## Inertia-gravity wave generation: a WKB approach

##### Abstract

The dynamics of the atmosphere and ocean are dominated by slowly evolving,
large-scale motions. However, fast, small-scale motions in the form of inertia-gravity
waves are ubiquitous. These waves are of great importance for the circulation of the
atmosphere and oceans, mainly because of the momentum and energy they transport
and because of the mixing they create upon breaking. So far the study of inertia-gravity
waves has answered a number of questions about their propagation and dissipation, but
many aspects of their generation remain poorly understood.
The interactions that take place between the slow motion, termed balanced or
vortical motion, and the fast inertia-gravity wave modes provide mechanisms for
inertia-gravity wave generation. One of these is the instability of balanced flows to
gravity-wave-like perturbations; another is the so-called spontaneous generation in
which a slowly evolving solution has a small gravity-wave component intrinsically
coupled to it.
In this thesis, we derive and study a simple model of inertia-gravity wave
generation which considers the evolution of a small-scale, small amplitude perturbation
superimposed on a large-scale, possibly time-dependent °ow. The assumed spatial-scale
separation makes it possible to apply a WKB approach which models the perturbation
to the flow as a wavepacket. The evolution of this wavepacket is governed by a set of
ordinary differential equations for its position, wavevector and its three amplitudes. In
the case of a uniform flow (and only in this case) the three amplitudes can be identifed
with the amplitudes of the vortical mode and the two inertia-gravity wave modes. The
approach makes no assumption on the Rossby number, which measures the time-scale
separation between the balanced motion and the inertia-gravity waves.
The model that we derive is first used to examine simple time-independent flows,
then flows that are generated by point vortices, including a point-vortex dipole and
more complicated flows generated by several point vortices. Particular attention is also
paid to a flow with uniform vorticity and elliptical streamlines which is the standard
model of elliptic instability. In this case, the amplitude of the perturbation obeys a
Hill equation. We solve the corresponding Floquet problem asymptotically in the limit
of small Rossby number and conclude that the inertia-gravity wave perturbation grows
with a growth rate that is exponentially small in the Rossby number. Finally, we apply
the WKB approach to a flow obtained in a baroclinic lifecycle simulation. The analysis
highlights the importance of the Lagrangian time dependence for inertia-gravity wave
generation: rapid changes in the strain field experienced along wavepacket trajectories
(which coincide with fluid-particle trajectories in our model) are shown to lead to
substantial wave generation.