  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.
