1. Approaches to Turbulence in High-Energy-Density Experiments R. Paul Drake University of Michigan 2007 TMBW, Trieste Work supported by the U.S. Department of Energy under grants DE-FG52-07NA28058, DE-FG52-04NA00064, by the Naval Research Laboratory under contract NRL N00173-06-1-G906, and by other grants and contracts

2. Page 22007 August Turbulent Mixing and Beyond Who contributed; where we are going Key collaborators (others listed later) –Michigan: C.C. Kuranz, E.C. Harding, M. Grosskopf –LLNL: J.F. Hansen, H.F. Robey, B.A. Remington, A. Miles –Rochester: J. Knauer –Florida State: T. Plewa –NRL: Yefim Aglitskiy Outline –How lasers accomplish hydrodynamic experiments –Shock-driven Richtmyer Meshkov –Turbulence? –Blast-wave-driven instabilities & why not yet turbulent –Steepness of shear appears to matter (blast-waves, jets) –An experiment to create a steep shear layer –An experiment having a steep shear layer and turbulence –Some speculation regarding requirements for quick turbulence

3. Page 32007 August Turbulent Mixing and Beyond High-Energy-Density Physics The study of systems in which the pressure exceeds 1 Mbar (= 0.1 Tpascal = 10 12 dynes/cm 2 ), and of the methods by which such systems are produced. This also matters for astrophysics Direct reasons –pressures > 1 Mbar are important in planets –High-temperature dense matter is important in stars Indirect reasons –Dynamics of Mach >> 2, very high Re systems –T >> 1 eV (10 4 K) and/or  = (8  p/B 2 ) >> 1 systems –Radiation hydrodynamic systems We work in my group with high Mach number hydrodynamic and radiative hydrodynamic systems

4. Page 42007 August Turbulent Mixing and Beyond The Omega laser is our major tool at present Target chamber at Omega laser There are lots of kJ, ns lasers around the world Omega is the most capable for experiments. Omega is at the Laboratory for Laser Energetics, affiliated with the University of Rochester. Its primary mission is inertial fusion research.

5. Page 52007 August Turbulent Mixing and Beyond Here is what such lasers do to a material The laser is absorbed at less than 1% of solid density From Drake, High-Energy-Density Physics, Springer (2006) Rad xport, high-v plas Hydro, rad hydro

6. Page 62007 August Turbulent Mixing and Beyond Shock waves establish the regime of an experiment

7. Page 72007 August Turbulent Mixing and Beyond The Mach number in these experiments is effectively infinite Sound speeds are ~ 1 km/sec –Exact value depends on upstream “preheating” Shock velocities are ~ 50 km/sec Mach number terms in shock jump relations are fundamentally present as 1/M 2 Implications –Density jump is ~ (  – 1)/(  – 1) –Post-shock temperature is ~ –Here , Z are post-shock values A complication is that they are temperature-dependent.

8. Page 82007 August Turbulent Mixing and Beyond Here is a drawing of a typical target for hydrodynamics experiments on lasers Precision structure inside a shock tube Experiment design: Carolyn Kuranz

9. Page 92007 August Turbulent Mixing and Beyond Ungated imaging provides improved resolution in such experiments (side-on schematic) 4 backlighter beams Delayed 10-30 ns X-ray photons X-ray film Target Titanium/Scandium backlit pinhole 20 mm Au shield protects film from overexposure Drive beams t=0 Credit: Carolyn Kuranz

10. Page 102007 August Turbulent Mixing and Beyond This shows images of actual targets built at Michigan Acrylic cone Gold cone Laser-driven surface Side view 1 mm Targets: Mike Grosskopf, Donna Marion, Robb Gillespie, UROP team

11. Page 112007 August Turbulent Mixing and Beyond XTVS YTVS Alignment in Omega Chamber

12. Page 122007 August Turbulent Mixing and Beyond Main target positioner Backlighter stalk Backlighter positioner Acrylic shield What a shot looks like in the chamber…

13. Page 132007 August Turbulent Mixing and Beyond We have greatly improved the resolution and signal-to-noise in the data Also, the UM-led team was the first group to get simultaneous orthogonal physics data Mid- 1990’s data Recent data and analysis: Carolyn Kuranz Aug. 2005 Dec. 2006 Grid squares have 43 µm openings Eggcrate + 424 µm sine wave, 10 to 20 µm tapered pinhole, 25 ns Eggcrate mode only, 20 to 50 µm stepped pinhole, 17 ns

14. Page 142007 August Turbulent Mixing and Beyond Theoretical considerations for these experiments One likes to imagine that these systems are described by the Euler equations But do these accurately describe the physical system?  = density v = velocity p = pressure  = constant (adiabatic index)

