Nonlinear critical layer development of forced wave packets in geophysical shear flows

We investigate the nonlinear development of a forced wave packet in the presence of a critical layer in a shear flow. Two different geophysical flows are considered: vertically propagating internal gravity wave packets in a stratified shear flow and Rossby wave packets propagating toward the equator in a zonal flow. Most previous analyses of these phenomena have dealt with spatially periodic, monochromatic waves. These studies observed that, in the initial linear stages, the disturbance is absorbed at the critical layer, but subsequently, the linear theory breaks down and nonlinear phenomena such as wave breaking and reflection result. For a more realistic representation of wave activities in the atmosphere, we employ a forcing in the form of a spatially localized wave packet, rather than a monochromatic wave. We solve the nonlinear equations numerically using a pseudo-spectral Fourier approximation and a high order compact finite difference scheme.; It is found that the spatial localization delays the onset of the nonlinear breakdown in the critical layer, the absorption of the disturbance continues for large time and there is an outward flux of momentum in the zonal or horizontal direction. For the Rossby wave packet problem, we also derive an approximate analytical solution for the special case of long waves. According to this solution, the total length of the packet increases with time, as seen in the numerical simulations. In the gravity wave packet problem, the horizontal extent of the packet increases with time, but there appears to be a different mechanism for this: the part of the disturbance that spreads out is centered at the zero wave number. The region over which the packet interacts with the mean flow increases in length with time. We observe also that the prolonged absorption of the disturbance stabilizes the solution to the extent that it is always convectively stable; the local Richardson number remains positive well into the nonlinear regime. In this sense, our results differ from those in the case of monochromatic forcing in which significant regions with negative Richardson number appear.