1. Big Bear Solar Observatory of NJIT
Spatio-Temporal Dynamics of Magnetic Fields and Flaring Productivity of Active Regions
Valentina I. Abramenko
Big Bear Solar Observatory of NJIT

2. Introduction
@ )These two wonderful movies (the courtesy of Richard Nightingale) make a good introduction into the topic I will talk about.
Subphotospheric convection creates filigree of sunspots and magnetic structures observed in the photosphere. Motions of of magnetic elements (footpoints of magnetic flux tubes) in the photosphere twist and braid magnetic field lines in the corona. (Here we see how the rotation of the sunspot is reflected in the twisting and braiding of coronal loops).
The photosphere plays a role of a chain joint between sub-photospheric processes of magnetic field generation and coronal processes of magnetic energy dissipation. And the last one may occur in an explosive manner as flares in active regions.
So, important aspects of flare-related physics may be hidden in the photosphere.
The 2005 Joint Assembly / AGU,SEG, NABS and SPD/AAS, May 23-27, 2005, New Orleans, LA, USA

3. Introduction Essential properties of the photospheric plasma:
Magnetized plasma in a turbulent state (Parker 1979),
very intermittent (or, in other words, multifractal) medium
(Lawrence et al. 1993, Abramenko et al. 2002),
where the magnetic helicity may have an inverse cascade
(Biskamp 1993)

4. Introduction Situation in the photosphere – flaring in the corona:
Magnetic helicity transport in the photosphere
(Rust & LaBonte 2003; Georgoulis – present meeting; Chae2001; Romano 2005 ).
These properties suggest several ways to study relation between: Situation in the photosphere - and– flaring in the corona .
One way is to analyze magnetic helicity transport in the photosphere as an early indicator of the buitd up of energy in the corona (Rust, LaBonte, Georgoulis, Chae, Romano).
Another way is to analyze statistical properties of different quantities derived from vector-magnetograms, such as magnetic flux density, electric currents, current helicity etc., using large data sets in order to distinguish between flaring and flare-quiet active regions.
One more possible approach to analyze the statistical organization is to calculate a fractal dimension of the magnetic field and to treat it as a measure of complexity (Tarbell, Balke , Meunier. ) Recently, McAteer, Gallagher and Ireland showed that there is a correspondence between the fractal dimension and flaring in an active region.
Statistical properties of electric currents, current helecity,
magnetic flux, etc., derived from vector-magnetograms
(Leka & Barnes 2004).
Fractal dimensions of the photospheric magnetic fields
(Tarbell et al. 1990; Balke et al. 1993; Meunier 1999;
McAteer, Gallagher & Ireland 2005).

5. Introduction
Situation in the photosphere – flaring in the corona: we propose to analyze
1. Distribution functions of the magnetic flux in elements of the magnetic field in active regions
(Abramenko & Longcope, ApJ, 2005)
We propose some other ways to analyze the situation. First, we propose , to analyze the distribution functions of the magnetic flux in magnetic elements of an active region.
Second, analysis of the degree of multifractality of the magnetic structures. This topic is present in our poster here.
And third, analysis of the turbulent state on the basis of power spectrum of the magnetic field. The last topic will be discussed in the present talk.
2. Multifractality (intermittency) of the photospheric magnetic
fields – poster SP41B-04
(Abramenko, Yurchyshyn, Wang, Goode, ApJ 577, 2002).
3. Turbulence state of the photospheric magnetic field as derived
from magnetic power spectrum – present talk.

6. Observational Data: SoHO/MDI high resolution magnetograms
I analyzed line-of-sight magnetograms for 16 active regions obtained by MDI in high resolution mode.
Between them there were emerging and stable ones, flare-productive and flare-quiet ones.
This is the strongest flare occurred in an AR. The last column shows the soft X-ray flare index, which is a quantitative estimation of the flare productivity. The active regions are listed in the order of decreasing soft X-ray flare index.
A couple of words about how the flare index was calculated.

