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

I. The stochastic wave equation

LESIA, Observatoire de Paris, PSL Research University, CNRS, Université Pierre et Marie Curie, Université Paris Diderot, 92195 Meudon, France
e-mail: jordan.philidet@obspm.fr

Received: 7 June 2021
Accepted: 3 September 2021

Abstract

Context. The development of space-borne missions such as CoRoT and Kepler now provides us with numerous and precise asteroseismic measurements that allow 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. The aim of this series of papers is to build 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 timescale. In this first paper we derive 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 a 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 of 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 turbulence-oscillation coupling, and can therefore be used to constrain the properties of the turbulence itself with the help of asteroseismic observations. This will be the subject of the rest of the papers in this series.