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\begin{titlepage}
\title{Multiscale Analysis of Magnetograms and Dopplerograms}

\author{Lead Investigator: A. Ruzmaikin (JPL/CSUN) \\
\and
Team Members: A.C. Cadavid, and J. K. Lawrence (CSUN) \\
\and
SOI Coordinators: Ted Tarbell (Lockheed)}

\date{}

\maketitle
\end{titlepage}

\newpage
\begin{abstract}
Our scientific objective is to study the ways in which magnetic fields are spread across the 
surface of the Sun and how they are gathered into local concentrations. We hope to identify 
various transport mechanisms of fields in the turbulent, convective photospheric plasma as 
well as the spatial and temporal scales at which these mechanisms operate. This will include
exploration of new approaches to the problem of solar turbulent diffusion. We propose to 
match the turbulent diffusive models to time series of high-resolution Doppler and magnetic 
MDI images of both solar active regions and quiet Sun areas.  Supplementary comparison 
with contemporaneous correlation tracking and Extreme-Ultraviolet  Imaging Telescope 
data will be important as well.
\end{abstract}

\newpage
\section{Investigation Plan}
The work will focus on two main issues:

\subsection*{Multifractal Analysis and Noise}
The objective is to study the structural properties of magnetic and velocity fields at the small 
scales at which magnetic fields are expected to be concentrated into flux tubes or evolve 
into more complex forms with a continuous spectrum of scales. We have previously 
demonstrated that the line-of-sight fields in solar magnetic images possess self-similar 
scaling (Lawrence et al. 1993, Cadavid et al. 1994, Lawrence et al. 1995). Preliminary 
studies indicate the same result for Doppler images (Ruzmaikin et al. 1994). Here the 
situation is complicated by the need to remove the effects of p-mode oscillations; MDI 
Doppler image time series should be optimal for accomplishing this. From these results we 
can calculate the filling factor in various regions and can determine the appropriate 
technique for integration of functions using the multifractal measure.  Furthermore, the
multifractal spectra of the magnetic and Doppler images are such as to indicate the presence 
of fields concentrations (singularities). By combining multifractal and two-dimensional 
wavelet analysis, we can identify locations of the singularities of the multifractal spectrum 
and can investigate whether they correspond to regions of maximum energy dissipation.
The X-ray images from Yohkoh (TRACE) and Extreme-Ultraviolet Imaging Telescope data 
from SOHO itself taken at the same time will be ideal for this study.

Gaussian noise in digital, photoelectric images has been found to have a characteristic 
scaling signature. Thus, as a byproduct of the multifractal analysis it is possible to quantify 
the amount and distribution of such noise in each image.

\subsection*{The Study of Turbulent Diffusion}
A second avenue of investigation involves observations of the photospheric velocity field. 
The goal is to characterize the random components of motions of photospheric magnetic 
features in quiet Sun and in the enhanced network and plage near active regions. This 
includes not only estimation of a diffusion coefficient for the motions, but also of possible 
subdiffusive or superdiffusive properties and effective fractal dimensions of the random 
walks (Lawrence 1991; Schrijver, et al. 1992; Lawrence \& Schrijver 1993; Schrijver 
1994; Ruzmaikin et al. 1995).