15. Page 152007 August Turbulent Mixing and Beyond Is it sufficient to use only hydrodynamics? General Fluid Energy Equation: Material Energy Flux  m Smaller or Hydro-like Typ. small Or Ideal MHD

16. Page 162007 August Turbulent Mixing and Beyond Typical parameters in these experiments U ~ 10 km/s Range 1 to < 1,000 Driving scale for turbulence  ~ 100 µm Range 10 µm to 1 mm Kinematic viscosity ~ 10 -5 m 2 /s Range 10 -6 to 10 -4 Re ~ 10 5 Range 10 4 to 10 7 Kolmogorov scale ~ 0.02 µm Viscosity ~ Diffusion (Schmidt # ~1). RT & RM typically quenched on scales of 1 to a few µm (Robey 2004) Pe and Pe rad are both >>> 1 The plasmas are well localized (fluid models are OK) See: D.D. Ryutov, et al. ApJ 518, 821 (1999); ApJ Suppl. 127, 465 (2000); Phys. Plasmas 8, 1804 (2001)

17. Page 172007 August Turbulent Mixing and Beyond Consider what we mean by “turbulence” Is turbulence –Growth of structures beyond the nonlinear saturation of existing modes? –The development of strong mixing as indicated by supra-linear growth of a mixing layer in time? –The appearance of an inertial range in the fluctuation spectrum? –Something else? Different communities use different definitions For turbulence corresponding to strong mixing –Dimotakis (2000) argued that the necessary condition was a sufficient separation of the Taylor microscale and the dissipation scale

18. Page 182007 August Turbulent Mixing and Beyond Does the large Re lead to “turbulence”? HEDP experiments typically have Re of 10 4 to 10 6 The Dimotakis picture. Credit: Zhou 2003

19. Page 192007 August Turbulent Mixing and Beyond The experimental Re is > 10 5, so these systems should be turbulent? Let’s look at Richtmyer-Meshkov Practical experiments are “heavy to light” From Drake, High-Energy-Density Physics, Springer (2006)

20. Page 202007 August Turbulent Mixing and Beyond At high Mach number, RM does not become turbulent by almost any definition Hard experiments –Difficult to sustain the shock Glendinning et al. (Phys. Plas., 2003 ) –The spikes rapidly overtake the shock, distorting it and limiting their development –Initial amplitude 5% Sensible result: –Incompressible Meyer-Blewett amplitude growth (d  /dt) –Can exceed the shock separation velocity data simulation

21. Page 212007 August Turbulent Mixing and Beyond Do we see turbulence in scaled hydrodynamic experiments relevant to supernovae? SN 1987A –A core-collapse supernova –Early high-Z x-ray lines with large Doppler shifts –Early glow from radioactive heating –The issue is the post-core-collapse explosive behavior In 19 years of simulations –Only one (Kifonidis, 2006) makes fast enough high-Z material –3D simulations coupling all the interfaces where initial conditions matter are not feasible –Experiments will be able to address this fully within a decade –We now address a single interface SN1987A, WFPC2, Hubble Kifonidis, 2003

22. Page 222007 August Turbulent Mixing and Beyond This type of system involves a blast-wave- driven interface Laser ablation drives shock wave for 1 ns Front surface rarefaction overtakes shock wave by 2 ns, forming planar blast wave Blast wave crosses interface, followed by deceleration and rarefaction From Drake, High-Energy-Density Physics, Springer (2006)

23. Page 232007 August Turbulent Mixing and Beyond Boundary conditions in time & space matter for scaling Interface velocity vs time Pressure and density star lab Conclusion: scaling is possible

24. Page 242007 August Turbulent Mixing and Beyond Experimental image of a turbulent flow at Re = 3x10 4 Numerical image of an unstable, though non-turbulent flow at Re(sim) ~ 10 3 ; Re = 10 10 Kelvin-Helmholtz roll-up Laminar flow All simulations have too much numerical viscosity to produce the observed turbulent state. But it is an open question whether the flows in supernovae actually become turbulent There is a turbulence angle to these experiments and supernovae Van Dyke, Album of Fluid Motion Kifonidis et al., 2003

25. Page 252007 August Turbulent Mixing and Beyond t = 8 nst = 12 nst = 14 ns Data from a 2D single-mode perturbation having an initial perturbation  = 50 µm, a  = 2.5 µm, ka  = 0.3 The perturbation growth is well into the non-linear regime and Re > 3 x 10 4 by 7 ns, but the system does not appear to become turbulent within 14 (or 20) ns shock spikes bubbles Our Rayleigh-Taylor experiments produce time- dependent, high-Reynolds-number systems.

