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

TL;DR

This paper derives a simplified hydrostatic wave equation to model the propagation of large-scale internal tides through quasi-geostrophic flow, validated by numerical comparisons with more complex equations.

Contribution

It introduces a novel asymptotic model that captures internal tide propagation without restrictions on stratification or spatial scales, extending previous models.

Findings

The hydrostatic wave equation accurately predicts internal tide propagation.

Model validity decreases with strong quasi-geostrophic flow or near-inertial frequencies.

Numerical comparisons confirm the model's applicability and limitations.

Abstract

Starting from the hydrostatic Boussinesq equations, we derive a time-averaged `hydrostatic wave equation' that describes the propagation of inertia-gravity internal waves through quasi-geostrophic flow. The derivation uses a multiple-time-scale asymptotic method to isolate wave field evolution over intervals much longer than a wave period, assumes that the wave field has a well-defined and non-inertial frequency such as that of the mid-latitude semi-diurnal lunar tide, neglects nonlinear wave-wave interactions and makes no restriction on either the background density stratification or the relative spatial scales between the wave field and quasi-geostrophic flow. 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 and illustrates the manners of model failure when the quasi-geostrophic flow is too strong and the wave frequency is too close to inertial.