Linear Stability Theory Of Zonal Shear Flows With A Free Surface

We investigate the linear stability of zonal shear flows with a free surface (like the flow in oceans) taking into account the buoyancy effects at the deformed free surface. The necessary and sufficient conditions for the existence of neutral and unstable modes are established, and a semicircle theorem for unstable modes is deduced. An upper bound on the growth rate of an unstable mode is established. We have given a sufficient condition for stability that has a close resemblance with Miles\u27 criterion for stability of a continuously stratified shear flow. The number of distinct unstable modes is shown not to exceed the number of neutral modes. The stability characteristics of smoothly-varying unbounded shear-layer flow profiles for long-wave disturbances are investigated. Finally, stability of a linear zonal flow velocity profile and a zonal Bickley jet-flow profile are investigated. Some of these results are new even in the special barotropic case (with no free-surface effect). © 2001 OPA (Overseas Publishers Association) N.V. Published by license under the Gordon and Breach Science Publishers imprint, a member of the Taylor & Francis Group