26. Page 262007 August Turbulent Mixing and Beyond The question is why these experiments did not enter a turbulent state One answer is time dependence For turbulence corresponding to strong mixing –Dimotakis (2000) argued that the necessary condition was a sufficient separation of the Taylor microscale and the dissipation scale –Zhou et al. (2003) and Robey et al. (2003) argued that sufficient time was also needed The role of large Re is to allow diffusive laminar boundary layers to become large enough to separate driving scales from dissipation scales This also takes time –Miles et al. (2004, 2005) argued from simulations that interacting structures could produce turbulent conditions, but not soon enough to be observed in the Omega experiments

27. Page 272007 August Turbulent Mixing and Beyond The Robey/Zhou picture. Credit: Robey 2003 Here is the Robey/Zhou picture The point is –It can take longer to establish the needed boundary layers than it does to reach the threshold value of Re –Zhou et al. 2003 show that this picture applies successfully to numerous cases

28. Page 282007 August Turbulent Mixing and Beyond This picture is consistent but not universal Consistency: –Dimotakis requires that the Liepmann-Taylor scale exceed the inner viscous scale, or 5  Re -1/2 > 50  Re -3/4 –Robey/Zhou requires the viscous boundary layer, 5( t) 1/2 > 50  Re -3/4 –These are consistent if t =  /U For typical turbulence  /U = l /u, a large-eddy turnover time, so both these arguments are consistent with development of an inertial range and strong mixing in one eddy turnover time I will refer to this as “quick turbulence”, distinct from e.g. long- term RT Issues –Perhaps  /U ≠ l /u. –Sometimes velocity shear is gentle …

29. Page 292007 August Turbulent Mixing and Beyond The path to quick turbulence at shear layers begins with Kelvin Helmholtz (KH) Experiments and some simulations do not show Kelvin-Helmholtz growth along Rayleigh-Taylor spikes We certainly do not see it, especially in our improved 3D experiments However, simulations of our experiments show that the velocity shear is too shallow to allow KH growth

30. Page 302007 August Turbulent Mixing and Beyond High-energy-density jet experiments produce supersonic jets They create a significant bow shock –Surrounds the jet with a cocoon of shocked material –Weakens the velocity shear These experiments see a lot of structure at the head of the jet, but with many possible causes 4.0 mm dia. 700 µm 125 µm 300 µm dia. hole 0.1 g/cc RF foam Titanium 4 µm CH ablator Target dimensions Laser Ablation

31. Page 312007 August Turbulent Mixing and Beyond Astro & HEDP Jets show less Kelvin Helmoltz than laboratory jets in gasses Van Dyke, Album of Fluid Motion (1982) Turbulent jet in gas HH34 Astro Jet Burrows, Hester, Morse, WFPC2 Hubble Z-pinch Jet Lebedev et al. Ap J Laser-driven Jet Foster et al. Ap J The difference is most likely in L u.

32. Page 322007 August Turbulent Mixing and Beyond This motivates HEDP experiments to study shear flow effects directly Target Cross Sectional Views Expanding fluid bubble Gold “knife-edge” t =4ns t=0 t >4ns CH ablator Al driver 1st fluid Cold foam Shocked foam 2nd fluid Drive Beams KH here? Credit: Eric Harding

33. Page 332007 August Turbulent Mixing and Beyond Initial experiments show we can get data The edge that clips the flow is too close to laser-driven material, Complex laser-heating above the rippled surface greatly complicates results The next experiments will have an improved design Experiments at NRL Credit: Eric Harding

34. Page 342007 August Turbulent Mixing and Beyond One experiment having shear at a boundary definitely produces turbulence … but not in a way that lets one diagnose details The experiment involves blast- wave-driven mass stripping from a sphere Early experiments used Cu in plastic; recent experiments use Al in foam

35. Page 352007 August Turbulent Mixing and Beyond This has found application Experimental results used to help interpret Chandra data from the Puppis A supernova remnant Well-scaled experiments have deep credibility Una Hwang et al., Astrophys. J. (2005)

36. Page 362007 August Turbulent Mixing and Beyond Observations of the Al/foam case continued until mass stripping had destroyed the cloud Hansen et al., ApJ 2007

37. Page 372007 August Turbulent Mixing and Beyond The stripping is clearly turbulent, consistent with the necessary conditions The turbulent model is based on “Spalding’s law of the wall”, Spalding (1961) Parameters Re ~ 10 5 to 10 6 U ~ 10 km/s ~ 10 -5 to 10 -6 m 2 /s  ~ Sphere ~ 60 µm radius  /U ~ 6 ns ~ rollup time (data) Robey/Zhou time is ~ 1 ns Remaining Mass (µg) Time (ns) Hansen et al., ApJ 2007

38. Page 382007 August Turbulent Mixing and Beyond The HEDP results taken together suggest a condition on steepness of shear The Dimotakis and Robey/Zhou models have turbulence begin when fluctuations in the inertial range exist primarily within viscous diffusion layers Something must establish these fluctuations, almost certainly beginning with Kelvin Helmholtz (KH), followed by some secondary instability In the standard model from Chandrasekhar, a linear velocity scale length H creates a KH threshold of k th ~ 1.4/H The KH growth rate to a factor of two is

