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Turbulence & dynamos

Turbulence & dynamos. Axel Brandenburg (Nordita/Stockholm). Kemel+12. K äpylä +12. Ilonidis+11. Warnecke+11. Brandenburg+11. Pencil code. Started in Sept. 2001 with Wolfgang Dobler High order (6 th order in space, 3 rd order in time) Cache & memory efficient

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Turbulence & dynamos

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  1. Turbulence & dynamos Axel Brandenburg (Nordita/Stockholm) Kemel+12 Käpylä+12 Ilonidis+11 Warnecke+11 Brandenburg+11

  2. Pencilcode • Started in Sept. 2001 with Wolfgang Dobler • High order (6th order in space, 3rd order in time) • Cache & memory efficient • MPI, can run PacxMPI (across countries!) • Maintained/developed by ~80 people (SVN) • Automatic validation (over night or any time) • 0.0013 ms/pt/step at 10243 , 2048 procs • http://pencil-code.googlecode.com • Isotropic turbulence • MHD, passive scl, CR • Stratified layers • Convection, radiation • Shearing box • MRI, dust, interstellar • Self-gravity • Sphere embedded in box • Fully convective stars • geodynamo • Other applications • Chemistry, combustion • Spherical coordinates

  3. Scaling properties Wlad Lyra on Kraken Babkovskaia et al (2011)

  4. Evolution of code size User meetings: 2005 Copenhagen 2006 Copenhagen 2007 Stockholm 2008 Leiden 2009 Heidelberg 2010 New York 2011 Toulouse 2012 Helsinki 2013 Lund 2014 Gottingen

  5. Increase of revision number

  6. Pencil h-index Red line gives the diagonal to see the crossing

  7. Automatic validation tests

  8. Increase in # of auto tests

  9. Credit

  10. Hyperviscous, Smagorinsky, normal height of bottleneck increased Haugen & Brandenburg (PRE, astro-ph/0402301) onset of bottleneck at same position Inertial range unaffected by artificial diffusion

  11. Flow of energy

  12. Helicity, as pseudoscalar • Change of sign about the equator • Magnetic fields can also be chiral • Helicity important for solar dynamo  zero at the equator? Not necissarily! (Chatterjee et al 2011) north equator south

  13. Parker’s (1955) dynamo wave New loop Differential rotation (faster inside) Cyclonic convection; Buoyant flux tubes Equatorward migration  a-effect

  14. Major steps in dynamo theory • Larmor 19 suggests a dynamo might work • Cowling 33 anti-dynamo theorem • (2-D, axisymmetry), Larmor 34, Cowling 45 • Parker 55 cyclonic convection, waves • Herzenberg 58 first existence proof of dynamo • Steenbeck, Krause, Rädler 66  a effect • Pouquet, Frisch, Leorat 76  magn a effect

  15. aW dynamo: self-excited oscillator 2 relevant componets x y z eigenvalues oscillatory traveling wave exponential growth

  16. Get the same from turbulence simulation kf/k1=1 kf/k1=3

  17. Turbulent dynamos • 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

  18. 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

  19. Nowadays simples test examplethe Roberts (1970) flow • Kinematic 2-flow • But B-field is 2-D (Cowling anti-dynamo theorem)

  20. Dynamos withEuler Potentials • B = gradax gradb • A = a gradb, so A.B=0 • Take non-helical flows • Agreement for early t • For smooth fields, not for d-correlated initial fields • Exponential growth (A) • Algebraic decay (EP) Brandenburg (2010, MNRAS 401, 347)

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

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

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

  24. Another twist: bi-helical fieldssignature of a effect Magnetic helicity spectrum

  25. Bi-helical magnetic field Magnetic helicity spectrum Pouquet, Frisch, & Leorat (1976)

  26. Magnetic helicity measures linkage of flux Therefore the unit is Maxwell squared

  27. Expansion in eigenfunctions of curl Fourier space Expand into longitudinal and polarized contributions so spectra

  28. Realizability condition Spectra Realizability condition Shell-integrated spectra Energies in positively and negatively polarized waves (Obtained just from the spectra)

  29. Inverse cascade of magnetic helicity argument due to Frisch et al. (1975) and Initial components fully helical: and  k is forced to the left

  30. Decay of helical field: inverse cascade Important applications to early Universe: EW & QCD phase transitions • Inverse cascade on large scales • Forward cascade on small scales Initial slope E~k4 Christensson et al. (2001, PRE 64, 056405)

  31. Nonhelical & helical turbulence Dynamos in both cases: non-magnetic solutions do not exist …when conductivity high enough With helicity: gradual build-up of large-scale field

  32. Inverse cascade

  33. Non-helical vs helical • Similar at intermediate k • k3/2 in both cases • Super-equip. • Difference: LS field at k=1 • Here PrM=1

  34. Low PrMissue for small-scale dynamos • 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

  35. Re-appearence at low PrM Gap between 0.05 and 0.2 ? Is just because of bottleneck effect Iskakov et al (2007) Haugen & Brandenburg (2006)

  36. Nonlinear small-scale dynamo also works Does survive for PrM < 0.1 Brandenburg (2011, ApJ 741:92) Related to bottleneck in kinematic regime Velocity roughest there. Magnetic peak further left for smaller PrM

  37. Helical dynamos at low PrMalways work • Energy dissipation via Joule • Viscous dissipation weak • Can increase Re substantially! Brandenburg (2009, ApJ)

  38. Solar dynamo • Motions from • Convection instability • Magnetorotational inst. • Supernova forcing • Dynamo instability • Stretch-twist-fold • Turbulent dynamo

  39. White light image ofyesterday Tips of icebergs: Magnetic flux concentrations in magnetogram!

  40. Cycic Maunder mininum: 10Be record

  41. Brun, Brown, Browning, Miesch, Toomre

  42. Ghizaru, Charbonneau, Racine, … • Cycle now common! • Activity from bottom of CZ • but at high latitudes

  43. Käpylä, Mantere, Brandenburg (2012) Pencilcode

  44. Dynamo wave from simulations Kapyla et al (2012)

  45. Importance of solar activity

  46. Historic space weather events • 21 Dec 1806 Humbold: erratic compass var. • 1 Sep 1859 Carrington flare: telegraphs disr.

  47. More recent weather events • 2 Aug 1972: life-threatening particle expos. • Apollo 16: 16 Apr 1972 • Apollo 17: 7 Dec 1972 • 13 Mar 1989: Quebec blackout • Aug 1989: halt of all trading in Toronto • Story in New Scientist of 9 Sept • Now routinely rerouting transpolar routes

  48. Need a holistic approach Warnecke, Brandenburg, Mitra (2011, A&A, 534, A11)

  49. Next level in mean-field concept a effect, turbulent diffusivity, Yoshizawa effect, etc Turbulent viscosity and other effects in momentum equation

  50. Negative effective magnetic pressure instability • Gas+turb. press equil. • B increases • Turb. press. Decreases • Net effect?

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