An improved model of near-inertial wave dynamics

Abstract

The YBJ equation (Young & Ben Jelloul, J. Marine Res., vol. 55, 1997, pp. 735–766) provides a phase-averaged description of the propagation of near-inertial waves (NIWs) through a geostrophic flow. YBJ is obtained via an asymptotic expansion based on the limit $\mathit{Bu}\rightarrow 0$, where $\mathit{Bu}$ is the Burger number of the NIWs. Here we develop an improved version, the YBJ+ equation. In common with an earlier improvement proposed by Thomas, Smith & Bühler (J. Fluid Mech., vol. 817, 2017, pp. 406–438), YBJ+ has a dispersion relation that is second-order accurate in $\mathit{Bu}$. (YBJ is first-order accurate.) Thus both improvements have the same formal justification. But the dispersion relation of YBJ+ is a Padé approximant to the exact dispersion relation and with $\mathit{Bu}$ of order unity this is significantly more accurate than the power-series approximation of Thomas et al. (2017). Moreover, in the limit of high horizontal wavenumber $k\rightarrow \infty$, the wave frequency of YBJ+ asymptotes to twice the inertial frequency $2f$. This enables solution of YBJ+ with explicit time-stepping schemes using a time step determined by stable integration of oscillations with frequency $2f$. Other phase-averaged equations have dispersion relations with frequency increasing as $k^{2}$ (YBJ) or $k^{4}$ (Thomas et al. 2017): in these cases stable integration with an explicit scheme becomes impractical with increasing horizontal resolution. The YBJ+ equation is tested by comparing its numerical solutions with those of the Boussinesq and YBJ equations. In virtually all cases, YBJ+ is more accurate than YBJ. The error, however, does not go rapidly to zero as the Burger number characterizing the initial condition is reduced: advection and refraction by geostrophic eddies reduces in the initial length scale of NIWs so that $\mathit{Bu}$ increases with time. This increase, if unchecked, would destroy the approximation. We show, however, that dispersion limits the damage by confining most of the wave energy to low $\mathit{Bu}$. In other words, advection and refraction by geostrophic flows does not result in a strong transfer of initially near-inertial energy out of the near-inertial frequency band.