Holmboe instability in non-Boussinesq fluids

We perform the stability analysis for stratified shear flows whose density transition layer is much thinner than, and possibly, displaced with respect to, the velocity shear layer for which Holmboe instability along with the well-known Kelvin-Helmholtz (KH) instability is known to be present. Here, we provide a more complete picture of stability characteristics of stratified shear flows with taking into account the effects of non-negligible density increment for which the classical Boussinesq approximation is no longer valid. It is shown that, in addition to the Kelvin-Helmholtz and Holmboe instabilities for which two unstable modes exist, there is another instability with a single unstable mode so that the unstable waves excited by this instability mechanism propagate only in one direction. Depending on the physical parameters, this unstable mode may not be captured by the stability analysis under the Boussinesq approximation. With a better understanding of the instability mechanisms with including the non-Boussinesq effects, we could validate some of previous experimental results and provide new evidences to observations that have not been fully explained. The results are also expected to be useful in designing laboratory experiments to observe Holmboe waves and estimating their wavelengths and phase speeds.