1. Distribution of the magnetic flux in elements of the magnetic field in an active region Valentyna Abramenko Big Bear Solar Observatory, NJIT

2. INTRODUCTION Turbulent flows in the photospheric plasma Braiding and intertwining of magnetic flux tubes Heating of the corona (Parker, 1996) Magnetic coupling between the photosphere and the corona

3. INTRODUCTION Information on the dynamics and statistical parameters of the photospheric magnetic field is necessary when Analyzing processes in the corona: Thomas & Stanchfield 2000, Bogdan et al. 2003, Gudiksen & Nordlund 2002, Handy & Schrijver 2001, Bewcher ey al. 2002, Moore,Falconer,Porter & Hathaway 2003 Modeling of interaction between turbulent plasma and the magnetic field below the photosphere: Schrijver et al. 1997, Fan, Abbett & Fisher 2003, Longcope, McLeish Fisher 2003

4. INTRODUCTION The distribution function of the magnetic flux content in flux concentrations in the photosphere: Previous studies - Wang, J.X., Wang H., Tang, Lee, Zirin 1995, Sol. Phys 160 the flux range: (0.01-10)10 18 Mx; quiet sun areas Power Law with the index –1.67 (intranetwork) and –1.27(network) Schrijver, Title, van Ballegooijen, Hagenaar, Shine 1997 the flux range:(0.7-5)10 18 Mx; quiet sun areas Exponential approximation Schrijver, Title, Hagenaar, Shine 1997, Sol.Phys.175 ( the flux range: (0.7-150)10 18 Mx; quiet sun; plage areas outside sunspots Exponential approximation with the varying index Abramenko – present study the flux range: (0.2-250)10 18 Mx; active region Lognormal + Power Law

5. Observational data: Processing: - A 3-point running mean procedure; - An absolute value of the magnetic field density, i.e. an unsigned flux. 248 high resolution SOHO/MDI magnetograms of active region NOAA 9077 obtained on July 14, 2000 between 06:26UT and 11:00UT. The entire area of the active region (145x145 arcsec or 250x250 pixels) was analyzed.

6. Two codes to determine flux concentrations: The Circle code The Maximum-gradient code

7. Probability Distribution Function

13. m and s 2 are the mean and the variance of the Gaussian distribution of log(  ) Lognormal Distribution Function: The expected value: The variance :

15. Probability Distribution Function

16. Discussion Discussion I. Longcope, Mc Leish & Fisher (2003, ApJ 599) : a viscoelastic theory the distribution function a viscoelastic theory of interaction between turbulent flows and fibril magnetic fields. The theory is based on an assumption of a back-reaction of fibrils on the plasma flow. All aspects of the viscous back-reaction depend on the distribution function of the magnetic flux in fibrils. log-normally-distributed flux tubes An ensemble of log-normally-distributed flux tubes will provide viscous back-reaction larger than predictions based on exponential distributions.

17. Discussion Discussion II.What does the lognormal distribution imply? product When a random variable u is a product of a large number of independent random variables: u=u 1 · u 2 · u 3 · … · u n · …, sum then log(u) is a sum of a large number of independent variables: log(u)=log(u 1 )+ log(u 2 )+log(u 3 ) + …+ log( u n ) +… And log(u) produces a normal (Gaussian) distribution. Then it is said that u has a lognormal distribution. fragmentation process. This kind of random variables arises, in particular, through the fragmentation process.

18. Discussion Discussion III In the solar photosphere and convective zone: - fragmentation through the turbulent diffusion; - concentration at convergence points of the flows. Petrovay&Moreno-Insertis(1997): erosion in inhomogeneous and/or non-stationary situation, turbulent diffusion dominates over concentration causing a turbulent erosion of magnetic flux tubes. Simon & Leighton (1964): erosion observations of a gradual disintegration of sunspots due to erosion of penumbral boundaries. Bogdan, Gilmar, Lerche, Howard (1988): Fragmentation observed lognormal distribution of areas of sunspot umbra. Fragmentation of magnetic elements may be the essential process in the formation of an observed magnetic structure.