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# Volume 26, Number 5, 2021 Special Issue: 200th birthday of Hermann von Helmholtz

 Dritschel D. G.,  Sokolovskiy M. A.,  Stremler M. A. Celebrating the 200th Anniversary of the Birth of Hermann Ludwig Ferdinand von Helmholtz (31.08.1821–08.09.1894) Abstract Citation: Dritschel D. G.,  Sokolovskiy M. A.,  Stremler M. A., Celebrating the 200th Anniversary of the Birth of Hermann Ludwig Ferdinand von Helmholtz (31.08.1821–08.09.1894), Regular and Chaotic Dynamics, 2021, vol. 26, no. 5, pp. 463-466 DOI:10.1134/S1560354721050014
 Dritschel D. G. Ring Configurations of Point Vortices in Polar Atmospheres Abstract This paper examines the stability and nonlinear evolution of configurations of equalstrength point vortices equally spaced on a ring of constant radius, with or without a central vortex, in the three-dimensional quasi-geostrophic compressible atmosphere model. While the ring lies at constant height, the central vortex can be at a different height and also have a different strength to the vortices on the ring. All such configurations are relative equilibria, in the sense that they steadily rotate about the $z$ axis. Here, the domains of stability for two or more vortices on a ring with an additional central vortex are determined. For a compressible atmosphere, the problem also depends on the density scale height $H$, the vertical scale over which the background density varies by a factor $e$. Decreasing $H$ while holding other parameters fixed generally stabilises a configuration. Nonlinear simulations of the dynamics verify the linear analysis and reveal potentially chaotic dynamics for configurations having four or more vortices in total. The simulations also reveal the existence of staggered ring configurations, and oscillations between single and double ring configurations. The results are consistent with the observations of ring configurations of polar vortices seen at both of Jupiter’s poles [1]. Keywords: vortex dynamics, point vortices Citation: Dritschel D. G., Ring Configurations of Point Vortices in Polar Atmospheres, Regular and Chaotic Dynamics, 2021, vol. 26, no. 5, pp. 467-481 DOI:10.1134/S1560354721050026
 Stremler M. A. Something Old, Something New: Three Point Vortices on the Plane Abstract The classic problem of three point vortex motion on the plane is revisited by using the interior angles of the vortex triangle, $\theta_{j}$, $j=1,2,3$, as the key system variables instead of the lengths of the triangle sides, $s_j$, as has been used classically. Similar to the classic approach, the relative vortex motion can be represented in a phase space, with the topology of the level curves characterizing the motion.  In contrast to the classic approach, the alternate formulation gives a compact, consistent phase space representation and facilitates comparisons of vortex motion in a co-moving frame. This alternate formulation is used to explore the vortex behavior in the two canonical cases of equal vortex strength magnitudes, $\Gamma_{1} = \Gamma_{2} = \Gamma_{3}$ and $\Gamma_{1} = \Gamma_{2} = -\Gamma_{3}$. Keywords: vortex dynamics, point vortices, three-vortex problem, potential flow Citation: Stremler M. A., Something Old, Something New: Three Point Vortices on the Plane, Regular and Chaotic Dynamics, 2021, vol. 26, no. 5, pp. 482-504 DOI:10.1134/S1560354721050038
 Reinaud J. N. Three-dimensional Quasi-geostrophic Staggered Vortex Arrays Abstract We determine and characterise relative equilibria for arrays of point vortices in a three-dimensional quasi-geostrophic flow. The vortices are equally spaced along two horizontal rings whose centre lies on the same vertical axis. An additional vortex may be placed along this vertical axis. Depending on the parameters defining the array, the vortices on the two rings are of equal or opposite sign. We address the linear stability of the point vortex arrays. We find both stable equilibria and unstable equilibria, depending on the geometry of the array. For unstable arrays, the instability may lead to the quasi-regular or to the chaotic motion of the point vortices. The linear stability of the vortex arrays depends on the number of vortices in the array, on the radius ratio between the two rings, on the vertical offset between the rings and on the vertical offset between the rings and the central vortex, when the latter is present. In this case the linear stability also depends on the strength of the central vortex. The nonlinear evolution of a selection of unstable cases is presented exhibiting examples of quasi-regular motion and of chaotic motion. Keywords: quasi-geostrophy, point vortex dynamics, equilibria, vortex arrays Citation: Reinaud J. N., Three-dimensional Quasi-geostrophic Staggered Vortex Arrays, Regular and Chaotic Dynamics, 2021, vol. 26, no. 5, pp. 505-525 DOI:10.1134/S156035472105004X
