The growth and decay of a convective boundary layer (CBL) over a surface with a constant surface temperature that develops into a linear stratification is studied, and a mathematical model for this system is derived. The study is based on direct numerical simulations with four different Reynolds numbers; the two simulations with the largest Reynolds numbers display Reynolds number similarity, suggesting that the results can be extrapolated to the atmosphere. Because of 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 derived model provides characteristic scales for bulk properties of the CBL. Even though the system is unsteady, self-similar vertical profiles of buoyancy, buoyancy flux, and 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.
van Heerwaarden, C. C., & Mellado, J. P. (2016). Growth and Decay of a Convective Boundary Layer over a Surface with a Constant Temperature. Journal of the Atmospheric Sciences, 73(5), 2165-2177. https://doi.org/10.1175/JAS-D-15-0315.1