Abstract
Abstract Renzini wrote an influential critique of “overshooting” in mixing-length theory (MLT), as used in stellar evolution codes, and concluded that three-dimensional fluid dynamical simulations were needed. Such simulations are now well tested. Implicit large eddy simulations connect large-scale stellar flow to a turbulent cascade at the grid scale, and allow the simulation of turbulent boundary layers, with essentially no assumptions regarding flow except the number of computational cells. Buoyant driving balances turbulent dissipation for weak stratification, as in MLT, but with the dissipation length replacing the mixing length. The turbulent kinetic energy in our computational domain shows steady pulses after 30 turnovers, with no discernible diminution; these are caused by the necessary lag in turbulent dissipation behind acceleration. Interactions between coherent turbulent structures give multi-modal behavior, which drives intermittency and fluctuations. These cause mixing, which may justify use of the instability criterion of Schwarzschild rather than the Ledoux. Chaotic shear flow of turning material at convective boundaries causes instabilities that generate waves and sculpt the composition gradients and boundary layer structures. The flow is not anelastic; wave generation is necessary at boundaries. A self-consistent approach to boundary layers can remove the need for ad hoc procedures of “convective overshooting” and “semi-convection.” In Paper II, we quantify the adequacy of our numerical resolution in a novel way, determine the length scale of dissipation—the “mixing length”—without astronomical calibration, quantify agreement with the four-fifths law of Kolmogorov for weak stratification, and deal with strong stratification.
Highlights
The standard treatment of convection in stellar evolution theory is “mixing-length theory” (Böhm-Vitense 1958, MLT), which uses a semi-empirical, “engineering” approach, based upon an approximate model due to Prandtl (Prandtl 1925; Clayton 1968; Kippenhahn & Weigert 1990; Hansen et al 2004)
Basing our analysis on 3D simulations, which involve minimal assumptions for this problem, we focus on the question: why does MLT work at all, and what is still missing? It has become traditional to complain about the flaws of MLT, but as Renzini (1987) emphasized, MLT works surprisingly well in some respects
25 In our simulations, we find that convective mixing is efficient inside the convection zone, but that composition gradients may develop on the nonconvective side of boundaries
Summary
The standard treatment of convection in stellar evolution theory is “mixing-length theory” (Böhm-Vitense 1958, MLT), which uses a semi-empirical, “engineering” approach, based upon an approximate model due to Prandtl (Prandtl 1925; Clayton 1968; Kippenhahn & Weigert 1990; Hansen et al 2004). It is local, requires calibration, and has little connection to modern methods used by the turbulence community (e.g., Pope 2000; Davidson 2004).
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