intermittency Large Scales L 1 K Dissipation Small Scales η 1
7 Large Scale intermittency
8 Dissipation range intermittency I Nonlocal energy transfer leads to a constant scale dependent flatness in the inertial range.
9 Dissipation rate intermittency II Nonlocal energy transfer leads to an exponential dependent flatness in the dissipation range.
10 Dissipation range intermittency III Illustration : Gaussian noise (left) versus low Reynolds number turbulence (right) 10-2 10-3 Fourier Wavelet E(k) 10-4 10-5 10 3 10 2 F u (k x ) F u (k y ) F u (k z ) 10-6 10 0 10 1 10 2 k 10 3 10 2 F u (k x ) F u (k y ) F u (k z ) 10 1 10 1 10 0 10 0 10 1 10 2 10 0 10 0 10 1 10 2 k j k j WB, Liechtenstein and Schneider PRE 2007 The nonlinear transfer between modes is enough to create non-gaussian statistics in the dissipation range. See also Kraichnan, Phys. Fluids 1967.
11 Inertial range intermittency This type of intermittency is a correction of Kolmogorov s 1941 theory, taken into account by Kolmogorov s 1962 theory.
12 Kolmogorov s 1941 theory (K41) L η
12 Kolmogorov s 1941 theory (K41) L η Kinetic Energy E(K) ǫ Large Scales L 1 K Dissipation Small Scales η 1
12 Kolmogorov s 1941 theory (K41) Kinetic Energy E(K) Large Scales L 1 ǫ K Dissipation Small Scales η 1 L η Kolmogorov 1941 If L 1 k η 1 then E(k) = C K ǫ 2/3 k 5/3
13 Kolmogorov s 1962 theory (K62) Replace E(k) ǫ 2/3 k 5/3 by E(k) ǫ 2/3 k 5/3 or in terms of structure functions, D LL (r) = [(u(x + r) u(x)) r/r] 2 ǫ 2/3 r 2/3 The dependence on ǫ 2/3 induces corrections if the spatial distribution of ǫ is a function of r.
14 K62, continued If ǫ, the energy flux, is only significant in a small part of the volume α, so that ǫ 2/3 = ǫ 2/3 α(r) 1/3 (2)
15 If we assume α(r) (r/l) µ we find ǫ 2/3 r µ/3 et ( D LL (r) ǫ 2/3 r 2/3 r ) µ/3 r (2+µ)/3. (3) L With µ depending on statistical distribution of ǫ.
16 Higher order structure functions D n (r) = [(u(x + r) u(x)) r/r] n D n (r) (ǫr) n/3 (r/l) µ(1 n/3) (4) µ is of order 0.1 Example : skewness S(r) D 3 (r)/d 2 (r) 3/2 r 1 3ζ 2/2 r 0.05 (5) Multifractal : r 0.04, She-Lévêque comparable.
17 Scale dependent skewness D LLL (r)/[d LL (r)] 3/2 lim R λ lim R λ EDQNM K41, Multifractal K41 0.4 EDQNM Windtunnel Multifractal Sk(r) 0.2 r -0.045 Both in agreement with power-law correction 0.1 10-4 10-3 10-2 10-1 10 0 10 1 r
18 Scale dependent skewness D lll (r)/[d ll (r)] 3/2 0.4 EDQNM 2500 MF 2500 Modane 2500 EDQNM 25000 -S(r) 0.2 0.1 10 0 10 1 10 2 10 3 10 4 10 5 r/η WB, Chevillard, Scott, Rubinstein, ArXiv 2011 In EDQNM this power-law vanishes for R λ
19 R λ = 2500 is already large...
20 Conclusion on intermittency Different types of intermittency Large-scale and small scale intermittency not incompatible with K41 At real-world Reynolds numbers, K62 and K41 are not easily distinguishable
21 Part 2. Lagrangian intermittency in drift-wave turbulence Why consider lagrangian statistics?
22 Energy cascade is Lagrangian Why being interested in the Lagrangian dynamics? The energy cascade is in essence a Lagrangian mechanism. The Eulerian observer cannot see the difference between sweeping and internal distortion (see also Kraichnan 1964). It is therefore a more natural framework (but not necessarily more convenient) to study turbulence.
23 L Timescales in turbulence T int 1. T int = LU 1 k 0 U τ ν 2. ν 2 x j νk 2 l k τ ν = (νk 2 ) 1 k 2 3. τ E = l k U 1 τ E τ L τ E (ku) 1 k 1 4. τ L = l k u(k) 1 u(k) ke(k) τ L ( k 3 E(k) ) 1/2 k 2/3
24 Intermittency in the Lagrangian reference frame Following a particle, multi-scale behavior is captured by time-increments. δu L i (t, τ) = u L i (t + τ) u L i (t) (6) Small-time limit acceleration force on fluid particle. limδu L i (t, τ) = limτ τ 0 τ 0 ( u L i (t + τ) u L i (t) ) τ = τa L i (t) (7) Intermittent behavior of this quantity analogous to dissipation range intermittency in the Eulerian framework.
