The theory and numerical simulation of non-local mixing-length convection.

Abstract

Local convection theory makes the unphysical prediction that turbulent mixing terminates at the Schwarzschild stability boundary, and existing non-local convection theories have been criticized by Renzini (1987). Since the size of convecting cores bears upon stellar structure and evolution, a self-consistent treatment of non-local convection is needed. We have developed a theory of non-local mixing-length convection based upon a Boltzman transport theory for subsonic, turbulent fluid elements. The momentum and thermal energy excesses of fluid elements are dissipated on the scale of a mixing length. The distribution function, f(t,z,v,T), which is the mass density per velocity-temperature phase space volume, evolves according to the Boltzmann equation. The minimal non-local theory is obtained by taking moments of the Boltzmann equation, up to third order. The local limit of the moment equations reduces to standard mixing-length theory. We extend this moment method to local convection in a composition stratified fluid by considering the evolution of the distribution function, f(t,z,v,T,μ), in velocity-temperature-molecular weight phase space. The stability criteria for convection, semiconvection, and salt-finger instability are derived. To determine closure approximations and evaluate the validity of the moment theory, we have developed an algorithm called Generalized Smooth Particle Hydrodynamics (GSPH) that numerically simulates convection. The vertical structure of the background fluid is calculated by SPH averaging of particles on a grid. Forces on particles are calculated from the background grid and from the local deviations between particles and grid. Particles move vertically only, but the local deviation forces, which account for turbulent losses of momentum and energy, arise from horizontal interactions. GSPH simulations show that the fourth moments are approximately proportional to squares of the second moments in unstable regions, with a proportionality constant between 2 and 4. With this closure approximation, we show that solutions of the moment equations agree well with GSPH results. The closure relations lead to nearly correct second moments, even in overshooting regions where the closure approximations are poor. GSPH simulations of convective overshooting in plane parallel and spherical geometry typically give overshooting distances in the range dₒᵥₑᵣ ≈ 1 - 2ℓ(M). We discuss improvements that we would like to make to the GSPH code and to the analytic work to obtain more precise answers that are directly relevant to realistic stars.