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

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.