Distribution of the magnetic Distribution of the magnetic flux in elements of the flux in elements of the

magnetic field in an active magnetic field in an active region region

Valentyna Abramenko

Big Bear Solar Observatory, NJIT

INTRODUCTIONINTRODUCTION

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

INTRODUCTIONINTRODUCTION

Magnetic coupling between the photosphere and the

corona

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

INTRODUCTIONINTRODUCTIONThe 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)•1018 Mx; quiet sun areasPower Law with the index –1.67 (intranetwork) and –1.27(network)

Schrijver, Title, van Ballegooijen, Hagenaar, Shine 1997 the flux range:(0.7-5)•1018 Mx; quiet sun areasExponential approximation

Schrijver, Title, Hagenaar, Shine 1997, Sol.Phys.175 (the flux range: (0.7-150)•1018 Mx; quiet sun; plage areas outside sunspots

Exponential approximation with the varying index

Abramenko – present studythe flux range: (0.2-250)•1018 Mx; active region

Lognormal + Power Law

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.

Two codes to determine flux concentrations:

The Circle code The Maximum-gradient code

Probability Distribution Function

Probability Distribution Function

Probability Distribution Function

Probability Distribution Function

Probability Distribution Function

Probability Distribution Function

m and s2 are the mean and the variance of the Gaussian distribution of log()

Lognormal

Distribution

Function:

The expected value:

The variance :

Probability Distribution Function

Discussion Discussion

I. Longcope, Mc Leish & Fisher (2003, ApJ 599) :

a viscoelastic theorya 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 functionthe distribution function of the magnetic flux in fibrils.

An ensemble of log-normally-distributed flux tubeslog-normally-distributed flux tubes will provide viscous back-reaction larger than predictions based on exponential distributions.

Discussion Discussion

II. What does the lognormal distribution imply?

When a random variable u is a productproduct of a large number of independent random variables:

u=u1 · u2 · u3 · … · un · …,

then log(u) is a sumsum of a large number of independent variables:

log(u)=log(u1 )+ log(u2 )+log(u3 ) + …+ log( un ) +…

And log(u) produces a normal (Gaussian) distribution. Then it is said that u has a lognormal distribution.

This kind of random variables arises , in particular, through the fragmentation process.fragmentation process.

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): in inhomogeneous and/or non-stationary situation, turbulent diffusion

dominates over concentration causing a turbulent erosionerosion of magnetic flux tubes.

Simon & Leighton (1964): observations of a gradual disintegration of sunspots due to erosionerosion of

penumbral boundaries.

Bogdan, Gilmar, Lerche, Howard (1988): observed lognormal distribution of areas of sunspot umbra. FragmentationFragmentation of

magnetic elements may be the essential process in the formation of an observed magnetic structure.