The effect of shear discontinuities and critical levels on orographic gravity wave drag

There are still many aspects about the fundamental dynamics of mountain waves that need to be clarified. These aspects are interesting on their own right, but may also find applications in improving existing gravity wave drag parameterizations. One of these aspects concerns the effects of wind shear and its variations, and of critical levels, on mountain wave drag. Existing drag parameterizations do not take vertical wind shear into account, and recent proposals for wave momentum flux expressions for wind profiles with directional shear assume that critical levels absorb the waves completely.

In this study, the effects of shear discontinuities and critical levels are addressed using linear theory for relatively low Richardson numbers (Ri), where these effects are expected to have an appreciable impact on the drag. We consider inviscid, non-rotating, hydrostatic and uniformly stratified atmospheric flow over a mesoscale mountain. In order to isolate the effects under consideration, very simple idealized wind profiles are adopted: the wind velocity is assumed to vary linearly up to a certain level, and to become constant above it.

It is found that shear discontinuities may either enhance or reduce the surface drag, depending on whether there is constructive or destructive interference between upward and downward propagating waves in the layer beneath the shear discontinuity. The surface drag is also found to be highly sensitive to the existence (or not) of critical levels between the surface and the shear discontinuity. Here we must distinguish between the cases of unidirectional shear and directional shear flow. While for unidirectional shear, all wavenumbers in the mountain waves' spectrum either have or do not have a critical level, for directional shear some wavenumbers have critical levels whereas others do not.

It is seen that, the greater the amount of critical levels that the waves encounter, the closer the drag is to its predicted value for a shear of unlimited extent. At one extreme, in the case of unidirectional shear with a critical level, the drag behaves as a function of Ri not too differently from the case of a shear that extends indefinitely. At the other extreme, for forward unidirectional shear, the drag displays large oscillations as a function of Ri. In situations with directional wind shear, the drag has an intermediate behavior.

The dependence of the drag on the fraction of wavenumbers filtered by critical levels creates an asymmetry between flows with positive and negative shear. This asymmetry is not predicted by linear theory for a shear extending indefinitely in hydrostatic conditions. Since many theoretical studies on gravity waves assume constant-shear flows, the present study highlights the importance of taking into account the shear discontinuities, or shear variations, that always exist in practice.

Nonlinear effects are found to considerably amplify the drag, but many qualitative features of the linear drag behavior remain. However, differences between linear and nonlinear results are substantially enhanced by the existence of shear, being much more pronounced than for a constant wind velocity