The study consists of two parts. First, a reference case with a full water surface is studied, and afterward this case is compared against mixed water-land surfaces. The study is entirely based on direct numerical simulations, to eliminate uncertainties related to the performance of the surface model under free convective conditions.

In the first part, we focus on the reference case that describes a full water surface. We have simulated the system with four different Reynolds numbers; Reynolds number similarity has been shown for the two with the largest numbers, suggesting that the results can be extrapolated to the atmosphere. Due to the interplay of the growing CBL and the gradually decreasing surface buoyancy flux, the system has a complex time evolution in which integrated kinetic energy, buoyancy flux and dissipation peak and subsequently decay. The model that we derived for this case provides characteristic scales for bulk properties of the CBL. Even though the system is unsteady, self-similar vertical profiles of buoyancy, buoyancy flux and the velocity variances are recovered. There are two important implications for atmospheric modeling. First, the magnitude of the surface buoyancy flux sets the time scale of the system, thus over a rough surface the roughness length is a key variable. Therefore, the performance of the surface model is crucial in large-eddy simulations of convection over water surfaces. Second, during the phase in which kinetic energy decays, the integrated kinetic energy never follows a power law, because the buoyancy flux and dissipation balance until the kinetic energy has almost vanished. Therefore, the applicability of power law decay models to the afternoon transition in the atmospheric boundary layer is questionable; the presented model provides a physically sound alternative.

In the second part, we study a setting with alternating stripes of open water and snow-covered surfaces. Here, we study three cases: stripes that have a width that is considerably less than the height of the system, stripes that have a similar width as the height of the system and stripes that are much larger than that. The introduction of heterogeneity changes the system considerably. The presence of a colder surface adjacent to the water stripes enhances the buoyancy flux and therefore lowers the characteristic time scale of the system considerably. Surprisingly, the width of the stripes has little influence on the buoyancy flux. The kinetic energy on the other hand, is a strong function of the width of the stripes. A large stripe size introduces a large-scale circulation in the system that not only strongly enhances the kinetic energy, but also introduces large oscillations in the integrated kinetic energy in the boundary layer. We aim to introduce the effects of heterogeneity into our derived model by means of the non-dimensional parameter that describes the ratio of the stripe size to the height of CBL.