Supersonic Turbulence in Shock Bound Slabs. Doris Folini and Rolf Walder, CRAL, ENS Lyon, France Visualization by Jean Favre, CSCS Manno, Switzerland

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1 Supersonic Turbulence in Shock Bound Slabs Doris Folini and Rolf Walder, CRAL, ENS Lyon, France Visualization by Jean Favre, CSCS Manno, Switzerland

2 Outline shock bound slabs: why study? plane parallel isothermal shock bound slabs boundary of slab turbulence in slab self similarity structure functions, modeled observed summary / conclusions

3 Shock Bound Slabs: another toy model for molecular clouds? cloud formation? star formation (IMF)? (driven) turbulence within cloud? Orion: a giant molecular cloud Distance: ~ 470pc Mass: ~ 105 M Betelgeuse Diameter: ~ 30pc Orion nebula Riegel Orion: stellar mass distribution stellar mass (L. Hillenbrand, Caltech) IRAS (12, 60, 100 ; x ~240x200 pc) Visible (S.Kohle & T.Credner)

4 From observations and (3D periodic box) simulations molecular clouds are supersonically turbulent: observations supersonic rms velocities the turbulence must be driven: if not decay within a sound crossing time higher star formation rate than observed the driving occurs at large scales: observations large scales dominate velocity field and structure of low density gas 3D box models driving wavelength sets structure size Our focus, somewhat complementary to 3D box: study one possibility of a more natural forcing study interplay confining shocks turbulence

5 Model problem: left flow, supersonic turbulent interaction zone 2D (3D) plan parallel isothermal colliding flows right flow, supersonic Computations done with A MAZE: ideal hydro AMR following evolution of growing interaction zone

6 left flow, supersonic turbulent interaction zone 2D plan parallel isothermal colliding flows right flow, supersonic Interplay: logarithm of density Mu = 22 Oblique confining shocks force turbulence turbulence forces shocks to be oblique

7 Higher upstream Mach number confining shocks have more narrow, steeper wiggles with larger amplitude Angle of obliqueness 0.8 Shock length : lsh ~ M 0.6 'Auto correlation: lcorr ~ Mu lcdl (Folini & Walder, A&A, 2006)

8 upstream kinetic energy flux logarithm of density kinetic energy flux entering slab Mu = 22 feff = feff (Mrms) = 1 Mrms 0.6 (Figure includes asymmetric runs with Ml Mr) driving efficiency feff goverened by Mrms, indicating back coupling of driving and turbulence turbulence forces shocks to be oblique

9 scaling laws for mean quantities (like Mrms): dimensional analysis suggests self similarity Dimensional considerations: feff Numerical simulations confirm: 1 = 0 2 = 1 1= 1 Numerical simulations yield (2D): = (1/ 1)1/

10 Numerical simulations, Mach number and density (2D): Predicted : Mrms / Mu = const. m / u = const. (independent of Mu!) Mrms / Mu ~ lcdl m / u ~ lcdl

11 Second order effects I: slight decrease in Mrms Predicted : Mrms / Mu = const. Observed : 15% decrease as lcdl goes from 10 to 70 Mrms / Mu ~ lcdl Why the deviation? viscous dissipation? sub grid scale model (MILES) not appropriate? time scale of turbulence decay 'non culprits' : y extent of domain and spatial discretization

12 Second order effects II: no convergence so far finer grids (factor 2) smaller (15%) Mrms Possible reasons? finer grids more / better resolved shocks enhanced total dissipation in shocks back coupling between Mrms and feff amplifies effect sub grid scale model (MILES) sensitive to grid spacing? [MILES, monotone integrated large eddy simulation; Boris et al. 1992; Porter et al. 1992, 1994; Garnier et al. 1999]

13 predicted by self similarity & confirmed by simulations: column integrated dissipation independent from lcdl Ediss / Edriv Possible explanation: if self similar, all length scales proportional to each other distance between shocks proportional to lcdl number of shocks within CDL column constant column integrated dissipation (by shocks) constant

14 Density for three different times, three different shell sizes lcdl time Structure size increases with lcdl hypothesis A: wiggling of shocks effective driving wave length scale of turbulence (Mac Low, 1999, 3d box) hypothesis B: small scale structures decay first larger structures in center of CDL (Smith et al., 2000)

15 Divergence for two different Mach numbers, same lcdl Mu = 33 Mu = 11 Structure size increases with decreasing upstream Mach number Structure size increases with lcdl

16 velocity in slab clearly anisotropic x velocity, 6 symmetric runs y velocity, 6 symmetric runs y x (Walder & Folini, 2000, ApSS, 274) sound speed ~ cm/s

17 width of density pdf levels off with large Mrms PVS98 PNJ97 2D slabs Passot & Vázquez Semadeni, 1998, 1d data: ln M rms Padoan, Nordlund, & Jones, 1997, 3d data: ln ln 1 M rms / 4 Federath et al. 2009: / m = b Mrms (b depends on driving) F09: b2d,comp = F09: b2d,sol =

18 2D slabs 3D slabs?

19 same upstream Mach number: 3D more turbulent 2D m(2d) 3/2 m(3d) Mrms(2D) 0.8 Mrms(3D) 3D

20 3D slabs: plane parallel isothermal symmetric

21 velocity structure functions (3D) 1.6 log10(sp) Mu = 42 Mu = 32 Mu = 21 Mu = 11 longitudinal & transverse directions: no clear difference best agreement with Schmidt et al Dubrulle log10(s3) p=1: p=2: p=3: p=4: p=5: 3D slabs 0.40 / / / / 1.33 Schmidt et al SL B K

22 red diamonds: 3D slabs black squares: Gustafsson et al (Orion) magenta/green circles: Hily Blant et al (Polaris / Taurus) blue diamonds: Schmidt et al blue/black stars: Dubrulle 1994 Gustafsson et al., 2006, Orion p=1: p=2: p=3: p=4: p=5: 3D slabs 0.40 / / / / 1.33 Schmidt et al SL B K

23 S3 not well represented by single power law in 3D slabs and Orion (Gustafsson et al, 2006) S3 from 3D slab Observed S3, Orion S3 from 3D periodic box run Gustafsson et al., 2006

24 Summary / Conclusions confining shocks (driving) interior turbulence driving more efficient in 3D and for larger M u Mrms thicker slab / smaller Mu larger scale interior structure mean quantities: self similar, governed by Mu density pdf: width levels off with increasing Mrms Sp: no single power law, small exponents implications for molecular clouds? velocities of colliding flows M 4 M u rms naturally obtain non single power law structure functions