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

The spontaneous generation of inertia-gravity waves by balanced motion is investigated in the limit of small Rossby number K 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(/), with a constant 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 K 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.