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1. Vortex pairs and dipoles

   https://doi.org/10.1134/S1560354713010140  

   Llewellyn Smith, Stefan G.; Nagem, Raymond J. (January 2013, Regular and Chaotic Dynamics)

   null (Ed.)
2. A new canonical reduction of three-vortex motion and its application to vortex-dipole scattering

   https://doi.org/10.1063/5.0208538  

   Anurag, Atul; Goodman, Roy H; O'Grady, Ellison K (June 2024, Physics of Fluids)

   We introduce a new reduction of the motion of three point vortices in a two-dimensional ideal fluid. This proceeds in two stages: a change of variables to Jacobi coordinates and then a Nambu reduction. The new coordinates demonstrate that the dynamics evolve on a two-dimensional manifold whose topology depends on the sign of a parameter κ2 that arises in the reduction. For κ2>0, the phase space is spherical, while for κ2<0, the dynamics are confined to the upper sheet of a two-sheeted hyperboloid. We contrast this reduction with earlier reduced systems derived by Gröbli, Aref, and others in which the dynamics are determined from the pairwise distances between the vortices. The new coordinate system overcomes two related shortcomings of Gröbli's reduction that have made understanding the dynamics difficult: their lack of a standard phase plane and their singularity at all configurations in which the vortices are collinear. We apply this to two canonical problems. We first discuss the dynamics of three identical vortices and then consider the scattering of a propagating dipole by a stationary vortex. We show that the points dividing direct and exchange scattering solutions correspond to the locations of the invariant manifolds of equilibria of the reduced equations and relate changes in the scattering diagram as the circulation of one vortex is varied to bifurcations of these equilibria. 

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3. The Diabatic Rossby Vortex: Growth Rate, Length Scale, and the Wave–Vortex Transition

   https://doi.org/10.1175/JAS-D-22-0022.1  

   Kohl, Matthieu; O’Gorman, Paul A. (October 2022, Journal of the Atmospheric Sciences)

   Abstract In idealized simulations of moist baroclinic instability on a sphere, the most unstable mode transitions from a periodic wave to an isolated vortex in sufficiently warm climates. The vortex mode is maintained through latent heating and shows the principal characteristics of a diabatic Rossby vortex (DRV) that has been found in a range of different simulations and observations of the current climate. Currently, there is no analytical theory for DRVs or understanding of the wave–vortex transition that has been found in warmer climates. Here, we introduce a minimal moist two-layer quasigeostrophic model with tilted boundaries capable of producing a DRV mode, and we derive growth rates and length scales for this DRV mode. In the limit of a convectively neutral stratification, the length scale of ascent of the DRV is the same as that of a periodic moist baroclinic wave, but the growth rate of the DRV is 54% faster. We explain the isolated structure of the DRV using a simple potential vorticity (PV) argument, and we create a phase diagram for when the most unstable solution is a periodic wave versus a DRV, with the DRV emerging when the moist static stability and meridional PV gradients are weak. Last, we compare the structure of the DRV mode with DRV storms found in reanalysis and with a DRV storm in a warm-climate simulation. Significance Statement Past research has identified a special class of midlatitude storm, dubbed the diabatic Rossby vortex (DRV), which derives its energy from the release of latent heat associated with condensation of water vapor and as such goes beyond the traditional understanding of midlatitude storm formation. DRVs have been implicated in extreme and poorly predicted forms of cyclogenesis along the east coast of the United States and the west coast of Europe with significant damage to property and human life. The purpose of this study is to develop a mathematical theory for the intensification rate and length scale of DRVs to gain a deeper understanding of the dynamics of these storms in current and future climates. 

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4. Synthesis of Vortex Rossby Waves. Part III: Rossby Waveguides, Vortex Motion, and the Analogy with Midlatitude Cyclones

   https://doi.org/10.1175/JAS-D-20-0197.1  

   Gonzalez III, Israel; Willoughby, H. E. (August 2021, Journal of the Atmospheric Sciences)

   Abstract Vortex Rossby waves (VRWs) affect tropical cyclones’ (TCs’) motion, structure, and intensity. They propagate within annular waveguides defined by a passband between Ω_1D, the Doppler-shifted frequency of a one-dimensional VRW, and zero. Wavenumber-1 VRWs cause TC motion directly and have wider waveguides than wavenumbers ≥ 2. VRWs forced with fixed rotation frequency propagate away from the forcing. Initially outward-propagating waves are Doppler shifted to zero at a critical radius, where they are absorbed. Initially inward-propagating waves are Doppler shifted to Ω_1D, reflect from a turning point, propagate outward, and are ultimately absorbed at the critical radius. Between the forcing and the turning radii, the VRWs have standing-wave structure; outward from the forcing they are trailing spirals. They carry angular momentum fluxes that act to accelerate the mean flow at the forcing radius and decelerate it at the critical radius. Mean-flow vorticity monopoles are inconsistent with Stokes’s theorem on a spherical Earth, because a contour enclosing the monopole’s antipode would have nonzero circulation but would enclose zero vorticity. The Rossby waveguide paradigm also fits synoptic-scale Rossby waves in a meridionally sheared zonal flow. These waves propagate within a waveguide confined between a poleward turning latitude and an equatorward critical latitude. Forced waves are comma-shaped gyres that resemble observed frontal cyclones, with trailing filaments equatorward of the forcing latitude and standing waves poleward. Even neutral forced Rossby waves converge westerly momentum at the latitude of forcing. 

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5. Instability of a dusty vortex

   https://doi.org/10.1017/jfm.2022.687  

   Shuai, Shuai; Jeswin Dhas, Darish; Roy, Anubhab; Kasbaoui, M. Houssem (October 2022, Journal of Fluid Mechanics)

   We investigate the effect of inertial particles dispersed in a circular patch of finite radius on the stability of a two-dimensional Rankine vortex in semi-dilute dusty flows. Unlike the particle-free case where no unstable modes exist, we show that the feedback force from the particles triggers a novel instability. The mechanisms driving the instability are characterized using linear stability analysis for weakly inertial particles and further validated against Eulerian–Lagrangian simulations. We show that the particle-laden vortex is always unstable if the mass loading $M>0$ . Surprisingly, even non-inertial particles destabilize the vortex by a mechanism analogous to the centrifugal Rayleigh–Taylor instability in radially stratified vortex with density jump. We identify a critical mass loading above which an eigenmode $$m$$ becomes unstable. This critical mass loading drops to zero as $$m$$ increases. When particles are inertial, modes that fall below the critical mass loading become unstable, whereas modes above it remain unstable but with lower growth rates compared with the non-inertial case. Comparison with Eulerian–Lagrangian simulations shows that growth rates computed from simulations match well the theoretical predictions. Past the linear stage, we observe the emergence of high-wavenumber modes that turn into spiralling arms of concentrated particles emanating out of the core, while regions of particle-free flow are sucked inward. The vorticity field displays a similar pattern which leads to the breakdown of the initial Rankine structure. This novel instability for a dusty vortex highlights how the feedback force from the disperse phase can induce the breakdown of an otherwise resilient vortical structure. 

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