  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, {\Delta}s, determined from the 3D RHD models, are matched with the mixing
length parameter, {\alpha}_MLT, from 1D hydrostatic atmosphere models with
identical microphysics (opacities and equation-of-state). We also derive the
mass mixing length, {\alpha}_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 {\alpha}_MLT (s_bot) = 1.98. For different stellar parameters,
{\alpha}_MLT varies systematically in the range of 1.7 - 2.4. In particular,
{\alpha}_MLT decreases towards higher effective temperature, lower surface
gravity and higher metallicity. We find equivalent results for {\alpha}_MLT
({\Delta}s). 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, {\alpha}_m, and find that it exhibits qualitatively a similar
variation with stellar parameter (between 1.6 and 2.4) with a solar value of
{\alpha}_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,
{\alpha}_m and related quantities vary a lot.
