Coupling between turbulence and solar-like oscillations: a combined Lagrangian PDF/SPH approach. I. The stochastic wave equation

Context. The ever-increasing quality of asteroseismic measurements offers a unique opportunity to use the observed global acoustic modes to infer the physical properties of stellar interiors. In solar-like oscillators, the finite lifetime of the modes allows their amplitudes and linewidths to be estimated, which provide invaluable information on the highly turbulent motions at the top of the convective envelope. But exploiting these observables requires a realistic theoretical framework for the description of the turbulence–oscillation coupling. Aims. The first paper of this series established a linear stochastic wave equation for solar-like p-modes, correctly taking the effect of turbulence thereon into account. In this second paper, we aim at deriving simultaneous expressions for the excitation rate, damping rate, and modal surface effect associated with any given p-mode, as an explicit function of the statistical properties of the turbulent velocity field. Methods. We reduce the stochastic wave equation to complex amplitude equations for the normal oscillating modes of the system. We then derive the equivalent Fokker-Planck equation that governs the evolution of the probability density function jointly associated with the real amplitudes and phases of all the oscillating modes of the system simultaneously. The effect of the finite-memory time of the turbulent fluctuations (comparable to the period of the modes) on the modes themselves is consistently and rigorously accounted for, by means of the simplified amplitude equation formalism. This formalism accounts for mutual linear mode coupling in full, and we then turn to the special single-mode case. This allows us to derive evolution equations for the mean energy and mean phase of each mode, from which the excitation rate, the damping rate, and the modal surface effect naturally arise. Results. The expressions obtained here (1) are written as explicit functions of the statistical properties of turbulence, thus allowing for any prescription thereof to be tested against observations, (2) include the contribution of the turbulent dissipation more realistically, and (3) concern the excitation rate, the damping rate, and the modal surface effect of the modes simultaneously. We show that the expression for the excitation rate of the modes is identical to previous results obtained through a different modelling approach, thus supporting the validity of the formalism presented here. We also recover the fact that the damping rate and modal surface effect correspond to the real and imaginary part of the same single complex quantity. We explicitly separate the different physical contributions to these observables, in particular the turbulent pressure contribution and the joint effect of the pressure-rate-of-strain correlation and the turbulent dissipation. We show that the former dominates for high-frequency modes and the latter for low-frequency modes. To illustrate the usefulness of this formalism, we apply it to a simplified case where we can quantify the relative importance of these two contributions, and in particular the threshold between the two frequency regimes, as a function of the turbulent frequency and the degree of anisotropy of both the Reynolds-stress tensor and the dissipation of turbulent energy. Conclusions. The formalism developed in these first two papers, applied to the case of a simplified Lagrangian stochastic model for proof-of-concept purposes, indeed proves to be viable, relevant, and useful for addressing the issue of turbulence–oscillation coupling in the context of solar-like oscillators. It opens the door to subsequent studies physically more appropriate to the stellar case. It will also allow, once mode coupling is included (i.e. by going beyond the single-mode case), for a realistic description of mode-mode scattering and its influence on mode damping, mode frequency, and the energy distribution across the solar p-mode eigenspectrum.