1. Turbulence and Chemistry P. Lesaffre Thanks: E. Falgarone (Image mosaic: eta carina by HST)‏

2. Outline (A) Turbulence => Chemistry (A) Turbulence => Chemistry  1. Turbulent Mixing  2. Reaction enhancement (B) Chemistry => Turbulence (B) Chemistry => Turbulence  1. Indirect action via heating/cooling function  2. Direct action on MHD via electron fraction (C) Turbulence Chemistry (C) Turbulence Chemistry  An example: MHD turbulence in accretion disks

3. 5/5/2008 Arcachon (A 1) Turbulent Mixing

4. 5/5/2008 Arcachon “Chaotic” Mixing “Blinking” vortex mapping : “Blinking” vortex mapping :  Stretching  Folding Patterns are transferred to smaller and smaller scales Patterns are transferred to smaller and smaller scales The distribution function remains unchanged The distribution function remains unchanged (Credits: PJS Franks)

5. 5/5/2008 Arcachon “Chaotic” Mixing “Blinking” vortex mapping : “Blinking” vortex mapping :  Stretching  Folding Patterns are transferred to smaller and smaller scales Patterns are transferred to smaller and smaller scales The distribution function remains unchanged The distribution function remains unchanged (Credits: PJS Franks)

6. 5/5/2008 Arcachon “Chaotic” Mixing “Blinking” vortex mapping : “Blinking” vortex mapping :  Stretching  Folding Patterns are transferred to smaller and smaller scales Patterns are transferred to smaller and smaller scales The distribution function remains unchanged The distribution function remains unchanged (Credits: PJS Franks)

7. 5/5/2008 Arcachon “Chaotic” Mixing “Blinking” vortex mapping : “Blinking” vortex mapping :  Stretching  Folding Patterns are transferred to smaller and smaller scales Patterns are transferred to smaller and smaller scales The distribution function remains unchanged The distribution function remains unchanged (Credits: PJS Franks)

8. 5/5/2008 Arcachon The Kolmogrov cascade Wave number Amplitude Kinetic energy Injection Viscous Dissipation Compress ====> Strain ====>

9. 5/5/2008 Arcachon The Mixing cascade Wave number Amplitude Chemical patterns InjectionMolecular diffusion Compress ====> Strain ====>

10. 5/5/2008 Arcachon 2D incompressible mixing (experiment) Jullien et al (2002)‏ : A cylinder of dye is stretched and folded A cylinder of dye is stretched and folded When the patterns are small enough, molecular diffusion proceeds When the patterns are small enough, molecular diffusion proceeds The final state is homogeneous The final state is homogeneous

11. 5/5/2008 Arcachon Turbulent Mixing Is a two-stages process : Transfer of patterns from large to small scales Transfer of patterns from large to small scales Molecular diffusion mixes the small scales Molecular diffusion mixes the small scales What is the effective rate of diffusion ? Diffusion coefficient = Length x Velocity example: molecular diffusion = mean free path x sound speed

12. 5/5/2008 Arcachon Taylor diffusion Taylor (1921) relates diffusion to the r.m.s distance run by tracer particles in a given time t Taylor (1921) relates diffusion to the r.m.s distance run by tracer particles in a given time t S=auto-correlation function in time of velocity v Small times Small times (ballistic) : (ballistic) : Large times Large times (Brownian) : (Brownian) : T is the integral time scale of turbulence. Of the order of the large eddies turnover time scale. t S(t) T 0 1

13. 5/5/2008 Arcachon Taylor diffusion Effective turbulent diffusion coefficient : D ~ large eddy size x r.m.s velocity D ~ large eddy size x r.m.s velocity => Turbulent mixing is VERY efficient. Observations show r.m.s velocity is of the order Observations show r.m.s velocity is of the order of the sound speed in the ISM The effective correlation length depends on more subtle properties of the interstellar turbulence... The effective correlation length depends on more subtle properties of the interstellar turbulence... Note: At a specified scale L, the effective diffusion coefficient is D = L times v r.m.s at that scale.

14. 5/5/2008 Arcachon Intermittency Intermittent turbulence Kolmogorov (1941)

15. 5/5/2008 Arcachon Intermittency Intermittent turbulence Kolmogorov (1941) Homogeneous properties of turbulence Homogeneous properties of turbulence Gaussian statistics Gaussian statistics Intermittency : Intermittency : Local variation of turbulent properties Local variation of turbulent properties Coherent structures Coherent structures Exponential wings in some probability distributions (Levy flights : Martinell et al 2006)‏ Exponential wings in some probability distributions (Levy flights : Martinell et al 2006)‏

16. 5/5/2008 Arcachon Turbulent Mixing Summary Is a two-stages process : Is a two-stages process :  Transfer of patterns from large to small scales  Molecular diffusion smoothes the small scales Transfer proceeds in a large scale eddy turnover time : mixing is VERY EFFICIENT. Transfer proceeds in a large scale eddy turnover time : mixing is VERY EFFICIENT. BUT: BUT:  Diffusion model valid only for large time scales  At small scales, diffusion coefficient is not homogeneous => diffusion model valid only for large length scales  It applies to averaged quantities : the distribution function is unknown.

