Bern, MHD, and shear Axel Brandenburg (Nordita, Copenhagen) Collaborators: Nils Erland Haugen (Univ. Trondheim) Wolfgang Dobler (Freiburg  Calgary) Tarek Yousef (Univ. Trondheim) Antony Mee (Univ. Newcastle) Ideal vs non-ideal simulations Pencil code Application to the sun Near-surface shear layer

2 (i) Can we trust ideal hydro? Porter, Pouquet, Woodward (1998, Phys. Fluids, 10, 237) Majority of public astrophysics codes are “inviscid”

3 Direct vs hyper at With hyperdiffusivity Normal diffusivity Biskamp & Müller (2000, Phys Fluids 7, 4889)

4 Ideal hydro: should we be worried? Why this k -1 tail in the power spectrum? –Compressibility? –PPM method –Or is real?? Hyperviscosity destroys entire inertial range? –Can we trust any ideal method? Needed to wait for direct simulations

5 Hyperviscous, Smagorinsky, normal Inertial range unaffected by artificial diffusion Haugen & Brandenburg (PRE 70, , astro-ph/041266) height of bottleneck increased onset of bottleneck at same position

6 Relation to ‘laboratory’ 1D spectra Dobler, et al (2003, PRE 68, )

7 256 processor run at Haugen et al. (2003, ApJ 597, L131)

8 (ii) Helical dynamo saturation with hyperdiffusivity for ordinary hyperdiffusion ratio 5 3 =125 instead of 5 PRL 88,

9 (iii) Small scale dynamo: Pm dependence?? Small Pm=  : stars and discs around NSs and YSOs Here: non-helically forced turbulence Schekochihin Haugen Brandenburg et al (2005) k Cattaneo, Boldyrev

10 (iv) Does compressibility affect the dynamo? Direct simulation,  =5 Direct and shock-capturing simulations,  =1 Shocks sweep up all the field: dynamo harder? -- or artifact of shock diffusion?  Bimodal behavior!

Supersonic shock turblence Gustafsson et al. (2006, A&A, in press)

12 LES conclusions Hydro: LES does a good job, but hi-res important –the bottleneck is physical –hyperviscosity does not affect inertial range Helical MHD: hyperresistivity exaggerates B-field Prandtl number does matter! –LES for B-field difficult or impossible! Fundamental questions  idealized simulations important at this stage!

13 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 ~20 people (CVS!) Automatic validation (over night or any time) Max resolution so far , 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

14 (i) Higher order – less viscosity

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

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

17 Vector potential B=curlA, advantage: divB=0 J=curlB=curl(curlA) =curl2A Not a disadvantage: consider Alfven waves B-formulation A-formulation 2 nd der once is better than 1 st der twice!

18 Comparison of A and B methods

19 Wallclock time versus processor # nearly linear Scaling 100 Mb/s shows limitations Gb/s no limitation

20 Forced LS dynamo with no stratification Open bc critical Helicity losses Shear relevant to the sun azimuthally averaged Actual field: (i) kinematic phase (ii) late phase

21 Current helicity and magn. hel. flux Bao & Zhang (1998), neg. in north, plus in south (also Seehafer 1990) Berger & Ruzmaikin (2000) S N DeVore (2000) (for BR & CME)

22 Helicity fluxes at large and small scales Negative current helicity: net production in northern hemisphere Mx 2 /cycle Brandenburg & Sandin (2004, A&A 427, 13) Helicity fluxes from shear: Vishniac & Cho (2001, ApJ 550, 752) Subramanian & Brandenburg (2004, PRL 93, 20500)

23 Application to the sun: spots rooted at r/R=0.95 Benevolenskaya, Hoeksema, Kosovichev, Scherrer (1999) Pulkkinen & Tuominen (1998)  =  AZ  =(180/  ) (1.5x10 7 ) (2  ) =360 x 0.15 = 54 degrees!

24 Simulations of near-surface shear Unstable layer in 0<z<1 0 o latitude 4x4x1 aspect ratio 512x512x256 Prograde pattern speed, but rather slow (Green & Kosovichev 2006)

25 Simulations of near-surface shear 4x4x1 aspect ratio 512x512x256 0 o lat 15 o lat negative uyuz stress  negative shear

Convection with rotation Inv. Rossby Nr. 2  d/u rms =4

Horizontal flow pattern Stongly retrograde motions Plunge into prograde shock y x

28 Arguments against and in favor? Flux storage Distortions weak Problems solved with meridional circulation Size of active regions Neg surface shear: equatorward migr. Max radial shear in low latitudes Youngest sunspots: 473 nHz Correct phase relation Strong pumping (Thomas et al.) 100 kG hard to explain Tube integrity Single circulation cell Too many flux belts* Max shear at poles* Phase relation* 1.3 yr instead of 11 yr at bot Rapid buoyant loss* Strong distortions* (Hale’s polarity) Long term stability of active regions* No anisotropy of supergranulation in favor against Tachocline dynamosDistributed/near-surface dynamo Brandenburg (2005, ApJ 625, 539)

Is magnetic buoyancy a problem? Stratified dynamo simulation in 1990 Expected strong buoyancy losses, but no: downward pumping Tobias et al. (2001)

Is magnetic buoyancy a problem? Stratified dynamo simulation in 1990 Expected strong buoyancy losses, but no: downward pumping Brandenburg et al. (1996)

Is magnetic buoyancy a problem? Stratified dynamo simulation in 1990 Expected strong buoyancy losses, but no: downward pumping Tobias et al. (2001)

32 Conclusions Shearflow turbulence: likely to produce LS field –even w/o stratification (WxJ effect, similar to Rädler’s  xJ effect) Stratification: can lead to  effect –modify WxJ effect –but also instability of its own SS dynamo not obvious at small Pm Application to the sun? –distributed dynamo  can produce bipolar regions –  perhaps not so important? –solution to quenching problem? No:  M even from WxJ effect Mx 2 /cycle