Deriving the Young-Ben Jelloul model of near-inertial waves by Whitham averaging

TL;DR

This paper derives the Young-Ben Jelloul model for near-inertial waves using Whitham averaging of the hydrostatic-Boussinesq equations, providing a variational framework that clarifies its conservation laws and underlying structure.

Contribution

It presents a novel derivation of the Young-Ben Jelloul model via Whitham averaging, linking it to a variational principle and systematic conservation law derivation.

Findings

Model derived from variational averaging of hydrostatic-Boussinesq equations.

Conservation laws of the model obtained systematically.

Provides a new theoretical foundation for understanding near-inertial wave modulation.

Abstract

Oceanic near-inertial waves - internal waves with frequencies close to the local Coriolis frequency $f_0$ - are strongly influenced by the presence of mean currents. To study this influence, Young and Ben Jelloul (1997) derived an asymptotic model that describes the slow modulation of the amplitude of these waves about their rapid oscillation at frequency $f_0$. Here we show that this model can be obtained within a variational framework, by (Whitham) averaging the Lagrangian of the hydrostatic-Boussinesq equations over the wave period $2\pi/f_0$. The derivation leads to a variational formulation of the Young-Ben Jelloul model from which its conservation laws can be obtained systematically.