  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 (we relate temperature and density fluctuations via
coefficients); the procedure (CABS, Convection Algorithms Based on Simulations)
is terrestrially testable and is amenable to systematic improvement. We develop
a turbulent kinetic energy equation which contains both nonlocal and time
dependent terms, and is appropriate if the convective transit time is shorter
than the evolutionary time scale. The interpretation of mixing-length theory
(MLT) as generally used in astrophysics is incorrect; MLT forces the mixing
length to be an imposed constant. Direct tests show that the damping associated
with the flow is that suggested by Kolmogorov. The eddy size is approximately
the depth of the convection zone, and this dissipation length corresponds to
the "mixing length". New terms involving local heating by turbulent dissipation
should appear in the stellar evolution equations. The enthalpy flux
("convective luminosity") is directly connected to the buoyant acceleration,
and hence the velocity scale. MLT tends to systematically underestimate this
velocity scale. Quantitative comparison with a variety of 3D simulations
reveals a previously recognized consistency. Examples of application to stellar
evolution will be presented in subsequent papers in this series.
