The Solar Dynamo NSO Solar Physics Summer School Tamara Rogers, HAO June 15, 2007 Sunspots on Solar Disk PSPT (CaK) PSPT (blue)

Regions of strong magnetic field (3000 Gauss) About 20000km diameter Lifetime of a few weeks X-ray Activity over sunspot cycle Yohkoh X-ray images Joys law QuickTime and a

Cinepak decompressor are needed to see this picture. Summary of Observations Butterfly diagram Equatorward propagation of activity starting from 35 degrees latitude over 11 years (individual lifetimes of sunspots ~ a few weeks) Hales polarity law Opposite polarity of bipolar groups in north and south

hemisphere Polarity in individual hemisphere changes every 11 years Joys law Bipolar groups are tilted to east-west direction Leading polarity closer to equator Tilt angle increases with latitude What is a dynamo? dynamo is a process by which kinetic energy of fluid motion is converte

nto magnetic energy. By this process a magnetic field can maintain itsel gainst ohmic dissipation Why study the dynamo? s the source of all magnetic activity on the Sun and likely most other tars (although the process of the dynamo is different in massive or very ow mass stars) Why a dynamo?

is possible that a diffusing primordial field is responsible for the magnetism observed: the diffusion time for a poloidal field of is pproximately 109 years, so this is not strictly ruled out. However, n oscillating primordial field would likely be observed by helioseismology unless of course the oscillations took place in the tachocline or deep nterior, regions not sampled well by helioseismology). The (Magneto-) Hydrodynamic Equations Terms:

oloidal - field in the oroidal - field in the eridional - flow in the Azimuthal - flow in the direction direction

direction direction 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

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

How to make an axisymmetric dynamo Can make toroidal field (B) from poloidal field (A, also Br ,B )with differential rotation effect 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

effect This alpha effect is fundamentally 3D so how do we put it into 2D equations? The alpha effect Alpha is meant to represent the twisting an induction effect due to turbulent motions but we dont want to solve for turbulence (hard!) so we will parametrize it In CZ

In the simplest approximation In bulk of convection zone (in N.H.), rising fluid Note: in elements produce + alpha effect (negative subadiabatic vorticity, positive radial velocity) in tachocline regions the above - alpha effect effect has opposite

sign. Signs are all reversed in southern This alpha effect is fundamentally 3D so how do we hemisphere put it into 2D equations? Mean Field Electrodynamics*assume flow and field are nearly axisymmetric with small scale turbulence low field and field are 2D

axisymmetric bstituting 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

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 The Dynamo Wave

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): If alpha effect is in tachocline: At low latitudes: (from helioseismology) Need a negative (-) alpha effect for

equatorward propagation (good) If alpha effect is in bulk of CZ: At low latitudes: Need a negative (-) alpha effect for equatoward propagation (bad) The Role of the Tachocline he tachocline provides the proper sign for the alpha effect to produce dynamo wave that propagates toward the equator at low latitudes - goo

lace for the alpha effect he radial shear in the tachocline provides ideal place for Omega effect. he remarkable coherence of sunspots (Hales Law and Joys Law) require field strong enough to resist shredding by turbulent motions in the onvection zone. Such a field strength can only be generated in the achocline where the Parker instability is less efficient Cartoon schematic of dynamo process QuickTime and a

YUV420 codec decompressor are needed to see this picture. 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) Can get remarkably periodic Solutions (even 11 years) due to Poleward propagating component amplitude is solutions of the alpha-Omega dynamo equations too high (compared with observations) (-) alpha in tachocline gives equatorward propagation (observed)

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

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 The Flux Transport Dynamo Again, can get nice periodic solutions with equatorward

propagation at low latitude, but poleward branch is bad (typical). 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 t 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

n a sphere (remember: expensive). This has been done for the Earths dynamo and is being done for the Sun Earths Dynamo QuickTime and a YUV420 codec decompressor are needed to see this picture. QuickTime and a Video decompressor

are needed to see this picture. Solar Dynamo QuickTime and a Cinepak decompressor are needed to see this picture. No reversal and certainly no butterfly diagram or equatorward propagation of

toroidal fieldbut this model does not have a tachocline LOTS OF WORK TO BE DONE The Induction Equation