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We study stimulated generation – the transfer of energy from balanced flows to existing internal waves – using an asymptotic model that couples barotropic quasi-geostrophic flow and near-inertial waves with $$\text{e}^{\text{i}mz}$$ vertical structure, where $$m$$ is the vertical wavenumber and $$z$$ is the vertical coordinate. A detailed description of the conservation laws of this vertical-plane-wave model illuminates the mechanism of stimulated generation associated with vertical vorticity and lateral strain. There are two sources of wave potential energy, and corresponding sinks of balanced kinetic energy: the refractive convergence of wave action density into anti-cyclones (and divergence from cyclones); and the enhancement of wave-field gradients by geostrophic straining. We quantify these energy transfers and describe the phenomenology of stimulated generation using numerical solutions of an initially uniform inertial oscillation interacting with mature freely evolving two-dimensional turbulence. In all solutions, stimulated generation co-exists with a transfer of balanced kinetic energy to large scales via vortex merging. Also, geostrophic straining accounts for most of the generation of wave potential energy, representing a sink of 10 %–20 % of the initial balanced kinetic energy. However, refraction is fundamental because it creates the initial eddy-scale lateral gradients in the near-inertial field that are then enhanced by advection. In these quasi-inviscid solutions, wave dispersion is the only mechanism that upsets stimulated generation: with a barotropic balanced flow, lateral straining enhances the wave group velocity, so that waves accelerate and rapidly escape from straining regions. This wave escape prevents wave energy from cascading to dissipative scales.
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An improved model of near-inertial wave dynamics
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.
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- Award ID(s):
- 1657041
- PAR ID:
- 10169189
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 876
- ISSN:
- 0022-1120
- Page Range / eLocation ID:
- 428 to 448
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2019.557
