The transient emission of propagating gravity waves by a stably stratified shear layer

By analysing the transient evolution of an initial perturbation, it is shown that a stably stratified shear layer emits gravity waves having well defined dynamical characteristics. For instance, the outgoing waves systematically have phase lines tilting against the shear, their vertical momentum flux has a sign opposite to that of the shear and their amplitude increases when the flow stability decreases. Those features are commonly observed in numerical simulations of gravity waves generated by convection. It is shown that they are related to the singular vectors of the system that have fast energy growth and fast energy decay within a finite time. In this work, the singular vectors are computed using a linear gravity-wave model, its adjoint and an iterative Lanczos algorithm. For a given shear layer, characterized by a minimum Richardson number Ri and a depth d, these perturbations show that emission of gravity waves by a stratified shear layer essentially occurs for waves with horizontal wave number close to a critical value k_c = √Ri/d. The importance of the singular vectors on the dynamics of more general initial conditions is also tested by making few ensembles of numerical simulations with stochastic initial conditions imposed inside the shear layer. The amplitude of the momentum fluxes of the outgoing waves as a function of Ri is also evaluated systematically. It gives a relationship between the efficiency of gravity-wave emission and Ri that could be used in convective gravity-wave parametrization schemes.