The S tagger-grid: A grid of 3D stellar atmosphere models: III. the relation to mixing length convection theory

Aims. We investigate the relation between 1D atmosphere models that rely on the mixing length theory and models based on full 3D radiative hydrodynamic (RHD) calculations to describe convection in the envelopes of late-type stars. Methods. The adiabatic entropy value of the deep convection zone, s_bot, and the entropy jump, Δs, determined from the 3D RHD models, were matched with the mixing length parameter, α_MLT, from 1D hydrostatic atmosphere models with identical microphysics (opacities and equation-of-state). We also derived the mass mixing length parameter, α_m, and the vertical correlation length of the vertical velocity, C[v_z,v_z], directly from the 3D hydrodynamical simulations of stellar subsurface convection. Results. The calibrated mixing length parameter for the Sun is α^o_MLT (S_bot) = 1.98. αMLT sbot()=1.98. For different stellar parameters, α_MLT varies systematically in the range of 1.7-2.4. In particular, α_MLT decreases towards higher effective temperature, lower surface gravity and higher metallicity. We find equivalent results for α^o_MLT (ΔS)αMLT Δs. In addition, we find a tight correlation between the mixing length parameter and the inverse entropy jump. We derive an analytical expression from the hydrodynamic mean-field equations that motivates the relation to the mass mixing length parameter, α_m, and find that it qualitatively shows a similar variation with stellar parameter (between 1.6 and 2.4) with the solar value of α^o_m = 1.83.αm =1.83. The vertical correlation length scaled with the pressure scale height yields 1.71 for the Sun, but only displays a small systematic variation with stellar parameters, the correlation length slightly increases with T_eff. Conclusions. We derive mixing length parameters for various stellar parameters that can be used to replace a constant value. Within any convective envelope, α_m and related quantities vary strongly. Our results will help to replace a constant α_MLT.