The pressure force induced by neutral, turbulent flow over small-scale hills is considered. The influence that the level of turbulence closure has on liner, analytical expressions for the force is evaluated. The two-dimensional linear expressions are then extended to three dimensions and a heuristic extension to moderate and steep slopes, for which nonlinear effects are important, is proposed. The predictions of these various expressions are compared with results from a three-dimensional numerical model employing a 1 1/2-order turbulence-closure model, and very good agreement is found.
The numerical-model results supports the idea that, sufficiently far above the hills, the areally averaged velocity profile varies approximately logarithmically with height but with an enhanced, or effective, roughness length and a friction velocity equal to the square root of the total surface force per unit area and unit density. Using the theoretical results for the pressure force, discussed above, and approximate expression for the effective roughness length is obtained that agrees well with values from the model, and also matches with existing expressions that are valid for two-dimensional hills with either very low or very large slopes. Qualitative agreement is also found with the limited number of observed values of roughness length in hilly terrain. These results suggest that the effective roughness length is an accurate and convenient method of parametrizing the drag due to subgrid-scale orography in large-scale numerical-weather-prediction and climate models.