Mode excitation may be investigated using the kinetic energy equation for the modes. For radial modes this is the equation for the horizontally averaged variables (indicated by an overbar),

where is the gas pressure, is the turbulent pressure, and is the viscous stress tensor, all averaged over horizontal planes. Likewise is the horizontally averaged mass density, and is the density weighted average vertical velocity (Nordlund and Stein, 2001). Integrate over the spatial domain and time. If there is no net mass displacement the buoyancy work term vanishes. If there is no displacement on the boundaries (e.g., at the bottom), or if the pressure fluctuations vanish on the boundaries (e.g., at the top), then the integral of the divergence term vanishes. The remaining term is the work integral, where is the displacement and is the velocity. Here denotes the “pseudo-Lagrangian” fluctuation, that is the fluctuation in the frame moving with the average radial velocity in which the net vertical mass flux vanishes (cf. Nordlund and Stein, 2001). The pressure fluctuations and displacement can be split into modal and convective parts. The gas pressure fluctuations can further be split into an adiabatic part (ad) proportional to the density fluctuation, and the remaining non-adiabatic part (non-ad). The dominant term is the product of the turbulent pressure plus non-adiabatic gas pressure fluctuations with the divergence of the mode displacement, so the change in the mode amplitude in a time interval is, where , the mode energy per unit surface area (at r = R), is Here is the mode mass and is the mode surface velocity amplitude. The ensemble average of the change in mode energy for small changes in amplitude over all phases between and is proportional to The result for the energy increase per unit area in time interval is where is the imaginary part of the sum of the pseudo-Lagrangian fluctuations of the turbulent pressure (Reynolds stress) and non-adiabatic gas pressure (entropy) at frequency , is the displacement eigenfunction for the mode with frequency , and , where is the time interval over which the expression is evaluated. This factor appears when the discrete Fourier transform is normalized such that the sum of the squares of the absolute values of the discrete amplitudes equals the average of the squares of the fluctuations in time (Parseval’s theorem). If the power spectral density were normalized per unit frequency interval, then the factor would disappear and the total power would be the time integral of the squared amplitude rather than its average. The appearance of in the denominator makes the expression independent of the arbitrary normalization of the mode eigenfunctions . The terms in this expression can then be evaluated using the results of realistic convection simulations (Figure 36) which yields good agreement with observations, or using a model of the convective turbulence properties.Several people have derived equations for p-mode excitation using analytic expressions for the convective kinetic energy and entropy spectra, e.g., (Balmforth, 1992; Goldreich et al., 1994; Samadi and Goupil, 2001). These authors’ formulas and Equation (50) are similar. In all, the excitation rate is proportional to the turbulent pressure and entropy induced pressure fluctuations times the gradient of the displacement eigenfunction divided by the mode mass. The main difference is that Equation (50) contains the absolute value squared of the product of the pressure fluctuations with the mode compression, while the others contain the the absolute value squared of the mode compression times the sum of the absolute values squared of the turbulent pressure and entropy fluctuation contributions. This difference results from the assumption (explicit or implicit) that there are no correlations between the turbulence at different scales and that the mode compression, , does not change on the length scale of the turbulence and so can be removed from the integral over the local turbulence. Neither of these assumptions is correct and is not necessary. Chaplin et al. (2005) have derived an analytic expression for the mode amplitude where the pressure-mode compression interaction is retained.

Aside from the above simplifying assumptions, there is the question of characterizing the turbulence properties. It is generally assumed that the convective energy spectrum is separable into independent spatial and temporal factors. Convection simulations have revealed that this is not possible. The convection velocity spectrum, is found to have the form (Georgobiani et al., 2006)

The width and exponent for the vertical velocity are shown in Figure 37.Samadi et al. (2003a,b) investigated in detail the effects of different assumptions about the turbulence properties on the p-mode excitation rate. They found that the mode driving is particularly sensitive to the turbulence anisotropy factor and to the dynamic properties of the turbulence as represented by the temporal part of the turbulence spectrum. Using the simulation results, fit by analytic expressions for these factors, increases the excitation rate and the contribution of turbulent pressure relative to that of entropy fluctuations.

What physics determines the spectrum of the p-mode driving? Mode driving (Figure 36) decreases at low frequencies for two reasons: first, the mode mass (inertia) increases with decreasing frequency (Figure 38), and second the mode compression decreases with decreasing frequency (Figure 39). The mode driving decreases at high frequencies because convection is a low frequency process (see Figure 33), so the power in the pressure fluctuations decreases at high frequencies (Figure 40).

In the Sun and cool stars the contributions from the Reynolds stresses and entropy fluctuations are comparable at the peak driving frequencies, while the Reynolds stress excitation dominates at lower frequencies where the excitation occurs over a larger range of depths. In hotter stars and giants the Reynolds stress excitation dominates at all frequencies.

P-mode excitation occurs close to the solar surface where the turbulent velocities and entropy fluctuations are largest (Figure 41). As the frequency increases the driving is more and more confined to a shallow layer just below the surface.

Jacoutot et al. (2008b) and Jacoutot et al. (2008a) have investigated the influence of the presence of magnetic fields on the excitation of p-modes. They find that regions with magnetic field strengths typical of the peripheries of active regions have enhanced excitation of p-modes, particularly at high frequencies. This is consistent with the observations of so-called “acoustic halos” (Braun et al., 1992; Brown et al., 1992; Hindman and Brown, 1998; Jain and Haber, 2002) around active regions.

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