Turbulent Dynamos and Small-Scale Activity in the Sun

and Stars

George H. FisherDave Bercik

Chris Johns-KrullLauren Alsberg

Bill Abbett

The large scale solar magnetic field evolves in a “solar cycle” dynamo on time scales of years

On small scales, the solar magnetic field appears, evolves, and disappears over much shorter time scales

Is the small scale magnetic field on the Sun (and other stars) the lint from the clothes in the solar washing

machine, or is it generated by its own dynamo mechanism?

Why should the Sun or any star generate magnetic fields?

The first term in the expansion of the ideal MHD electric field represents the stretching of magnetic field by velocity shear – this is the driving term for dynamos in stars.

Magnetic fields will grow until they are dissipated resistively or until balanced by a back-reaction from the Lorentz force.

For the solar cycle, the driving velocity shear is believed to come from differential rotation

Differential rotation will act to stretch out an initially poloidal (N-S or radial) magnetic field into the azimuthal (toroidal) direction.

The toroidal field erupts, is twisted by the Coriolis force, and generates a new poloidal field of the opposite sign

Current solar cycle dynamo models (Dikpati & Gilman) do a good job of matching solar cycle behavior and can even

predict 1-2 cycles into the future

Simulated solar cycles (N+S)

Observed cycles

Given the qualitative success of rotation-inspired solar dynamo models, one might guess that stars that rotated

more quickly would have a more vigorous dynamo

And that guess is born out, as shown here in a well-known paper by Skumanich from 1972. This led to the view thatall stellar magnetic activity could be related to stellar rotation.

But there were some hints that rotation was not the whole story on solar and stellar magnetic dynamos

• Studies of the distribution of small scale magnetic flux on the Sun seem to show that the flux levels are roughly independent of solar cycle period

• Stellar activity indicators (X-rays, chromospheric and transition region radiation) seem so show a “basal” level of emission that was presumed to be a signature of acoustic, rather a magnetic origin for atmospheric heating.

The advent of cheap, high speed computing during the past 5-10 years has made it possible to directly address magnetic field generation in dynamos via 3D numerical MHD simulations

• Fausto Cattaneo (1999) demonstrated via a 3D MHD simulation of Boussinesq convection that a small-scale disordered magnetic field can be generated efficiently by turbulent convection.

• As part of the Solar MURI project, we purchased a 24-node Beowulf cluster devoted to MHD simulation, making such simulations practical.

• Bill Abbett and Yuhong Fan (HAO) developed an anelastic MHD code suitable for modeling small portions of the solar or stellar interior. The big advantage over the Boussinesq approximation of Cattaneo is that the anelastic approximation allows for the steep gravitational stratification, necessary to describe stellar convective envelopes.

Our beowulf cluster, grizzly, which cost about $60K

• Grizzly was purchased in 2002, and consists of 24 nodes of xeon dual processor machines, connected by 2 network interfaces (1Gb + 100Mb)

• Grizzly consumes roughly 4KW of power and AC load

• Grizzly is noisier than hell

• Getting a home for Grizzly within SSL was the most difficult task. It resulted in our present server room.

• MURI paid $8000 to have the server room rewired to accommodate Grizzly and the other servers now in there

Quantitative studies of magnetic dynamos on other stars requires a quantitative knowledge of the relationship between

magnetic fields and “activity” indicators such as X-ray flux:(Pevtsov et al. 2003, ApJ 598, 1387)

We have performed our own simulations of small-scale magnetic fields driven by convective turbulence in a stratified model convection zone

without rotation, starting from a small seed field. The magnetic energy grows by 12 orders of magnitude, and saturates at a level of roughly 7% of the kinetic energy in convective motions. This simulation took about 6 CPU

months of computing time.

What does the generated magnetic field look like? Here is a movie showing “magnetograms” movies of the vertical

component of the field in 2 slices of the atmosphere, near the bottom and near the top:

Here is a snapshot showing volume renderings of the entropy and the magnetic

field strength in the convective dynamo

simulation at a time after saturation:

This movie shows the time evolution of a volume rendering of the magnetic field strength in the convective dynamo

after saturation has occurred

What is the distribution of magnetic flux with depth and time during

saturation?

What is the ratio of magnetic to kinetic energy density with depth, and what are the levels of the

magnetic energy fluctuation with time?

How do we connect our simulation results to real data for the Sun and stars?

• We must first convert the dimensionless units of the anelastic MHD code to real (cgs) units corresponding to the convective envelopes of real stars: (1) demand that stellar surface temperature and density match those of model stellar envelopes, (2) Demand that the convective energy flux in the simulation match the stellar luminosity divided by the stellar surface area. We use mixing length theory to connect energy flux to the unit of velocity in the simulation. After applying these assumptions, we can scale a single simulation to the convective envelopes of main-sequence stars from spectral types F to M.

• To convert magnetic quantities from the simulations to observable signatures, we use the empirical relationship between magnetic flux and X-ray radiance (from Pevtsov et al) to predict surface X-ray fluxes for main-sequence stars

So how does our convective dynamo model compare to observed X-ray fluxes in main-sequence stars?

The convective dynamo model does an excellent job of predicting thelower limit of X-ray emission for slowly rotating stars, and for predicting the amount of magnetic flux observed on the Quiet Sun during solar minimum.

Where do we go from here?

• Investigate how rotation affects a convective dynamo (Bercik, Alsberg, Fisher, Abbett,…)

• Develop global 3D (spherical) models to understand what happens on larger scales and in fully convective stars (SANMHD – Bercik,…)

• Investigate quantitative coronal response to convective dynamo mechanisms (Bercik, Fisher, Lundquist,…)