Effect of variation in density on the stability of bilinear shear currents with a free surface

We perform the stability analysis for a free surface fluid current modeled as two finite layers of constant vorticity, under the action of gravity and absence of surface tension. In the same spirit as Taylor ["Effect of variation in density on the stability of superposed streams of fluid," Proc. R. Soc. London, Ser. A 132, 499 (1931)], a geometrical approach to the problem is proposed, which allows us to present simple analytical criteria under which the flow is stable. A strong destabilizing effect of stratification in density is perceived when the results are compared with those obtained for the physical setting where the vorticity interface is also a density interface separating two immiscible fluids with constant densities. In contrast with the homogeneous case, the stratified bilinear shear current is mostly unstable and can only be stabilized when the background current in the upper layer is constant.