PROPAGATION OF NEAR-INERTIAL OSCILLATIONS THROUGH A GEOSTROPHIC FLOW

The method of multiple time scales is used to obtain an approximate de scription of the linear propagation of near-inertial oscillations (NIO s) through a three-dimensional geostrophic flow. This 'NIO equation' u ses a complex held, M(x, y, z, t), related to the demodulated horizont al velocity by M-z = exp (if(0)t)(u + iv), when f(0) is the inertial f requency. The three processes of wave dispersion, advection by geostro phic velocity and refraction (geostrophic vorticity slightly shifts th e local inertial frequency) are all included in the formulation. The N IO equation has an energy conservation law, so that there is no transf er of energy between NIOs and the geostrophic flow in the approximatio n scheme. As an application, the NIO equation is used to examine propa gation of waves through a field of smaller scale, geostrophic eddies. The spatially local zeta/2 frequency shift, identified by earlier WKB calculations (zeta is the vertical vorticity of the geostrophic eddies ), is not expressed directly in the wave field: the large-scale NIO sa mples regions of both positive and negative zeta so that there is canc ellation. Instead, the zeta/2 frequency shift is rectified to produce an average dispersive effect. The calculation predicts that an NIO wit h infinite horizontal scale has a frequency shift -Kf(0)m(2)/N-2 where K is average kinetic energy density of the geostrophic eddies, m the vertical wavenumber of the NIO, f(0) the inertial frequency and N the buoyancy frequency. Because of the dependence of the frequency shift o n m(2), there is an effective vertical dispersion, whose strength is p roportional to the eddy kinetic energy. This process greatly increases the vertical propagation rate of synoptic scale NIOs.