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1. A generalized wave-vortex decomposition for rotating Boussinesq flows with arbitrary stratification

   https://doi.org/10.1017/jfm.2020.995  

   Early, Jeffrey J. ; Lelong, M.P. ; Sundermeyer, M.A. ( April 2021 , Journal of Fluid Mechanics)

   null (Ed.)

   The energetically independent linear wave and geostrophic (vortex) solutions are shown to be a complete basis for velocity and density variables $(u,v,w,\rho )$ in a rotating non-hydrostatic Boussinesq fluid with arbitrary stratification and non-periodic vertical boundaries. This work extends the familiar wave-vortex decomposition for triply periodic domains with constant stratification. As a consequence of the decomposition, the fluid can be unambiguously separated into decoupled linear wave and geostrophic components at each instant in time, without the need for temporal filtering. The fluid can then be diagnosed for temporal changes in wave and geostrophic coefficients at each unique wavenumber and mode, including those that inevitably occur due to nonlinear interactions. We demonstrate that this methodology can be used to determine which physical interactions cause the transfer of energy between modes by projecting the nonlinear equations of motion onto the wave-vortex basis. In the particular example given, we show that an eddy in geostrophic balance superimposed with inertial oscillations at the surface transfers energy from the inertial oscillations to internal gravity wave modes. This approach can be applied more generally to determine which mechanisms are involved in energy transfers between wave and vortices, including their respective scales. Finally, we show that the nonlinear equations of motion expressed in a wave-vortex basis are computationally efficient for certain problems. In cases where stratification profiles vary strongly with depth, this approach may be an attractive alternative to traditional spectral models for rotating Boussinesq flow. 

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2. Wave-averaged balance: a simple example

   https://doi.org/10.1017/jfm.2020.1032  

   Kafiabad, Hossein A. ; Vanneste, Jacques ; Young, William R. ( March 2021 , Journal of Fluid Mechanics)

   null (Ed.)

   In the presence of inertia-gravity waves, the geostrophic and hydrostatic balance that characterises the slow dynamics of rapidly rotating, strongly stratified flows holds in a time-averaged sense and applies to the Lagrangian-mean velocity and buoyancy. We give an elementary derivation of this wave-averaged balance and illustrate its accuracy in numerical solutions of the three-dimensional Boussinesq equations, using a simple configuration in which vertically planar near-inertial waves interact with a barotropic anticylonic vortex. We further use the conservation of the wave-averaged potential vorticity to predict the change in the barotropic vortex induced by the waves. 

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3. An improved model of near-inertial wave dynamics

   https://doi.org/10.1017/jfm.2019.557  

   Asselin, Olivier ; Young, William R. ( October 2019 , Journal of Fluid Mechanics)

   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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4. Stimulated generation: extraction of energy from balanced flow by near-inertial waves

   https://doi.org/10.1017/jfm.2018.308  

   Rocha, Cesar B. ; Wagner, Gregory L. ; Young, William R. ( July 2018 , Journal of Fluid Mechanics)

   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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5. Penetration of Wind-Generated Near-Inertial Waves into a Turbulent Ocean

   https://doi.org/10.1175/JPO-D-19-0319.1  

   Asselin, Olivier ; Young, William R. ( June 2020 , Journal of Physical Oceanography)

   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.

    

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