Solar oscillations and convection. I. Formalism for radial oscillations

We present a formalism for investigating the interaction between p-mode oscillations and convection by analyzing realistic, three-dimensional simulations of the near-surface layers of the solar convection zone. By choosing suitable definitions for fluctuations and averages, we obtain a separation that retains exact equations. The equations for the horizontal averages contain one part that corresponds directly to the wave equations for a one-dimensional medium, plus additional terms that arise from the averaging and correspond to the turbulent pressure gradient in the momentum equation and the divergence of the convective and kinetic energy fluxes in the internal energy equation. These terms cannot be evaluated in closed form, but they may be measured in numerical simulations. The additional terms may cause the mode frequencies to shift, relative to what would be obtained if only the terms corresponding to a one-dimensional medium were retained-most straightforwardly by changing the mean stratification and more subtly by changing the effective compressibility of the medium. In the presence of time-dependent convection, the additional terms also have a stochastic time dependence, which acts as a source of random excitation of the coherent modes. In the present paper, we derive an expression for the excitation power and test it by applying it to a numerical experiment of sufficient duration for the excited modes to be spectrally resolved.