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# Volume 21, Number 3, 2016 Special issue on Vortex Dynamics. To the memory of Hassan Aref and Vyacheslav Meleshko

 O'Neil K. A. Point Vortex Equilibria Related to Bessel Polynomials Abstract The method of polynomials is used to construct two families of stationary point vortex configurations. The vortices are placed at the reciprocals of the zeroes of Bessel polynomials. Configurations that translate uniformly, and configurations that are completely stationary, are obtained in this way. Keywords: point vortex, equilibrium, polynomial method Citation: O'Neil K. A., Point Vortex Equilibria Related to Bessel Polynomials, Regular and Chaotic Dynamics, 2016, vol. 21, no. 3, pp. 249-253 DOI:10.1134/S1560354716030011
 Ryzhov E. A.,  Koshel K. Parametric Instability of a Many Point-vortex System in a Multi-layer Flow Under Linear Deformation Abstract The paper deals with a dynamical system governing the motion of many point vortices located in different layers of a multi-layer flow under external deformation. The deformation consists of generally independent shear and rotational components. First, we examine the dynamics of the system’s vorticity center. We demonstrate that the vorticity center of such a multi-vortex multi-layer system behaves just like the one of two point vortices interacting in a homogeneous deformation flow. Given nonstationary shear and rotational components oscillating with different magnitudes, the vorticity center may experience parametric instability leading to its unbounded growth. However, we then show that one can shift to a moving reference frame with the origin coinciding with the position of the vorticity center. In this new reference frame, the new vorticity center always stays at the origin of coordinates, and the equations governing the vortex trajectories look exactly the same as if the vorticity center had never moved in the original reference frame. Second, we studied the relative motion of two point vortices located in different layers of a two-layer flow under linear deformation. We analyze their regular and chaotic dynamics identifying parameters resulting in effective and extensive destabilization of the vortex trajectories. Keywords: point vortex, parametric instability, shear flow, chaotic dynamics Citation: Ryzhov E. A.,  Koshel K., Parametric Instability of a Many Point-vortex System in a Multi-layer Flow Under Linear Deformation, Regular and Chaotic Dynamics, 2016, vol. 21, no. 3, pp. 254-266 DOI:10.1134/S1560354716030023
 Okulov V. L. An Acentric Rotation of Two Helical Vortices of the Same Circulations Abstract The aim of this paper is to test the possibility of a secondary solution of the acentric rotation of helical vortex pairs with the same pitch, sign and strength. The investigation addresses the three-dimensional vortex dynamics of thin vortex filaments. As a result of the current investigation, this secondary solution with acentric vortex positions in the helical pairs is found. This fact was not discussed in previous studies, and the existence of the new equilibrium solution for the helical vortex pairs is an original result. Keywords: vortex dynamics, helical vortex, vortex pair, equilibrium rotation Citation: Okulov V. L., An Acentric Rotation of Two Helical Vortices of the Same Circulations, Regular and Chaotic Dynamics, 2016, vol. 21, no. 3, pp. 267-273 DOI:10.1134/S1560354716030035
 Duarte R.,  Carton X.,  Poulin F. J. The Dynamics of a Meandering Coastal Jet in the Lee of a Cape Abstract In a barotropic model of an oceanic channel, bounded to the north by a straight coast indented by a Gaussian cape, the evolution of a coastal jet is studied numerically. In the absence of the cape, the barotropic instability of the jet is determined. In the presence of the cape, a regular row of meanders develops downstream of this feature, and becomes stationary for a particular range of parameters. The relevant parameters are the velocity and width of the jet, size of the cape, and beta effect. The formation of meanders occurs first via the instability of the jet, then via the generation of vorticity anomalies at the cape, which are advected both downstream by the flow and offshore by the radiation of Rossby waves. Once the meanders are established, they remain stationary features if the propagation velocity of the meanders (due to the dipolar effect at the coast) opposes the jet velocity and the phase speed of the wave on the vorticity front. Finally, a steady state of a regular row of meanders is also obtained via a matrix method and is similar to that obtained in the time-dependent case. Keywords: two-dimensional fluid, vorticity equation, nonlinear dynamics, effect of boundaries, Rossby waves, numerical modeling, steady state Citation: Duarte R.,  Carton X.,  Poulin F. J., The Dynamics of a Meandering Coastal Jet in the Lee of a Cape, Regular and Chaotic Dynamics, 2016, vol. 21, no. 3, pp. 274-290 DOI:10.1134/S1560354716030047
