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Numerical simulations of astrophysical dynamos

Numerical simulations of astrophysical dynamos. 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. Axel Brandenburg ( Nordita, Stockholm ). Struggle for the dynamo.

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Numerical simulations of astrophysical dynamos

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  1. Numerical simulations of astrophysical dynamos • 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 Axel Brandenburg (Nordita, Stockholm)

  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 (l~Rm-1) • Steenbeck, Krause, Rädler (1966): awdynamo • Many papers on this since 1970 • Kazantsev (1968): small-scale dynamo • Essentially unnoticed, simulations 1981, 2000-now

  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 PrM=n/h

  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 curl2nB • ampitude by (k/kf)2n-1 • Euler potentials with artificial diffusion • Da/Dt=hDa, Db/Dt=hDb, B = grada xgradb

  5. Dynamos withEuler Potentials • B = gradax gradb • A = a gradb, 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. Reasons for disagreement • because dynamo field is helical? • because field is three-dimensional? • none of the two: it is because h is finite

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

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

  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 a(x,y,z,t) and b(x,y,z,t)

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

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

  12. Helical dynamo saturation with hyperdiffusivity for ordinary hyperdiffusion ratio 53=125 instead of 5 PRL 88, 055003

  13. Test-field results for a and htkinematic: independent of Rm (2…200) Sur et al. (2008, MNRAS)

  14. Rm dependence for B~Beq • l is small  consistency • a1 and a2 tend to cancel • making a small • h2 small

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

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

  17. Connection with MRI turbulence 5123 w/o hypervisc. Dt = 60 = 2 orbits • No large scale field • Too short? • No stratification?

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

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

  20. Small-scale vs large-scale dynamo

  21. Low PrM dynamoswith helicity do work • Energy dissipation via Joule • Viscous dissipation weak • Can increase Re substantially!

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

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

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

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

  26. LS from Fluctuations of aij and hij Incoherent a effect (Vishniac & Brandenburg 1997, Sokoloff 1997, Silantev 2000, Proctor 2007)

  27. Conclusions • Cycles w/ stratification • Test field method robust • Even when small scale dynamo • a a0 and h ht0 • Rotation and shear: h*ij • WxJ not (yet?) excited • Incoherent a works

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