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

Context. The inclusion of convection in stellar evolution models, mostly based on mixing length theories, lacks realism, especially near convective-radiative interfaces. Furthermore, the interaction of convection with oscillations is poorly understood, giving rise to surface effects that currently prevent us from accurately predicting seismic frequencies, and therefore from fully exploiting the asteroseismic data of low-mass stars. Aims. Our aim was 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. Methods. 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. The PDF equation was established so that the underlying transport equations for all first- and second-order moments of the turbulent flow are identical to the exact ones stemming from first principles. We then developed 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. Results. We established a model for the evolution of turbulent properties along Lagrangian trajectories applicable to stellar turbulent convection, with the minimum 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 the 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 that is not possible in grid-based large-eddy simulations, in particular the turbulent pressure, Reynolds stress tensor, internal energy variance and convective flux.