An asymptotic model for the propagation of oceanic internal tides through quasi-geostrophic flow

• Published in: Journal of fluid mechanics 2017-10, Vol.828, p.779-811
• Main Authors: Wagner, G. L., Ferrando, G., Young, W. R.
• Format: Article
• Language: English
• Subjects: Asymptotic methods; Boussinesq approximation; Boussinesq equations; Dynamics; Eddies; Energy; Energy transfer; Fluid mechanics; Geostrophic flow; Gravitation; Gravitational waves; Gravity; Inertia; Internal tides; Internal waves; Lunar tides; Mathematical models; Nonlinear waves; Ocean currents; Ordinary differential equations; Propagation; Tidal dynamics; Tidal energy; Tides; Vortices; Wave dynamics; Wave equations; Wave frequency; Wave period; Wave propagation
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2017.509

We derive a time-averaged ‘hydrostatic wave equation’ from the hydrostatic Boussinesq equations that describes the propagation of inertia–gravity internal waves through quasi-geostrophic flow. The derivation uses a multiple-scale asymptotic method to isolate wave field evolution over intervals much longer than a wave period, assumes the wave field has a well-defined non-inertial frequency such as that of the mid-latitude semi-diurnal lunar tide, assumes that the wave field and quasi-geostrophic flow have comparable spatial scales and neglects nonlinear wave–wave dynamics. As a result the hydrostatic wave equation is a reduced model applicable to the propagation of large-scale internal tides through the inhomogeneous and moving ocean. A numerical comparison with the linearized and hydrostatic Boussinesq equations demonstrates the validity of the hydrostatic wave equation model and illustrates how the model fails when the quasi-geostrophic flow is too strong and the wave frequency is too close to inertial. The hydrostatic wave equation provides a first step toward a coupled model for energy transfer between oceanic internal tides and quasi-geostrophic eddies and currents.