3D Simulations and MLT: I. Renzini's Critique

TL;DR

This paper critically evaluates the limitations of mixing-length theory in stellar evolution, demonstrating that advanced 3D simulations and Reynolds averaging can address key issues and improve modeling of stellar convection.

Contribution

It introduces a self-consistent approach to boundary layers in stellar convection, reducing reliance on ad hoc procedures and advancing the understanding of turbulence in stellar models.

Findings

3D simulations approach required accuracy for stellar convection modeling.

Reynolds averaging captures essential boundary layer dynamics.

Simulations suggest a self-consistent boundary layer structure.

Abstract

Renzini (1987) wrote an influential critique of mixing-length theory (MLT) as used in stellar evolution codes, and concluded that three-dimensional (3D) fluid dynamical simulations were needed to clarify several important issues. We have critically explored the limitations of the numerical methods and conclude that they are approaching the required accuracy. Implicit large eddy simulations (ILES) automatically connect large scale turbulence to a Kolmogorov cascade below the grid scale, allowing turbulent boundary layers to remove singularities that appear in the theory. Interactions between coherent structures give multi-modal behavior, driving intermittency and fluctuations. Reynolds averaging (RA) allows us to abstract the essential features of this dynamical behavior of boundaries which are appropriate to stellar evolution, and consider how they relate static boundary conditions (Richardson, Schwarzschild or Ledoux). We clarify several questions concerning when and why MLT works, and does not work, using both analytical theory and 3D high resolution numerical simulations. The composition gradients and boundary layer structure which are produced by our simulations suggest a self-consistent approach to boundary layers, removing the need for ad hoc procedures for 'convective overshooting' and `semi-convection'. In a companion paper we quantify the adequacy of our numerical resolution, determine of the length scale of dissipation (the `mixing length') without astronomical calibration, quantify agreement with the four-fifths law of Kolmogorov for weak stratification, and extend MLT to deal with strong stratification.