The Propagation of Gravity–Inertia Waves and Lee Waves under a Critical Level

Abstract

As is well known, the linear dynamic equations for gravity-inertia waves are characterized by three singular levels, one being the critical level at which flow speed and wave speed are equal, and the other two at which the flow speed is equal to ±f/k, where f is the Coriolis parameter and k the wave number, herein called Rossby singularities. This article discusses the propagation of two-dimensional gravity-inertial disturbances, both monochromatic and with continuous spectrum (i.e., lee waves), in a direction toward all of these singular levels. The study is conducted by analysis, which provides closed-form solutions to the linear equations, and by numerical simulation, which confirms the analysis and also exhibits nonlinearities where these are significant. It is found that the Rossby singularity produces nonlinear reflection of a monochromatic wave, and comparisons are made with the case of the pure gravity wave (f = 0) reflected by a critical level. Unlike that situation, in the present problem the momentum flux is also singular at the reflecting level. However, this is no longer the case when the disturbance contains a continuous spectrum, as in a lee wave produced by a smooth isolated ridge. In this case, the problem is essentially linear, and a relatively simple analytic approximation to the solution is presented and verified by simulation. The critical level acts as a lid but produces no singular effects. However, certain types of forcing profiles are identified that, despite being themselves of small amplitude, do in fact lead to nonlinearities in the field.