The method of multiple time scales is used to obtain an approximate description of the linear propagation of near-inertial oscillations (NIOs) through a three-dimensional geostrophic flow. This ‘NIO equation’ uses a complex field, M(x, y, z, t), related to the demodulated horizontal velocity by Mz = exp (if0t)(u + iv), where f0 is the inertial frequency. The three processes of wave dispersion, advection by geostrophic velocity and refraction (geostrophic vorticity slightly shifts the local inertial frequency) are all included in the formulation. The NIO equation has an energy conservation law, so that there is no transfer of energy between NIOs and the geostrophic flow in the approximation scheme. As an application, the NIO equation is used to examine propagation of waves through a field of smaller scale, geostrophic eddies. The spatially local ζ/2 frequency shift, identified by earlier WKB calculations (ζ is the vertical vorticity of the geostrophic eddies), is not expressed directly in the wave field: the large-scale NIO samples regions of both positive and negative ζ so that there is cancellation. Instead, the ζ/2 frequency shift is rectified to produce an average dispersive effect. The calculation predicts that an NIO with infinite horizontal scale has a frequency shift −Kf0m2/N2 where K is average kinetic energy density of the geostrophic eddies, m the vertical wavenumber of the NIO, f0 the inertial frequency and N the buoyancy frequency. Because of the dependence of the frequency shift on m2, there is an effective vertical dispersion, whose strength is proportional to the eddy kinetic energy. This process greatly increases the vertical propagation rate of synoptic scale NIOs.
Young, W. R., and Mahdi B. Jelloul. 1997. "Propagation of near-inertial oscillations through a geostrophic flow." Journal of Marine Research 55, (4). https://elischolar.library.yale.edu/journal_of_marine_research/2242