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\section*{ \large\bf Diagnosis of the Structure of the Active and Quiet Regions
From Observations of High-Frequency Modes and From Time-Distance
Seismology}

\noindent
{\bf Lead Investigator:} R. Nigam\\
{\bf Other Team Members:} R.N. Bracewell, T.L. Duvall, A.G. Kosovichev,
G. Price, P.H. Scherrer & T.D. Tarbell\\

\noindent
{\large\bf Abstract/Technical Summary}

\noindent
Study stratification and magnetic effects of the upper reflection layer 
by inferring the wave speed and the acoustic cut-off frequency from 
observations of high-degree high-frequency modes and from time-distance
diagrams using the SOI Dynamics data.\\

\subsection*{Investigation Plan}

The upper wave reflection layer is of particular interest in helioseimology
because the oscillation frequencies are very sensitive to the structure of this
layer and its variations with the solar cycle (e.g. Ronan {\it et al.},
1994). The variations are particularly large near the acoustic 
cut-off frequency, $\omega_{ac}$, thus providing potential for 
probing structural
changes in the subphotospheric layer due to the magnetic field.
However, the physics of the observed frequency shifts near $\omega_{ac}$
have not been understood.

In this project, we propose a complex study of the structure of the
upper reflection layer using measurements of frequencies of modes
of high angular degree (cf Milford {\it et al.} 1993) 
and the mean time-distance diagrams (Jefferies, {\it et al}, 1994)
in both active and quiet regions, and developing theoretical
models of propagation of acoustic waves excited by a localized
source, in the subphotospheric zone and in the atmosphere.

For the observational part, we plan to use the Dynamics data.
The main problem of the data analysis will be to obtain
accurate measurements of high-degree, high-$\nu$ mode
frequencies separately in quiet and active regions.
Currently, such measurements have been made only for the full disk.
The time-distance diagrams will be also obtained separately
for the quiet and active regions, and for several different
frequency bands.

The theoretical research will focus on studying the equation of
propagation of acoustic waves in the presence of magnetic
field with a right hand side stochastic source term
\begin{equation}
V'' + \frac{(\omega^2 - \omega_{ac}^2)}{(c_S^2 + V_a^2)} V = F_{\rm src},
\end{equation}
where $V$ is the velocity perturbation, $c_S$ is the sound speed,
$V_a$ is the Alfven speed, $\omega$ is the wave frequency, 
$\omega_{ac}$ is the acoustic cut-off frequency.
Using this equation, we plan to study the theoretical spectrum of eigenmodes
below and above $\omega_{ac}$, the sensitivity of the spectrum to
variations of  $\omega_{ac}$, $c_S$ and $V_a$, and the amplitude
spectrum of the modes excited by a localized source. 
The equation will be solved by numerical (Milford {\it et al}, 1992) 
and asymptotic methods.
Various absorbing and non-reflecting boundary conditions will be
formulated at the upper boundary which would avoid the spurious
numerical reflections present in the Sommerfield radiation condition.
From the asymptotic analysis it would be possible to create maps of the
phase shifts due to the inhomogeneous medium in the presence of magnetic
field. In short, the phase shift as a function of latitude, longitude
and magnetic field will be calculated and an analogue to Duvall's law
will be determined. The phase shift can be theoretically related to
time-distance methods, from which it should be possible to determine
the shift from observations.
These results will be used to study variations of the frequency
difference $\omega_{n,l} - \omega_{n-1,l}$ in the presence of
magnetic field. The difference will be compared with the observational
results. When the stochastic source term is included in the right hand
side of equation (1), the spectrum becomes non-stationary lending itself
to a time-frequency analysis in the form of a wavelet transform. A simplified
physical model to mimic the observations will be developed. Including the source
term entails in doing an eigenfunction expansion for the reduced wave equation (1)
before applysing the outgoing radiation boundary conditions.
The theoretical results will be also used to estimate the
reflection coefficient above the acoustic cut-off frequency, and
explain the physical phenomena of mode leakage and p-mode ridges above
the acoustic cut-off frequency.

\begin{thebibliography} {}
\bibitem{} Jefferies, S.M., Osaki, Y., Shibahashi, H., Duvall, T.L., Jr.,
 Harvey, J.W., Pomerantz, M.A., 1994, {\it Astrophys. J.}, {\bf 434}, 795-800
\bibitem{}  Milford,P.N., Frank, Z., Gough, D.O.,  Kosovichev, A.G.,  
and Scherrer,P.H., 1993 
in: Proc. GONG 1992 Conference, {\it Seismic investigation
of the sun and stars}, ed. T.Brown, Astr. Soc. Pac., San Francisco, 97-100
\bibitem{} Ronan, R.S., Cadora, K., LaBonte, B.J., 1994, {\it Solar Phys.},
{\bf 150}, 389-392
\end{thebibliography}
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