A FORMULATION OF CONVECTION FOR STELLAR STRUCTURE AND EVOLUTION CALCULATIONS WITHOUT THE MIXING-LENGTH THEORY APPROXIMATIONS .1. APPLICATION TO THE SUN

One long-standing goal of numerical simulations of turbulent, compressible convection has been the removal of the mixing-length theory (MLT) of convective energy transport from stellar structure and evolution calculations; in this paper, we provide a thorough description of how such a goal may be achieved using a statistical formulation of the energy fluxes and velocities occurring in a convective region. The bases of our technique are approximate relationships between dynamic and thermodynamic properties derived by Chan & Sofia from three-dimensional numerical models of compressible, stratified convection. Approximate relationships for convective fluxes and velocities may be written in a form which resembles the MLT formulation for convective fluxes and velocities, making the implementation of our technique into stellar structure and evolution codes straightforward. We apply our technique to the Sun. A solar model generated using our revised treatment of convection is compared to a solar model generated using MLT. In the application of MLT, the mixing-length parameter (alpha) is adjusted so that almost any combination of input physics can yield a model whose radius matches the measured values of R.. Such freedom in the value of alpha can compensate for all uncertainties and approximations inherent in the model such as opacities, chemical abundances, treatments of the solar atmosphere, and the equation of state. Our revised treatment of convection, on the other hand, does not include any free parameters. This makes our models extremely sensitive to the accuracy of the treatments of the above processes in the sense that gross errors or approximations fail to yield a solar model. We must stress that this work provides neither an evaluation of the mixing-length parameter (alpha) nor a modification of the mixing-length formalism for some aspect of turbulence; rather, we replace MLT entirely and then apply our new treatment of convection to a solar model. All model results (using both the revised treatment of convection and MLT) are compared to measurements of the solar luminosity and the solar radius, and the depth of the solar convection zone as inferred from helioseismology.