Dynamo action in shear flow turbulence 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

Dynamos & shear flow turbulence2 Turbulence in astrophysics Gravitational and thermal energy –Turbulence mediated by instabilities convection MRI (magneto-rotational, Balbus-Hawley) Explicit driving by SN explosions –localized thermal (perhaps kinetic) sources Which numerical method should we use? Korpi et al. (1999), Sarson et al. (2003) no dynamo here…

Dynamos & shear flow turbulence3 (i) Turbulence in ideal hydro Porter, Pouquet, Woodward (1998, Phys. Fluids, 10, 237)

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

Dynamos & shear flow turbulence5 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

Dynamos & shear flow turbulence6 3 rd order hyper: inertial range OK Different resolution: bottleneck & inertial range Traceless rate of strain tensor 3 rd order dynamical hyperviscosity  3 Hyperviscous heat Haugen & Brandenburg (PRE 70, )

7 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

Dynamos & shear flow turbulence8 Bottleneck effect: 1D vs 3D spectra Compensated spectra (1D vs 3D) Why did wind tunnels not show this?

Dynamos & shear flow turbulence9 Relation to ‘laboratory’ 1D spectra Dobler, et al (2003, PRE 68, )

Dynamos & shear flow turbulence10 (ii) Energy and helicity Incompressible: How  diverges as  0 Inviscid limit different from inviscid case! surface terms ignored

Dynamos & shear flow turbulence11 Magnetic case How J diverges as   0 Ideal limit and ideal case similar!

Dynamos & shear flow turbulence12 Dynamo growth & saturation Significant field already after kinematic growth phase followed by slow resistive adjustment

Dynamos & shear flow turbulence13 Helical dynamo saturation with hyperdiffusivity for ordinary hyperdiffusion ratio 5 3 =125 instead of 5 PRL 88,

Dynamos & shear flow turbulence14 Slow-down explained by magnetic helicity conservation molecular value!! ApJ 550, 824

15 Connection with  effect: writhe with internal twist as by-product  clockwise tilt (right handed)  left handed internal twist Yousef & Brandenburg A&A 407, 7 (2003) both for thermal/magnetic buoyancy

Dynamos & shear flow turbulence16 (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

17 (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!

Dynamos & shear flow turbulence18 Overview 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!

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

Dynamos & shear flow turbulence20 (i) Higher order – less viscosity

Dynamos & shear flow turbulence21 (ii) High-order temporal schemes Main advantage: low amplitude errors 3 rd order 2 nd order 1 st order 2N-RK3 scheme (Williamson 1980)

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

Dynamos & shear flow turbulence23 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!

Dynamos & shear flow turbulence24 Comparison of A and B methods

Dynamos & shear flow turbulence processor run at

Dynamos & shear flow turbulence26 Structure function exponents agrees with She-Leveque third moment

Dynamos & shear flow turbulence27 Wallclock time versus processor # nearly linear Scaling 100 Mb/s shows limitations Gb/s no limitation

Dynamos & shear flow turbulence28 Sensitivity to layout on Linux clusters yprox x zproc 4 x 32  1 (speed) 8 x 16  3 times slower 16 x 8  17 times slower Gigabit uplink 100 Mbit link only 24 procs per hub

29 Why this sensitivity to layout? All processors need to communicate with processors outside to group of 24 16x8

Dynamos & shear flow turbulence30 Use exactly 4 columns Only 2 x 4 = 8 processors need to communicate outside the group of 24  optimal use of speed ratio between 100 Mb ethernet switch and 1 Gb uplink 4x32

Dynamos & shear flow turbulence31 Pre-processed data for animations

Dynamos & shear flow turbulence32 Simulating solar-like differential rotation Still helically forced turbulence Shear driven by a friction term Normal field boundary condition

33 Forced LS dynamo with no stratification geometry here relevant to the sun no helicity, e.g. azimuthally averaged neg helicity (northern hem.) Rogachevskii & Kleeorin (2003, 2004)

34 Wasn’t the dynamo supposed to work at the bottom? 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, June 1 isse)

Dynamos & shear flow turbulence35 In the days before helioseismology Angular velocity (at 4 o latitude): –very young spots: 473 nHz –oldest spots: 462 nHz –Surface plasma: 452 nHz Conclusion back then: –Sun spins faster in deaper convection zone –Solar dynamo works with d  /dr<0: equatorward migr

Dynamos & shear flow turbulence36 Application to the sun: spots rooted at r/R=0.95 Benevolenskaya, Hoeksema, Kosovichev, Scherrer (1999) Pulkkinen & Tuominen (1998) –Overshoot dynamo cannot catch up  =  AZ  =(180/  ) (1.5x10 7 ) (2  ) =360 x 0.15 = 54 degrees!

Is magnetic buoyancy a problem? compressible stratified dynamo simulation in 1990 expected strong buoyancy losses, but no: downward pumping

38 Lots of surprises… 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