3.2 Wave excitation

It is generally agreed that solar oscillations are excited by near surface turbulent convection (see, e.g., Goldreich et al., 1994

). The two main approaches to modeling wave excitation have been numerical convection simulations and analytical models based on mixing length convection models.

Goldreich et al. (1994 ) used a mixing length model of solar convection to compute the energy input rates for modes with angular degree less than 60. The model energy input rates were very similar to the observed rates. In the Goldreich et al. (1994 ) model the main source of wave excitation was entropy fluctuations. Numerical simulations of near-surface turbulent convection have also been able to explain the observed frequency dependence of the energy input and damping rates (see, e.g., Stein and Nordlund, 2001 ). In the Stein and Nordlund (2001 ) model, the main source of wave excitation is Reynold’s stresses (turbulent pressure) near the boundaries of granules. Samadi et al. (2003a ) compared wave excitation in a 3d numerical simulation and 1d mixing length based models. The numerical simulation gave about five times more energy input into the p-modes than did the mixing length model. In the numerical simulations of Samadi et al. (2003a ), excitation by entropy fluctuations dominates over excitation by Reynold’s stresses. Samadi et al. (2003b ) used a 3d numerical simulation to study the covariance function of the near-surface turbulent velocity and found that the temporal covariance was not Gaussian. As we will discuss in Section 3.4, this covariance is important for computing the power spectrum of solar oscillations.

As both the numerical convection simulations and the analytical convection models become more developed, it seems likely that they will converge and produce a definitive answer as to the source of solar oscillations.

In order to model the driving of solar oscillations by turbulent convection we add a source term $S$ to the right hand side of Equation (4 ), to obtain

The function $S$ can be thought of as one realization of a stochastic process (granulation). We will later show that the physical quantities that we are interested in, e.g., power spectra, cross-covariances, or ingression-egression correlations, can be written in terms of the covariance of the source function. Following Gizon and Birch (2002

), we define the source covariance matrix as

The indices $i$ and $j$ refer to components of the vector valued source function $S$ . The operator $E$ takes the expectation value of the expression in brackets. When the source model is translation invariant in the horizontal directions and stationary in time, we can write the source covariance as a function of the horizontal separation $' x - x$ , the time difference $' t- t$ , and the two depths $z$ and $' z$ ,

In this case it is convenient to write the source covariance in terms of a Fourier-domain source covariance,

For any particular type of source model we can obtain the corresponding form for the source covariance $M$ .