On a new non-Boussinesq instability in stratified lakes and oceans
Lakes and many other geophysical flows are shallow, density stratified, and contain a free-surface. Conventional studies on stratified shear instabilities make Boussinesq approximation. Free-surface arising due to large density variations between air and water cannot be taken into consideration under this approximation. Hence the free-surface is usually replaced by a rigid-lid, and therefore has little effect on the stability of the fluid below it. In this paper we have performed non-Boussinesq linear stability analyses of a double circulation velocity profile prevalent in two-layered density stratified lakes. One of our analyses is performed by considering the presence of wind, while the other one considers quiescent air. Both analyses have shown similar growth rates and stability boundaries. We have compared our non-Boussinesq study with a corresponding Boussinesq one. The maximum non-Boussinesq growth rate is found to be an order of magnitude greater than the maximum Boussinesq growth rate. Furthermore, the stability curves in these two studies are very different. The non-Boussinesq instability as well as the Boussinseq one can become three dimensional in some sub-ranges of the bulk Richardson number. An analytical study has also been conducted on a simple broken-line profile representing double circulation in order to complement the numerical stability analysis. The analytical growth rates are in good agreement with the non-Boussinesq numerical growth rates observed in continuous profiles. The physical mechanism behind this non-Boussinesq instability is the resonant interaction between a surface vorticity-gravity wave existing at the free-surface (air-water interface), and an interfacial vorticity-gravity wave existing at the pycnocline (warm and cold water interface). We expect similar kind of non-Boussinesq instability to occur in the upper layer of oceans.