Effective Roughness and Displaced Mean Flow over Complex Terrain

Abstract

AbstractVia 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 ($$z_\text {0,eff}$$ z 0,eff ), friction velocity ($$u _{*\text {,eff}}$$ u ∗ ,eff ), and displacement height ($$d_\text {eff}$$ d eff ) characterise the drag exerted by the terrain. Simulations and spectral analyses reveal that the terrain statistics—and consequently $$d_\text {eff}$$ d eff , $$u _{*\text {,eff}}$$ u ∗ ,eff and $$z_\text {0,eff}$$ z 0,eff —can change significantly with flow direction, including flow in opposite directions. Previous studies over scaled or simulated fractal surfaces reported $$z_\text {0,eff}$$ z 0,eff to depend on the standard deviation of terrain elevation ($$\sigma _h$$ σ h ), but over real terrains we find $$z_\text {0,eff}$$ z 0,eff varies with standard deviation of terrain slopes ($$\sigma _{\Delta h/\Delta x}$$ σ Δ h / Δ x ). Terrain spectra show the dominant scales contributing to $$\sigma _{\Delta h/\Delta x}$$ σ Δ h / Δ 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 $$z_\text {0,eff}$$ z 0,eff on $$\sigma _{\Delta h/\Delta x}$$ σ Δ h / Δ x is consistent with fractal terrain having $$\sigma _{\Delta h/\Delta x} \propto \sigma _h$$ σ Δ h / Δ x ∝ σ h , as well as classic theory for individual hills. We obtain relationships for $$z_\text {0,eff}$$ z 0,eff , $$d_\text {eff}$$ d eff , and $$u _{*\text {,eff}}$$ u ∗ ,eff in terms of $$\sigma _{\Delta h/\Delta x}$$ σ Δ h / Δ x , finding that $$d_\text {eff}$$ d eff acts as a characteristic length scale within $$z_\text {0,eff}$$ z 0,eff . Considering flow in opposite directions, use of upslope statistics did not improve $$z_\text {0,eff}$$ 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 $$z_\text {0,eff}$$ z 0,eff formulations developed herein, and provide recommendations for practical use.