We present a conceptual model for the diurnal cycle of the dry atmospheric boundary layer (ABL). It may serve as a framework for future numerical studies on the transitional dynamics that characterize the ABL over land. The conceptual model enables us to define expressions for relevant physical scales as a function of the most prominent forcing parameters and the low degree of complexity facilitates a dimensionless description. This is useful to help generalize boundary layer dynamics that occur on a diurnal time scale. Further, the model’s application for numerical studies is illustrated herein with two examples: a single-column-model study that assesses the effect of wind forcing on the main characteristics of the diurnal cycle, and a large-eddy-simulation study on the daily evolution of turbulence under weak-wind-forcing conditions. The results from these studies sketch the general evolution of the present set of diurnal-cycle systems in more detail. We discuss how the setups are able to reproduce well-known dynamical features of the ABL and also highlight limitations, where the simple conceptual system is unable to describe realistic ABL behavior. We conclude that the present conceptual model has an interesting balance between model-system complexity and physical realism, such that it is useful for future idealized studies on the diurnal cycle of the ABL.

We perform direct numerical simulation of the Couette flow as a model for the stable boundary layer. The flow evolution is investigated for combinations of the (bulk) Reynolds number and the imposed surface buoyancy flux. First, we establish what the similarities and differences are between applying a fixed buoyancy difference (Dirichlet) and a fixed buoyancy flux (Neumann) as boundary conditions. Moreover, two distinct parameters were recently proposed for the turbulent-to-laminar transition: the Reynolds number based on the Obukhov length and the "shear capacity," a velocity-scale ratio based on the buoyancy flux maximum. We study how these parameters relate to each other and to the atmospheric boundary layer. The results show that in a weakly stratified equilibrium state, the flow statistics are virtually the same between the different types of boundary conditions. However, at stronger stratification and, more generally, in nonequilibrium conditions, the flow statistics do depend on the type of boundary condition imposed. In the case of Neumann boundary conditions, a clear sensitivity to the initial stratification strength is observed because of the existence of multiple equilibriums, while for Dirichlet boundary conditions, only one statistically steady turbulent equilibrium exists for a particular set of boundary conditions. As in previous studies, we find that when the imposed surface flux is larger than the maximum buoyancy flux, no turbulent steady state occurs. Analytical investigation and simulation data indicate that this maximum buoyancy flux converges for increasing Reynolds numbers, which suggests a possible extrapolation to the atmospheric case.