Exponential Smallness of Inertia–Gravity Wave Generation at Small Rossby Number

Abstract

Abstract This paper discusses some of the mechanisms whereby fast inertia–gravity waves can be generated spontaneously by slow, balanced atmospheric and oceanic flows. In the small Rossby number regime relevant to midlatitude dynamics, high-accuracy balanced models, which filter out inertia–gravity waves completely, can in principle describe the evolution of suitably initialized flows up to terms that are exponentially small in the Rossby number ɛ, that is, of the form exp(−α/ɛ) for some α > 0. This suggests that the mechanisms of inertia–gravity wave generation, which are not captured by these balanced models, are also exponentially weak. This has been confirmed by explicit analytical results obtained for a few highly simplified models. These results are reviewed, and some of the exponential-asymptotic techniques that have been used in their derivation are presented. Two types of mechanisms are examined: spontaneous-generation mechanisms, which generate exponentially small waves from perfectly balanced initial conditions, and unbalanced instability mechanisms, which amplify unbalanced initial perturbations of steady flows. The relevance of the results to realistic flows is discussed.