Density currents in two-layer shear flows

An earlier two-fluid model of an idealized density-current in low-level shear is extended to include variable upper-level shear. Far-field solutions are determined, based on the conservation of mass, momentum and vorticity, and the conservation of Bernoulli function (energy) along streamlines for inviscid flows. It is found that the upper-level and low-level shears play similar roles in controlling the depth of steady-state density-currents. In most cases, large positive upper-level shear supports a deeper density-current and steeper front, and therefore a stronger updraught. It is also found that when the low-level shear is weak and upper-level shear occupies about half the domain depth, larger positive shear can result in a shallower rather than a deeper density-current. This behaviour was not found for either constant shear flow or flows with only low-level shear. The behaviour is understood by examining the flow structure and flow-force components as a function of the upper-level shear. Furthermore, by allowing the upper-level shear to vary, an overturning Row is permitted ahead of the density current. This was not possible in the earlier model in which the upper-level flow was assumed to be constant. The present extension allows us to draw closer analogues between the model solutions and the circulation patterns found in typical squall-lines in sheared enviromnents. Time-dependent numerical experiments are conducted for a range of upper- and low-level shears. The depth and the propagation speed of simulated density-currents are found to agree very well with predictions by the idealized theoretical model. This verifies the validity of the theoretical model. In addition, numerical experiments with identically zero low-level shear but differing upper-level shears suggest that the deeper shear is as important as the low-level shear in determining the uprightness of the upward branch of inflow. In fact, the presence of positive low-level inflow shear may not be essential. Such results are also supported by the theoretical model, and may have important implications for our understanding of squall-line dynamics.