Alexei Kritsuk. CalSpace IGPP/UCR Conference Palm Springs, March 27-30, 2006 c Alexei Kritsuk

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1 Supersonic Turbulence in Molecular Clouds 1 Alexei Kritsuk University of California, San Diego In collaboration with Rick Wagner, Mike Norman, & Paolo Padoan, UCSD CalSpace IGPP/UCR Conference Palm Springs, March 27-30, 2006

2 Outline Motivation: supersonic turbulence in molecular clouds controls initial conditions for star formation, Re 10 8 Methodology Turbulent structures at high Re-solution Statistical properties of supersonic turbulence Summary

3 Spitzer: Mountains of Creation in W5, Cassiopeia.. Credit: NASA/JPL-Caltech/L. Allen (Harvard-Smithsonian CfA) 3

4 Our niche High-resolution simulations of supersonic turbulence Intermittency, dissipative structures, and scaling relations for supersonic turbulence Interplay of interstellar turbulence and star formation Turbulence and hierarchical structure of molecular clouds in the context of multiphase ISM 4

5 Methodology Input 3D Euler equations Periodic box; Cartesian mesh Quasi-isothermal EOS (γ = 1.001) Driven turbulence (Mach 6, solenoidal driving force with a constant pattern) High-order Godunov-type method (PPM), strong shocks are two-zones-wide Uniform grids 64 3, 128 3, 256 3, 512 3, with PPM Also and simulations with the ZEUS solver for comparison Structured AMR with 1 2 levels of refinement by 4 on shocks & shear [Kritsuk et al. 2006, ApJL 638, L25] Engine ENZO code for cosmology and astrophysics. Public since March New release is planned for Output Visualizations & statistical properties of turbulent structures 5

6 Simulations First, a uniform grid simulation with resolution was evolved for 5t dyn to stir the gas within the box. Then, we doubled the resolution and evolved the simulation for another 5t dyn on a grid of points. The resulting dataset contains 170 snapshots, 6TB of data. 6.6 rms Mach number as a function of time (1024PPM) M v,rms t/t dyn t dyn L 2M rms c s, L = 1 is the box size, and c s = 1 is the speed of sound. 6

7 Reynolds numbers What Re do we get at Mach 6 with PPM and points? Taylor scale λ 5E mean specific kinetic energy and Ω Taylor-scale Reynolds number Re λ λu 1 ν one-dimensional rms velocity u 1 = Ω x, where E 1 2 u2 = 1 2 M 2 18 is the 1 2 ω is the mean enstrophy. 2 3 = 0.72 MN4/3 Ω 1/2 141, where the E, and the effective kinematic viscosity ν 2 1/3 MN 4/ is estimated assuming the Kolmogorov scale ( ) 1/4 ν η 3 ε x N 1 u and assuming the mean energy dissipation rate ε 3 0 l, 0 with the velocity of the order of the rms Mach number, u 0 u rms = M 6, at the outer scale l 0 L/2 = 0.5. Integral-scale Reynolds number Re l 0u 0 ν = (N/2) 4/

8 Results Structures in Physical Space 8

9 Turbulent structures: PPM vs. Zeus Density (log) slices from two simulations with resolution points PPM Zeus 9

10 Turbulent structures: PPM vs. Zeus Density power spectra for two snapshots with resolution points Density power spectra at : PPM vs. Zeus PPM HD Zeus HD log P(k) log k 10

11 Turbulent structures: PPM vs. Zeus Density slices from PPM and Zeus simulations PPM512 Zeus

12 Turbulent structures: PPM vs. Zeus Density power spectra for the snapshots from PPM512 and Zeus1024 runs Density power spectra: PPM vs. Zeus PPM 512 Zeus log P(k) log k 12

13 Turbulent structures: HD vs. MHD Density slices from two simulations with resolution points Zeus HD Zeus MHD 13

14 Turbulent structures: HD vs. MHD Density power spectra for two snapshots with resolution points Density power spectra at : HD vs. MHD Zeus HD Zeus MHD log P(k) log k

15 Substructure in large-scale dense filaments 15 Large-scale shocks are fragmented, consistent with observations of molecular gas by Brunt (2003)

16 Nested Mach cones in a thick density slice 16 Mach angles of large-scale structures in this image correspond to relative motion with M 2 3

17 Density Dilatation Vorticity Gas Density 17

18 Density Dilatation Vorticity Velocity Divergence (MPEG animation) 18

19 Density Dilatation Vorticity Specific Vorticity 19

20 Results PDFs, Power Spectra, and Structure Functions 20

21 Time evolution of the maximum density Is turbulence self-similar or is it intermittent? Maximum density as a function of time (1024PPM) ρ max / t/t dyn ρ max Maximum density, subject to high-pass filtering t/t dyn

22 Density statistics: PDF With an isothermal EOS, the local Mach number does not correlate with density lognormal density PDF [Nordlund & Padoan 1998; Passot & Vázquez-Semadeni 1998]. We get σ 2 = ln(1 + b 2 M 2 ) with b Watch MPEG animation here. Compare with b 0.5 from low-resolution MHD simulations [Padoan et al.1997; Nordlund & Padoan 1998]. 0 Time-average gas density PDF (1024PPM, Mach 6) <PDF> 4tdyn b=0.2602(5) log 10 PDF log 10 (ρ)

