1. Numerical simulations of astrophysical dynamos Axel Brandenburg (Nordita, Stockholm) Dynamos: numerical issues Alpha dynamos do exist: linear and nonlinear Constrained by magnetic helicity: need fluxes How high do we need to go with Rm? MRI and low PrM issue

2. 2 Struggle for the dynamo Larmor (1919): first qualitative ideas Cowling (1933): no  antidynamo theorem Larmor (1934): vehement response –2-D not mentioned Parker (1955): cyclonic events, dynamo waves Herzenberg (1958): first dynamo –2 small spinning spheres, slow dynamo ( ~R m -1 ) Steenbeck, Krause, Rädler (1966):  dynamo –Many papers on this since 1970 Kazantsev (1968): small-scale dynamo –Essentially unnoticed, simulations 1981, 2000-now

3. 3 Mile stones in dynamo research 1970ies: mean-field models of Sun/galaxies 1980ies: direct simulations Gilman/Glatzmaier: poleward migration 1990ies: compressible simulations, MRI –Magnetic buoyancy overwhelmed by pumping –Successful geodynamo simulations 2000- magnetic helicity, catastr. quenching –Dynamos and MRI at low Pr M = n/h

4. 4 Easy to simulate? Yes, but it can also go wrong 2 examples: manipulation with diffusion Large-scale dynamo in periodic box –With hyper-diffusion curl 2n B –ampitude by (k/k f ) 2n-1 Euler potentials with artificial diffusion –D  /Dt= hD , D  /Dt= hDb, B = grad a x grad b

5. Dynamos with Euler Potentials B = grad a x grad b A = a grad b, so A.B=0 Here: Robert flow –Details MNRAS 401, 347 Agreement for t<8 –For smooth fields, not for d -correlated initial fields Exponential growth (A) Algebraic decay (EP)

6. 6 Reasons for disagreement because dynamo field is helical? because field is three-dimensional? none of the two: it is because  is finite

7. 7 Is this artificial diffusion kosher? Make h very small, it is artificial anyway, Surely, the R term cannot matter then?

8. 8 Problem already in 2-D nonhelical Works only when  and  are not functions of the same coordinates Alternative possible in 2-D Remember:

9. 9 Method of choice? No, thanks It’s not because of helicity (cf. nonhel dyn) Not because of 3-D: cf. 2-D decay problem It’s really because  (x,y,z,t) and  (x,y,z,t)

10. 10 Other good examples of dynamos Helical turbulence (B y ) Helical shear flow turb. Convection with shear Magneto-rotational Inst. Käpyla et al (2008)

11. 11 One big flaw: slow saturation (explained by magnetic helicity conservation) molecular value!!

12. 12 Helical dynamo saturation with hyperdiffusivity for ordinary hyperdiffusion ratio 5 3 =125 instead of 5 PRL 88, 055003

13. 13 Test-field results for  and  t kinematic: independent of Rm (2…200) Sur et al. (2008, MNRAS)

14. 14 R m dependence for B~B eq (i) l is small  consistency (ii) a 1 and a 2 tend to cancel (iii) making a small (iv) h 2 small

15. 15 Boundaries instead of periodic Brandenburg, Cancelaresi, Chatterjee, 2009, MNRAS

16. 16 Boundaries instead of periodic Hubbard & Brandenburg, 2010, GAFD

17. 17 Connection with MRI turbulence 512 3 w/o hypervisc.  t = 60 = 2 orbits No large scale field (i)Too short? (ii)No stratification?

18. 18 Low Pr M issue Small-scale dynamo: R m.crit =35-70 for Pr M =1 (Novikov, Ruzmaikin, Sokoloff 1983) Leorat et al (1981): independent of PrM (EDQNM) Rogachevskii & Kleeorin (1997): R m,crit =412 Boldyrev & Cattaneo (2004): relation to roughness Ponty et al.: (2005): levels off at Pr M =0.2

19. 19 Re-appearence at low Pr M Iskakov et al (2005) Gap between 0.05 and 0.2 ?

20. 20 Small-scale vs large-scale dynamo

21. 21 Low Pr M dynamos with helicity do work Energy dissipation via Joule Viscous dissipation weak Can increase Re substantially!

22. 22 Comparison with stratified runs Brandenburg et al. (1995) Dynamo waves Versus Buoyant escape

23. 23 Phase relation in dynamos Mean-field equations: Solution:

24. 24 Unstratified: also LS fields? Lesur & Ogilvie (2008) Brandenurg et al. (2008)

25. 25 Low Pr M issue in unstratified MRI Käpylä & Korpi (2010): vertical field condition

26. 26 LS from Fluctuations of  ij and  ij Incoherent  effect (Vishniac & Brandenburg 1997, Sokoloff 1997, Silantev 2000, Proctor 2007)

27. 27 Conclusions Cycles w/ stratification Test field method robust –Even when small scale dynamo –    0 and    t0 Rotation and shear:  * ij –WxJ not (yet?) excited –Incoherent  works