The Influence of Gravity Gradient on the Inertialess Stratified Flow and Vortex Structure over an Obstacle in a Narrow Channel

The current study examines the influence of a varying gravity field and its interaction with density stratification. This represents a novel area in baroclinic flow analysis. The classical vortex and internal wave structures in stratified flows are shown to be significantly modified when gravity varies with height. Vortices may shift, stretch, or weaken depending on the direction and strength of gravity variation, and internal waves develop asymmetries or damping that are not present under constant gravity. We examine the influence of gravity variation on the flow of both homogeneous and density-stratified fluids in a channel with topography consisting of a Gaussian obstacle lying at the bottom of the channel. The flow is without inertia, induced by the translation of the top plate. Both the density and gravity are assumed to vary linearly with height, with the minimum density at the moving top plate. The narrow-gap approach is used to generate the flow field in terms of the pressure gradient along the top plate, which, in turn, is obtained in terms of the bottom topography and the three parameters of the problem, namely, the Froude number and the density and gravity gradients. The resulting stream function is a fifth-order polynomial in the vertical coordinate. In the absence of stratification, the flow is smooth, affected rather slightly by the variable topography, with an essentially linear drop in the pressure induced by the contraction. For a weak stratified fluid, the streamlines become distorted in the form of standing gravity waves. For a stronger stratification, separation occurs, and a pair of vortices generally appears on the two sides of the obstacle, the size of which depends strongly on the flow parameters. The influence of gravity stratification is closely coupled to that of density. We examine conditions where the coupling impacts the pressure and the velocity fields, particularly the onset of gravity waves and vortex flow. Only a mild density gradient is needed for flow separation to occur. The influence of the amplitude and width of the obstacle is also investigated.