## Abstract

In this study, nonlinear effects of barotropic instability in a downstream varying easterly jet are studied and compared with previous linear model results of Tupaz and others. The barotropic vorticity equation with Rayleigh friction and forcing is solved with finite differences. The initial mean flow is an easterly Bickley jet whose maximum speed and half-width vary downstream; the half-width ranges from 500 to 1200 km and the maximum speed is 30 m s^{−1}. The time-independent forcing makes the initial mean flow, which is unstable in the central jet region, a steady-state solution to the vorticity equation. A disturbance with wavenumber 10, which is predicted to be locally unstable and most dominant based on linear model results, is added to the initial mean flow. The equation is then integrated numerically for 450 days.

The solutions may be separated into two phases: 1) an initial adjustment phase which consists of several ∼50-day cycles wherein an initial wavenumber 10 disturbance grows rapidly in the jet region, and then the disturbance energy shifts to a slightly longer wavelength and decays before the next cycle; and 2) a quasi-equilibrium phase which is achieved after 350 days. Fourier analysis of the disturbance streamfunction at each point during a typical interval in the adjustment phase shows two dominant modes with periods near 3.35 days and 3.58 days, respectively. After entering the quasi-equilibrium phase, a 4-day oscillation develops in the kinetic energy and the main periods of the streamfunction become 4 and 2 days, respectively. The former is the dominant mode and the latter is the result of the nonlinear self-interaction by the former. The frequency of the dominant mode is equal to the frequency of the most unstable mode from a parallel flow calculation based on the outflow region mean flow. However, in most of the unstable region, it is much less than the most unstable local frequency inferred from the parallel flow solution.

The dominant mode in the quasi-equilibrium phase propagates through the modified mean flow essentially as a linear wave, and its behavior can be compared with the linear model results. However, its maximum growth rate is 25% larger than the highest local growth rate for the parallel flow solution. This “enhancement effect” is also larger than was found by Tupaz and others. In addition, there is a hysteresis effect wherein the growth rate curve and the phase structure from the full model are shifted downstream relative to the parallel flow solution, similar to the linear model results. On the other hand, the wavelength is generally short in the jet region and much longer in the outer regions, opposite to the wavelength variation in Tupaz and others. With the help of a generalized Rossby wave formula, it is shown that two effects determine the downstream variation of the disturbance wavelength: 1) the variation of the latitudinal integral of the mean zonal wind and 2) the variation of the latitudinal integral of the mean absolute vorticity gradient. Due to the difference in disturbance scale, the second effect dominates in the quasi-equilibrium phase of this study while the first effect dominates the linear model used by Tupaz and others.