Abstract
By
analysing the transient evolution of an initial perturbation, it is
shown that a stably sratified 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 kc= square root of 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.