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.

通过对跨越多个大陆的各种复杂地形的雷诺平均 Navier-Stokes 模拟的速度和应力场进行分析，在中性条件下，我们发现了位移的面积平均对数风速剖面。对应的有效粗糙度长度（\(z_\text {0,eff}\)）、摩擦速度（\(u _{*\text {,eff}}\)）和位移高度（\(d_\text { eff}\) ) 表征地形施加的阻力。模拟和光谱分析揭示了地形统计，因此\(d_\text {eff}\)、\(u _{*\text {,eff}}\)和\(z_\text {0,eff}\ )— 可随流动方向发生显着变化，包括相反方向的流动。以前对缩放或模拟分形表面的研究报告了\(z_\text {0,eff}\)取决于地形高程的标准偏差 ( \(\sigma _h\) )，但在真实地形上我们发现\(z_\文本 {0,eff}\)随地形坡度的标准差( \(\sigma _{\Delta h/\Delta x}\) ) 而变化。地形光谱显示对\(\sigma _{\Delta h/\Delta x}\)有贡献的主要尺度在\(\sim \) 1-10 km 范围内变化，在较小尺度上的幂律行为对应于所使用的分形地形在早期的作品中。\(z_\text {0,eff}\)的依赖关系关于\(\sigma _{\Delta h/\Delta x}\)与具有\(\sigma _{\Delta h/\Delta x} \propto \sigma _h\)的分形地形以及经典理论一致对于个别山丘。我们根据\ ( \ _ _ _ sigma _{\Delta h/\Delta x}\)，发现\(d_\text {eff}\)充当\(z_\text {0,eff}\)内的特征长度尺度。考虑到相反方向的流动，使用上坡统计并没有改善\(z_\text {0,eff}\)预测；庇护效应可能需要更复杂的治疗。我们的发现影响了实际应用和研究，包括微气象流动、计算流体动力学、大气模型耦合以及中尺度和气候建模。我们讨论了本文开发的\(z_\text {0,eff}\)公式的局限性，并为实际使用提供了建议。