Linear prediction of mountain wave drag in the case of generic wind profiles

In order to develop physically sound parameterizations of orographic gravity wave drag for large-scale numerical models, it is useful to study the variation of this force, exerted by mountains on a stratified atmosphere, with the various parameters of the flow. Among these, shear is one of the most important. Although the drag has been calculated analytically for particularly simple non-uniform wind profiles, and numerically for more complex wind profiles, a general framework to understand which characteristics of the wind variation with height affect the drag, and in what way, has been lacking.

This study uses the linearized, hydrostatic equations of motion to derive closed form analytical expressions for the drag induced by an axisymmetric mountain and a 2D ridge, for generic wind profiles. This analytical approach enables one to understand more clearly the individual physical processes affecting the drag, through the functional dependence on the various parameters. The proposed model solves the Taylor-Goldstein equation with variable coefficients using a WKB approximation. Formally, this approach assumes that the wind velocity varies over a scale that is much larger than the vertical wavelength of the internal gravity waves. However, in practice, it is shown that the model is accurate even when this condition is not satisfied or is only marginally satisfied. In order for the wind variation with height to have any impact on the drag, it is necessary that the WKB solution be extended to second order in a small perturbation parameter that is inversely proportional to the square root of the Richardson number of the flow, Ri. That is why previous analytical treatments using the first-order WKB solution (which is the most widely known) have failed to capture this effect.

Subject to the WKB assumption, the drag depends on the first and second vertical derivatives of the wind velocity at the surface. For simple types of flow, such as a wind that varies linearly with height or a wind that turns with height at a constant rate, maintaining its magnitude, it is found that the drag on an axisymmetric mountain varies proportionally to the inverse of the Richardson number of the flow. But whereas in the first case the drag decreases as Ri decreases, in the second case the drag increases as Ri decreases. It turns out that the effect of a non-zero first derivative of the wind velocity at the surface is always to decrease the drag, while the effect of a negative second derivative (such as exists in a turning wind) is to increase the drag by a considerably larger amount. This explains differences in behavior observed in previous numerical simulations. The model also predicts that the drag may not be aligned with the surface wind.

When results from the analytical model are compared with those obtained using non-linear, non-hydrostatic numerical models (albeit for approximately linear and hydrostatic conditions), good quantitative agreement is found, even for Ri of order 1. The analytical drag expressions derived for flow over a 2D ridge, although much simpler than those derived for an axisymmetric mountain, display qualitatively the same type of dependence on the first and second derivatives of the wind velocity and on Ri. However, the coefficients multiplying the corrections to the drag due to the wind variation are larger by a factor of 4/3, due to geometrical effects (the wind is forced to flow over a ridge, while it can go over or around a 3D mountain). In both cases, these corrections are independent of the exact shape of the orography, provided that this is axisymmetric or slab-symmetric. This feature adds much relevance to the present calculations.