Turbulent convection in stellar interiors. II. the velocity field

We analyze stellar convection with the aid of three-dimensional (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 (Convection Algorithm 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 timescale. 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 (ε_K ρ(u′)³ _rms/ℓ _D, where ℓ_D is the size of the largest eddy and (u′)_rms is the local rms turbulent velocity). This eddy size is approximately the depth of the convection zone ℓ_CZ in our simulations, and corresponds in some respects to the mixing length of MLT. New terms involving the local heating due to turbulent dissipation should appear in the stellar evolutionary equations, and are not guaranteed to be negligible. The enthalpy flux (stellar "convective luminosity") is directly connected to the buoyant acceleration, and hence to the scale of convective velocity. MLT tends to systematically underestimate the velocity scale, which affects estimates of chromospheric and coronal heating, mass loss, and wave generation. Quantitative comparison with a variety of 3D simulations reveals a previously unrecognized consistency. Extension of this approach to deal with rotational shear and MHD is indicated. Examples of application to stellar evolution will be presented in subsequent papers in this series.