Barotropic instability of a zonal jet:: From nondivergent perturbations on the β plane to divergent perturbations on a sphere

The linear instability of divergent perturbations that evolve on a cos(2) mean steady zonal jet embedded in a zonal channel on the beta plane and on a rotating sphere is studied for zonally propagating wavelike perturbations of the shallow-water equations. The complex phase speeds result from the imposition of the no-flow boundary conditions at the channel walls on the numerical solutions of the linear differential equations for the wave latitude-dependent amplitude. In addition, the same numerical method is applied to the traditional problem of linear instability of nondivergent perturbations on the beta plane where results reaffirm the classical, analytically derived, features. For these nondivergent perturbations, the present study shows that the growth rate increases monotonically with the jet maximal speed and that the classical result of a local maximum at some finite westward-directed speed results from scaling the growth rates on the jet's speed. In contrast to nondivergent perturbations, divergent perturbations on the beta plane have no short-wave cutoff, and so the nondivergent solution does not provide an estimate for the divergent solution, even when the ocean is 1000 km deep (i.e., when the speed of gravity waves exceeds 10 Mach). For realistic values of the ocean depth, the growth rates of divergent perturbations are smaller than those of nondivergent perturbations, but with the increase in the ocean depth they become larger than those of nondivergent perturbations. For both perturbations, a slight asymmetry exists between eastward- and westward-flowing jets. The growth rates of divergent perturbations on a sphere are similar to those on the beta plane for the same values of the model parameters, but the asymmetry between eastward and westward jets is more conspicuous on a sphere. The value of g'H' (g' is the reduced gravity; H' is the equivalent mean layer thickness), which is filtered out in nondivergent theory, determines for divergent perturbations the relative magnitude of zonal velocity, meridional velocity, and height but has little effect on the growth rates.