Jacques Vanneste, University of Edinburgh

The atmosphere and ocean can be be thought of as two-time-scale systems in which the dominant slow motion (termed balanced motion) is weakly coupled to fast inertia-gravity waves. A long-standing issue in geophysical fluid dynamics has then been to quantify the coupling between the two types of motion. The spontaneous generation of waves by slowly evolving flows has, in particular, received a great deal of attention, from the mid-1980s work by Lorenz on the atmospheric slow manifold to recent series of high-resolution numerical simulations. In this talk, I discuss asymptotic approaches to this problem. In the rotation-dominated regime on which I focus, there is a strict time-scale separation between balanced motion and fast waves. As a result, the wave generation is exponentially small in the relevant small parameter. Exponential-asymptotics techniques can be applied to describe the spontaneous generation of the waves and estimate their amplitude. I review several results of this type obtained for simple finite-dimensional models of the atmosphere, and for particular solutions of the fluid equations.