17. 5/5/2008 Arcachon Turbulent mixing in the ISM Prestellar cores: Boland & de Jong (1982), Xie et al (1994,1995) => Limits depletion on grain Prestellar cores: Boland & de Jong (1982), Xie et al (1994,1995) => Limits depletion on grain HIM/WIM interfaces : Begelman & Fabian (1990), Esquivel et al (2006) => Changes atomic line ratios HIM/WIM interfaces : Begelman & Fabian (1990), Esquivel et al (2006) => Changes atomic line ratios Protostellar disks : Ilgner & Nelson (2006) => CN Protostellar disks : Ilgner & Nelson (2006) => CN Diffuse clouds interface : Duley et al (1992), Lesaffre et al (2007) => H2 mixed with hot gas produces new molecules (CH+). Warm H2. Diffuse clouds interface : Duley et al (1992), Lesaffre et al (2007) => H2 mixed with hot gas produces new molecules (CH+). Warm H2. PDR with turbulent mixing ? Jets, ablation flows ? PDR with turbulent mixing ? Jets, ablation flows ? => Many potential observational signatures IRAS

18. 5/5/2008 Arcachon Turbulent mixing in the ISM C + + H 2 -> CH + + H C + + H 2 -> CH + + H Activation temperature : 4640 K ; IRAS L=10 -3 pc L=10 -2 pc L=mean free path CH + Lesaffre, Gerin, Hennebelle (2007) First, a 1D diffuse cold/hot interface is generated thanks to a self-absorbed interstellar radiation field (from the right) Then, run full hydro+chemistry simulations with an effective diffusion coefficient : D = L x sound speed

19. 5/5/2008 Arcachon (A 2) Reaction rate enhancement

20. 5/5/2008 Arcachon Example 1 : Wrinkling of a combustion flame Zingale et al (2005)‏ : Zingale et al (2005)‏ : Follow a flame front in a gravitational field Follow a flame front in a gravitational field Corrugation instability (Landau-Darrieus)‏ Corrugation instability (Landau-Darrieus)‏ Rayleigh-Taylor fingers Rayleigh-Taylor fingers => The reactive surface is considerably enhanced. The front moves faster. (See also Roepke et al 2002)

21. 5/5/2008 Arcachon Example 1 : Wrinkling of a combustion flame Zingale et al (2005)‏ : Zingale et al (2005)‏ : Follow a flame front in a gravitational field Follow a flame front in a gravitational field Corrugation instability (Landau-Darrieus)‏ Corrugation instability (Landau-Darrieus)‏ Rayleigh-Taylor fingers Rayleigh-Taylor fingers => The reactive surface is considerably enhanced. The front moves faster. (See also Roepke et al 2002)

22. 5/5/2008 Arcachon Example 1 : Wrinkling of a combustion flame Zingale et al (2005)‏ : Zingale et al (2005)‏ : Follow a flame front in a gravitational field Follow a flame front in a gravitational field Corrugation instability (Landau-Darrieus)‏ Corrugation instability (Landau-Darrieus)‏ Rayleigh-Taylor fingers Rayleigh-Taylor fingers => The reactive surface is considerably enhanced. The front moves faster. (See also Roepke et al 2002)

23. 5/5/2008 Arcachon Example 1 : Wrinkling of a combustion flame Zingale et al (2005)‏ : Zingale et al (2005)‏ : Follow a flame front in a gravitational field Follow a flame front in a gravitational field Corrugation instability (Landau-Darrieus)‏ Corrugation instability (Landau-Darrieus)‏ Rayleigh-Taylor fingers Rayleigh-Taylor fingers => The reactive surface is considerably enhanced. The front moves faster. (See also Roepke et al 2002)

24. 5/5/2008 Arcachon Example 2 : Turbulence dissipation Intermittent turbulence Dissipation of momentum at the end of the cascade provides heat => reaction rate enhancement Dissipation of momentum at the end of the cascade provides heat => reaction rate enhancement Due to intermittence, this heating is localised Due to intermittence, this heating is localised (see B. Godard's presentation) (see B. Godard's presentation)

25. 5/5/2008 Arcachon Chemical enhancement Summary Stretching and folding AND chemical instabilities increase the contact surface Stretching and folding AND chemical instabilities increase the contact surface Turbulent mixing of temperature Turbulent mixing of temperature Momentum dissipation provides local heating (cf B. Godard's presentaion)‏ Momentum dissipation provides local heating (cf B. Godard's presentaion)‏ Velocity drifts in MHD turbulence (not necessarily in shocks)‏ Velocity drifts in MHD turbulence (not necessarily in shocks)‏

