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

Abstract

Context. The development of space-borne missions such as CoRoT or Kepler now provides us with numerous and precise asteroseismic measurements, which allows us to put better constraints on our theoretical knowledge of the physics of stellar interiors. In order to utilise the full potential of these measurements, however, we need a better theoretical understanding of the coupling between stellar oscillations and turbulent convection. Aims. This series of papers aims at building a new formalism specifically tailored to study the impact of turbulence on the global modes of oscillation in solar-like stars. In building this formalism, we circumvent some fundamental limitations inherent to the more traditional approaches, in particular the need for separate equations for turbulence and oscillations, and the reduction of the turbulent cascade to a unique length and time scale. This first paper aims at deriving a linear wave equation that directly and consistently contains the turbulence as an input to the model, and therefore naturally contains the information on the coupling between the turbulence and the modes, through the stochasticity of the equations. Methods. We use a Lagrangian stochastic model of turbulence based on Probability Density Function methods to describe the evolution of the properties of individual fluid particles through stochastic differential equations. We then transcribe these stochastic differential equations from a Lagrangian frame to an Eulerian frame, more adapted to the analysis of stellar oscillations. We combine this method with Smoothed Particle Hydrodynamics, where all the mean fields appearing in the Lagrangian stochastic model are estimated directly from the set of fluid particles themselves, through the use of a weighting kernel function allowing to filter the particles present in any given vicinity. The resulting stochastic differential equations on Eulerian variables are then linearised. As a first step, the gas is considered to follow a polytropic relation, and the turbulence is assumed anelastic. Results. We obtain a stochastic, linear wave equation governing the time evolution of the relevant wave variables, while at the same time containing the effect of turbulence. The wave equation generalises the classical, unperturbed propagation of acoustic waves in a stratified medium (which reduces to the exact deterministic wave equation in the absence of turbulence) to a form that, by construction, accounts for the impact of turbulence on the mode in a consistent way. The effect of turbulence consists in a non-homogeneous forcing term, responsible for the stochastic driving of the mode, and a stochastic perturbation to the homogeneous part of the wave equation, responsible for both the damping of the mode and the modal surface effects. Conclusions. The stochastic wave equation obtained here represents our baseline framework to properly infer properties of turbulenceoscillation coupling, and can therefore be used to constrain the properties of the turbulence itself with the help of asteros...

Coupling between turbulence and solar-like oscillations: A combined Lagrangian PDF/SPH approach. II - Mode driving, damping and modal surface effect

“…This series of papers aims to investigate Lagrangian stochastic models of turbulence for the study of the coupling between solar-like oscillations and turbulent convection, as an alternative to more traditional methods based on mixing-length theory for example. In Paper I (Philidet et al 2021) we established a linear stochastic wave equation designed to govern the oscillations while encompassing the full effect of the turbulence thereon. In this second paper we exploit this wave equation to obtain stochastic equations that govern the coupled evolution of the complex amplitudes of all the modes of the system simultaneously.…”

“…In this section we summarise the results obtained in Philidet et al (2021), hereafter referred to as Paper I. Paper I established a stochastic linear wave equation for solar-like acoustic modes, which consistently encompass the effect of the turbulence, and therefore naturally contains the information on the coupling between turbulence and oscillations.…”

“…In this series of papers, we investigate a novel modelling approach designed to address the issue of turbulence-oscillation coupling in the context of solar-like oscillators, based on Lagrangian stochastic models of turbulence. In the first paper of this series (Philidet et al 2021), we used a Lagrangian stochastic model to obtain a linear stochastic wave equation that is representative of solar-like p-modes and simultaneously contains the effect of the turbulence on the modes. This linear stochastic wave equation constitutes our baseline theoretical framework for quantifying turbulence-oscillation coupling.…”

“…This paper is structured as follows. In Section 2 we recall the linear stochastic wave equation we obtained in the first paper of this series (Philidet et al 2021), and we give a succinct physical interpretation of the various terms that it exhibits. In Section 3 we detail how this stochastic wave equation can be reduced to coupled equations for the complex amplitude of all the individual normal modes of the system, through the application of the simplified amplitude equation formalism, and we explicitly derive this amplitude equation in the case of solar-like p-modes.…”