Reflection of hydrostatic gravity waves in a stratified shear flow. Part I: theory.

Abstract

Continuous partial reflection of linear hydrostatic gravity waves that propagate through a stratified shear flow is examined. The complex reflection coefficient R satisfies a Riccati equation, which is a first-order nonlinear differential equation. It is shown that (R) < 1 since critical levels and overreflection are not considered. In this case the conservation of wave action flux may be expressed as a relationship between (R) and El-1, where E is the wave energy and l a characteristic inverse vertical length scale of the background state. It is demonstrated that R for a layered model represents a limiting solution of the Riccati equation. A general solution is also derived, under the assumption that the characteristic scale l is directly proportional to the inverse scale height of the characteristic impedance associated with a stratified shear flow. It is shown that the vanishing of (R) at a specific level is analogous to the vanishing of (R) in a three layer model, when the characteristic impedances in the top and the bottom layers satisfy a matching condition. Finally, various properties of the reflection coefficient are displayed for a particular background state. The extension of the theory to encompass other types of wave motion is indicated. -Author