An analytical model for thermal convection and convective rolls with a linear eddy viscosity

Abstract

The aim of this work is to present a theoretical scheme of dry thermal convection and convective rolls in atmospheric boundary layer with a linear eddy viscosity for a steady-state stationary case. For this theoretical study, a few approximations valid for atmospheric boundary layer were used, such as the Boussinesq approximation, and also, the second-order spatial derivative of velocity has been neglected. Analytical models for dry air thermal updraft velocity are presented and wavelength between convective rolls and aspect ratio of rolls have been derived. Using these results, it is shown how the transition between cellular-type convection and roll-type convection depends on thermal stratification of boundary layer, the temperature excess of convective cell at ground level and other meteorological parameters. The analytical results are in agreement with both observational and numerical studies for horizontal convective rolls existent in scientific literature. In addition, it is shown how this presented model can be applied for layers with non-linear eddy viscosity and how it can be used in future studies, and what its limitations are.