1. The Solar Dynamo NSO Solar Physics Summer School Tamara Rogers, HAO June 15, 2007

2. Regions of strong magnetic field (3000 Gauss) About 20000km diameter Lifetime of a few weeks PSPT (blue) PSPT (CaK) Sunspots on Solar Disk

3. Yohkoh X-ray images X-ray Activity over sunspot cycle

7. Joy’s law

10. Butterfly diagram –Equatorward propagation of activity starting from 35 degrees latitude over 11 years (individual lifetimes of sunspots ~ a few weeks) Hale’s polarity law –Opposite polarity of bipolar groups in north and south hemisphere –Polarity in individual hemisphere changes every 11 years Joy’s law –Bipolar groups are tilted to east-west direction –Leading polarity closer to equator –Tilt angle increases with latitude Summary of Observations

11. What is a dynamo? A dynamo is a process by which kinetic energy of fluid motion is converted into magnetic energy. By this process a magnetic field can maintain itself against ohmic dissipation Why study the dynamo? It’s the source of all magnetic activity on the Sun and likely most other stars (although the process of the dynamo is different in massive or very low mass stars) Why a dynamo? It is possible that a diffusing primordial field is responsible for the magnetism observed: the diffusion time for a poloidal field of is approximately 10 9 years, so this is not strictly ruled out. However, an oscillating primordial field would likely be observed by helioseismology (unless of course the oscillations took place in the tachocline or deep interior, regions not sampled well by helioseismology).

12. The (Magneto-) Hydrodynamic Equations

13. Terms: Poloidal - field in the direction Toroidal - field in the direction Meridional - flow in the direction Azimuthal - flow in the direction

14. Cowlings Theorem Assume an axisymmetric poloidal field, any such field must have a neutral point where: Because of the assumption of axisymmetry the neutral point must circle the rotation axis on this line the poloidal field must equal zero, however the toroidal current does not But we also know These are in contradiction==>assumptions are not consistent A steady state axisymmetric fluid flow can not maintain an axisymmetric magnetic field. The flow and field must be 3D or time dependent or both

15. Induction Equation becomes in spherical coordinates Axisymmetric Field, Axisymmetric Flow Poloidal field Toroidal field No source Term!! This is just another way to illustrate Cowling’s theorem: an axisymmetric flow CANNOT maintain an axisymmetric field--NO 2D DYNAMOS!!!

16. How to make an axisymmetric dynamo Need a way to make poloidal field (A) from toroidal field (B) - Parker (1955) pointed out that a rising field could be twisted by the Coriolis force producing poloidal field from toroidal field Can make toroidal field (B) from poloidal field (A, also B r,B ) with differential rotation effect This alpha effect is fundamentally 3D so how do we put it into 2D equations?

17. The alpha effect In the simplest approximation In bulk of convection zone (in N.H.), rising fluid elements produce + alpha effect (negative vorticity, positive radial velocity) in tachocline - alpha effect In CZ Note: in subadiabatic regions the above effect has opposite sign. Signs are all reversed in southern hemisphere This alpha effect is fundamentally 3D so how do we put it into 2D equations? Alpha is meant to represent the twisting an induction effect due to turbulent motions but we don’t want to solve for turbulence (hard!) so we will parametrize it

18. Mean Field Electrodynamics *assume flow and field are nearly axisymmetric with small scale turbulence Flow field and field are 2D axisymmetric Substituting these decompositions into Ohms law and doing the proper averaging Expand Electromotive Force in Taylor Series, keep only first two terms Induction equation then becomes In general alpha and beta should be tensors, in practice they are not

19. Axisymmetric flow+field with Mean Field approximation “ Dynamos” These models are also called “kinematic” which means that the flow is specified and not allowed to evolve in response to the field

20. Solutions of the dynamo equations allow wave solutions (Parker 1955) who suggested that a latitudinally propagating wave was the source of the sunspot cycle Dynamo waves travel in the direction (Parker-Yoshimura sign rule): At low latitudes: Need a negative (-) alpha effect for equatorward propagation (good) (from helioseismology) If alpha effect is in tachocline: If alpha effect is in bulk of CZ: At low latitudes: Need a negative (-) alpha effect for equatoward propagation (bad) The Dynamo Wave

21. The Role of the Tachocline The tachocline provides the proper sign for the alpha effect to produce a dynamo wave that propagates toward the equator at low latitudes - good place for the alpha effect The radial shear in the tachocline provides ideal place for Omega effect. The remarkable coherence of sunspots (Hale’s Law and Joy’s Law) require a field strong enough to resist shredding by turbulent motions in the convection zone. Such a field strength can only be generated in the tachocline where the Parker instability is less efficient Cartoon schematic of dynamo process

22. Typical Solutions - Kinematic dynamos There are numerous models called by different names: Babcock-Leighton, Interface, Flux Transport, etc. They vary mainly in where the effect occurs: Get VASTLY different results depending on what you specify for alpha (both in radius and latitude) (-) alpha in tachocline gives equatorward propagation (observed) Poleward propagating component amplitude is too high (compared with observations) Can get remarkably periodic Solutions (even 11 years) - due to solutions of the alpha-Omega dynamo equations

23. alpha-Omega dynamos + Meridional Circulation Take previous alpha-Omega mean field equations (which only had differential rotation) and add a meridional circulation - same profile of alpha as previously Again, get VASTLY different results depending on assumptions

24. Typical Solutions - Kinematic dynamos The Flux Transport Dynamo Unlike typical dynamos, the flux transport dynamo relies on meridional circulation to bodily advect the toroidal field, instead of a dynamo wave

25. The Flux Transport Dynamo Again, can get nice periodic solutions with equatorward propagation at low latitude, but poleward branch is bad (typical).

26. Whats Wrong with these Models? Mean field theory requires that there be clear scale separation (i.e. that the mean quantities (B, u) are much larger than the fluctuating quantities (B’,u’) -- observations and simulations show that there is a range of spatial scales with no clear distinction between “large” and “small” There is no feedback. The MHD equations are COUPLED, the flow affects the field which affects the flow which affects the energy… They are solving 2 equations out of 7!! *Only keep first two terms in series expansion of induction term *The models are HIGHLY PARAMETRIZED, alpha is not known empirically it can be “tuned” to reproduce the results you want (like the butterfly diagram). What we really need (and want) to do is to solve the full MHD equations in a sphere (remember: expensive). This has been done for the Earth’s dynamo and is being done for the Sun

27. Earth’s Dynamo

28. Solar Dynamo No reversal and certainly no butterfly diagram or equatorward propagation of toroidal field…but this model does not have a tachocline LOTS OF WORK TO BE DONE

29. The Induction Equation