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

Pencil code Started in Sept 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 ~80 people (SVN) Automatic validation (over night or any time)  s/pt/step at , 2048 procs 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

Scaling properties 3 Wlad Lyra on KrakenBabkovskaia 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

Pencil h-index 6 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 Inertial range unaffected by artificial diffusion Haugen & Brandenburg (PRE, astro-ph/ ) height of bottleneck increased onset of bottleneck at same position

11 Flow of energy

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

13 Parker’s (1955) dynamo wave Differential rotation (faster inside) Cyclonic convection; Buoyant flux tubes Equatorward migration New loop    - 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   effect Pouquet, Frisch, Leorat 76  magn  effect

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

16 Get the same from turbulence simulation k f /k 1 =1 k f /k 1 =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 magnetic helicity, catastr. quenching –Dynamos and MRI at low Pr M = 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 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

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

Dynamos with Euler Potentials B = grad a x grad b A = a grad b, 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 (B y ) 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 5 3 =125 instead of 5 PRL 88,

24 Another twist: bi-helical fields signature of  effect Magnetic helicity spectrum

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

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

30 Decay of helical field: inverse cascade Inverse cascade on large scales Forward cascade on small scales Christensson et al. (2001, PRE 64, ) Initial slope E~k 4 Important applications to early Universe: EW & QCD phase transitions

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 k 3/2 in both cases Super-equip. Difference: LS field at k=1 Here Pr M =1

34 Low Pr M issue for small-scale dynamos Small-scale dynamo: R m.crit =35-70 for Pr M =1 (Novikov, Ruzmaikin, Sokoloff 1983) Leorat et al (1981): independent of Pr M (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

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

Nonlinear small-scale dynamo also works 36 Does survive for Pr M < 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 Pr M always work Energy dissipation via Joule Viscous dissipation weak Can increase Re substantially! Brandenburg (2009, ApJ)

38 Solar dynamo a)Motions from i.Convection instability ii.Magnetorotational inst. iii.Supernova forcing b)Dynamo instability i.Stretch-twist-fold ii.Turbulent dynamo

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

40 Cycic Maunder mininum: 10 Be record

Brun, Brown, Browning, Miesch, Toomre 41

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

Pencil code Käpylä, Mantere, Brandenburg (2012)

44 Dynamo wave from simulations Kapyla et al (2012)

45 Importance of solar activity

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

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

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?

51 Setup 3-D box, size (2  ) 3, isothermal MHD Random, nonhelical forcing at k f /k 1 =5, 15 or 30 Stratified in z,  ~exp(-z/H), H=1,  =535 Periodic in x and y stress-free, perfect conductor in z Weak imposed field B 0 in y Run for long times: what happens? Turnover time  to =(u rms k f ) -1, turb diff  td =(  t k 1 2 ) -1 Is longer by factor 3(k f /k 1 ) 2 = = 675 Average B y over y and  t=80  to

52 Larger scale separation: 30 instead of 15

53 Basic mechanism Anelastic: descending structure  compression B amplifies Growth rate

54 Alternative sunspot origins Theories for shallow spots: (i) Collapse by suppression of turbulent heat flux (ii) Negative pressure effects from vs B i B j Kosovichev et al. (2000)

Sunspot decay 55

Turbulent sunspot origins?

Spruit paper 57

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

59 Much stronger with vertical fields Gas+turb. press equil. B increases Turb. press. Decreases Net effect?

Self-assembly of a magnetic spot Minimalistic model 2 ingredients: –Stratification & turbulence Extensions –Coupled to dynamo –Compete with rotation –Radiation/ionization 60

512 3 vs resolution Rm/Re dependence? Here 40/80 and 95/190 Originally 18/36. 61

Surface-filling magnetic activity Rm/Re dependence? Here 40/80 and 95/190 Originally 18/36. 62

Imposed vs. self-assembly Appearance of sunspot when coupled to radiation Can be result of self- assembly when ~1000 G field below surface 63 Stein & Nordlund (2012) Rempel et al. (2009)

Conclusions Turbulence can possess surprising effects –Large scale dynamos –Magnetic flux concentrations Those can be predicted and understood with mean-field MHD 64