Allowing for the effects of form drag induced by unresolved small-scale hills in Numerical Weather Prediction models has led to significant improvements in model performance. However, the approach is based on the results of process studies of flow over hills which have almost exclusively concentrated on flow normal to two-dimensional hills, or ridges. The few studies which have considered three-dimensional hills have mainly studied isotropic, circularly symmetric hills. Virtually no attention has been given to the arguably more physically relevant, intermediate case of anisotropic three-dimensional hills. This case may, in an idealized way, be characterized by flow over an ellipsoid with lengths along the horizontal major and minor axes denoted by Lx and Ly, respectively. The usual ridge case is then retrieved by letting the aspect ratio a=Lx/Ly go to infinity and the isotropic case has a=1.

This study presents results from a large number of simulations designed to investigate the variation of the form drag on ellipsoidal hills with both aspect ratio and wind direction. Neutral conditions are considered first. As expected, the magnitude of the drag on isotropic hills is found to be independent of wind direction. However, a key finding is how quickly the results converge, as a is increased from unity, onto those obtained in the two-dimensional limit. Values of a of around 1.5 are already large enough to introduce a significant directional dependence to the form drag, and the a=3 results are very close to those obtained in the two-dimensional simulations. Based on the numerical results, a simple parametrization of the drag is proposed.

For flow over low hills in stable conditions, the variation of the drag with wind direction is found to be similar to that obtained in neutral conditions, until the boundary layer becomes stable enough to admit wave-like solutions. Rather counter-intuitively, rotating the incident wind to be more parallel to a ridge may lead to an increased pressure force on that ridge if it switches the solution into this regime. For cases with waves, the surface drag is found to be sensitive to the presence or otherwise of critical lines. The implications for NWP model parametrizations are discussed.