Coupling between Turbulence and Solar-like Oscillations: a combined Lagrangian PDF/SPH approach. I — The stochastic wave equation. (arXiv:2109.05983v1 [astro-ph.SR])

<a href="http://arxiv.org/find/astro-ph/1/au:+Philidet_J/0/1/0/all/0/1">J. Philidet</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Belkacem_K/0/1/0/all/0/1">K. Belkacem</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Goupil_M/0/1/0/all/0/1">M.-J. Goupil</a>

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. 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.

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 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.

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. 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.

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 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.

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