Analytical solutions for algebraic growth of disturbances in a stably stratified shear flow

Abstract

We investigate analytically the short-time response of disturbances in a density-varying Couette flow without viscous and diffusive effects. The complete inviscid problem is also solved as an initial value problem with a density perturbation. We show that the kinetic energy of the disturbances grows algebraically at early times, contrary to the well-known algebraic decay at time tending to infinity. This growth can persist for arbitrarily long times in response to sharp enough initial perturbations. The simplest in our three-stage study is a model problem forced by a buoyancy perturbation in the absence of background stratification. A linear growth with time is obtained in the vertical velocity component. This model provides an analogy between the transient mechanism of kinetic energy growth in a two-dimensional density-varying flow and the lift-up mechanism of the three-dimensional constant density flow. Next we consider weak stable background stratification. Interestingly, the lowest order solution here is the same as that of the model flow. Our final study shows that a strong background stratification results in a sub-linear growth with time of the perturbation. A framework is thus presented where two-dimensional streamwise disturbances can lead to large transient amplification, unlike in constant density flow where three dimensions are required.