26. 5/5/2008 Arcachon A promising technique: Local Large Eddy Simulations Car engines and SNe Ia Car engines and SNe Ia A local sub-grid model accounts for A local sub-grid model accounts for  turbulent diffusion  buoyancy  energy transfer to small scales  compressive dissipation  viscous dissipation SNe Ia Schmidt, Niemeyer, Hillebrandt (2006) Schmidt, Niemeyer, Hillebrandt (2006)‏ Direct Simulation Sub-grid model : resolution/7 isoenergy surfaces

27. 5/5/2008 Arcachon Principle of Large Eddy Simulations (LES)‏ Wave number Amplitude Kinetic energy Injection Viscous Dissipation Grid scale Sub-Gridmodel

28. 5/5/2008 Arcachon (B) Chemistry => Turbulence: (B1) Indirect effect : Chemical influence on the heating-cooling function

29. 5/5/2008 Arcachon Thermal/Chemical Instabilities Exothermic reactions can drive turbulence (example: convection in stars, fire places)‏ Exothermic reactions can drive turbulence (example: convection in stars, fire places)‏ Flame-front instabilities can drive turbulence Flame-front instabilities can drive turbulence Condensation front instabilities can drive turbulence (Hiroshi & Inutsuka 2006)‏ Condensation front instabilities can drive turbulence (Hiroshi & Inutsuka 2006)‏ Thermal instability in the ISM is driven by changes in the chemical composition (Audit & Hennebelle 2004)‏ Thermal instability in the ISM is driven by changes in the chemical composition (Audit & Hennebelle 2004)‏ Runaway formation of H2 ? Runaway formation of H2 ?

30. 5/5/2008 Arcachon (B) Chemistry => Turbulence: (B2) Direct effect : Influence of the ionisation degree on MRI turbulence

31. 5/5/2008 Arcachon MRI turbulence in shearing boxes Effect of the resistivity The electron fraction determines the efficiency of the gas/B field coupling The electron fraction determines the efficiency of the gas/B field coupling Dead zones: Magnetic field and turbulent momentum transport vanish at low magnetic Reynolds number Dead zones: Magnetic field and turbulent momentum transport vanish at low magnetic Reynolds number Active zones: magnetised, efficient transport, Ohmic heating Active zones: magnetised, efficient transport, Ohmic heating Sano, Stone (2002) =log Re' M Active Dead

32. 5/5/2008 Arcachon MRI turbulence in shearing boxes Effect of the Prandtl number Fromang, Papaloizou, Lesur, Heinemann (2007) =log Re' M Active Dead Sustained Turbulence ? Resistivity / Viscosity Prandtl number = Resistivity / Viscosity

33. 5/5/2008 Arcachon (C) Chemistry Turbulence: The example of magnetised accretion disks

34. 5/5/2008 Arcachon Stratified shearing box simulations R=1 AU (fast recombination rate) Time Height Momentum Transport Dead zone (low ionisation degree) Active zone Ilgner & Nelson (2008) 3DMHD + simple ionisation / recombination network Ionising field (X rays) mimicked by a z-dependent ionising rate mimicked by a z-dependent ionising rate

35. 5/5/2008 Arcachon Stratified shearing box simulations R=3 AU (moderate recombination) Time Height Momentum Transport Dead zone (low ionisation degree) Active zone Ilgner & Nelson (2008) Ionising field (X rays) mimicked by a z-dependent ionising rate mimicked by a z-dependent ionising rate Ilgner & Nelson (2008) Competition between recombination and mixing of e -

36. 5/5/2008 Arcachon Stratified shearing box simulations R=7 AU (slow recombination rate) Time Height Momentum Transport No more dead zone... No more dead zone... Ilgner & Nelson (2008) Turbulence self-sustained by mixing and chemistry Ionising field (X rays)

37. 5/5/2008 Arcachon Future Work Stratified shearing boxes with accurate temperature Stratified shearing boxes with accurate temperature Detailed post-processed chemistry in tracer particles Detailed post-processed chemistry in tracer particles  molecular chemistry  ionisation degree  grain surface chemistry and charging  chondrules processing Inter particle diffusion Inter particle diffusion Zeus3d + N-body tracker Zeus3d + N-body tracker Stratified MHD simulations Stratified MHD simulations 4H x 4H x 8H box 4H x 4H x 8H box Thermal diffusion : Thermal diffusion : opacity law K ~ T 2

38. 5/5/2008 Arcachon Summary Turbulent diffusion is a complex phenomenon Turbulent diffusion is a complex phenomenon  which transfers patterns to small scales  increases the contact surface bewteen reactants  is inhomogeneous on short times and lengths It changes the paths of reaction It changes the paths of reaction => potential observational signatures Chemistry is a central actor of dynamics Chemistry is a central actor of dynamics Numerical methods could prove useful Numerical methods could prove useful  local L.E.S.  tracer post processing in D.N.S.

39. 5/5/2008 Arcachon 2D incompressible mixing & reaction Arratia & Gollub (2005) Straining field Product Concentration evolution (Time -->) A + B --> C

40. 5/5/2008 Arcachon Universal time dependence of the mean product concentration (as predicted by Karolyi & Tel 2005) 2D incompressible mixing & reaction Normalised time