A re-examination of the inertial levels in vertically sheared rotating stratified flows

One of the preferential location for the breakdown of balanced dynamics are inertial levels. Across these, balanced disturbances become inertia-gravity waves (IGWs) in the linear approximation. This transition is associated with ageostrophic modes of baroclinic instability and with the emission of GWs from potential vorticity anomalies. Here we also consider a complementary problem and analyse how an incident IGW yields an exponentially decaying response between inertial levels. This exponential response is close to a quasi-geostrophic solution between the two inertial levels. This result, supported by exact and approximate solutions, gives a heuristic interpretation to the absorption of IGWs by critical layers in terms of tunelling. In this interpretation, the factor exp(-π (J(1+ν²)-0.25) ^1/2 ) where J is the Richardson number and ν the ratio between the zonal and meridional wavenumbers found by Booker and Bretherton~(1967) is simply the decay in amplitude of the evanescent solution between the lowest and the upper inertial levels. For values of the Richardson number J<1 we also show how a small incident IGW can yield very large disturbances between the inertial levels, triggering some form of inertial instabilities.