This paper presents a simplified model of the process through which a geostrophic flow enhances the vertical propagation of near-inertial activity from the mixed layer into the deeper ocean. The geostrophic flow is idealized as steady and barotropic with a sinusoidal dependence on the north-south coordinate; the corresponding streamfunction takes the form ψ = - Ψ cos (2αy). Near-inertial oscillations are considered in linear theory and disturbances are decomposed into horizontal and vertical normal modes. For this particular flow, the horizontal modes are given in terms of Mathieu functions. The initial-value problem can then be solved by projecting onto this set of normal modes. A detailed solution is presented for the case in which the mixed layer is set into motion as a slab. There is no initial horizontal structure in the model mixed layer; rather, horizontal structure, such as enhanced near-inertial energy in regions of negative vorticity, is impressed on the near-inertial fields by the pre-existing geostrophic flow. Many details of the solution, such as the rate at which near-inertial activity in the mixed layer decays, are controlled by the nondimensional number, Y = 4 Ψf0/H2mixN2mix, where f0 is the inertial frequency, Hmix is the mixed-layer depth, and Nmix is the buoyancy frequency immediately below the base of the mixed layer. When Y is large, near-inertial activity in the mixed layer decays on a time-scale HmixNmix/α2Ψ3/2f01/2. When Y is small, near-inertial activity in the mixed layer decays on a time-scale proportional to N2mixH2mix/α2Ψ2f0.
Balmforth, N. J., Stefan G. Llewellyn Smith, and W. R. Young. 1998. "Enhanced dispersion of near-inertial waves in an idealized geostrophic flow." Journal of Marine Research 56, (1). https://elischolar.library.yale.edu/journal_of_marine_research/2261