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 b e incorrect because of the MLT's inadequacies in describing convection and because of the need to include the strong coupling between radiat ion and convection in modeling this region. However, it is not known t o what extent and precisely in what ways the MLT approximation distort s the structure of the highly superadiabatic peak in the outer convect ion zone. The purpose of this paper is to compare the statistical resu lts of a more realistic three-dimensional numerical simulation of shal low convection to the predictions of the MLT. The simulations differ f rom the previous simulations of Chan & Sofia (1989) in that they inclu de a treatment of radiative transfer (in the diffusion approximation). The layers are superadiabatic and exhibit a sharp peak in the tempera ture gradient. The results we derive from this simulation provide much more information than conventional one-dimensional theories of convec tive energy transport. We attempt to analyze or condense the informati on 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 us e 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 f unctions of the mixing length. In these MLT relations, the ratio of mi xing length to the local pressure scale is assumed to be constant over the entire convection zone, including the region of high superadiabat icity where convective energy transfer becomes less efficient.