Simple Stability Limits for Vertically Propagating Unstable Modes in a tanh(z) Velocity Profile with a Rigid Lower Boundary

Abstract

Previous studies have revealed a number of unstable modes in addition to the Kelvin-Helmholtz instability associated with a velocity shear in the presence of a rigid lower boundary. These additional modes occupy regions in the (α,Ri0) plane which are largely distinct from that occupied by the Kelvin-Helmholtz mode. In this note, we demonstrate that the location of the various unstable modes can be explained in part in terms of the modal structure above and below the velocity shear and the necessary condition for dynamical instability, Ric < 1/4, obtained by Miles (1961). An extension of the stability analysis of Lalas and Einaudi (1976) to a larger shear height reveals that all unstable modes belong to one of two mode families. Finally, we discuss the implication of these results for additional modes or families of modes and the possible roles of various modes in atmospheric dynamics.