23 Density statistics: 3D power spectrum Time-average density power spectrum (170 snapshots within 4t dyn ) slope(0.6,1.3) = -1.07(1) slope(1.4,1.7) = (2) 1024PPM Mach 6 log <P(k)> log k 23 At high Mach numbers the density power spectrum is shallower than in the transonic case. The inertial range slope at Mach 6 is Compare with 1.08 for a Mach 3.4 simulation at with a more diffusive solver [Kim & Ryu 2005]. The spectrum gets shallower upon collisions of strong shocks, when the PDF s high density wing rises above its average lognormal representation. Watch MPEG animation here. There is some pileup of power at λ 40 x, see the next two slides.

24 Density statistics: a convergence test Time-average density power spectrum (37 snapshots within 1.3t dyn ) slope(0.6,1.3) = -0.91(3) slope(1.2,1.7) = -0.89(1) 512PPM Mach 6-1 log <P(k)> log k 24 At the resolution of 512 3, the bottleneck effect at high wave numbers and turbulence forcing at low wave numbers leave essentially no room for the unaffected inertial range in k-space, even though the spectrum does show a nice power law. The slope of 0.90 at is significantly shallower than 1.07 at Spectral slope estimates based on low resolution simulations, e.g. [Kim & Ryu 2005], should be taken with a grain of salt since the uncertainty is, perhaps, larger than 15%.

25 Velocity statistics: 3D power spectrum Compensated velocity power spectrum (time-average over 4t dyn ) slope(0.6,1.1) = -1.95(2) slope(1.2,1.7) = (3) 1024PPM Mach 6 log k 5/3 <E t (k)> log k Bottleneck phenomenon: numerical viscosity increases turbulence level in the inertial interval of scales [Falkovich, 1994]. Inertial range slope is The first real measurement for supersonic case! Watch MPEG animation here. NB: possible caveats discussed later in the talk.

26 Velocity statistics: 3D power spectrum The velocity field is decomposed into solenoidal ( v = 0) and dilatational ( v = 0) parts Time-average velocity power spectra (1024PPM Mach 6; <...> 4tdyn ) 0-1 Total Solenoidal Dilatational log <E t,s,c (k)> log k At Mach 6 the solenoidal component contributes about 68% of the total power. For comparison, in div-free driven transonic turbulence the dilatational component saturates at a level of 10% [Porter et al. 2002].

27 Velocity statistics: 3D power spectrum solenoidal k 5/3 dilatational k 2 Compensated power spectrum for solenoidal velocity (<...> 4tdyn ) Compensated power spectrum for dilatational velocity (<...> 4tdyn ) slope(0.6,1.1) = -1.92(2) slope(1.2,1.7) = (3) 1024PPM Mach slope(0.6,1.1) = -2.02(2) slope(1.2,1.7) = (3) 1024PPM Mach 6 0 log k 5/3 <E s (k)> log k 2 <E c (k)> log k log k Both components exhibit the bottleneck effect in the same range of scales. The inertial range slope for the dilatational velocity field 2.02 ± , as in Burgulence [Frisch & Beck 2001].

28 Velocity statistics: structure functions The velocity structure functions (SFs) are defined as 28 S p (l) u(x + l) u(x) p l ζ(p), where u l for longitudinal and u l for transverse SFs, and the spatial average is taken over all positions x within the box.

29 Velocity statistics: structure functions We first build the PDFs for velocity differences and then compute the moments (SFs). PDFs for Transverse Velocity Differences (PPM1024) -2 log 10 PDF(n x; δu t ) n= log 10 δu t

30 Velocity statistics: 2nd order structure functions At Mach 6 the best fit values for log 10 x [ 1.5, 0.5] give an average of ζ(2) This slope is substantially steeper than both 2 3 [Kolmogorov 1941] and 0.74 [Boldyrev 2002]. S 2 (l) δu(l) 2 l ζ(2) nd order longitudinal and transverse velocity SFs (1024PPM) 0.952(4) 0.977(8) <L> <T> 30 1 log 10 S 2 (x) log 10 x

31 Velocity statistics: 3rd order structure functions Unlike in the incompressible limit [Kolmogorov 1941] or in transonic regime [Porter et al. 2002], at Mach 6 ζ(3) The relative 2nd order exponent Z 2 ζ(2)/ζ(3) 0.75 is close to the predicted value of 0.74 [Boldyrev (2002)] S 3 (l) δu(l) 3 l ζ(3) 3rd order longitudinal and transverse velocity SFs (1024PPM) <L> <T> 31 2 log 10 S 3 (x) log 10 x

32 Velocity statistics: high-order structure functions Magnification of the large amplitude tail of the PDF With pairs per PDF, we get reasonably good statistics reliable estimates for high-order moments, perhaps, up to p = PDFs for Transverse Velocity Differences (PPM1024) p= log 10 PDF(n x; δu t ) n= log 10 δu t The four fences indicate the positions where the integrand of the pth order moment has its peak.

