Nonlinear Evolution of Forced Rossby Waves in a Barotropic Atmosphere Part II: Initial-value Problems

Abstract

The nonlinear temporal evolution of topographically forced Rossby waves and unstable perturbations examined in Part I is investigated analytically and numerically by using a barotropic quasi-geostrophic inviscid model.
For the near-resonant flow, a low-order spectral model as in Charney and DeVore (1979) is dealt with analytically. In the linearly unstable region (topographic instability), the evolution of the wave amplitude depends on the sign of the initial zonal flow perturbation: If the zonal flow is decelerated initially, the wave amplifies in a nonlinear oscillation at the expense of the initial zonal kinetic energy through the form-drag; otherwise, the wave amplitude decreases. The period of this oscillation decreases with the increase of the amplitude of the initial forced wave. In the linearly neutral region, on the other hand, the behavior of the motion depends on the magnitude of the initial perturbation. For small perturbations waves show small variations around the steady solution and evolve large trajectories like as in the unstable region for larger perturbations. These properties in the low-order model are also justified in a full nonlinear spectral model with many variables. The relationship between the zonal flow and the forced wave in this evolution shows that the theory of Tung and Lindzen (1979a, b) for the linear resonant growth of the forced Rossby wave is not applicable to this amplification of the planetary waves. It is also found that the results obtained in our nonlinear model essentially confirm the evolution of the waves in the weakly nonlinear theory of Plumb (1979; 1981a, b) for the near-resonant forced Rossby wave.
On the other hand, for the off-resonant flow, the result of some numerical integrations of the full nonlinear spectral equation shows that the time-variation of planetary waves is produced by the unstable perturbation with a horizontal scale different from that of the forced Rossby wave; this evolution is completely different from that in the near-resonasnt region.