 The study shows how truncations at second order can be used to model and elucidate the dynamics of turbulent atmospheric flows, specifically in convective boundary layers and large-scale waves. While these truncations can capture some aspects of the flow, they fail to capture important turbulent transport terms or strongly non-linear interactions that generate small-scale filaments. Higher-order closures are needed for more accurate representation of these processes in climate models. This article was authored by Faridate Charle to P. O. Schneider, Bettina Meyer and others.