We will use a self-similar (power-law) form for the velocity field. This approximation is 
valid in the ``intermediate asymptotic'' range of scales between granular and supergranular 
sizes. The idea is to find the two basic parameters of the time-spatial spectrum of this 
velocity field: the exponent $\beta $ in the spatial power spectra $E(k)\propto k^{-\beta }$, 
and the exponent $z$ of the spatial scale dependence of correlation times $\tau $ in the non- 
linear cascade ($\tau \propto k^{-z}$). In the intermediate asymptotic regime these two 
numbers define the character of turbulent diffusion of passive large-scale scalar quantities, 
i.e. the exponent in the dependence of the squared distance on time $\langle d^2\rangle 
\propto t^h$ (Avellaneda and Majda 1992; Ruzmaikin et al. 1995). For the normal 
diffusion $h=1/2$; $h>1/2$ for an enhanced diffusion; and $h<1/2$ for a slow diffusion. 
Although scale-frequency studies in this range have been conducted previously for Doppler 
images taken at La Palma, Kitt Peak, Big Bear, and Huairou observatories (Title et al. 
1989; Chou et al. 1991) the parameter $z$, which characterizes the time correlation of
every Fourier mode, was not determined in these studies. (In the previous studies the 
characteristic time was defined to be the correlation lifetime determined by measuring the 
1/e width of the temporal autocorrelation function averaged over spatial scales.) In the 
application to the turbulent diffusion problem, this correlation time carries essential 
information about the velocity field. The definition of the time parameter and finding it
(which is not easy task), together with the characterization of the type of turbulent 
transport, are the essential differences between these earlier studies and the studies 
proposed here.

This investigation is rendered especially interesting by the discovery (Avellaneda and Majda 
1992) that different regions of the two-dimensional ``$\beta -z$ diagram'' contain very 
different kinds of diffusion. For example, when $\beta <1-z$, called region I, the long 
range spatial correlation is weak, and the familiar Fickian diffusion occurs. On the other 
hand, when $\beta >3-2z$, called region III, long range spatial correlations are very strong.
Here the diffusion is independent of the temporal correlation, but is non-stationary and 
superdiffusive, with an effective diffusion coefficient growing linearly in time. Region II, 
with $1-z<\beta <3-2z$, is intermediate and characterized by superdiffusive but stationary 
transport.  The special case of the Kolmogorov turbulence with $\beta =5/3$ and $z=2/3$
lies exactly on the boundary between regions II and III.

The type of turbulent diffusion occurring in the solar photosphere has not yet been 
established. Our goal is to estimate the parameters $\beta $ and $z$ from a time series of 
Doppler images and with the help of these parameters determine the type of the solar 
diffusion (normal, fast, or slow). For the same purpose we will consider high-order 
correlations in the observed velocity field. If the diffusion is normal the even statistical 
moments of displacements can be expressed through the second moment. For super or sub
diffusion the statistical moments higher than the second are independent quantities, and 
they have own scaling exponents.

The two parameters $\beta $ and $z$ enter into many aspects of the transport of quantities 
advected by the turbulent motions. One such aspect is the fractal dimension of turbulent 
diffusion fronts, that is, of lines or surfaces advected by the small-scale motions of the 
fluid, for example the set defined by $B=0$ or $B=const$. We plan to study two-
dimensional cuts across the field distributions, so we are interested in advected line
elements. For the Avellaneda and Majda model the fractal dimension of the diffusion front 
is directly related to the spectral exponents $\beta $ and $z$. The fractal dimensions of the 
level sets can be computed by a two-dimensional box counting or by a correlation 
procedure with randomized box locations (Lawrence et al., 1993). Another aspect of this 
calculation is the relative diffusion of pairs of fluid markers. This will yield information 
analogous to the familiar ``Richardson 4/3 law'' which holds for hydrodynamic turbulence.

One difficulty with analysis of the Doppler images is that they provide information only 
about the line-of-sight velocities while it is the transverse velocities that presumably control 
the motions of fields across the solar surface. To make the connection between these 
quantities it is necessary to gain information on such structural properties of the motions
as aspect ratios of convective cells on different scales. Such information could be accessed 
by combining analysis of the Doppler images with correlation tracking of the transverse 
motions.

We hope also to evaluate the ``diffusive'' contribution to  the time evolution of the global 
field distribution. Differences between quiet Sun and plage behaviors will provide 
information about the back-reactions of enhanced fields on the plasma motions and may 
lead to understanding of the stability of plage structures. The only data set usable for this 
purpose up to now has been that of Schrijver and Martin (1990). Time series of seeing-free 
MDI magnetograms in which magnetic concentrations can be identified and their motions 
followed over periods of days (gaps are possible) will be crucial in improving the accuracy 
of such studies.

