In this paper, we investigate the meridional propagation of a forced Rossby wave packet towards a critical layer in a zonal shear flow by solving the linearized barotropic vorticity equation. The forcing is applied north of the critical layer. Two approaches are employed for solving this problem. First, an analytic solution valid for large time is derived, using Fourier and Laplace transform techniques and asymptotic approximations. This solution exhibits the modification due to the wave packet of the solution obtained by Warn and Warn (1976) [Warn, T., Warn H., 1976. On the development of a Rossby wave critical level. J. Atmos. Sci., 33, 2021-2024.] in the monochromatic case. A numerical investigation is then carried out using a finite difference absorbed a the critical layer and the total momentum transferred to the mean flow as a result of the absorption is observed to be proportional to the length scale of the wave packet. We also consider the case of a north-south mean flow with a longitudinally propagating wave packet forced to the east or west of the critical layer. The monochromatic version of this problem has been used before (Gesler, J.E., Dickinson, R.E., 1975. Critical level absorption of barotropic Rossby waves in a north-south flow. J. Geophys. Res., 80, 3805-3811.) to examine the interaction of western boundary currents and oceanic Rossby waves.

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Journal Dynamics of Atmospheres and Oceans
Campbell, L, & Maslowe, S.A. (S. A.). (1998). Forced Rossby wave packets in barotropic shear flows with critical layers. Dynamics of Atmospheres and Oceans, 28(1), 9–37. doi:10.1016/S0377-0265(98)00044-X