3D Simulations and MLT. I. Renzini’s Critique

Abstract

Renzini (1987) wrote an influential critique of mixing-length theory (Böhm-Vitense 1958, 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 Woodward et al. 2015). 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.

Relative importance of convective uncertainties in massive stars

“…Therefore, the boundary is not a stiff location but it bends and stretches (Cristini et al 2017). Consequently, the chemical composition gradient is erased and the boundary becomes more similar to the Schwarzschild boundary (Meakin & Arnett 2007;Arnett et al 2019). This is an initial value problem; the convective boundary location of the growing instability starts as the Ledoux location and moves to the Schwarzschild location on a finite time-scale.…”

“…Nevertheless, it is still an issue to large uncertainties, among them the treatment of convective boundaries, and the stellar community only starts to understand these details of convection (e.g. Arnett et al 2018Arnett et al , 2019.…”

Challenges in 2D Stellar Modeling

“…In particular, a better model for convection is needed to replace the current MLT theory, as well as a better understanding of angular momentum transport inside stars. This is an active area of research, and several groups are working on an updated convection model to replace MLT in 1D models (Garaud et al, 2010;Canuto, 2011;Arnett et al, 2019). Magnetic fields have been included in 1D models, but the details of their 2D influence on chemical mixing and angular momentum transport cannot be ignored.…”

“…Multi-dimensional models are common in simulations of shorttimescale events inside stars. Several groups have modeled the internal convection of stars (Woodward et al, 2018;Arnett et al, 2019), but even these simulations often don't include rotation. Typically, convection simulations model a part of a star, either a wedge or a shell.…”

“…Of course, many groups are working on the 2D and 3D properties of stellar convection and turbulence (Davis et al, 2018;Arnett et al, 2019), but these simulations usually cover only a small fraction of the star, and cover a timescale that is short compared to the evolutionary timescale. The ESTER code described above (Espinosa Lara and Rieutord, 2013;Rieutord et al, 2016) models the stellar structure in 2D, and includes the effects of meridional circulation and rotation, but is not capable of modeling the longer timescales required to describe the evolution in 2D.…”