Mikhail Sokolovskiy
Publications:
Carton X., Morvan M., Reinaud J. N., Sokolovskiy M. A., L’Hegaret P., Vic C.
Vortex Merger near a Topographic Slope in a Homogeneous Rotating Fluid
2017, vol. 22, no. 5, pp. 455478
Abstract
The effect of a bottom slope on the merger of two identical Rankine vortices is investigated in a twodimensional, quasigeostrophic, incompressible fluid.
When two cyclones initially lie parallel to the slope, and more than two vortex diameters away from the slope, the critical merger distance is unchanged. When the cyclones are closer to the slope, they can merge at larger distances, but they lose more mass into filaments, thus weakening the efficiency of merger. Several effects account for this: the topographic Rossby wave advects the cyclones, reduces their mutual distance and deforms them. This alongshelf wave breaks into filaments and into secondary vortices which shear out the initial cyclones. The global motion of fluid towards the shallow domain and the erosion of the two cyclones are confirmed by the evolution of particles seeded both in the cyclones and near the topographic slope. The addition of tracer to the flow indicates that diffusion is ballistic at early times. For two anticyclones, merger is also facilitated because one vortex is ejected offshore towards the other, via coupling with a topographic cyclone. Again two anticyclones can merge at large distance but they are eroded in the process. Finally, for taller topographies, the critical merger distance is again increased and the topographic influence can scatter or completely erode one of the two initial cyclones. Conclusions are drawn on possible improvements of the model configuration for an application to the ocean. 
Kurakin L. G., Ostrovskaya I. V., Sokolovskiy M. A.
On the Stability of Discrete Tripole, Quadrupole, Thomson’ Vortex Triangle and Square in a Twolayer/Homogeneous Rotating Fluid
2016, vol. 21, no. 3, pp. 291334
Abstract
A twolayer 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 steadystate 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 oneparameter orbit of a steadystate rotation of a vortex multipole, and the $\mathcal{G}$stability is the stability of a threeparameter invariant set $O_\mathcal{G}$, formed by the orbits of a continuous family of steadystate 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.

Gryanik V. M., Sokolovskiy M. A., Verron J.
Dynamics of hetonlike vortices
2006, vol. 11, no. 3, pp. 383434
Abstract
Studies of the properties of vortex motions in a stably stratified and fast rotating fluid that can be described by the equation for the evolution of a potential vortex in the quasigeostrophic approximation are reviewed. Special attention is paid to the vortices with zero total intensity (the socalled hetons). The problems considered include selfmotion of discrete hetons, the stability of a solitary distributed heton, and the interaction between two finitecore hetons. New solutions to the problems of three or more discrete vortices with a heton structure are proposed. The existence of chaotic regimes is revealed. The range of applications of the heton theory and the prospects for its future application, particularly in respect, to the analysis of the dynamic stage in the development of deep ocean convection, are discussed.

Sokolovskiy M. A., Verron J.
Dynamics of three vortices in a twolayer rotating fluid
2004, vol. 9, no. 4, pp. 417438
Abstract
The problem of studying the motion of three vortex lines with arbitrary intensities in an unbounded twodimensional finitethickness layer of a homogeneous fluid is known [25], [9], [28], [1] to belong to the class of integrable problems. However, a complete classification of possible motions was constructed only recently [10], [28], [41]. In [40], [39], [20] a generalization is given for twolayer rotating fluid in the particular case determined by the conditions of (i) zero total circulation of vortices, and (ii) the equality of the intensities of two vortices. Here, the first of these restrictions is lifted.

Sokolovskiy M. A., Verron J.
Dynamics of Triangular TwoLayer Vortex Structures with Zero Total Intensity
2002, vol. 7, no. 4, pp. 435472
Abstract
The problem of three vortex lines in a homogeneous layer of an ideal incompressible fluid is generalized to the case of a twolayer liquid with constant density values in each layer. For zerocomplexmomentum systems the theory of the roundabout twolayer tripole is built. When the momentum is different from zero, based on the phase portraits in trilinear coordinates, a classification of possible relative motions of a system composed of three discrete (or point) vortices is provided. One vortex is situated in the upper layer, and the other two in the lower layer; their total intensity is zero. More specifically, a model of a twolayer tripole is constructed, and existence conditions for stationary solutions are found. These solutions represent a uniform translational motion of the following vortex structures: 1) a stable collinear configuration triton, a discrete analog of the vortex structure modon+rider, 2) an unstable triangular configuration. Features of the absolute motion of the system of three discrete vortices were studied numerically.
We compared the dynamics of a system of three point vortices with the dynamics of three finitecore vortices (vortex patches). In studying the evolution of the vortex patch system, a twolayer version of the Contour Dynamics Method (CDM) was used. The applicability of discretevortex theory to the description of the finitesize vortex behavior is dicussed. Examples of formation of vortical configurations are given. Such configurations appear either after merging of vortices of the same layer or as a result of instability of the twolayer vortex structure. 
Sokolovskiy M. A., Verron J.
Fourvortex motion in the two layer Approximation: integrable case
2000, vol. 5, no. 4, pp. 413436
Abstract
The problem of four vortex lines with zero total circulation and zero impulse on a unlimited fluid plane, as it is known [1,3,4,16], is reduced to a problem of three point vortices and is integrated in quadratures. In the given work these results are transferred on a case of four vortices in a twolayer rotating liquid. The analysis of phase trajectories of relative motion of vortices is made, and the singularities of absolute motion on an example of a headon, offcenter collision of two twolayer vortex pairs are studied. In particular, the new class of quasistationary solutions for the given type of motions is obtained. The problems of interaction of the distributed (or finitecore) twolayer vortices are discussed.
