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Here we have considered the effects of shallowness of the domain as well asthe air-water free surface on the stratified shear instabilities of the fluidunderneath. First, we numerically solve the non-Boussinesq Taylor-Goldsteinequation for smooth velocity and density profiles of a model shear layer with afree surface. When the depth of the fluid is relatively shallow compared to theshear layer thickness, the surface gravity waves existing at the free surfacecome closer to the waves existing in the shear layer. This can lead to resonantwave interactions, making the flow unstable to more varieties of modalinstabilities. 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 whichdispersion diagrams are constructed. Through these diagrams we have underpinnedthe resonantly interacting waves leading to each type of instability. Two newinstabilities have been found; one of them, referred to as the surface gravity- interfacial gravity (SG-IG) mode, arises due to the interaction between asurface gravity wave and an interfacial gravity wave, and would be absent ifthere is no internal density stratification. The other one - the surfacegravity - lower vorticity (SG-LV) mode, which arises due to the interactionbetween a surface gravity wave and the lower vorticity wave, surpassesKelvin-Helmholtz instability to become the most unstable mode. Except forHolmboe instability, remarkable differences are observed in all otherinstabilities occurring in shallow domains when the air-water interface isreplaced by a rigid lid. We infer that the rigid-lid approximation is valid forlarge vertical domains. We have also shown that if shear is absent at the freesurface, our problem can be modeled using a Boussinesq type approximation.

Effect of free surface on submerged stratified shear instabilities