Revisiting the relation between momentum and scalar roughness lengths of urban surfaces

Large-Eddy Simulations (LESs) of neutral flow over regular arrays of cuboids are conducted to explore connections between momentum (z_0m) and scalar (z_0s) roughness lengths in urban environments, and how they are influenced by surface geometry. As LES resolves the obstacles but not the micro-scale boundary layers attached to them, the aforementioned roughness lengths are analyzed at two distinct spatial scales. At the micro-scale (roughness of individual facets, e.g., roofs), it is assumed that both momentum and scalar transfer are governed by accepted arguments for smooth walls that form the basis for the LES wall-model. At the macro-scale, the roughness lengths are representative of the aggregate effects of momentum and scalar transfer over the resolved roughness elements of the whole surface, and hence they are directly computed from the LES. The results indicate that morphologically based parametrizations for macro-scale z_0m are adequate overall. The relation between the momentum and scalar macro-roughness values, as conventionally represented by (Formula presented.) and assumed to scale with (Formula presented.) (where Re_∗ is a roughness Reynolds number), is then interpreted using surface renewal theory (SRT). SRT predicts n = 1/4 when only Kolmogorov-scale eddies dominate the scalar exchange, whereas n = 1/2 is predicted when large eddies limit the renewal dynamics. The latter is found to better capture the LES results. However, both scaling relations indicate that z_0s decreases when z_0m increases for typical urban geometries and scales. This is opposite to how their relation is usually modelled for urban canopies (i.e., z_0s/z_0m is a fixed value smaller than unity).