In this paper, we obtain some nonoscillatory theories of the functional differential equation (r(t)ψ(x(t))x (t)) + f(t, x(t), x(σ(t))) = 0, t ≥ t 0 , where r ∈ C 1 ([t 0 , ∞); (0, ∞)), ψ∈ C 1 (R, R) and f ∈ C([t 0 , ∞) × R × R, R).
A new analytical model was developed to predict the gravity wave drag (GWD) induced by an isolated 3-dimensional mountain, over which a stratified, non-rotating non-Boussinesq sheared flow is impinged. The model is confined to small amplitude motion and assumes the ambient velocity varying slowly with height. The modified Taylor-Goldstein equation with variable coefficients is solved with a Wentzel-Kramers-Brillouin (WKB) approximation, formally valid at high Richardson numbers. With this WKB solution, generic formulae of second order accuracy, for the GWD and surface pressure perturbation (both for hydrostatic and non-hydrostatic flow) are presented, enabling a rigorous treatment on the effects by vertical variations in wind profiles. In an ideal test to the circular bell-shaped mountain, it was found that when the wind is linearly sheared, that the GWD decreases as the Richardson number decreases. However, the GWD for a forward sheared wind (wind increases with height) decreases always faster than that for the backward sheared wind (wind deceases with height). This difference is evident whenever the model is hydrostatic or not.
Using a three-dimensional nonhydrostatic mesoscale numerical model (MM5), the evolution and structures of baroclinic waves with and without surface drag in case of dry and moist atmosphere are simulated, with special emphases on the effects of surface drag on the low-level frontal structure and frontogenesis. There are two different effects of surface drag on the low-level frontogenesis in the dry case. On one hand, the surface drag weakens the low-level frontogenesis and less inclined to develop the baroclinic wave due to the dissipation. But on the other hand, the surface drag induces a strong ageostrophic flow, which prolongs the low-level frontogenesis and finally leads to the enhancement of cold front. Compared with the no surface drag case, the surface drag increases the frontal slope espe- cially in the boundary layer, where the front is almost vertical to the surface, and then enhances the prefrontal vertical motion. All these conclusions expanded the analytical theory of Tan and Wu (1990). In the moist atmosphere, the influence of surface drag on frontal rainbands is also obvious. The surface drag weakens the convection, and reduces the energy dissipation near the surface when the initial relative humidity is relatively weak. At this time, the confluence induced post-frontal updrafts moves across the cold front and reinforces the prefrontal convection, which is beneficial to the maintenance of the rainband in cold sector. Given the enhancement of relative humidity, the moist convection domi- nates the low-level frontogenesis while the retardation of surface drag on energy dissipation is not obvious, therefore the effects of surface drag on the low-level frontogenesis and precipitation are re- duced.
