Exponentially Small Inertia–Gravity Waves and the Breakdown of Quasigeostrophic Balance

Abstract

The spontaneous generation of inertia?gravity waves by balanced motion is investigated in the limit of small Rossby number ? ? 1. Particular (sheared disturbance) solutions of the three-dimensional Boussinesq equations are considered. For these solutions, there is a strict separation between balanced motion and inertia?gravity waves for large times. This makes it possible to estimate the amplitude of the inertia?gravity waves that are generated spontaneously from perfectly balanced initial conditions. It is shown analytically using exponential asymptotics, and confirmed numerically, that this amplitude is proportional to ??1/2 exp(?α/?), with a constant α > 0 and a proportionality constant that are given in closed form. This result demonstrates the inevitability of inertia?gravity wave generation and hence the nonexistence of an invariant slow manifold; it also exemplifies the remarkable, exponential, smallness of the wave generation for ? ? 1. The importance of the singularity structure of the balanced motion for complex values of time is emphasized, and some general implications of the results are discussed.