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

-Numerical setup

-Harmonic source

-Local cooling event (non-harmonic source)

-Some analysis

We aim to develop a numerical “toolbox” for helioseismological studies

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 setup

The simulation domain

We look at the level ~600 km below the upper boundary

The source is located ~200 km below this level

Modified Christensen-Dalsgaard's standard Model, pressure equilibrium.

The model profile

density temperature

sound speed

convection

<0: no

>0: yes

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.

Convective instability

1=const=5/3

Convective instability is suppressed:

Harmonic pressure perturbation (cf. Tong et al. 2003):

Ttp /2sin

p is the pressure perturbation amplitude

t – real time

T=5.5 min

Source #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.

Evolution of pressure perturbation #1

Synthetic time-distance diagram (the cut of p/p0 is taken at about 600 km below the upper boundary of the domain).

Time-distance diagram #1

Localized cooling event causing local convective instability, mass inflow and sound waves extinction

1

03logtanh1

ttd

where timescale 1=120 s

Power spectrum of the source

Source #2

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.

Velocity field around the source

Evolution of pressure perturbation #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.

Time-distance diagram #2

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.

Single non-harmonic source, 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.

Multiple non-harmonic sources, some analysis

To-do list

•Better boundaries are necessary

•Non-uniform grid (and possibility of 3D)

•Magnetic field