Abstract
When atmospheric storms pass over the ocean, they resonantly force near-inertial waves (NIWs), internal waves with a frequency close to the local Coriolis frequency $f$. It has long been recognised that the evolution of NIWs is modulated by the ocean's mesoscale eddy field. This can result in NIWs being concentrated into anticyclones which provide an efficient pathway for NIW propagation to depth. Here we analyse the eigenmodes of NIWs in the presence of mesoscale eddies and heavily draw on parallels with quantum mechanics. Whether the eddies are effective at modulating the behaviour of NIWs depends on the wave dispersiveness $\varepsilon ^2 = f\lambda ^2/\varPsi$, where $\lambda$ is the deformation radius and $\varPsi$ is a scaling for the eddy streamfunction. If $\varepsilon \gg 1$, NIWs are strongly dispersive, and the waves are only weakly affected by the eddies. We calculate the perturbations away from a uniform wave field and the frequency shift away from $f$. If $\varepsilon \ll 1$, NIWs are weakly dispersive, and the wave evolution is strongly modulated by the eddy field. In this weakly dispersive limit, the Wentzel–Kramers–Brillouin approximation, from which ray tracing emerges, is a valid description of the NIW evolution even if the large-scale atmospheric forcing apparently violates the requisite assumption of a scale separation between the waves and the eddies. The large-scale forcing excites many wave modes, each of which varies on a short spatial scale and is amenable to asymptotic analysis analogous to the semi-classical analysis of quantum systems. The strong modulation of weakly dispersive NIWs by eddies has the potential to modulate the energy input into NIWs from the wind, but we find that this effect should be small under oceanic conditions.