Free Magnetic Energy in Solar Active Regions above the Minimum-Energy Relaxed State

(Regnier, S., Priest, E.R. 2007 ApJ)

Use magnetic field extrapolations to calculate the amount of free magnetic energy available for processes such as solar

flares.

In the solar corona above an active region, we can neglect the effects of gravity and pressure with respect to those from the magnetic field.

Then we have

For currents along the field lines, this becomes

There are three types of fields that satisfy the above equations:

1.A potential field, j=0 everywhere

2.A linear force-free field, where α is constant everywhere

3.A non-linear force-free field, where α= α(x, y ,z)

The Woljter-Taylor theory states that in a weak but finite resistive regime, the free magnetic energy available from a relaxation process is the energy above the linear force-free field that satisfies the boundary conditions and has the same helicity as the observed field. Heyvaerts & Priest (1984) extended the theory to coronal fields.

The authors calculate the magnetic energy in each configuration with

They then calculate the free energy using

Regions studied

AR 8151, a decaying region AR 8210, a newly emerged region

AR 9077, Bastille Day flare AR 10486, Halloween events of ‘03

Actual photospheric field data obtained from the MSO/IVM

(Mees Solar Observatory / Imaging Vector Magnetograph)

Conclusions

• Free magnetic energy can vary by at least 2 orders of magnitude. This is because it depends greatly on the total magnetic flux and on the arrangement of the polarities.

• By noting the calculated free magnetic energy to the activity level associated with the various regions, they found that the value computed with the linear force-free configuration as the reference field gives a better estimate of the energy available for flaring and associated processes.

Nonlinear force-free models for the solar corona 1. Two active regions with very different structure

(Regnier, S., Priest, E.R. 2006 A&A)

Use the same extrapolation methods described in the previous paper to investigate how the various current density

distributions change the geometry of the field lines, where and how much magnetic energy is stored, and the amount of

magnetic helicity.

The vertical current density, Jz, is used to obtain a suitable α (α =Jz/Bz)

Again, AR 8151 was a decaying active region at the time of observation. Other studies of the region found twisted flux tubes with various numbers of turns and handedness.

AR 8210 was a newly emerged active region with a complex topology. It consisted of a strongly negative sunspot surrounded by several positive polarities. However, studies of this region have not found any twisted flux tubes.

First let’s see the effects of current density on the shape of the field lines.

In AR 8151, high current densities strongly influence the geometry of the field lines.

The angle of the polarity inversion line with respect to the top of the loops changes.

The locations of the footpoints are quite different.

The chosen field lines seem to indicate that the nlff field lines top out lower than the potential ones. However, statistical analysis revealed that the nlff

field lines are generally taller, longer, and stronger.

In AR 8210, the weak current densities modify the geometry only slightly. Even the heights seem the same.

This field reconstruction does not produce any twisted flux bundles, which agrees with observations.

The authors claim that the low influence of increased current along the lines means that these field lines can store magnetic energy, since an input of current density would modify the field strength, but not the configuration.

AR 8151

Free magnetic energy is mostly in he mid-corona, at heights where the twisted flux tubes are found.

Calculated energy budget of 2.6 x 1031 erg (40% of the total energy in nlff) is enough to trigger a small flare.

AR 8210

Free magnetic energy is mostly in the low corona, near the photosphere.

Calculated energy budget of 2.4 x 1031 erg (only 2.5% of the total energy in nlff) is still enough to trigger a small flare. Several C-class flares occurred around the time of observation.

(Note the logarithmic nature of the plot. )

References

Regnier, S., Priest, E.R. 2007, Free Magnetic Energy in Solar Active Regions above the Minimum-Energy Relaxed State, preprint, accepted into ApJ

Regnier, S., Priest, E.R. 2006 A&A 468,701