The Solar DynamoNSO Solar Physics Summer School

Tamara Rogers, HAOJune 15, 2007

• Regions of strong magnetic field (3000 Gauss)• About 20000km diameter• Lifetime of a few weeks

PSPT (blue)PSPT (CaK)

Sunspots on Solar Disk

Yohkoh X-ray images

X-ray Activity over sunspot cycle

Joy’s law

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• 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

What is a dynamo?A dynamo is a process by which kinetic energy of fluid motion is convertedinto magnetic energy. By this process a magnetic field can maintain itselfagainst ohmic dissipation

Why study the dynamo?It’s the source of all magnetic activity on the Sun and likely most otherstars (although the process of the dynamo is different in massive or verylow 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 109 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).

The (Magneto-) Hydrodynamic Equations

Terms:

Poloidal - field in the direction

Toroidal - field in the direction

Meridional - flow in the direction

Azimuthal - flow in the direction

Cowlings TheoremAssume an axisymmetric poloidal field, any suchfield must have a neutral point where:

Because of the assumption of axisymmetry the neutral point must circle the rotation axis on thisline the poloidal field must equal zero, however the toroidal current does not

But we also know

These are in contradiction==>assumptionsare 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

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 axisymmetricflow CANNOT maintain an axisymmetric field--NO 2D DYNAMOS!!!

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 Br ,B ) with differential rotation

€

θ

effect

effect

This alpha effect is fundamentally 3D so how do weput it into 2D equations?

The alpha effect

In the simplest approximation

In bulk of convection zone (in N.H.), rising fluidelements produce + alpha effect (negative vorticity, positive radial velocity) in tachocline- alpha effect

In CZ

Note: in subadiabaticregions the above effect has opposite sign. Signs are all reversed in southern hemisphereThis 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

Mean Field Electrodynamics*assume flow and field are nearly axisymmetric with small scaleturbulence

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

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

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

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The Role of the Tachocline

The tachocline provides the proper sign for the alpha effect to producea dynamo wave that propagates toward the equator at low latitudes - goodplace 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) requirea 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 schematicof dynamo process

Typical Solutions - Kinematic dynamos

There are numerous models called by different names: Babcock-Leighton, Interface, Flux Transport, etc. They vary mainlyin where the effect occurs:

Get VASTLY different results depending onwhat you specify for alpha (both in radiusand latitude)

(-) alpha in tachocline gives equatorward propagation (observed)

Poleward propagating component amplitude is too high (compared with observations)

Can get remarkably periodicSolutions (even 11 years) - due tosolutions of the alpha-Omega dynamo equations

alpha-Omega dynamos + Meridional CirculationTake previous alpha-Omega mean field equations (which only had

differential rotation) and add a meridional circulation - same profileof alpha as previously

Again, get VASTLY different results dependingon assumptions

Typical Solutions - Kinematic dynamosThe 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 fluctuatingquantities (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 butterflydiagram).

What we really need (and want) to do is to solve the full MHD equationsin a sphere (remember: expensive). This has been done for the Earth’s dynamo and is being done for the Sun

Earth’s Dynamo

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Solar Dynamo

No reversal and certainly nobutterfly diagram or equatorward propagation of toroidal field…but this model does not have a tachoclineLOTS OF WORK TO BE DONE

The Induction Equation