  We compare 1D nonlocal turbulent convection models with 3D hydrodynamic
numerical simulations. We study the validity of closure models and turbulent
coefficients by varying the Prandtl number, the P$\acute{e}$clet number, and
the depth of the convection zone. Four closure models of the fourth-order
moments are evaluated with the 3D simulation data. The performance of the
closure models varies among different cases and different fourth-order moments.
We solve the dynamic equations of moments together with equations of the
thermal structure. Unfortunately, we cannot obtain steady-state solutions when
these closure models of fourth-order moments are adopted. The numerical
solutions of the down-gradient approximations of the third-order moments, on
the other hand, are robust. We calibrate the coefficients of the 1D
down-gradient model from the 3D simulation data. The calibrated coefficients
are more robust in the cases of deep convection zones. Finally we have compared
the 1D steady-state solutions with the 3D simulation results. The 1D model has
captured many features appearing in the 3D simulations : (1)
$\nabla-\nabla_{a}$ has a U-shape with a minimum value at the lower part of the
convection zone. (2) There exists a bump for $\nabla-\nabla_{a}$ near the top
of the convection zone when the P$\acute{e}$clet number is large. (3) The
temperature gradient can be sub-adiabatic due to the nonlocal effect. Apart
from these similarities, the prediction on the kinetic energy flux, however, is
unsatisfactory.
