Acoustic wave propagation in the solar
subphotosphere
S. Shelyag, R. Erdélyi, M.J. Thompson
Solar Physics and upper Atmosphere Research Group, Department of
Applied Mathematics, University of Sheffield, Sheffield, UK
Outline
We aim to develop a numerical “toolbox” for helioseismological
studies
-Numerical setup
-Harmonic source
-Local cooling event (non-harmonic source)
-Some analysis
The simulation setup
Full 2-dimensional HD
Cartesian geometry
Total Variation Diminishing spatial discretization scheme
Fourth order Runge-Kutta time discretization
The simulation domain: 150 Mm wide and 52.6 Mm deep, 600x4000 grid points
The upper boundary of the domain is near the temperature minimum
Two boundary regions of 1.3 Mm each at the top and bottom boundaries
The main part of the domain is 50 Mm deep
The simulation domain
We look at the level ~600 km below the upper boundary
The source is located ~200 km below this level
The model profile
density
temperature
convection
sound speed
<0: no
>0: yes
Modified Christensen-Dalsgaard's standard Model, pressure equilibrium.
Convective instability
Convective instability is suppressed:
1=const=5/3
This approach has advantage, because the waves, while propagating through the
quiescent medium, can be observed more clearly, undisturbed by convective
fluid motions far from the source.
Source #1
Harmonic pressure perturbation (cf. Tong et al. 2003):
p  sin 2t / T 
p is the pressure perturbation amplitude
t – real time
T=5.5 min
Evolution of pressure perturbation #1
Consecutive snapshots of pressure deviation p in the simulated domain after the
harmonic perturbation has been introduced. High order acoustic modes produced by
interference of the lower ones can be noticed in the upper part of the domain on the
latest snapshots.
Time-distance diagram #1
Synthetic time-distance diagram (the cut of p/p0 is taken at about 600 km below
the upper boundary of the domain).
Source #2
Localized cooling event causing
local convective instability, mass
inflow
and
sound
waves
extinction

t  t0 

d  1  tanh  log 3 
1 

where timescale 1=120 s
Power spectrum of the source
Velocity field around the source
The behavior of the source in time
can be understood as two stages. In
the beginning, the source creates
expanding inflow and the pressure
and temperature drop. At the
second stage, due to an increased
temperature
gradient,
two
convective cells surrounding the
source are developed.
Evolution of pressure perturbation #2
Time-distance diagram #2
Time-distance diagram produced with the non-harmonic source. The picture is
covered by the flows caused by the source.
Time-distance diagram #2
Pressure cut with high-pass frequency filtering applied. The filtering revealed seismic
traces similar to the ones shown for the harmonic source.
Single non-harmonic source, some analysis
The power spectrum of the time-distance diagram generated by a single perturbation
source. The p-modes are visible up to high orders. The theoretically calculated p1 mode
is marked by two dashed lines.
Multiple non-harmonic sources, some analysis
The power spectrum of a large number of sources randomly distributed along a
selected depth and time. The features, connected with fluid motions caused by these
sources, and the high order p-modes faint with the growth of the number of random
sources. The p1 mode is marked in the same way as before.
To-do list
•Better boundaries are necessary
•Non-uniform grid (and possibility of 3D)
•Magnetic field