Over-reflection and instability

Stability problems involving parallel shear flow are considered in the context of wave-reflection problems. It is found that if an unstable solution exists and its growth rate is sufficiently small, the growth rate can be connected to the reflection coefficient through a formula as if the unstable growth were the direct result of repeated over-reflections. If the stability problem under consideration has time symmetry, then for every growing solution there exists a corresponding decaying solution. It is shown that a consistent formula can also be derived for the decaying solution, and the existence of at least one critical layer in the corresponding wave-reflection problem is needed in order to account for both the growing and decaying solutions. As an application of these concepts, the small-scale non-geostrophic instabilities found by Stone (1970) are identified to be associated with over-reflection of inertia-gravity waves.