Including Advection in Boundary Condition Models of Momentum and Heat for Heterogeneous Stratified Boundary Layers

Horizontal surface heterogeneity and non-stationarity invalidate the assumptions inherent to Monin-Obukhov similarity theory (MOST). These impacts can be enhanced under stratified conditions prevalent in the nocturnal boundary layer. A critical shortcoming in equilibrium models, including MOST, modifications to MOST, or power-law formulations, is the assumption that advection is negligible in the momentum and thermal energy equations. When these models are applied instantaneously and locally, as in surface boundary conditions in large-eddy simulations, their underlying assumptions are highly suspect. Yet, these models are routinely employed because of a lack of clear alternatives for atmospheric flows. In this study, the filtered versions of the momentum and thermal energy equations are examined using data from eight direct numerical simulation (DNS) experiments. Five of these simulations were conducted with homogeneous surface forcing and changing levels of imposed stratification. The remaining three simulations were performed using a single moderately-stable stratification with heterogeneous surface forcing through surface patches of varying size. Each of the terms in the momentum and energy equations are evaluated as functions of filter scale, distance in the surface layer from the ground, and static stability for their contribution to the overall budget for each simulation in the DNS database. Although vertical transport dominates as expected, contributions from horizontal advection and pressure forces are of increasing importance for smaller filter scales, increasing static stability, and larger surface heterogeneities. Analysis of these budgets suggests possible formulations for reduced-order models that weaken some assumptions used in commonly employed surface boundary conditions.