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

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. The adiabatic entropy value of the deep convection zone, s_bot, and the entropy jump, Δs, determined from the 3D RHD models, are matched with the mixing length parameter, α_MLT, from 1D hydrostatic atmosphere models with identical microphysics (opacities and equation-of-state). We also derive the mass mixing length, 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. The calibrated mixing length parameter for the Sun is α_MLT (s_bot) = 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_MLT (Δs). Also, 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, α_m, and find that it exhibits qualitatively a similar variation with stellar parameter (between 1.6 and 2.4) with a solar value of α_m = 1.83. The vertical correlation length scaled with the pressure scale height yields for the Sun 1.71, but displays only a small systematic variation with stellar parameters, the correlation length slightly increasing with Teff. 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 a lot.