Instability of a boundary layer flow on a vertical wall in a stably stratified fluid

The stability of a horizontal boundary layer flow on a vertical wall in a viscous stably stratified fluid is considered in this work. A temporal stability analysis is performed for a tanh velocity profile as a function of the Reynolds number $Re=UL/{\it\nu}$ and the Froude number $F=U/(LN)$ where $U$ is the main stream velocity, $L$ the boundary layer thickness, $N$ the buoyancy frequency and ${\it\nu}$ the kinematic viscosity. The diffusion of density is neglected. The boundary layer flow is found to be unstable with respect to two instabilities. The first one is the classical viscous instability which gives rise to Tollmien–Schlichting (TS) waves. We demonstrate that, even in the presence of stratification, the most unstable TS wave remains two-dimensional and therefore independent of the Froude number. The other instability is three-dimensional, inviscid in nature and associated with the stratification. It corresponds to the so-called radiative instability. We show that this instability appears first for $Re\geqslant Re_{c}^{(r)}\approx 1995$ for a Froude number close to 1.5 whereas the viscous instability develops for $Re\geqslant Re_{c}^{(v)}\approx 3980$. For large Reynolds numbers, the radiative instability is also shown to exhibit a much larger growth rate than the viscous instability in a large Froude number interval. We argue that this instability could develop in experimental facilities as well as in geophysical situations encountered in ocean and atmosphere.