The Solar Dynamo

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
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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?
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?

Its 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 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 Toroidal - field in the direction direction
Meridional - flow in the Azimuthal - flow in the
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
Induction 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 elements produce + alpha effect (negative vorticity, positive radial velocity) in tachocline - alpha effect
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?
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
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
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

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 (Hales Law and Joys 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
QuickTime and a YUV420 codec decompressor are neede d 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) (-) 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
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 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 Earths dynamo and is being done for the Sun
Earths Dynamo
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QuickTime and a Video d ecompressor are neede d to see this picture.
Solar Dynamo
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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

The Solar Dynamo

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The Solar Dynamo

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The Solar Dynamo

Sunspots on Solar Disk

X-ray Activity over sunspot cycle

Yohkoh X-ray images

QuickTime an d a Cinepak decompressor are need ed to see this p icture .

Hales polarity law

Tilt angle increases with latitude

Why study the dynamo?

The (Magneto-) Hydrodynamic Equations

Meridional - flow in the Azimuthal - flow in the

But we also know

Axisymmetric Field, Axisymmetric Flow

Induction Equation becomes in spherical coordinates

How to make an axisymmetric dynamo

The alpha effect

In the simplest approximation

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

Mean Field Electrodynamics

Flow field and field are 2D

Induction equation then becomes

Axisymmetric flow+field with Mean Field approximation

The Dynamo Wave

If alpha effect is in tachocline: At low latitudes:

If alpha effect is in bulk of CZ: At low latitudes:

The Role of the Tachocline

Typical Solutions - Kinematic dynamos

alpha-Omega dynamos + Meridional Circulation

Again, get VASTLY different results depending on assumptions

Typical Solutions - Kinematic dynamos The Flux Transport Dynamo

The Flux Transport Dynamo

Whats Wrong with these Models?

QuickTime and a Video d ecompressor are neede d to see this picture.

QuickTime and a Cinepak decompressor are neede d to se e this picture.

The Induction Equation

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