Interaction between Rossby and Gravity Waves in a Simple Analytical Model

The interaction between Rossby and gravity waves is examined in a vertical-zonal two-dimensional model, in which the basic state has an upward gradient of buoyancy at a lower level and a downward gradient of horizontal vorticity at a higher level. Because of the gradients, there exist westward-propagating upper Rossby and westward-and eastward-propagating lower gravity waves, where the propagation is relative to the fluid. The initial value problem for the disturbance is analytically solved. The temporal evolution of the analytical solution from an initial value shows the following characteristics. Resonant interaction between the westward-propagating upper Rossby wave and the eastward-propagating lower gravity wave is possible, in the same way as between two counter-propagating Rossby waves in barotropic and baroclinic problems. On the boundary in the parameter space between the unstable region, where resonant exponential growth occurs, and the stable region, where the solution oscillates, the marginal solution grows as a linear function of time. As in other instability problems, the marginal linear growth in the present model is not trivial. For small horizontal wave numbers, the westward-propagating gravity wave makes a non-negligible contribution to the resonant interaction between the westward-propagating Rossby wave and the eastward-propagating gravity waves.