Beyond Mixing-length Theory: a step toward 321D

W. David Arnett, Casey Meakin, Maxime Viallet, Simon W. Campbell, John Lattanzio, Miroslav Moćak

Published 2015-03-01Version 1

We present a physical basis for algorithms to replace mixing-length theory (MLT) in stellar evolutionary computations. The 321D procedure is based on three-dimensional (3D) time-dependent solutions of the Navier-Stokes equations, including the Kolmogorov cascade as a sub-grid model of dissipation (implicit large eddy simulations; ILES). We use Reynolds-averaged Navier-Stokes (RANS) averaging to make 3D simulation data concise, and use 3D simulations to give RANS closure. We sketch a simple algorithm, which is non-local and time-dependent, with both MLT and the Lorenz convective roll as particular subsets of solutions. The damping length is determined from a balance between the large-scale driving and damping at the Kolmogorov scale. We find that (1) braking regions (boundary layers in which mixing occurs) automatically appear {\it beyond} the edges of convection as defined by the Schwarzschild criterion, (2) dynamic (non-local) terms imply a non-zero turbulent kinetic energy flux (unlike MLT), (3) the effects of composition gradients on flow are important, and (4) convective boundaries in neutrino-cooled stages differ in nature from those in photon-cooled stages. The 321D approach may be easily generalized, and allows connections with modern research on turbulent flow of solar and terrestrial fluids and plasmas. Calibration to astronomical systems is unnecessary, so the approach can be predictive rather than merely descriptive. Implications for solar abundances, helioseismology, asteroseismology, nucleosynthesis yields, supernova progenitors and core collapse are indicated.