 Kurakin L. G.,  Ostrovskaya I. V. Resonances in the Stability Problem of a Point Vortex Quadrupole on a Plane Abstract A system of four point vortices on a plane is considered. Its motion is described by the Kirchhoff equations. Three vortices have unit intensity and one vortex has arbitrary intensity $\varkappa$. We study the stability problem for the stationary rotation of a vortex quadrupole consisting of three identical vortices located uniformly on a circle around a fourth vortex. It is known that for $\varkappa> 1$ the regime under study is unstable, and in the case of $\varkappa <-3$ and $0 <\varkappa <1$ the orbital stability takes place. New results are obtained for $-3 <\varkappa <0$. It is found that, for all values of $\varkappa$ in the stability problem, there is a resonance $1:1$ (diagonalizable case). Some other resonances through order four are found and investigated: double zero resonance (diagonalizable case), resonances 1:2 and 1:3, occurring with isolated values of $\varkappa$. The stability of the equilibrium of the system reduced by one degree of freedom with the involvement of the terms in the Hamiltonian through degree four is proved for all $\varkappa \in (-3,0)$. Keywords: $N + 1$ vortex problem, point vortices, Hamiltonian equation, stability, resonances Citation: Kurakin L. G.,  Ostrovskaya I. V., Resonances in the Stability Problem of a Point Vortex Quadrupole on a Plane, Regular and Chaotic Dynamics, 2021, vol. 26, no. 5, pp. 526-542 DOI:10.1134/S1560354721050051
 de Marez C.,  Carton X. Interaction of an Upwelling Front with External Vortices: Impact on Cross-shore Particle Exchange Abstract Coastal upwellings, due to offshore Ekman transport, are more energetic at the western boundaries of the oceans, where they are intensified by incoming Rossby waves, than at the eastern boundaries. Western boundary upwellings are often accompanied by a local vortex field. The instability of a developed upwelling front and its interaction with an external vortex field is studied here with a three-dimensional numerical model of the hydrostatic rotating Navier-Stokes equations (the primitive equations). The baroclinic instability of the front leads to the growth of meanders with 100-200 km wavelength, in the absence of external vortex. On the $f$-plane, these waves can break into a row of vortices when the instability is intense. The $\beta$-effect is stabilizing and strongly decreases the amplitude of meanders. Simulations are then performed with a front initially accompanied by one or several external vortices. The evolutions in this case are compared with those of the unstable jet alone. On the $f$-plane, when an external vortex is close to the front, this latter sheds a long filament which wraps up around the vortex. This occurs over a period similar to that of the instability of the isolated front. Cyclones are more efficient in tearing such filaments offshore than anticyclones. On the $\beta$-plane, the filaments are short and turbulence is confined to the vicinity of the front. At long times, waves propagate along the front, thus extending turbulence alongshore. The initial presence of a vortex alley leads to a stronger destabilization of the front and to a larger cross-shore flux than for a single vortex, with many filaments and small vortices pushed far offshore. In the ocean, this cross-shore exchange has important consequences on the local biological activity. Keywords: coastal upwelling front, vortices, filaments, baroclinic instability, frontal waves, particle motion and tracking Citation: de Marez C.,  Carton X., Interaction of an Upwelling Front with External Vortices: Impact on Cross-shore Particle Exchange, Regular and Chaotic Dynamics, 2021, vol. 26, no. 5, pp. 543-561 DOI:10.1134/S1560354721050063
 Riccardi G.,  Dritschel D. G. Evolution of the Singularities of the Schwarz Function Corresponding to the Motion of a Vortex Patch in the Two-dimensional Euler Equations Abstract The paper deals with the calculation of the internal singularities of the Schwarz function corresponding to the boundary of a planar vortex patch during its self-induced motion in an inviscid, isochoric fluid. The vortex boundary is approximated by a simple, time-dependent map onto the unit circle, whose coefficients are obtained by fitting to the boundary computed in a contour dynamics numerical simulation of the motion. At any given time, the branch points of the Schwarz function are calculated, and from them, the generally curved shape of the internal branch cut, together with the jump of the Schwarz function across it. The knowledge of the internal singularities enables the calculation of the Schwarz function at any point inside the vortex, so that it is possible to check the validity of the map during the motion by comparing left and right hand sides of the evolution equation of the Schwarz function. Our procedure yields explicit functional forms of the analytic continuations of the velocity and its conjugate on the vortex boundary. It also opens a new way to understand the relation between the time evolution of the shape of a vortex patch during its motion, and the corresponding changes in the singular set of its Schwarz function. Keywords: two-dimensional vortex dynamics, contour dynamics, Schwarz function, complex analysis Citation: Riccardi G.,  Dritschel D. G., Evolution of the Singularities of the Schwarz Function Corresponding to the Motion of a Vortex Patch in the Two-dimensional Euler Equations, Regular and Chaotic Dynamics, 2021, vol. 26, no. 5, pp. 562-575 DOI:10.1134/S1560354721050075

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