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

TL;DR

The paper tackles the challenge of realistically modeling convection in stellar interiors and its coupling to oscillations, which is difficult for traditional Mixing-Length Theory and limited by grid-based simulations. It develops a Lagrangian PDF formalism using notional particles with SDEs for position, velocity, internal energy, and turbulent frequency, linked to a Fokker-Planck description that reproduces exact moment transport while avoiding closure of nonlinear advection. The authors implement a Monte Carlo, stand-alone code that couples particle realizations through a kernel-based mean field and apply a Corrective Smoothed Particle Method to ensure consistency, validating the approach with Poiseuille/Couette and Simplified Langevin Model benchmarks and demonstrating a 2D convective case with detailed energy flux budgets, upflow/downflow asymmetries, and turbulent diffusivity estimates. This framework yields time-dependent maps of turbulent pressure, Reynolds stresses, energy variance, and convective flux—facilitating direct study of wave-convection interactions and surface effects in asteroseismology, with clear paths to 3D extension and broader stellar applications.

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.

Figures (12)

• Figure 1: Incompressible Poiseuille flow benchmark case. The coloured dots show snapshots of the Poiseuille velocity profile at different times. The solid lines show the exact analytical solution given by Eq. \ref{['eq:exact_poiseuille']}. Relative errors remain under $1 \%$ throughout the simulation.
• Figure 2: Incompressible Couette flow benchmark case. The symbols are the same as in Fig. \ref{['fig:poiseuille']}, with the exact analytical solution being given by Eq. \ref{['eq:exact_couette']}. Relative errors remain under $0.6 \%$ throughout the simulation.
• Figure 3: Evolution of the spatially averaged turbulent kinetic energy $k$ with time, for the benchmark incompressible Simplified Langevin Model case. The dots show the simulation output, while the solid line shows the analytical solution given by Eq. \ref{['eq:decay_law_slm']}. Relative errors remain under $0.35 \%$ throughout the simulation.
• Figure 4: Evolution of the spatially averaged turbulent kinetic energy $k$ with time, for the compressible Simplified Langevin Model case, for different run parameters. Top: comparison of different kernel sizes $h$. Middle: comparison of different time steps $\Delta t$. Bottom: comparison of different particle numbers $N$.
• Figure 5: Snapshot of the particle properties for the convectively unstable simulation described in Sect. \ref{['sec:results']}. Each dot represents one particle, and the color code refers to the individual vertical velocity $u_x^\ast$ (top), specific internal energy $e^\ast$ (middle), and turbulent frequency $\omega^\ast$ (bottom) of each particle. Only $10,000$ particles are shown for readability, whereas the simulation contains $500,000$.