7. Soft X-ray Flare Index
The daily SRX flare index was first introduced by Anna Antalova in 1996 and later was applied by other authors to characterize the daily flare productivity of the Sun from the soft X-ray flux measured by GOES in the 1-8 A range. Discussions with Alexei and Dana helped me a lot in application of the idea to a single active region.
The flare index is constructed by weighing [weiin] flares of classes X,M,C as 100, 10, 1 and summing over all flares occurred in a given active region during its passage across the disk. The result is divided by the duration of the passage, tau (measured in days) . And we obtain the specific flare productivity per day.
For example, ….
A=1 corresponds to one flare of C1 per day.
For all ARs analyzed in this study, tau was no less than 9 days – a time period sufficient enough to display the ability for flaring.
For example, during 13 days an active region launched flares:
X5.2 , M1.2 , C6.0
A=( ) / 13 = 41.4
(in units 10 W m ) -6 -2

8. Magnetic Power Spectrum
Flare-quiet active region 0061 (A=2.6)
Flaring active region 9077 (A=120)
Here I show typical examples of the power spectrum for two active regions.
The first one is flare-quiet AR and it displays the Kolmogorov-type spectrum of the power index close to five thirds.
The second AR was very flare-productive one. It shows the spectrum steeper than the Kolmogorov spectrum.

9. Magnetic Power Spectrum: time-variations of the power index
Flare-quiet
active region 0061:
This is the time variations of the power index for the same Ars (black lines). And red lines show the GOES flux.
For the quiet active region the power index waves around the dashed line which shows the state of Kolmogorov turbulence.
This is the result for flaring AR. The power index is considerably above the K41 line.
Both above mentioned active regions were developed ones. It would be interesting to check what is going on in emerging active regions.
Flaring
active region 9077:

10. Magnetic Power Spectrum: emerging active region
This is an emerging and very flaring active region Three-days movie of 5 min cadence. The power spectrum for each magnetogram. And the bottom panel shows the time variations if the power index, alpha, and the X-ray Goes flux. Several M-class flares occurred during our observations. And several X-class flares – after.
At the very beginning of emergence, this magnetic structure possessed a very steep, non-Kolmogorov spectrum.The dotted line shows the power spectrum of the first magnetogram. As the active region emerges, we observe the lifting of the spectrum at all scales, with nearly the same magnitude of the power index .
It looks like we observe unraveling , disentangling emergence of a very complicated, very tangled magnetic structure generated under the photosphere.

11. Soft X-ray Flare Index versus Magnetic power index
This plot summarizes results obtained for 16 active regions. Each point corresponds to one active region.
The magnitude of the flare index A (the vertical axis) is usually below 5 when an active region launches C-class flare , A is between 5 and 25 when M flares occur. And A exceeds 30 when X-class flares were launched. Horizontal axis is the power index of the magnetic power spectrum as averaged from several magnetograms registered in the high resolution mode by SOHO/MDI.
One can see a good correlation between the power index and the flare index. The data points may be successively fitted by such an analytical curve (red line), the reduced chi-square test is about 1. Active regions which produced X-class flares possessed a steep power spectrum with alpha higher than 2. Flare-quiet active regions display a Kolmogorov-type spectrum.
The most interesting point is the situation with emerging active regions red circles) . For all of them, the power index determined from the first magnetogram (before the start of the flare activity) got into the red circles, in other words, it was very close to the averaged magnitude . This means that the magnitude of the power index, determined at the beginning of the emergence, seems to be related to the future flare productivity of AR. This finding shows the way to distinguish at the very early stage
those solar spots that are ``born bad'' and have a potential to produce powerful flares.
So, this plot can help much in forecasting on oncoming flare activity , especially for emerging magnetic structures.

12. The very steep non-Kolmogorov spectra of flaring active regions imply the inhomogeneous non-stationary turbulence regime when the energy transport rate along the spectrum may not be constant. And the energy dissipation displays an intermittent(burst-like) behavior as in time so in spatial domains.
The Kolmogorov-type spectra of quiet active regions might suggest a nearly stationary turbulent regime , which provides premises for smooth evolution without catastrophes.
This study has demonstrated that structural and dynamical characteristics of the magnetic field as measured in the photosphere are relevant to the intensity of non-stationary processes in the entire magnetic configuration.

13. Magnetic Power Spectrum: calculationsk y k x

14. Table 2: Emerging Active Regions

15. Table 1: Active Regions

16. Emerging of a Flare-quiet Active Region NOAA 9851

17. Emergence of a Flaring Active Region NOAA 0365