Dynamo action in shear flow turbulence

1. 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

2. 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 turbulence

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

4. Direct vs hyper at 5123 Biskamp & Müller (2000, Phys Fluids 7, 4889) Normal diffusivity With hyperdiffusivity

5. 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 40963direct simulations Dynamos & shear flow turbulence

6. 3rd order hyper: inertial range OK Different resolution: bottleneck & inertial range Haugen & Brandenburg (PRE 70, 026405) Traceless rate of strain tensor Hyperviscous heat 3rd order dynamical hyperviscosity m3 Dynamos & shear flow turbulence

7. Hyperviscous, Smagorinsky, normal height of bottleneck increased Haugen & Brandenburg (PRE 70, 026405, astro-ph/041266) onset of bottleneck at same position Inertial range unaffected by artificial diffusion

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

9. Relation to ‘laboratory’ 1D spectra Dobler, et al (2003, PRE 68, 026304) Dynamos & shear flow turbulence

10. (ii) Energy and helicity surface terms ignored Incompressible: How w diverges as n0 Inviscid limit different from inviscid case! Dynamos & shear flow turbulence

11. Magnetic case How J diverges as h0 Ideal limit and ideal case similar! Dynamos & shear flow turbulence

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

13. Helical dynamo saturation with hyperdiffusivity for ordinary hyperdiffusion ratio 53=125 instead of 5 PRL 88, 055003 Dynamos & shear flow turbulence

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

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

16. (iii) Small scale dynamo: Pm dependence?? Small Pm=n/h: stars and discs around NSs and YSOs Schekochihin Haugen Brandenburg et al (2005) Cattaneo, Boldyrev k Here: non-helically forced turbulence Dynamos & shear flow turbulence

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

18. 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! Dynamos & shear flow turbulence

19. Pencil Code • 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 ~20 people (CVS!) • Automatic validation (over night or any time) • Max resolution so far 10243 , 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

20. (i) Higher order – less viscosity Dynamos & shear flow turbulence

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

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

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

24. Comparison of A and B methods Dynamos & shear flow turbulence

25. 256 processor run at 10243 Dynamos & shear flow turbulence

26. Structure function exponents agrees with She-Leveque third moment Dynamos & shear flow turbulence

27. Wallclock time versus processor # nearly linear Scaling 100 Mb/s shows limitations 1 - 10 Gb/s no limitation Dynamos & shear flow turbulence

28. Sensitivity to layout onLinux 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 Dynamos & shear flow turbulence

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

30. 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 turbulence

31. Pre-processed data for animations Dynamos & shear flow turbulence

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

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

34. 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.) Wasn’t the dynamo supposed to work at the bottom? Tachocline dynamos Distributed/near-surface dynamo in favor against • 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 Brandenburg (2005, ApJ 625, June 1 isse)

35. In the days before helioseismology • Angular velocity (at 4o 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 dW/dr<0: equatorward migr Dynamos & shear flow turbulence

36. Application to the sun: spots rooted at r/R=0.95 Benevolenskaya, Hoeksema, Kosovichev, Scherrer (1999) Pulkkinen & Tuominen (1998) • Overshoot dynamo cannot catch up • Df=tAZDW=(180/p) (1.5x107) (2p 10-8) • =360 x 0.15 = 54 degrees! Dynamos & shear flow turbulence

37. 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 WxJ effect) • Stratification: can lead to a effect • modify WxJeffect • but also instability of its own • SS dynamo not obvious at small Pm • Application to the sun? • distributed dynamo  can produce bipolar regions • a perhaps not so important? • solution to quenching problem? No: aM even from WxJ effect 1046 Mx2/cycle