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The drag associated with resonant flow past 2D and 3D topographic obstacles is studied analytically using a simple linear hydrostatic model. The wind velocity is assumed to be constant up to a certain level, and to decrease linearly above. Sufficiently high Richardson numbers are considered in the shear region, so that the wave energy is essentially absorbed at the critical level. Up to the critical level, this flow is similar or approximately similar to that considered in previous studies of resonant flows (e.g. hyperbolic tangent profiles), but the dimensionless height of the orography is arbitrarily small.

A closed analytical expression for the drag is derived in the case of a 2D ridge, while the corresponding expression for an axisymmetric mountain involves the numerical evaluation of a simple 1D integral. As the critical level rises, the analytical model predicts that drag suffers modulations of its value, with a periodicity of half the vertical wavelength of the internal gravity waves. The amplitude of these modulations increases as the Richardson number decreases, so that for Ri=1, the drag changes by a factor of 2 for an axisymmetric mountain and by a factor of 3 for a 2D ridge. The model clearly shows that the 2 parameters controlling the drag are the Richardson number in the shear layer and the height of the discontinuity in the gradient of the ambient wind velocity (not the height of the critical level). At this height, the mountain wave is reflected partially, leading to the reinforcement or weakening of the pressure perturbation at the surface, depending on the ratio of the reflecting height to the vertical wavelength of the waves.

These results are consistent with recent studies on wave ducting where, however, the drag was not calculated. Predictions from the drag expressions proposed here are supported by simulations of a non-linear, non-hydrostatic mesoscale model, showing that, at small dimensionless heights, the modulation of the drag can be attributed almost totally to linear processes. For larger dimensionless heights, linear predictions are no longer useful quantitatively, but qualitatively the behavior of the drag does not change very much in the case of an axisymmetric mountain, while a drastic change in behavior is observed for flow past an infinite ridge.

The present results show that high-drag states exist even in flow over very gentle topography, and help to clarify the dynamics of the drag modulation (in particular emphasizing the importance of the level where the vertical velocity gradient is discontinuous). Although a hyperbolic tangent velocity profile has no discontinuity in its gradient, one can think of the zone where this profile has a maximum in negative curvature as being a sort of attenuated discontinuity, with an essentially equivalent reflecting effect. The present model also shows how the amplitude of the drag modulation depends on the Richardson number.