Elementary stratified flows with stability at low Richardson number

We revisit the stability analysis for three classical configurations of multiple fluid layers proposed by Goldstein ["On the stability of superposed streams of fluids of different densities," Proc. R. Soc. A. 132, 524 (1931)], Taylor ["Effect of variation in density on the stability of superposed streams of fluid," Proc. R. Soc. A 132, 499 (1931)], and Holmboe ["On the behaviour of symmetric waves in stratified shear layers," Geophys. Publ. 24, 67 (1962)] as simple prototypes to understand stability characteristics of stratified shear flows with sharp density transitions. When such flows are confined in a finite domain, it is shown that a large shear across the layers that is often considered a source of instability plays a stabilizing role. Presented are simple analytical criteria for stability of these low Richardson number flows.