cattaneo@flash.uchicago.edu

1. Chicago 2003 The solar dynamo(s) Fausto Cattaneo Center for Magnetic Self-Organization in Laboratory and Astrophysical Plasmas cattaneo@flash.uchicago.edu

2. Chicago 2003 The solar dynamo problem The solar dynamo is invoked to explain the origin magnetic activity Three important features: • Wide range of spatial scales. From global scale to limit of resolution • Wide range of temporal scales. From centuries to minutes • Solar activity is extremely well documented Models are strongly observationally constrained

3. Chicago 2003 Observations Hale’s polarity law suggests organization on global scale. Typical size of active regions approx 200,000Km Typical size of a sunspot 50,000Km Small magnetic elements show structure down to limit of resolution (approx 0.3")

4. Chicago 2003 Observations: large scale • Active regions migrate from mid-latitudes to the equator • Sunspot polarity opposite in two hemispheres • Polarity reversal every 11 years

5. Observations: large scale : stored in ice cores after 2 years in atmosphere : stored in tree rings after ~30 yrs in atmosphere 14 C 10 Be Chicago 2003 PROXY DATA OF SOLAR MAGNETIC ACTIVITY AVAILABLE Beer (2000) Wagner et al (2001) Cycle persists through Maunder Minimum (Beer et al 1998)

6. Chicago 2003 Observations: small scale Two distinct scales of convection (maybe more) • Supergranules: • not visible in intensity • 20,000 km typical size • 20 hrs lifetime • weak dependence on latitude • Granules: • strong contrast • 1,000km typical size • 5 mins lifetime • homogeneous in latitude

7. Chicago 2003 Observations: small scale Quiet photospheric flux • Network fields • emerge as ephemeral regions (possibly) • reprocessing time approx 40hrs • weak dependence on solar cycle • Intra-network magnetic elements • possibly unresolved • typical lifetime few mins

8. Chicago 2003 General dynamo principle Any three-dimensional, turbulent (chaotic) flow with high magnetic Reynolds number is (extremely) likely to be a dynamo. • Reflectionally symmetric flows: • Small-scale dynamo action • Disordered fields; same correlation length/time as turbulence • Generate but not • Non-reflectionally symmetric flows: • Large-scale dynamo; inverse cascade of magnetic helicity • Organized fields; correlation length/time longer than that of turbulence • Possibility of

9. Chicago 2003 Rotational constraints In astrophysics lack of reflectional symmetry associated with (kinetic) helicity  Coriolis force  Rotation Introduce Rossby radius Ro (in analogy with geophysical flows) • Motions or instabilities on scales Ro “feel’’ the rotation. • Coriolis force important  helical motions • Inverse cascades  large-scale dynamo action • Motions or instabilities on scales < Ro do not “feel” the rotation. • Coriolis force negligible  non helical turbulence • Small-scale dynamo action

10. Chicago 2003 Modeling: large-scale generation Dynamical ingredients • Helical motions: Drive the α-effect. Regenerate poloidal fields from toroidal • Differential rotation: (with radius and/or latitude) Regenerate toroidal fields from poloidal. Probably confined to the tachocline • Magnetic buoyancy: Removes strong toroidal field from region of shear. Responsible for emergence of active regions • Turbulence: Provides effective transport

11. Chicago 2003 Modeling: helical motions • Laminar vs turbulent α-effect: • Babcock-Leighton models. α-effect driven by rise and twist of large scale loops and subsequent decay of active regions. Coriolis-force acting on rising loops is crucial. Helical turbulence is irrelevant. Dynamo works because of magnetic buoyancy. • Turbulent models. α-effect driven by helical turbulence. Dynamo works in spite of magnetic buoyancy. • Nonlinear effects: • Turbulent α-effect strongly nonlinearly suppressed • Interface dynamos? • α-effect is not turbulent (see above) Cattaneo & Hughes

12. Chicago 2003 Modeling: differential rotation • Latitudinal differential rotation: • Surface differential rotation persists throughout the convection zone • Radiative interior in solid body rotation Schou et al. • Radial shear: • Concentrated in the tachocline; a thin layer at the bottom of the convection zone • Whys is the tachocline so thin? What controls the local dynamics? No self-consistent model for the solar differential rotation

13. Chicago 2003 Modeling: magnetic buoyancy What is the role of magnetic buoyancy? • Babcock-Leighton models: • Magnetic buoyancy drives the dynamo • Twisting of rising loops under the action of the Coriolis force generates poloidal field from toroidal field • Dynamo is essentially non-linear • Turbulent models: • Magnetic buoyancy limits the growth of the magnetic field • Dynamo can operate in a kinematic regime Wissink et al. Do both dynamos coexist? Recovery from Maunder minima?

14. Chicago 2003 Modeling: turbulence How efficiently is turbulent transport? • Babcock-Leighton models: Turbulent diffusion causes the dispersal of active regions. Transport of poloidal flux to the poles. • Interface models: Turbulent diffusion couples the layers of toroidal and poloidal generation • All models: • Turbulent pumping helps to keep the flux in the shear region • Turbulence redistributes angular momentum • Etc. etc. etc. Tobias et al.

15. Chicago 2003 Modeling: challenges No fully self-consistent model exists. • Self-consistent model must capture all dynamical ingredients (MHD, anelastic) • Geometry is important (sphericity) • Operate in nonlinear regime • Resolution issues. Smallest resolvable scales are • in the inertial range • rotationally constrained • stratified Need sophisticated sub-grid models

16. Chicago 2003 Modeling: small-scale generation cold g hot • Plane parallel layer of fluid • Boussinesq approximation • Ra=500,000; P=1; Pm=5 Simulations by Lenz & Cattaneo time evolution temperature

17. Chicago 2003 Rm Pm=1 Pm =1 IM Stars 103 simulations 102 Liquid metal experiments Re 103 107 Modeling: physical parameters • Dynamo must operate in the inertial range of the turbulence • Driving velocity is rough • How do we model MHD behaviour with Pm <<1

18. Chicago 2003 Modeling: kinematic and dynamical issues Pm=1 Pm=0.5 yes • Does the dynamo still operate? (kinematic issue) • Dynamo may operate but become extremely inefficient (dynamical issue) Re=550, Rm=550 no Re=1100, Rm=550

19. Chicago 2003 Modeling: magneto-convection • Relax requirement that magnetic field be self sustaining (i.e. impose a uniform vertical field) • Construct sequence of simulations with externally imposed field, 8 ≥ Pm ≥ 1/8, and S = = 0.25 • Adjust Ra so that Rm remains “constant” Simulations by Emonet & Cattaneo

20. Chicago 2003 Modeling: magneto-convection B-field (vertical) vorticity (vertical) Pm = 8.0 Pm = 0.125

21. Chicago 2003 Modeling: magneto-convection • Energy ratio flattens out for Pm < 1 • PDF’s possibly accumulate for Pm < 1 • Evidence of regime change in cumulative PDF across Pm=1 • Possible emergence of Pm independent regime

22. Chicago 2003 Summary Two related but distinct dynamo problems. • Large-scale dynamo • Reproduce cyclic activity • Reproduce migration pattern • Reproduce angular momentum distribution (CV and tachocline) • Needs substantial advances in computational capabilities • Small scale dynamo • Non helical generation • Small Pm  turbulent dynamo