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1. The motion of a buoyant vortex filament

   https://doi.org/10.1017/jfm.2018.795  

   Chang, Ching; Llewellyn Smith, Stefan G. (December 2018, Journal of Fluid Mechanics)

   We investigate the motion of a thin vortex filament in the presence of buoyancy. The asymptotic model of Moore & Saffman ( Phil. Trans. R. Soc. Lond. A, vol. 272, 1972, pp. 403–429) is extended to take account of buoyancy forces in the force balance on a vortex element. The motion of a buoyant vortex is given by the transverse component of force balance, while the tangential component governs the dynamics of the structure in the core. We show that the local acceleration of axial flow is generated by the external pressure gradient due to gravity. The equations are then solved for vortex rings. An analytic solution for a buoyant vortex ring at a small initial inclination is obtained and asymptotically agrees with the literature. 

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2. On the approximation of vorticity fronts by the Burgers–Hilbert equation

   https://doi.org/10.3233/asy-211724  

   Hunter, John K.; Moreno-Vasquez, Ryan C.; Shu, Jingyang; Zhang, Qingtian (August 2022, Asymptotic Analysis)

   This paper proves that the motion of small-slope vorticity fronts in the two-dimensional incompressible Euler equations is approximated on cubically nonlinear timescales by a Burgers–Hilbert equation derived by Biello and Hunter (2010) using formal asymptotic expansions. The proof uses a modified energy method to show that the contour dynamics equations for vorticity fronts in the Euler equations and the Burgers–Hilbert equation are both approximated by the same cubically nonlinear asymptotic equation. The contour dynamics equations for Euler vorticity fronts are also derived. 

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3. Experimental investigations of linear and nonlinear periodic travelling waves in a viscous fluid conduit

   https://doi.org/10.1017/jfm.2022.993  

   Mao, Yifeng; Hoefer, Mark A. (January 2023, Journal of Fluid Mechanics)

   Conduits generated by the buoyant dynamics between two miscible Stokes fluids with high viscosity contrast, a type of core–annular flow, exhibit a rich nonlinear wave dynamics. However, little is known about the fundamental wave dispersion properties of the medium. In the present work, a pump is used to inject a time-periodic flow that results in the excitation of propagating small- and large-amplitude periodic travelling waves along the conduit interface. This wavemaker problem is used as a means to measure the linear and nonlinear dispersion relations and corresponding periodic travelling wave profiles. Measurements are favourably compared with predictions from a fully nonlinear, long-wave model (the conduit equation) and the analytically computed linear dispersion relation for two-Stokes flow. A critical frequency is observed, marking the threshold between propagating and non-propagating (spatially decaying) waves. Measurements of wave profiles and the wavenumber–frequency dispersion relation quantitatively agree with wave solutions of the conduit equation. An upshift from the conduit equation's predicted critical frequency is observed and is explained by incorporating a weak recirculating flow into the full two-Stokes flow model. When the boundary condition corresponds to the temporal profile of a nonlinear periodic travelling wave solution of the conduit equation, weakly nonlinear and strongly nonlinear, cnoidal-type waves are observed that quantitatively agree with the conduit nonlinear dispersion relation and wave profiles. This wavemaker problem is an important precursor to the experimental investigation of more general boundary value problems in viscous fluid conduit nonlinear wave dynamics. 

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4. Interaction of near-inertial waves with an anticyclonic vortex

   https://doi.org/10.1175/JPO-D-20-0257.1  

   Kafiabad, Hossein A.; Vanneste, Jacques; Young, William R. (April 2021, Journal of Physical Oceanography)

   null (Ed.)

   Abstract Anticyclonic vortices focus and trap near-inertial waves so that near-inertial energy levels are elevated within the vortex core. Some aspects of this process, including the nonlinear modification of the vortex by the wave, are explained by the existence of trapped near-inertial eigenmodes. These vortex eigenmodes are easily excited by an initialwave with horizontal scale much larger than that of the vortex radius. We study this process using a wave-averaged model of near-inertial dynamics and compare its theoretical predictions with numerical solutions of the three-dimensional Boussinesq equations. In the linear approximation, the model predicts the eigenmode frequencies and spatial structures, and a near-inertial wave energy signature that is characterized by an approximately time-periodic, azimuthally invariant pattern. The wave-averaged model represents the nonlinear feedback of the waves on the vortex via a wave-induced contribution to the potential vorticity that is proportional to the Laplacian of the kinetic energy density of the waves. When this is taken into account, the modal frequency is predicted to increase linearly with the energy of the initial excitation. Both linear and nonlinear predictions agree convincingly with the Boussinesq results. 

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5. 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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