39. Page 392007 August Turbulent Mixing and Beyond A crude estimate of the impact of distributed shear Take the linear scale length of the velocity transition to be Integrating in time, the linear number of e-foldings is Assume 10 e-foldings needed and find minimum wavelength with this much growth Can’t quickly populate the turbulent spectrum for smaller wavelengths The KH curve must be below the inner viscous scale for KH driven turbulence Steeper shear layers are required at higher Re

40. Page 402007 August Turbulent Mixing and Beyond Conclusions High-energy lasers readily produce high-Mach-number, ionized, high-Reynolds-number flows. These flows may develop turbulence, but the nature of turbulent onset is typically important. Richtmyer-Meshkov development in such systems can be constrained by interaction with the shock. Driving Rayleigh Taylor long enough to see turbulence develop is challenging. The challenge in shear flows is to produce steep enough shear. –A fundamental experiment is in progress –Mass stripping of a shocked sphere was clearly turbulent The development of turbulence from shear layers in an eddy turnover time may involve a steepness constraint The next page shows the extended list of collaborators

41. Page 412007 August Turbulent Mixing and Beyond This work and this talk have involved contributions from many individuals My students and post docs at Michigan Bruce Remington, Harry Robey, Dmitri Ryutov, Adam Frank, Kent Estabrook, Sasha Velikovich, Riccardo Betti The present and past Omega NLUF collaborators Numerous Michigan collaborators at LLNL & NRL High Energy Density Laboratory Physicists around the world Harry Robey Dave ArnettDick McCrayRobert Rosner LLNL University of Arizona University of Colorado University of Chicago Ted Perry Romain Teyssier Jim Knauer Bruce Fryxell LLNL CEA Saclay, France LLE; Univ. of Rochester University of Chicago Dimitri Ryutov James GlimmJames Stone Alexei Khokhlov LLNL SUNY-Stony Brook Univ. of MarylandNaval Research Laboratory Jave Kane Omar Hurricane John Grove James Carroll LLNL LLNL LANLEastern Michigan Univ. Adam Frank Kim Budil Keisuke Shigemori Riccardo Betti University of Rochester LLNL Osaka University; LLNL University of Rochester, MIT Mary Jane Graham Christof Litwin Gail Glendinning Grant Bazan West Point University of ChicagoLLNL LLNL Serge Bouquet Michel Koenig Aaron Miles CEA BruyeresLULIUniv. Maryland

42. Page 422007 August Turbulent Mixing and Beyond Next consider systems driven by flowing plasma Ejecta-driven systems –Rarefactions drive nearly steady shocks –Supernova remnants –Experiments –Rarefactions often evolve into blast waves A rarefaction can produce flowing plasma that can drive instabilities

43. Page 432007 August Turbulent Mixing and Beyond Supernova remnants produce the instability driven by plasma flow in simulation, … 1D profile and 2D simulation Chevalier, et al. ApJ 392, 118 (1992)

44. Page 442007 August Turbulent Mixing and Beyond.. in observation, and in lab experiment Blast-wave driven lab result Remnant E0102

45. Page 452007 August Turbulent Mixing and Beyond But we do understand how to scale hydro. An important example: strongly shocked systems Consider two hydrodynamic systems driven by strong shocks Suppose their density structure is spatially identical: Suppose they are caused to evolve by a strong shock of speed v i The two systems will evolve identically on normalized time scales t/  i, with  i = C i H(r/h i ) Constant Shape function Normalizing dimension  i = h i / v i Example h i (cm) v i  i (km/s) SN 1987A 9x10 10 = ~ 1million 2000 450 s km Lab 0.0053 = 1/2 human 13 4 ns hair System 1 System 2 Details: D.D. Ryutov, et al., Ap.J. 518, 821 (1999)

46. Page 462007 August Turbulent Mixing and Beyond In other lab experiments, the instabilities moved material all the way to the shock 13 ns 17 ns13 ns 21 ns 17 ns modulated planar 21 ns Drake, Phys. Plasmas 2004

47. Page 472007 August Turbulent Mixing and Beyond We are now observing the role of complex initial conditions in spike penetration Interferogram of complex surface on component provided by GA (analysis: Kai Ravariere) Preliminary data on mix layer thickness Data and analysis: Carolyn Kuranz

48. Page 482007 August Turbulent Mixing and Beyond We collaborate with simulation groups to evaluate our results and validate codes Work with the FLASH Center (Chicago), to include 3D adaptive modeling, has now begun Preliminary FLASH simulation at 30 ns of recent experiments Left: Single eggcrate mode. Right: Two-mode system.