Modeling convection in the outer layers of the Sun: A comparison with predictions of the mixing-length approximation

The mixing-length theory (MLT) approximation (Vitense 1953) is used in most stellar evolution codes to describe the structure of the outer, highly superadiabatic, layers of the Sun. This procedure is known to be incorrect because of the MLT's inadequacies in describing convection and because of the need to include the strong coupling between radiation and convection in modeling this region. However, it is not known to what extent and precisely in what ways the MLT approximation distorts the structure of the highly superadiabatic peak in the outer convection zone. The purpose of this paper is to compare the statistical results of a more realistic three-dimensional numerical simulation of shallow convection to the predictions of the MLT. The simulations differ from the previous simulations of Chan & Sofia (1989) in that they include a treatment of radiative transfer (in the diffusion approximation). The layers are superadiabatic and exhibit a sharp peak in the temperature gradient. The results we derive from this simulation provide much more information than conventional one-dimensional theories of convective energy transport. We attempt to analyze or condense the information from the simulation to be compared with a traditional ''theory'' in an effort to establish how much a large eddy simulation can teach us about mean convective transport theories. In this paper we chose to use the mixing-length approximation for comparison. The standard mixing-length approximation predicts a few linear relationships between local thermodynamic and dynamic quantities, the coefficients of which are functions of the mixing length. In these MLT relations, the ratio of mixing length to the local pressure scale is assumed to be constant over the entire convection zone, including the region of high superadiabaticity where convective energy transfer becomes less efficient.