6.1 Wave propagation in the convection zone

Understanding how acoustic and surface gravity waves propagate in the solar convection zone is necessary to be able to use observations of these waves to determine the structure of the layers they propagate through (e.g., the sound speed variations, flow velocities, and magnetic fields). The Green’s function for linearized wave propagation determines how the waves respond to small perturbations (Skartlien, 2002Gizon and Birch, 2005). Green’s functions are the solution of the linearized wave equation with an impulse source in space and time, in an unperturbed mean background state without flows, density or temperature fluctuations (except for the mean vertical stratification) or magnetic fields,
[ ] ∂2-- 2 2 ∂t2 − c ∇ G (r,t) = S (r,t)δ(r − r0)δ(t − t0) , (43 )
where G is the Green’s function and S is the impulse source. The Green’s functions is then used to compute the response of various observable quantities to perturbations in the background state.

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., 2003cD’Silva, 1998). What is the interaction and coupling between different MHD wave modes (Cally and Goossens, 2007Rosenthal et al., 2002Bogdan et al., 2003). How do various types of perturbations and distribution of acoustic sources affect acoustic wave propagation (Tong et al., 2003b,aShelyag et al., 2006Cameron et al., 2007aParchevsky and Kosovichev, 2007Parchevsky 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., 2001Birch et al., 2001Birch and Felder, 2004Shelyag 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.

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Figure 33: Two-dimensional power spectra (ℓ − ν) of the vertical velocity in an 8.5 hour long convection simulation in a domain 48 Mm wide by 20 Mm deep (left) and Doppler velocity from MDI high-resolution observations (right). f- and p1-p4-modes are visible in the simulation data. The dashed curve is the theoretical f-mode ridge. Because of the finite width of the computational box, there are fewer modes which are individually identifiable in the simulation. The low frequency power is convection.
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Figure 34: Comparison the horizontal velocities at a depth of 2 – 3 Mm in the simulation (left) and obtained by inverting travel-times (right) for vertical velocities at a height of 200 km above the surface from the simulation. The small scale flows in the simulation have been removed by filtering.

Georgobiani et al. (2003Jump To The Next Citation Point) and Straus et al. (2006Jump To The Next Citation Point) 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 β = 1 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 k − ω 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 33View Image). 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 34View Image). 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 35View Image). 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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Figure 35: A slice through the center of the Born approximation travel-time kernel for a focus depth of 5 Mm in units of s Mm–3. A negative kernel relates the flow in the positive x-direction to a travel-time decrease.

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