\section{Data Analysis}
To determine the singularity spectrum we will use either a box-counting method (Cadavid, 
et al. 1994) or a multiplier probability distribution method (Lawrence, et al. 1995) together 
with a Monte Carlo technique for sampling that improves the counting statistics. The 
singular points of the multifractal spectrum will be identified on the images by using types
wavelet transforms that have been found to be ideally suited for this purpose (Arneodo, 
Grasseau \& Holschneider 1988; Lawrence, Cadavid \& Ruzmaikin 1994).

To characterize the turbulent diffusion we plan to use space-time spectra of the background 
velocity field. To eliminate the p-mode oscillations we have used a subsonic filter, $\omega 
/k<7$ km/s. The first step, after the filtering 5 min. oscillations, consists of finding the 
spectra for two correlators:
\[
\langle v(x+s,t+\tau )v(x,t)\rangle ,\quad \langle {\frac{dv(x+s,t+\tau )}{
d\tau }}v(x,t)\rangle 
\]
at $\tau =0$. The first one is the standard spatial power spectrum. The second spectrum 
includes the scale distribution of the correlation time. The next step is to find the slopes of 
the linear parts of the spectra plotted in log-log coordinates. The resulting two numbers 
define the character of the turbulent diffusion at least on the observed time interval. We then
calculate the energy spectrum as a function of $\omega $ and ${\bf k}$. The energy 
spectrum of the velocity fields is 
\begin{equation}
E(k)d^2k=\int {k<dv(k,t+\tau )dv^{*}(k,t)>dt}
\end{equation}
where $v(k,t)$ is the spatial FFT of the velocity, and the integral is taken over the whole 
time interval. The exponent $\beta $ is given by the fit to the straight line $E(k)\propto k^{-
\beta }$ in the self-similar part of the spectrum. An estimate for the decorrelation time 
exponent $z$ is found by use of the relation 
\begin{equation}
E(k,\omega =0)\propto k<v(k,0)v^{*}(k,0)>\propto k^{-\beta -z}.
\end{equation}

\section{Required Observations}
Series of the high-resolution line-of-sight velocity images at intervals of 60 sec or less for 
up to several hours time. Present ground-based data sets, with attendant seeing distortions, 
extend over only 4 hours. Quiet and active regions have to be taken separately.  In 
particular, this gives us an opportunity to estimate the back action of the large-scale 
magnetic field (which is present in active regions) on the diffusion. We are interested in the 
background continuous spectrum so that it is desirable to get rid of 5 min. oscillations by 
filtering. The space measurements are needed to avoid the atmospheric distortions intrinsic 
to the ground based observations and which make the analysis of the image time series 
difficult. Seeing-free measurements will make feasible the effective filtering of the 5 min. 
oscillations. Comparison with parallel ground based observations (at San Fernando and La 
Palma Observatories) of line-of-sight components of the magnetic and velocity fields will 
give the level and other characteristics of the distortions.

Series of MDI magnetograms at high resolution at intervals of a few hours and covering a 
given region of the Sun for several days. These should permit the identification of magnetic 
concentrations from frame to frame and permit tracking of their transverse velocities. 

\section{Outstanding Problems}
Passive scalar diffusion may be a good first approximation to the diffusion of the line-of-
sight component of the solar magnetic field. The solar large-scale magnetic field however is 
not a passive scalar. It is a vector which can affect the turbulent motions. We are in the 
process of developing theoretical models of magnetic field diffusion which take into 
account the back action of the fields on the motions.  Another extension, similar to a well-
known reaction-diffusion approach, is to include the effects of magnetic field reconnections 
and amplifications.  Possible observational implementations will be reported later.

\begin{thebibliography}{99}
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\bibitem{}  Ruzmaikin, A., C. Cadavid, G. Chapman, J. Lawrence, and S.
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\bibitem{}  C. J. Schrijver 1994, in R. Rutten and C. J. Schrijver, eds., 
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\end{thebibliography}

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