The description of stellar turbulent convection requires a minimum of five coupled, time-dependent, nonlocal, differential equations for the five variables: turbulent kinetic energy, turbulent potential energy, turbulent pressure, convective flux, and energy dissipation. Any fewer number of equations makes the model local. In this paper, we present the following results: 1. We derive the five coupled equations using a new turbulence model. The physical foundations and the turbulence statistics on which the model was tested are discussed. The model is able to reproduce the high Rayleigh number laboratory and direct numerical simulation data corresponding to medium-to-high values of the Peclet number (a measure of the efficiency of convection). 2. One of the major difficulties for any stellar convective model is the description of the low-efficiency, low Pe number region in which the physical timescale is no longer the turbulent timescale but the radiative one. No previous turbulence model has been able to incorporate these multiple timescales within the same framework properly. The present model does. 3. Overshooting is an unsolved problem in stellar structure. Its solution requires not only the above ingredients, but an additional one, a nonlocal model. This is because in the stably stratified region where ? - ?ad 0, the only source of energy is diffusion, a nonlocal process. We discuss why the expressions used thus far to describe diffusion terms are inadequate. We then present a model that was successfully tested against LES data on the convective planetary boundary layer. 4. We analyze the nonlocal models of Gough and Xiong and discuss the approximations that are required to derive them from the full set of equations. 5. We discuss a model that relates the up/down drafts filling factors found by DNS/LES to the skewness of the velocity field which can be computed from the turbulence model. The results from DNS/LES and this model can thus be cross-checked. 6. We show that the stationary, local limit of the model reproduces recent local models (independently derived) which have been successfully tested against a variety of astrophysical data. 7. We discuss the fact that if the dissipation is described by a local model with a mixing length l (as done by all authors thus far), the remaining nonlocal equations exhibit divergences which preclude a physical solution to be found. OV results based on this method may be a coincidence since they are arrived at by fine tuning a coefficient. 8. The role of compressibility is discussed.
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