25 Lagrangian acceleration from Experiments and DNS Yeung and Pope, DNS Warhaft and others, windtunnel Mordant, Pinton, Washing machine PDFs of Lagrangian acceleration are non Gaussian
26 What kind of statistics do we find in drift-wave turbulence?
27 Hasegawa-Wakatani model A minimum model for edge plasma turbulence Γ r = nu r «t ν 2 2 φ = ˆ 2 φ, φ + c(φ n), (8) «t D 2 n = [n, φ] u ln( n ) + c(φ n), (9) Defining the vorticity ω = 2 φ, assuming n κ exp( x)
28 Edge plasma : Hasegawa-Wakatani model Γ r = nu r ω = u (10) ω t + (u )ω = ν 2 ω + c(φ n), (11) n t + (u )n = D 2 n uκ + c(φ n). (12)
29 Edge plasma : Hasegawa-Wakatani 2D vorticity visualizations c=0.01 c=0.7 c=4 Hydrodynamic intermediate atmospheric 10 5 0 5 10 10 8 6 4 2 0 2 4 6 8 10 1.5 1 0.5 0 0.5 1 1.5 Pseudospectral simulations, resolution = 1024 2.
30 Lagrangian description 10 4 particles were injected, equally spaced, and their velocity and acceleration were monitored.
31 Lagrangian description : velocity increments PDF 10 4 10 3 10 2 10 1 10 0 10-1 10-2 10-3 10-4 10-5 u LX a LX 10-6 -20-15 -10-5 0 5 10 15 20 u LX (τ) PDF 10 3 10 2 10 1 10 0 10-1 10-2 10-3 10-4 10-5 u LX a LX 10-6 -15-10 -5 0 5 10 15 u LX (τ) WB, Kadoch, Neffaa, Schneider, Phys. D 2010 Heavy tails (algebraic) for the hydro case Exponential for the intermediate case
32 Dependence acceleration on flow regime PDF 10 6 10 4 10 2 10 0 10-2 10-4 10-6 c=0.01 c=0.05 c=0.1 c=0.7 c=2-20 -15-10 -5 0 5 10 15 20 a LX /σ alx
33 Algebraic tails for the hydro case PDF 10 2 10 1 10 0 10-1 10-2 10-3 10-4 10-5 10-6 10-7 R λ =129 R λ =195 R λ =399 R λ =671 10-2 10-4 a L -2 1 10-20 -15-10 -5 0 5 10 15 20 a LX /σ alx Note : in a point vortex study (Rast and Pinton, PRE 2009) algebraic tails a 5/3 were found.
34 Exponential tails for the intermediate case PDF 10 0 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 R λ =124 R λ =215 R λ =418 R λ =737 exp. -15-10 -5 0 5 10 15 a LX /σ alx Explanation?
35 P(a) of a Gaussian velocity field is given by an exponential pdf Gaussian velocity fields exponential pressure and pressure gradient. Lagrangian acceleration is dominated by pressure gradient a L = p + ν 2 u [c(n φ)], (13) 2 even non-intermittent velocity yields intermittent acceleration.
36 Different regimes explained by time correlations?
37 Long-living coherent structures (vortices), time correlations Vortical motion leads to long-time correlations of the norm of the acceleration. Quantified by C a (τ) = a(t)a(t + τ)dt
38 Yes, there is a qualitative correlation PDF 10 6 10 4 10 2 10 0 10-2 10-4 10-6 c=0.01 c=0.05 c=0.1 c=0.7 c=2-20 -15-10 -5 0 5 10 15 20 a LX /σ alx 1 0.8 0.6 0.4 0.2 0-0.2-0.4 1 0.5 0 C ax (τ/t*) 0 1 2 3 4 5 10-2 10-1 10 0 10 1 τ/t* c=0.01 c=0.05 c=0.1 c=0.7 c=2 C al (τ/t*) Kadoch, WB, Schneider, PRL 2010
39 Conclusions In turbulence, Reynolds number effects might be more important than inertial range intermittency In drift-wave turbulence the acceleration is non Gaussian For small c : algebraic tails, long time correlations, point vortex like behavior For large c : exponential tails, reminiscent of Gaussian velocity statistics. Proof for link time-correlations and intermittency References WB, Liechtenstein, Schneider PRE 2007 (dissipation range intermittency) WB, Chevillard, Scott, Rubinstein ArXiv 2011 (Skewness, anomalous scaling) WB, Kadoch, Neffaa, Schneider, Phys. D 2010 (HW, Lagrangian intermittency) Kadoch, WB, Schneider PRL 2010 (HW, Lagrangian intermittency)