Rossby wave interactions and instabilities in a rotating, two-layer fluid on a beta-plane. Part II: Stability

The stability of Rossby waves in a two-layer fluid on a β-plane is examined. There are four parameters in the problem, (a) M =UK2 /β where U is the velocity amplitude of the particular wave being considered and K is its wavenumber, (b) F = (Ka c)−2 where a c is the baroclinic radius of deformation, (c) the depth ratio δ = H 1/H 2 where H 1 and H 2 are the mean thicknesses of the upper and lower layer, respectively, and (d) the wave direction. For all values of M, a finite amplitude barotropic Rossby wave is always unstable whatever the wavenumber, and will lose its energy to barotropic disturbances in preference to baroclinic disturbances. Similarly, for all values of M, a finite amplitude baroclinic Rossby wave is unstable whatever the wavenumber. For large M, the stability problem reduces to a Rayleigh-type stability problem for a sinusoidal velocity distribution. This has been solved only for the “barotropic instability” case in the limit of small δ. For small F, the growth rate of the most unstable disturbance is of order UK, and as F increases, this maximum growth rate decreases until, for large F, it is of order UK/F. For small M and small δ, the disturbance consists of either two baroclinic waves, or a barotropic and a baroclinic wave, which interact resonantly with the primary wave. For small values of F, the pure mode triad is preferred, while for large F, the mixed mode triad is preferred. Some oceanographic applications are discussed. It is shown, for instance, that baroclinic eddies possessing horizontal length scales larger than a c should decay into both baroclinic and barotropic motions. It is suggested that these large scale eddies will transfer their energy to smaller des only.