Jacques Vanneste School of Mathematics, University of Edinburgh, UK J.Vanneste@ed.ac.uk
In the standard quasi-geostrophic regime, with small Rossby number and order-one Burger number, there is a formal time-scale separation between the slow balanced motion on the one hand, and the fast inertia-gravity waves of all spatial scales on the other hand. In principle, there are then no obstacles to the definition of a slow manifold and associated balanced model that are accurate to arbitrary (algebraic) order in the Rossby number. Therefore, one expects the phenomena not captured by (any) balanced models, in other words the inertia-gravity waves (IGWs) generated spontaneously, to have an amplitude that is exponentially small in the Rossby number. This has been shown to be the case in a few idealised models of geophysical fluids for which explicit analytic results can be derived. We will review these results, introduce some of the ideas used in the study of exponentially small phenomena, and discuss their relevance to the problem of IGW generation. Recent results on the generation of IGWs by a sheared vortex and by shear-flow instabilities will also be presented.