Effective Roughness and Displaced Mean Flow over Complex Terrain

Via analysis of velocity and stress fields from Reynolds-Averaged Navier–Stokes simulations over diverse complex terrains spanning several continents, in neutral conditions we find displaced areal-mean logarithmic wind speed profiles. The corresponding effective roughness length (z0,eff), friction velocity (u_∗,eff), and displacement height (d_eff) characterise the drag exerted by the terrain. Simulations and spectral analyses reveal that the terrain statistics—and consequently d_eff, u_{∗ ,eff}  and z_0,eff—can change significantly with flow direction, including flow in opposite directions. Previous studies over scaled or simulated fractal surfaces reported z0,eff to depend on the standard deviation of terrain elevation (σh), but over real terrains we find z_0,eff varies with standard deviation of terrain slopes (σ_Δh/Δx). Terrain spectra show the dominant scales contributing to σΔh/Δx vary from ∼ 1–10 km, with power-law behaviour over smaller scales corresponding to fractal terrain used in earlier works. The dependence of z0,eff on σ_Δh/Δx is consistent with fractal terrain having σ_Δh/Δx α σh, as well as classic theory for individual hills. We obtain relationships for z_0,eff, d_eff, and u∗ ,eff in terms of σ_Δh/Δx, finding that deff acts as a characteristic length scale within z_0,eff. Considering flow in opposite directions, use of upslope statistics did not improve z_0,eff predictions; sheltering effects likely require more sophisticated treatment. Our findings impact practical applications and research, including micrometeorological flow, computational fluid dynamics, atmospheric model coupling, and mesoscale and climate modelling. We discuss limitations of the z_0,eff formulations developed herein, and provide recommendations for practical use.