We have begun a new program to improve the treatment of turbulent convection in stars; it features Convection Algorithms Based on Simulations (CABS). We analyze stellar convection with the aid of 3D hydrodynamic simulations, introducing the turbulent cascade into our theoretical analysis. We devise closures of the Reynolds-decomposed mean field equations by simple physical modeling of the simulations. The procedure is testable and allows systematic improvement. The interpretation of mixing-length theory (MLT) as generally used in astrophysics is incorrect. Direct tests show that the damping associated with the turbulent cascade is that suggested by Kolmogorov  (εK≈(u′)3rms/lCZ), where lCZ is the depth of the convection zone. This implies that the mixing-length parameter α is a simple function of depth of the convection zone, and not freely adjustable. Appropriate adjustment of another parameter of MLT, the "geometric factor", can leave solar models almost unchanged, except for their velocity scale. This parameter adjusts the super-adiabatic region just below the photosphere, and represents a crude interpolation function for characterizing 3D hydrodynamic atmospheres.