Stability analysis of stratified shear flows with a monotonic velocity profile without inflection points. Part 2. Continuous density variation

Abstract

We investigate stability with respect to two-dimensional (independent of z) disturbances of plane-parallel shear flows with a velocity profile Vx=u(y) of a rather general form, monotonically growing upwards from zero at the bottom (y=0) to U0 as y → ∞ and having no inflection points, in an ideal incompressible fluid stably stratified in density in a layer of thickness ℓ, small as compared to the scale L of velocity variation. In terms of the ‘wavenumber k – bulk Richardson number J’ variables, the upper and lower (in J) boundaries of instability domains are found for each oscillation mode. It is shown that the total instability domain has a lower boundary which is convex downwards and is separated from the abscissa (k) axis by a strip of stability 0 < J < J0(−)(k) with minimum width J*=O(ℓ2/L2) at kL=O(1). In other words, the instability domain configuration is such that three-dimensional (oblique) disturbances are first to lose their stability when the density difference across the layer increases. Hence, in the class of flows under consideration, it is a three- not two-dimensional turbulence that develops as a result of primary instability.