33 Velocity statistics: structure functions Bottleneck Phenomenon? Same flattening at small scales is observed for high-order structure functions. The bottleneck corrections grow with the order p [Falkovich 1994] and influence the SFs in a nonlocal fashion [Dobler et al. 2003]. 33 Second order SFs are less susceptible to the bottleneck effect [Dobler et al. 2003], and within the inertial range their slope ( 0.96) agrees well with the energy spectrum slope. 3 Relative differential slopes for the transverse velocity SFs (1024PPM) p=1 23 ζ(p)/ζ(3) log 10 x

34 Velocity statistics: Hierarchical Structure model A model for the inertial range structures The HS model describes the inertial range dynamics and involves a hierarchy of functions F p (l) = S p+1 (l)/s p (l) 34 associated with higher intensity of fluctuations as p. She & Lévêque ( ) postulated a hierarchical symmetry relation with respect to a translation in p F p+1 (l) = A p F p (l) β F 1 β, where 0 β 1 measures the degree of intermittency (β 1 K41; β 0 Burgers shocks), A p is independent of l and may weakly depend on p. F characterizes the most intermittent structures and can be eliminated by considering the ratio F p+1 (l) F 2 (l) ( = A p Fp (l) A 1 F 1 (l) ) β The postulate proposed by She & Lévêque can be tested by plotting F p+1(l) F 2 (l) log-log plane. against F p(l) F 1 (l) in

35 Velocity statistics: Hierarchical Structure model The β-test If we measure average slopes for 32 x < l < 256 x, we get β = Compare with β [0.77, 0.82] for MHD turbulence at [Padoan et al., 2004]. 35 Notice a weak dependence of A p on p. 0.5 β test, transverse velocity (F p (x)=s p+1 (x)/s p (x), 1024PPM) 0.4 log 10 F p+1 (x)/f 2 (x) 0.3 p= β= log 10 F p (x)/f 1 (x)

36 Velocity statistics: Hierarchical Structure model Assuming F S γ 3, 36 She & Lévêque derived a general scaling formula for the relative exponents ζ(p)/ζ(3) = γp + C(1 β p ), with the normalization condition for the codimension of the support of the most singular dissipative structures C = (1 3γ)/(1 β 3 ). Then ζ(p)/ζ(3) χ(p, β) = γ(p 3χ(p, β)), where χ(p, β) (1 β p )/(1 β 3 ). Now we can plot ζ(p)/ζ(3) χ(p, β) vs. p 3χ(p, β) and check the validity of the assumption made above.

37 Velocity statistics: Hierarchical Structure model 1.2 The γ-test γ test (p=4,...,15; transverse velocity; ESS; 1024PPM) ζ p /ζ 3 -χ(p,β) p-3χ(p,β) γ=0.06 T γ = 0.06 could be an indication that the most intermittent structures in supersonic HD turbulence are more singular than in MHD turbulence (γ = 1/9 [Padoan et al., 2004]).

38 Velocity statistics: structure function exponents The HS model provides good fits to the differential exponents within the inertial interval. The scatter of the best-fit values of β and γ at different displacements l within the inertial range measures the uncertainty of the scaling. 38 The dimension of the support for the most singular dissipative structures, D = Caution! The high order statistics at large displacements can be affected by small anisotropies of the driving force [Porter et al. 2002]. ζ(p)/ζ(3) Relative scaling exponents for the transverse VSFs (1024PPM, ESS) K41 SL94 B p 90 x: γ = β = 0.67 C = x: γ = β = 0.63 C = x: γ = β = 0.58 C = 0.70

39 Density statistics: multifractal distributions Boldyrev et al. (2002) proposed that in isotropic supersonic turbulence the multifractal distributions for the density and velocity fields are related. 39 The quantities of practical interest are density correlators: [ρ(x + l)ρ(x)] m l ξ(m), where ξ(m) = ζ(2mp 0 ) 2mζ(p 0 ), and p 0 is obtained as a solution of ζ(2p + 2) = 2ζ(p), while ζ(p) represent scaling exponents of velocity structure functions n= PDFs for Density Correlator (PPM1024) log 10 PDF log 10 ρ(x)ρ(x+n x)

40 Density statistics: multifractal distributions At Mach 6 we are getting substantially shallower slopes for orders m = 1, 2, 3 than those predicted by Boldyrev et al. (2002). 40 log 10 <[ρ(r)ρ(r+x)] m > Density correlators (Mach 6, 1024PPM, samples) (-0.28) m= (-1.2) m=2-1.5 (-2.3) m= log 10 x

41 Summary 41 High Re[solution] is needed to reproduce the complex pattern of nested hierarchical structures forming in supersonic turbulent flows due to (non)linear instabilities. These instabilities may control the scaling properties of supersonic turbulence. The exponents of the 3rd order velocity structure functions ζ(3) 1 at Mach 6. The bottleneck effect strongly contaminates high-order velocity statistics. A higher than points resolution (and/or a good subgrid model) is needed to determine the scaling properties of turbulence in the inertial range. The fractal dimension of the support for dynamically important structures at Mach 6 D 2.1, but the uncertainty is large. To be measured with other techniques.