Linear Calculations of Orographic-Convective Flows in a Two-Layer Atmosphere

In an attempt to better understand the dynamics of orographic precipitation, we theoretically examine orographic-convective flows in the context of the response of a stably stratified atmosphere to thermal convective forcing and mechanical orographic forcing. Here, a two-layer atmosphere is considered in which the lower layer (troposphere) has a smaller stability and a basic-state wind with constant shear and the upper layer (stratosphere) has a larger stability and a uniform basic-state wind. Convective heating is specified in a deep layer of the troposphere. The equations governing small-amplitude perturbations in a two-dimensional, linear, hydrostatic, Boussinesq system are solved to obtain analytic solutions for perturbation horizontal and vertical velocities. Flows forced by convective forcing and orographic forcing are analyzed with different basic-state wind shears, tropopause heights, upper-layer static stabilities, and locations of convection relative to the mountain. Near-surface compensating downdraft upstream of the convective forcing suppresses the orographic uplift when the convective forcing is located at or downstream of the mountain center. Deep updraft forced by the convective forcing above the upslope of the mountain enhances the orographic uplift. The analysis of the ratio of the horizontal and vertical velocities forced by the convective forcing to those forced by the orographic forcing at the cloud base height upstream of the mountain shows that the relative sign and strength of orographic-convective flows are strongly dependent on various factors such the location of convection relative to the mountain. It is found that the reflectivity and phase change of vertically propagating internal gravity waves at the tropopause play key roles in controlling orographic-convective flows. The vertical fluxes of the horizontal momentum transported by vertically propagating internal gravity waves are calculated. The total momentum flux contains the components related to the nonlinear interaction between convectively and orographically forced waves. The importance of the nonlinear interaction terms in the total momentum flux is emphasized.