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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. 

An idealized storm scenario is examined in which a wind-generated inertial wave interacts with a turbulent baroclinic quasigeostrophic flow. The flow is initialized by spinning up an Eady model with a stratification profile based on observations. The storm is modeled as an initial value problem for a mixed layer confined, horizontally uniform inertial oscillation. The primordial inertial oscillation evolves according to the phase-averaged model of Young and Ben Jelloul. Waves feed back onto the flow by modifying the potential vorticity. In the first few days, refraction dominates and wave energy is attracted (repelled) by regions of negative (positive) vorticity. Wave energy is subsequently drained down into the interior ocean guided by anticyclonic vortices. This drainage halts as wave energy encounters weakening vorticity. After a week or two, wave energy accumulates at the bottom of negative vorticity features, that is, along filamentary structures at shallow depths and in larger anticyclonic vortices at greater depths. Wave feedback tends to weaken vortices and thus slows the penetration of waves into the ocean interior. This nonlinear effect, however, is weak even for vigorous storms.