First-order turbulence modelling for the stratified atmospheric boundary layer

This work concerns the first-order statistical modelling of atmospheric turbulence. Until now the first-order turbulence models fail to simulate many of the effects of stable stratification on turbulence and especially the growing of a nocturnal stable layer in the atmospheric boundary layer. An analysis of the statistical transport equations of the Reynolds tensor, the heat fluxes, and the temperature variance q², leads to a conceptual representation of the anisotropisation by the stable stratification, of the turbulence and of the exchanges between the kinetic energy and potential energy. This representation, schematised by an hydraulic analogy, is used to derive an extended non linear k-e model, including a transport equation for the temperature variance. The rationale is based on a limited expansion as a function of the turbulence time scale t=k/e. This approach, applied to the transport equations of a second-order model (Craft and Launder, 1991), leads to an original non linear explicit and t-quadratic model using the transport equations for k, e and q². At variance with the previous explicit first-order models that are limited to homogeneous quadratic terms for the dynamics, in this model the couplings between the dynamical, thermal and heterogeneous terms appear explicitly. The model has been first tested on a few atmospheric flows, as e.g. the time-development of the Boulder storm (Lilly et Zipser, 1972), but the most demonstrative exercice is the simulation of the W33-34 diurnal evolution of the boundary layer during the Wangara experiment (Clark et al., 1971). The model not only predicts accurately the day-time growing of the convective boundary layer, but also the nocturnal stable internal layer. This success is very noticeable because even the most sophisticated statistical models (second- or third-order closure) do not properly simulate the time evolution of the nocturnal stable layer.