1. 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

2. Outline -Numerical setup -Harmonic source -Local cooling event (non-harmonic source) -Some analysis We aim to develop a numerical “toolbox” for helioseismological studies

3. 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

4. The simulation domain We look at the level ~600 km below the upper boundary The source is located ~200 km below this level

5. Modified Christensen-Dalsgaard's standard Model, pressure equilibrium. The model profile density temperature sound speed convection <0: no >0: yes

6. 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:

7. Harmonic pressure perturbation (cf. Tong et al. 2003):  p is the pressure perturbation amplitude t – real time T=5.5 min Source #1

8. 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

9. Synthetic time-distance diagram (the cut of  p/p 0 is taken at about 600 km below the upper boundary of the domain). Time-distance diagram #1

10. Localized cooling event causing local convective instability, mass inflow and sound waves extinction where timescale  1 =120 s Power spectrum of the source Source #2

11. 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

12. Evolution of pressure perturbation #2

13. Time-distance diagram produced with the non-harmonic source. The picture is covered by the flows caused by the source. Time-distance diagram #2

14. 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

15. 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 p 1 mode is marked by two dashed lines. Single non-harmonic source, some analysis

16. 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 p 1 mode is marked in the same way as before. Multiple non-harmonic sources, some analysis

17. To-do list Better boundaries are necessary Non-uniform grid (and possibility of 3D) Magnetic field