1. Turbulence research at Nordita 1.Bottleneck effect 2.Magnetic fields (active vector) 3.Passive scalar diffusion Haugen & Brandenburg (2006, Phys. Fl. 18, 075106) Brandenburg & Subramanian (2005, Phys. Rep., 417, 1) Brandenburg (2001, ApJ 550, 824; and 2005, ApJ 625, 539) Brandenburg et al. (2004, Phys. Fl. 16, 1020)

2. 2 Nordita in Stockholm Nordita in Stockholm

3. 3 Nordita in Stockholm (i)~6 new post-docs Nordic+non-Nordic (ii)~2-3 new assist. profs (iii) ~5 programs/year (iv) visiting professors (1/2-2yr) (v)Nordita-days next August

4. 4 Pencil Code Started in Sept. 2001 with Wolfgang Dobler High order (6 th order in space, 3 rd order in time) Cache & memory efficient MPI, can run PacxMPI (across countries!) Maintained/developed by ~25 people (CVS!) Automatic validation (over night or any time) Max resolution so far 1024 3, 256 procs Isotropic turbulence – –MHD, passive scl, CR Stratified layers – –Convection, radiation Shearing box – –MRI, dust, interstellar Sphere embedded in box – –Fully convective stars – –geodynamo Other applications – –Homochirality – –Spherical coordinates

5. 5 (i) Higher order – less viscosity

6. 6 (ii) High-order temporal schemes Main advantage: low amplitude errors 3 rd order 2 nd order 1 st order 2N-RK3 scheme (Williamson 1980)

7. 7 Cartesian box MHD equations Induction Equation: Magn. Vector potential Momentum and Continuity eqns Viscous force forcing function (eigenfunction of curl)

8. 8 Wallclock time versus processor # nearly linear Scaling 100 Mb/s shows limitations 1 - 10 Gb/s no limitation

9. 9 Wallclock time versus processor # nearly linear Scaling 100 Mb/s shows limitations 1 - 10 Gb/s no limitation

10. 10 Pre-processed data for animations

11. 11 1. Bottleneck: surprise at higher res.

12. 12 Bottleneck effect in hydro at 1024 3 (Porter, Pouquet,& Woodward 1998)

13. 13 She-Jackson spectra

14. 14 Bottleneck effect: 1D vs 3D spectra

15. 15 Relation to ‘laboratory’ 1D spectra Parseval used:

16. 16 Longitudinal spectra

17. 17 Hyperviscous, Smagorinsky, normal Inertial range unaffected by artificial diffusion Haugen & Brandenburg (Phys. Fluids, astro-ph/041266) height of bottleneck increased with hyper onset of bottleneck at same position C S =0.20

18. 18 Hyperdiffusion bottleneck Biskamp & Müller (2000)

19. 19 Structure function exponents agrees with She-Leveque third moment

20. 20 2. Allow for B: small scale dynamo action Pr M =  non-helically forced turbulence Pr M = 

21. 21 Looks like k -3/2 at 1024 3 Still not large enough?! Spectra not on top of each other??  Different from case with imposed field!

22. 22 Integral quantities converged “Only” 30% of energy is magnetic But 70% of dissipation is Ohmic!

23. 23 Maybe no small scale “surface” dynamo? Small Pr M =  : stars and discs around NSs and YSOs Here: non-helically forced turbulence Schekochihin Haugen Brandenburg et al (2005) k When should we think of extrapolating to the sun? Implications for global models (w/strong SS field)

24. 24 Scale separation: inverse cascade No inverse cascade in kinematic regime Decomposition in terms of Chandrasekhar-Kendall-Waleffe functions Position of the peak compatible with LS field: force-free Beltrami

25. 25 3. Passive scalar diffusion

26. 26 System of mean field equations mean concentration flux equation Damped wave equation, wave speed (causality!)

27. 27 MTA - the minimal tau approximation 1) replace triple correlation by quadradatic 2) keep triple correlation 3) instead of now: 4) instead of diffusion eqn: damped wave equation i) any support for this proposal?? ii) what is tau?? (remains to be justified!)

28. 28 Wave equation: consequences >>1 (!) small tau i) i)late time behavior unaffected (ordinary diffusion) ii) ii)early times: ballistic advection (superdiffusive) large tauintermediate tau Illustration of wave-like behavior:

29. 29 Test 1: finite initial flux experiment Initial state: but with black: closure model red: turbulence sim.  direct evidence for oscillatory behavior! Dispersion Relation: Oscillatory for k 1 /k f <3 k f /k 1 St=  uk f 1.5 1.8 5.1 2.4

30. 30 Test 2: imposed mean C gradient >>1 (!) Convergence to St=3 for different Re

31. 31 Conclusions Turbulence increasingly important in astrophysics Bottleneck: asymptotically unimportant In practice: simulations not asymptotic: –hence important Dynamo action: details affected by bottleneck 10 46 Mx 2 /cycle