Interaction of Inertial Oscillations with a Geostrophic flow

In this work we investigate the interaction of wind excited fast inertial oscillations (IOs) in the upper ocean with a slow geostrophic balanced flow in the strong dispersion regime (SDR), Bu(n) ~ 1, Ro << 1, where Bu(n) is Burger number corresponding to the nth baroclinic mode and Ro is Rossby number based on the geostrophic flow, and study transition from SDR to weak dispersion regime (WDR), Bu(n) ~ Ro << 1. In SDR the initial IO field does not undergo amplitude modulation. As a consequence, all the interaction effects are O(Ro). The interaction results in the formation of new small scale inertial oscillations trapped in the upper ocean and gravity waves that propagate vertically into the deep ocean within a few inertial periods. The regular perturbation expansion used for SDR breaks down as Bu(n) is decreased by going to higher baroclinic modes and a modal transition from SDR to WDR takes place. We construct a simple 1D rotating shallow water model that captures SDR, WDR and also transition from SDR to WDR as Bu(n) is decreased from O(1) to O(Ro) for fixed Ro << 1. In the 1D model, we also identify a special subset of WDR -- very weak dispersion regime (vWDR), where Bu(n) ~ Ro^2 in which case the IO amplitude develops finer spatial scales due to weaker dispersion.