Stating the Linear Stability Problem for Plane-Parallel Flows of Ideal Homogeneous and Nonhomogeneous Fluids

Abstract

In the previous chapter we used the motion of a rigid body with a fixed point to get acquainted with two methods of studying linear and nonlinear stability of fixed points of a system. Those methods were based on linearization of the equations of motion and on the use of the first integrals of motion, respectively. The subject of the next few chapters will be the development and use of these methods in studying the stability of two-dimensional stationary flows of an ideal fluid. According to the theorem of H.B. Squire (see Lin, 1958) the most unstable modes develop along the plane of the flow, so one can ignore the three-dimensional perturbations and remain within the two-dimensional formulation of the problem.

We have already mentioned that stratification of a fluid rather than its compressibility plays the decisive role in the formation of baroclinic global atmospheric flows. So let us first study the linear stability of two-dimensional flows of an incompressible stratified fluid in a gravity field without taking into account the Coriolis forces. The results, as we shall see below, can be easily generalized to the global baroclinic geophysical flows. As a bonus, assuming the fluid density to be constant, we will obtain results on barotropic flows, and also generalize them to global geophysical flows.