The Stability of Currents in the Atmosphere and the Ocean: Part I

Abstract

The stability with respect to quasi-geostrophic disturbances of atmospheric and oceanic currents containing both horizontal and vertical shear is investigated for both a continuously stratified and a twolayer model. Certain necessary conditions for instability are derived. The potential vorticity gradient of the basic flow must be both positive and negative for instability to occur in the two-layer model. In the continuous model the condition for instability states that the potential vorticity gradient must either change sign or be balanced by surface terms incorporating the surface potential temperature gradient and topographical variations. For a large class of flows, the product of the zonal current and the potential vorticity gradient must be somewhere positive for instability to occur, and the maximum of this product bounds the growth rate. The square of the growth rate is bounded for an arbitrary flow field by the maximum value of the available mean kinetic and potential energy densities. The phase speed of unstable waves is within limits specified by the mean velocity field, the planetary vorticity gradient, and topographical variations across the stream. It is also shown that a wide class of flows in the two-layer model must be stable for sufficiently high zonal wave number.