Measures of statistical equilibrium in atmospheric boundary layers

Atmospheric boundary layers can reach a statistically steady state given a suitable forcing. Conditions that result in steady states are typically found in large-scale atmospheric circulation branches, such as trade wind boundary layers. In large-eddy simulations of boundary layers under steady forcing, canonical cases can approach steady state after relatively long times, e.g., several days. In this study, we examine how different flow statistics quantify the departure of the boundary layer from the equilibrium state. A collection of large-eddy simulations of subtropical marine atmospheric boundary layers with a typical range of sea surface temperature and large-scale divergence is used. First, we define the equilibrium condition based on the vertically integrated liquid water potential temperature equation. The departure from equilibrium is objectively defined in energetic terms as the energy per unit area in the column which is required to close the energy budget of the liquid water potential temperature. Subsequently, we compare typical flow statistics, including kinetic energy, liquid water path, cloud cover, and surface fluxes. The steady-state values of all flow statistics are determined based on the energy budget. Finally, the approach of different flow statistics towards equilibrium is quantified. The methodology can help quantify and assess the departure from statistical equilibrium using readily available flow statistics, as all the terms of the energy budget are not always observed or computed.