Effect of free surface on submerged stratified shear instabilities

Abstract

Here we have considered the effects of shallowness of the domain as well as the air-water free surface on the stratified shear instabilities of the fluid underneath. First, we numerically solve the non-Boussinesq Taylor-Goldstein equation for smooth velocity and density profiles of a model shear layer with a free surface. When the depth of the fluid is relatively shallow compared to the shear layer thickness, the surface gravity waves existing at the free surface come closer to the waves existing in the shear layer. This can lead to resonant wave interactions, making the flow unstable to more varieties of modal instabilities. To obtain a deeper understanding of the instability mechanisms, we have performed analytical studies with broken-line profiles. Furthermore, reduced order broken-line profiles have also been developed, based on which dispersion diagrams are constructed. Through these diagrams we have underpinned the resonantly interacting waves leading to each type of instability. Two new instabilities have been found; one of them, referred to as the surface gravity - interfacial gravity (SG-IG) mode, arises due to the interaction between a surface gravity wave and an interfacial gravity wave, and would be absent if there is no internal density stratification. The other one - the surface gravity - lower vorticity (SG-LV) mode, which arises due to the interaction between a surface gravity wave and the lower vorticity wave, surpasses Kelvin-Helmholtz instability to become the most unstable mode. Except for Holmboe instability, remarkable differences are observed in all other instabilities occurring in shallow domains when the air-water interface is replaced by a rigid lid. We infer that the rigid-lid approximation is valid for large vertical domains. We have also shown that if shear is absent at the free surface, our problem can be modeled using a Boussinesq type approximation.