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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