Effective Roughness and Displaced Mean Flow over Complex Terrain

Abstract

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 (z_text {0,eff}), friction velocity (u _{*text {,eff}}), and displacement height (d_text {eff}) characterise the drag exerted by the terrain. Simulations and spectral analyses reveal that the terrain statistics—and consequently d_text {eff}, u _{*text {,eff}} and z_text {0,eff}—can change significantly with flow direction, including flow in opposite directions. Previous studies over scaled or simulated fractal surfaces reported z_text {0,eff} to depend on the standard deviation of terrain elevation (sigma _h), but over real terrains we find z_text {0,eff} varies with standard deviation of terrain slopes (sigma _{Delta h/Delta x}). Terrain spectra show the dominant scales contributing to sigma _{Delta h/Delta x} vary from sim 1–10 km, with power-law behaviour over smaller scales corresponding to fractal terrain used in earlier works. The dependence of z_text {0,eff} on sigma _{Delta h/Delta x} is consistent with fractal terrain having sigma _{Delta h/Delta x} propto sigma _h, as well as classic theory for individual hills. We obtain relationships for z_text {0,eff}, d_text {eff}, and u _{*text {,eff}} in terms of sigma _{Delta h/Delta x}, finding that d_text {eff} acts as a characteristic length scale within z_text {0,eff}. Considering flow in opposite directions, use of upslope statistics did not improve z_text {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_text {0,eff} formulations developed herein, and provide recommendations for practical use.