Recently, numerical calculations of linearized wave propagation have been used to study properties for the more complex conditions of the solar atmosphere which is both inhomogeneous and dispersive. There are two important issues: first, because the solar atmosphere is dispersive, waves of different frequencies travel at different speeds (Jefferies et al., 1994), and second, the solar atmosphere’s inhomogeneities in temperature, magnetic field, and density are especially large close to the surface, so they are not small perturbations on a mean background state.
Numerical solutions for linear wave propagation in a mean background solar model with its structure modified so as to be convectively stable (sometimes with a given perturbation) have been used to study various helioseismic issues: How does the dispersive nature of acoustic waves affect their propagation (Tong et al., 2003c; D’Silva, 1998). What is the interaction and coupling between different MHD wave modes (Cally and Goossens, 2007; Rosenthal et al., 2002; Bogdan et al., 2003). How do various types of perturbations and distribution of acoustic sources affect acoustic wave propagation (Tong et al., 2003b,a; Shelyag et al., 2006; Cameron et al., 2007a; Parchevsky and Kosovichev, 2007; Parchevsky et al., 2008). What is the accuracy of time-distance inversion methods – ray tracing and the Born approximation in determining perturbation properties (Hung et al., 2001; Birch et al., 2001; Birch and Felder, 2004; Shelyag et al., 2007). Are there techniques for improving the signal/noise in the observations (Hanasoge et al., 2007). Recently, it has become possible to use realistic models of the near surface layers of the Sun from numerical simulations of solar surface convection to explore the properties of wave propagation and test the accuracy of local helioseismic methods. In such models, p-modes are naturally excited by the convective motions and entropy fluctuations. The mean atmospheric structure is different from the one-dimensional models of the solar atmosphere, which alters the resonant cavity. A complex hierarchy of fluctuations in temperature and sound speed, flow velocities and magnetic flux are present. Such simulations provide the needed test bed of known data against which to test and validate the methods of local helioseismology.
Georgobiani et al. (2003) and Straus et al. (2006) used such three-dimensional simulations to highlight the differences between measuring variables at a constant geometric height in the simulations and at a given local optical depth. The latter corresponds well with observations. Steiner et al. (2007) drove acoustic waves into a magneto-convection simulation from the bottom and then studied the coupling of acoustic and magnetic modes near the level, where magnetic pressure becomes equal to the gas pressure. They also calculated the wave travel times between different levels in the atmosphere corresponding to commonly observed lines and showed that such observations may be used to map the topography of the magnetic field. Georgobiani et al. (2007) showed that supergranulation scale (48 Mm wide by 20 Mm deep) simulations of quiet Sun surface convection of Stein et al. (2007a) possessed a diagram with f- and p1-p4-modes very similar to those observed by MDI and with travel-time maps that were also nearly the same (Figure 33). Zhao et al. (2007b) showed that the horizontal velocities inferred from the ray-tracing inversion of f-mode travel times are in good agreement with the flows actually in the simulation down to depths of about 4 Mm (Figure 34). The vertical velocity inversions, however, were poorly correlated with the actual flows, often having the opposite sign. Within a 48 Mm horizontal extent, the deepest that rays which return to the surface can penetrate is 6.5 Mm. The same set of simulation data has been used to test helioseismic holography by Braun et al. (2007). They found that Born approximation travel times were in good agreement with the model travel times at shallow depths but became less similar at deeper depths, because of near-surface contributions from nearby (and oppositely directed) flows in the near surface side lobes of the kernel functions (Figure 35). As a result, supergranule scale flows are essentially undetectable below depths of about 5 Mm. Longer-lived, larger-scale flows can be detected at greater depths.
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