Wind profile effects and critical layer filtering of the wave momentum flux in stratified flow over mountains

The question of how the gravity wave drag produced in stratified flow over mountains is affected by wind profile effects (i.e. shear and curvature) has been the object of recent investigations. These investigations employed a WKB approximation to treat local impacts on the surface drag of the first and second derivatives of the wind velocity for generic wind profiles. In order for these effects to be captured, it was necessary to extend the WKB method to 2nd order in the small perturbation parameter, since it was shown that corrections to the surface pressure that are asymmetric with respect to the orography only arise at this order.

While these results have some interest for parameterization purposes, especially when formulated for mountains with an elliptical cross section, the actual force that is exerted on the atmosphere, and therefore directly parameterized, is related with the wave momentum flux profile. It is the variation of this momentum flux with height that corresponds to the reaction force acting on the atmosphere that balances the surface gravity wave drag exerted on the mountain.

By Eliassen-Palm's theorem, and its more recent extension to directional shear flows, the momentum flux only varies in linear conditions due to the existence of critical levels, where the wave equation is singular. In previous linear studies addressing the momentum flux profiles, wind shear does not have an impact on the surface drag, and the gravity waves are totally absorbed at critical levels. This occurs because high Richardson numbers (Ri) were considered.

The present study aims to extend this approach to lower Ri, where the surface drag is modified by shear or curvature of the wind profile, and critical levels do not absorb the waves completely, but rather filter them. Again, a WKB approximation is employed to obtain the wave solutions for generic wind profiles, but now this method must be extended to third order. The calculations are performed for a circular mountain and tested for 3 idealized wind profiles with directional shear. Two of these wind profiles are linear, while in the third the wind turns with height at a constant rate, maintaining its magnitude. The use of a circular mountain considerably simplifies the calculations, while at the same time providing a wave forcing that is more or less isotropic. This is important for understanding how the different wavenumbers are affected by the critical levels. The flow is assumed to be inviscid, non-rotating and hydrostatic, with a constant Brunt Väisälä frequency.

Contour integration techniques enable us to obtain simple expressions for the momentum flux, correct to second order in the small perturbation parameter, and closed form analytical expressions for the momentum flux divergence. The momentum flux profiles are compared with results from numerical simulations, both for linear and nonlinear conditions, and found to provide considerable improvements upon existing theories, derived for higher Ri. Critical levels partially absorb the waves, but also change their phase, so that the momentum fluxes may change sign above these levels.