 Kurakin L. G.,  Ostrovskaya I. V.,  Sokolovskiy M. A. On the Stability of Discrete Tripole, Quadrupole, Thomson’ Vortex Triangle and Square in a Two-layer/Homogeneous Rotating Fluid Abstract A two-layer quasigeostrophic model is considered in the $f$-plane approximation. The stability of a discrete axisymmetric vortex structure is analyzed for the case when the structure consists of a central vortex of arbitrary intensity $\Gamma$ and two/three identical peripheral vortices. The identical vortices, each having a unit intensity, are uniformly distributed over a circle of radius $R$ in a single layer. The central vortex lies either in the same or in another layer. The problem has three parameters $(R, \Gamma, \alpha)$, where $\alpha$ is the difference between layer thicknesses. A limiting case of a homogeneous fluid is also considered. The theory of stability of steady-state motions of dynamic systems with a continuous symmetry group $\mathcal{G}$ is applied. The two definitions of stability used in the study are Routh stability and $\mathcal{G}$-stability. The Routh stability is the stability of a one-parameter orbit of a steady-state rotation of a vortex multipole, and the $\mathcal{G}$-stability is the stability of a three-parameter invariant set $O_\mathcal{G}$, formed by the orbits of a continuous family of steady-state rotations of a multipole. The problem of Routh stability is reduced to the problem of stability of a family of equilibria of a Hamiltonian system. The quadratic part of the Hamiltonian and the eigenvalues of the linearization matrix are studied analytically. The cases of zero total intensity of a tripole and a quadrupole are studied separately. Also, the Routh stability of a Thomson vortex triangle and square was proved at all possible values of problem parameters. The results of theoretical analysis are sustained by numerical calculations of vortex trajectories. Keywords: discrete multipole vortex structure, two-layer rotating fluid, stability Citation: Kurakin L. G.,  Ostrovskaya I. V.,  Sokolovskiy M. A., On the Stability of Discrete Tripole, Quadrupole, Thomson’ Vortex Triangle and Square in a Two-layer/Homogeneous Rotating Fluid, Regular and Chaotic Dynamics, 2016, vol. 21, no. 3, pp. 291-334 DOI:10.1134/S1560354716030059
 Prants S. V. A Lagrangian Study of Eddies in the Ocean Abstract A brief review of our results on the application of the Lagrangian approach to study observed and simulated eddies in the ocean is presented. It is shown by a few examples of mesoscale vortex structures in the North Western Pacific how to compute and analyze maps of specific Lagrangian indicators in order to study the birth, formation, evolution, metamorphoses and death of ocean eddies. The examples involve two-dimensional eddies observed in satellitederived velocity fields in the deep ocean and three-dimensional ones simulated in a regional numerical model of circulation with a high resolution. Keywords: ocean eddies, Lagrangian analysis, Lyapunov exponent Citation: Prants S. V., A Lagrangian Study of Eddies in the Ocean, Regular and Chaotic Dynamics, 2016, vol. 21, no. 3, pp. 335-350 DOI:10.1134/S1560354716030060
 Demina M. V.,  Kudryashov N. A. Multi-particle Dynamical Systems and Polynomials Abstract Polynomial dynamical systems describing interacting particles in the plane are studied. A method replacing integration of a polynomial multi-particle dynamical system by finding polynomial solutions of partial differential equations is introduced. The method enables one to integrate a wide class of polynomial multi-particle dynamical systems. The general solutions of certain dynamical systems related to linear second-order partial differential equations are found. As a by-product of our results, new families of orthogonal polynomials are derived. Keywords: multi-particle dynamical systems, polynomial solutions of partial differential equations, orthogonal polynomials Citation: Demina M. V.,  Kudryashov N. A., Multi-particle Dynamical Systems and Polynomials, Regular and Chaotic Dynamics, 2016, vol. 21, no. 3, pp. 351-366 DOI:10.1134/S1560354716030072
 Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S. The Dynamics of Vortex Sources in a Deformation Flow Abstract This paper is concerned with the dynamics of vortex sources in a deformation flow. The case of two vortex sources is shown to be integrable by quadratures. In addition, the relative equilibria (of the reduced system) are examined in detail and it is shown that in this case the trajectory of vortex sources is an ellipse. Keywords: integrability, vortex sources, reduction, deformation flow Citation: Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S., The Dynamics of Vortex Sources in a Deformation Flow, Regular and Chaotic Dynamics, 2016, vol. 21, no. 3, pp. 367-376 DOI:10.1134/S1560354716030084

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