Nonlinear effects in two-layer large-amplitude geostrophic dynamics. Part 2. The weak-beta case

This paper is a continuation of our study on nonlinear processes in large-a mplitude geostrophic (LAG) dynamics. Here, we examine the so-called weak-be ta models. These models arise when the intrinsic length scale is large enou gh so that the dynamics is geostrophic to leading order but not so large th at the beta-effect enters into the dynamics at leading order (but remains, nevertheless, dynamically non-negligible). In contrast to our previous anal ysis of strong-P LAG models in Part 1, we show that the weak-beta models al low for vigorous linear baroclinic instability. For two-layer weak-beta LAG models in which the mean depths of both layers are approximately equal, the linear instability problem can exhibit an ultr aviolet catastrophe. We argue that it is not possible to establish conditio ns for the nonlinear stability in the sense of Liapunov for a steady flow. We also show that the finite-amplitude evolution of a marginally unstable h ow possesses explosively unstable modes, i.e. modes for which the amplitude becomes unbounded in finite time. Numerical simulations suggest that the d evelopment of large-amplitude meanders, squirts and eddies is correlated wi th the presence of these explosively unstable modes. For two-layer weak-beta LAG models in which one of the two layers is substa ntially thinner than the other, the linear stability problem does not exhib it an ultraviolet catastrophe and it is possible to establish conditions fo r the nonlinear stability in the sense of Liapunov for steady flows. A fini te-amplitude analysis for a marginally unstable how suggests that nonlinear ities act to stabilize eastward and enhance the instability of westward flo ws. Numerical simulations are presented to illustrate these processes.