A new particle-based code for Lagrangian stochastic models applied to stellar turbulent convection

Abstract

The inclusion of convection in stellar evolution models lacks realism, especially near convective-radiative interfaces. Furthermore, the interaction of convection with oscillations prevent us from accurately predicting seismic frequencies, and therefore from fully exploiting the asteroseismic data of low-mass stars. We aim to develop a new formalism to model the one-point statistics of stellar convection, to implement it in a new numerical code, and to validate this implementation against benchmark cases. This new formalism is based on Lagrangian Probability Density Function (PDF) methods, where a Fokker-Planck equation for the PDF of particle-based turbulent properties is integrated in time. We then develop a Monte-Carlo implementation of this method, where the flow is represented by a large number of notional particles acting as realisations of the PDF. Notional particles interact with each other through the time- and space-dependent mean flow, which is estimated from the particle realisations through a scheme similar to Smoothed Particle Hydrodynamics. We establish a model for the evolution of turbulent properties along Lagrangian trajectories applicable to stellar turbulent convection, with only a minimal number of physical assumptions necessary to close the system. In particular, no closure is needed for the non-linear advection terms, which are included exactly through the Lagrangian nature of formalism. The numerical implementation of this new formalism allows us to extract time-dependent maps of the statistical properties of turbulent convection in a way which is not possible in grid-based large-eddy simulations, in particular the turbulent pressure, Reynolds stress tensor, internal energy variance and convective flux.