What is the smallest physically acceptable scale for 1D turbulence schemes?

In numerical weather prediction (NWP) models, at mesoscale, the sub-grid convective boundary-layer turbulence is dominated by the uni-dimensional (1D) vertical thermal production. In Large-Eddy Simulations (LES), the thermal plumes are resolved and the residual sub-grid turbulent motions are homogeneous and isotropic, thus three-dimensional (3D), resulting from the dynamical production. This article sets the critical horizontal resolution for which the usually 1D turbulence schemes of NWP models must be replaced by 3D turbulence schemes. LES from five dry and cumulus-topped free convective boundary layers and one forced convective boundary layer are performed. From these LES data, the thermal production and vertical and horizontal dynamical productions are calculated at several resolutions from LES to mesoscale. It appears that the production terms of both dry and cumulus-topped free convective boundary layers have the same behavior. A pattern emerges whenever data are ranked by the resolution scaled by the size of thermal plumes, (h+h(c), where h is the boundary-layer height and h(c) is the depth of the cloud layer). In free convective boundary layers, the critical horizontal resolution for which the horizontal motions must be represented is 0.5(h + h(c)). However, the critical horizontal resolution in the forced convective boundary layer case is 3(h + h(c)).