Forward modelling of sub-photospheric flows for time-distance helioseismology

Context. The results of forward modelling of acoustic wave propagation in a realistic solar sub-photosphere with two cases of steady horizontal flows are presented and analysed by the means of local helioseismology. Aims. This paper is devoted to an analysis of the influence of steady flows on the propagation of sound waves through the solar interior. Methods. The simulations are based on fully compressible ideal hydrodynamical modelling in a Cartesian grid. The initial model is characterised by solar density and pressure stratifications taken from the standard Model S and is adjusted in order to suppress convective instability. Acoustic waves are excited by a non-harmonic source located below the depth corresponding to the visible surface of the Sun. Numerical experiments with coherent horizontal flows of linear and Gaussian dependences of flow speed on depth are carried out. These flow fields may mimic horizontal motions of plasma surrounding a sunspot, differential rotation or meridional circulation. An inversion of the velocity profiles from the simulated travel time differences is carried out. The inversion is based on the ray approximation. The results of inversion are then compared with the original velocity profiles. Results. The results of forward modelling of acoustic wave propagation in a realistic solar sub-photosphere with two cases of steady horizontal flows are presented. The influence of steady flow on the propagation of sound waves through the solar interior is analysed. A time-distance analysis technique is applied to compute the direct observable signatures of the background bulk motions on travel times and phase shifts. This approach allows direct comparison with observational data. Further, we propose a method of obtaining the travel-time differences for the waves propagating in sub-photospheric solar regions with horizontal flows. The method employs directly the difference between travel-time diagrams of waves propagating with and against the background flow. Conclusions. The analysis shows that the flow speed profiles obtained from inversion based on the ray approximation differ from the original ones. The difference between the original and observed profiles is caused by the fact that the wave packets propagate along the ray bundle, which has a finite extent, and thus reach deeper regions of the sub-photosphere in comparison with ray theory. © ESO 2007.