1. Diffusion and accurate hydrodynamics in SPH and grid codes James Wadsley (McMaster) Tom Quinn (Washington), Fabio Governato (Washington), Hugh Couchman (McMaster) Disks 2012, Heidelberg

2. Test Agertz et al 2007 Code comparison paper from the proto AstroSim conference (2004) Gasoline (Wadsley, Stadel & Quinn 2004) Gadget (Springel) FLASH ENZO ART

3. Test Agertz et al 2007 Code comparison paper from the proto AstroSim conference (2004) PPM Piecewise Parabolic Method (Collela & Woodward 1987) SPH Smoothed Particle Hydrodynamics (Monaghan 1992, Springel & Hernquist 2002)

4. Basic Result: SPH blobs don’t break up Quantitative measure: Fraction of cloud remaining above 64% of initial density As cloud fragments – size R halves every Kelvin-Helmholtz time τ KH ~ R/v -Effective Kelvin-Helmholtz time halves repeatedly until cloud catches up with flow (v  0) SPH: Kelvin-Helmholtz time static

5. Basic Result: SPH blobs don’t break up Immediate SPH issue: Surface Tension present in arithmetic sum Pressure force (e.g. Monaghan 1992, Gasoline, Gadget, …) Suppresses Kelvin Helmholtz instabilities  Issue first identified by Ritchie & Thomas (2001)

6. SPH Kelvin Helmholtz fixed Ritchie & Thomas (2001) – smooth pressure not density and Geometric Density Average in Force: remove surface tension (pressure spike at density jump) Price (2008) -- smear density jumps Read, Hayfield & Agertz (2010), Read & Hayfield (2011), Abel (2011), Murante et al (2011) … modified SPH Solutions typically expensive (more accurate)

7. Key to alleviating SPH surface tension: Geometric Density Average in Force (GDForce): Morris (1996), Monaghan (1992) Ritchie and Thomas (2001), Can be derived from a Lagrangian: Monaghan & Rafiee (2012) see also Abel (2011) Standard Force Geometric Density Force Merging Cluster Test

8. Blob Test in Entropy (T 3/2 /ρ) Hi-Res ENZO Hi-Res Standard SPH

9. Blob Test in Entropy (T 3/2 /ρ) Hi-Res ENZO Hi-Res GDForce SPH

10. Blob Test in Entropy (T 3/2 /ρ) ENZO GDForce SPH Standard SPH t = 1.25 τ KH t = 3.75 τ KH t = 2.5 τ KH

11. Blob Test in Entropy (T 3/2 /ρ) ENZO GDForce SPH Standard SPH t = 1.25 τ KH t = 3.75 τ KH t = 2.5 τ KH Low Entropy Blobs Indestructible!

12. The second issue: Entropy mixing Cluster Comparison (Frenk et al 1999) Grid codes have entropy cores, SPH codes don’t (because they don’t mix) ENZO SPH Wadsley, Veeravalli & Couchman (2008)

13. How to get entropy cores? Shocks (while c s <v) Mix hot & cold cluster gas SPH can’t: Eulerian codes can (accidentally):

14. Subgrid Turbulent Mixing Fluid elements on a fixed (resolved) physical scale do exchange energy/entropy due to unresolved (turbulent) motions Turbulent diffusive heat flux

15. Ways to model turbulent diffusion: Lowest-order turbulent diffusion model:  Turb has units of velocity x length Smagorinksy model (1963): Assumes Prandtl number ~ 1 S ij = strain tensor of resolved flow, l S Smagorinsky length Incompressible grid models set l s 2 ~ 0.02  x 2 (Lilly 1967) For SPH we can try  Turb = C h 2 S C ~ 0.1 Wadsley, Veeravalli & Couchman (2008) Shen, Wadsley & Stinson (2010)

16. Ways to model turbulent diffusion: Lowest-order turbulent diffusion model: Wadsley, Veeravalli & Couchman (2008) Shen, Wadsley & Stinson (2010) Cluster Entropy Cores easily obtained in SPH with thermal diffusion included – solved in 2008

