Coupling between turbulence and solar-like oscillations: A combined Lagrangian PDF/SPH approach

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. 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 for 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. 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.
Accepted for publication in A&A. 23 pages, 2 figures