Using a physical space (i.e. non-modal) approach, we investigate interactions between
fast inertio-gravity (IG) waves and slow balanced flows in a shallow rotating fluid.
Specifically, we consider a train of IG waves impinging on a steady, exactly balanced
vortex. For simplicity, the one-dimensional problem is studied first; the limitations
of one-dimensionality are offset by the ability to define balance in an exact way.
An asymptotic analysis of the problem in the small-amplitude limit is performed to
demonstrate the existence of interactions. It is shown that these interactions are not
confined to the modification of the wave field by the vortex but, more importantly,
that the waves are able to alter in a non-trivial way the potential vorticity associated
with that vortex. Interestingly, in this one-dimensional problem, once the waves have
traversed the vortex region and have propagated away, the vortex exactly recovers
its initial shape and thus bears no signature of the interaction. Furthermore, we
prove this last result in the case of arbitrary vortex and wave amplitudes. Numerical
integrations of the full one-dimensional shallow-water equations in strongly nonlinear
regimes are also performed: they confirm that time-dependent interactions exist and
increase with wave amplitude, while at the final state the vortex bears no sign of the
interaction. In addition, they reveal that cyclonic vortices interact more strongly with
the wave field than anticyclonic ones.
(A)geostrophic adjustment of dipolar perturbations, formation of coherent structures and their properties, as follows from high-resolution numerical simulations with rotating shallow water model
