|Title||An improved model of near-inertial wave dynamics|
|Publication Type||Journal Article|
|Year of Publication||2019|
|Authors||Asselin O., Young W.R|
|Type of Article||Article|
|Keywords||dispersion; energy; flow; internal waves; mechanics; ocean; oscillations; physics; propagation; surface; wave-turbulence interactions|
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 , where 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 & Buhler (J. Fluid Mech., vol. 817, 2017, pp. 406-438), YBJ(+) has a dispersion relation that is second-order accurate in (YBJ is first-order accurate.) Thus both improvements have the same formal justification. But the dispersion relation of YBJ(+) is a Pade approximant to the exact dispersion relation and with 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 , the wave frequency of YBJ(+) asymptotes to twice the inertial frequency . This enables solution of YBJ(+) with explicit time-stepping schemes using a time step determined by stable integration of oscillations with frequency . Other phase-averaged equations have dispersion relations with frequency increasing as (YBJ) or (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 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 . 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.