This paper contains (1) a physical argument to show that the one-eddy MLT model underestimates the convective flux F-c in the high-efficiency regime, while it overestimates F-c in the low-efficiency regime, and (2) a new derivation of the F-c(MLT) using a turbulence model in the one-eddy approximation. (3) We forsake the one-eddy approximation and adopt the Kolmogorov spectrum to represent the turbulent energy spectrum. The resulting F-c > F-c(MLT) in the high-efficiency regime, and F-c < F-c(MLT) in the low-efficiency case, are in agreement with the physical arguments concerning the one-eddy MLT model. (4) By forsaking the Kolmogorov model and solving a two-point closure model, one obtains the CM model. The F-c(CM) satisfies (1). F-c(CM) corresponds to a ''tilt'' in efficiency space of F-c(MLT), an effect that cannot be achieved by changing alpha. We discuss the astrophysical tests of the CM model. (5) Concerning the laboratory turbulent convection, we show that the CM model provides a better fit than the MLT to recent high Rayleigh number (Ra) laboratory data on convection. (6) Concerning nonlocal convection, the most complete model available is the one-point closure model (Reynolds stress model), which entails five differential equations for the five second-order moments. We present the solution corresponding to the local, stationary case. The results are expressed analytically in terms of Ko (Kolmogorov constant), Pe (Peclet number), and S (convective efficiency). (7) We find that the superadiabatic temperature gradient is given by -partial derivative T/partial derivative r - c(p)(-1)g(r) where the renormalized g(r) = g(1 + g(-1)rho(-1) dpt(d)/dz) and p(t) is the turbulent pressure. This result, which follows naturally from the Reynolds stress approach, contrasts with previous empirical suggestions to include p(t). (8) We derive new expressions for the turbulence pressure using two different turbulence models and (9) we show that the often used Kolmogorov-Prandtl expression for the turbulent diffusivity is valid only in the high convective efficiency limit. We derive a new expression valid for arbitrary Peclet numbers. (10) We derive an expression for the flux conservation law, which includes F(KE), the flux of turbulent kinetic energy, a third-order moment for which we provide a new expression. (11) No model has thus far accounted for the influence on F-c due to the presence of a stable layer (radiative layer) bordering the convective zone. We work out the first such model, and (12) we discuss topics for future research.