17. Bottom Line on Diffusion & SPH Physical diffusion in needed in all simulations e.g. Metals should mix in galactic outflows ( Shen, Wadsley & Stinson 2010) With thermal diffusion in SPH: Get entropy cores in Galaxy Clusters Necessary to model entropy of mixing (Springel 2010) Better Kelvin-Helmholtz Better Blob results … (but not quite there) … Need Geometric Density Force AND Diffusion

18. Blob Test in Entropy (T 3/2 /ρ) Hi-Res ENZO GDForce + Turbulent Diffusion SPH

19. Blob Test in Entropy (T 3/2 /ρ) ENZO GDForce + Turbulent Diffusion SPH Standard SPH t = 1.25 τ KH t = 3.75 τ KH t = 2.5 τ KH

20. Is grid (PPM) the right answer? No: Numerical Diffusion Approximate, e.g. is not Galilean Invariant ENZO Moving flow t = 1.25 τ KH t = 3.75 τ KH t = 2.5 τ KH ENZO 1/2 – 1/2 ENZO Moving blob

21. Blob’s falling apart… t / τ KH GDForce + Diffusion Coefficient 0.1,0.03,0.01 Standard SPH Standard SPH + Diffusion GDForce SPH

22. Blob’s falling apart… (Log) t / τ KH GDForce + Diffusion Coefficient 0.1,0.03,0.01 Standard SPH Standard SPH + Diffusion GDForce SPH

23. Blob’s falling apart… (Log) t / τ KH GDForce + Diffusion Coefficient 0.1,0.03,0.01 Standard SPH Standard SPH + Diffusion GDForce SPH ENZO (1/2 each) ENZO Moving blob ENZO Moving flow (Agertz et al.)

24. Diffusion coefficient… Smash galaxy clusters together: Clean version of Frenk et al. ( 1999) with no substructure Results look the same but ENZO mixes more in the core (see left) than SPH+GDForce+Diffusion Gasoline diffusion coefficients: Also: Mass metallicity + IGM Metals Inner 100 kpc ENZO @ 4.5 Gyr GASOLINE @ 4.5 Gyr ENZO 1.0 0.1, 0.03, 0.01, no Diffusion 500 pc Gasoline

25. Diffusion coefficient… Smash smooth galaxy clusters together: Clean version of Frenk et al. ( 1999) with no substructure Results look the same but ENZO mixes more in the core (see left) than SPH+GDForce+Diffusion Gasoline diffusion coefficients: Also: Mass metallicity + IGM Metals Inner 100 kpc ENZO @ 4.5 Gyr GASOLINE @ 4.5 Gyr ENZO 1.0 0.1, 0.03, 0.01, no Diffusion 500 pc Gasoline

26. Galaxy Formation: Kaufmann et al (2008) blobs no more Toy Galaxy Model (cf. Kaufmann et al.), 20 kpc wide edge on view Conclusion: Blobs product of SPH surface tension effects (see also: Joung et al 2011) Standard SPH GDForce+Diffusion SPH

27. Galaxy Formation: Smoother disks & spiral structure Toy Galaxy Model (cf. Kaufmann et al.), ~ 12 kpc wide face on view Smoother Structure Standard SPH GDForce+Diffusion SPH

28. Galaxy Formation 10 11 Solar Mass Galaxy, z=0.6 (latest output) Initial indications: Cold flow mass only ~ 2% different Increased star formation. Need to understand how marginally resolved cooling works Standard SPH GDForce+Diffusion

29. Conclusions Geometric Density Average Force alleviates surface tension affects in SPH Well-characterized diffusion treatments necessary for all codes (SPH & Grid) Careful attention to resolution scale behaviour important (e.g. Instabilities: KH, Jeans, Thermal/Cooling…) Gasoline public release: including corrected force (blob free!), Stinson et al. star formation and feedback, cooling, planetesimal collisions – coming to google code this summer…

30. Sedov Test GDForce