Vladimir Yanovsky

Point vortices with a rational necklace: New exact stationary solutions of the two-dimensional Euler equation

In this paper we find a new class of explicit exact stationary solutions of the two-dimensional (2D) Euler equation which describe vortex patterns of necklace type with N+1-fold symmetry in rotational shear flow. The point vortex with a strength equal to ±4πN (where N-integer number) is situated in the center of the vortex structure. The vorticity distribution outside of the center is smooth and is described by a two-parametric family of rational functions which are known in explicit form for any N. In the centers of vortex satellites the vorticity does not have any singularity and remains of finite value. When N is increasing, the solutions describe the transition layer in 2D–rotational shear flow.

Singularities motion equations in 2-dimensional ideal hydrodynamics of incompressible fluid

In this Letter, we have obtained motion equations for a wide class of one-dimensional singularities in 2D ideal hydrodynamics. The simplest of them, are well known as point vortices. More complicated singularities correspond to vorticity point dipoles. It has been proved that point multipoles of a higher order (quadrupoles and more) are not the exact solutions of two-dimensional ideal hydrodynamics. The motion equations for a system of interacting point vortices and point dipoles have been obtained. It is shown that these equations are Hamiltonian ones and have three motion integrals in involution. It means the complete integrability of two-particle system, which has a point vortex and a point dipole.

On the link between the two-dimensional hydrodynamics and the three-dimensional (3-D) magnetostatic: A new method for obtaining a 3-D solution of the magnetostatic equilibrium

It is shown that a formal link relates two-dimensional nonstationary hydrodynamic solutions to the three-dimensional solutions of the magnetostatic equilibrium. This method is used for obtaining new exact analytical solutions of the magnetostatic equilibrium (flux rope type) from known hydrodynamics solutions (point vortex type).

Kinetic theory of the electron bounce instability in two dimensional current sheets—Full electromagnetic treatment

In the general context of understanding the possible destabilization of a current sheet with applications to magnetospheric substorms or solar flares, a kinetic model is proposed for studying the resonant interaction between electromagnetic fluctuations and trapped bouncing electrons in a 2D current sheet. Tur et al. [A. Tur et al., Phys. Plasmas 17, 102905 (2010)] and Fruit et al. [G. Fruit et al., Phys. Plasmas 20, 022113 (2013)] already used this model to investigate the possibilities of electrostatic instabilities. Here, the model is completed for full electromagnetic perturbations. Starting with a modified Harris sheet as equilibrium state, the linearized gyrokinetic Vlasov equation is solved for electromagnetic fluctuations with period of the order of the electron bounce period. The particle motion is restricted to its first Fourier component along the magnetic field and this allows the complete time integration of the non local perturbed distribution functions. The dispersion relation for electroma...

Point vortices in two dimensional-plasma hydrodynamics

An exact theory of point vortices in two dimensional (2D) electron-ion plasma hydrodynamics is presented. This theory is a logical generalization of the classical theory of point vortices in a 2D Euler equation. The existence of two types of point vortices is shown: ion and electron, and their structure is described in detail. Ion vortices interact over long distances, while electron vortices interact over short distances. A dynamic system is obtained, which describes the common motion of an arbitrary number of electron and ion vortices. The proposed theory can be used to construct finite dimensional dynamical models of plasma motion, as well as for the construction of finite dimensional statistical models of turbulence, transport processes and filaments.

Generation of Large-Scale Vortices

This section deals with the problem of the generation of large-scale vortex structures due to the impact of a small-scale force. Generally, it is believed that the small-scale force leads to the destruction of large-scale structures. Actually, this is not true. Often, the small-scale force engenders the generation of the large-scale structures. In this chapter we give some examples of large-scale instabilities that are caused by a small-scale force and generate nonlinear large-scale vortex structures. Most often, the small-scale forces have the property of helicity, but that is not obligatory. In addition, broken symmetry in the system is necessitated by the presence, for example, of stratification, convection, rotation, broken parity, or other additional factors.

Dynamics of Point Vortex Singularities

This chapter focuses on localized vortices in an incompressible fluid. We consider in detail a class of point vortices that can serve as an example of single hydrodynamic quasiparticles. The interaction and movement of even a small finite number of these vortices generate complex hydrodynamic flows. We develop the general theory of the motion of complex point vortex singularities. We also examine the interaction of dipole vortices with ordinary point vortices. In addition, we study the motion of point dipoles in a restricted domain. The first section presents the main properties of a large number of known localized vortices.

Influence of Potential Waves on Point Vortex Motion

In this chapter we will focus on the impact of waves on the motion of point vortices. These processes are very important for fluid dynamics. Waves and vortices are two key actors whose interaction determines all hydrodynamic phenomena. The previous chapter discussed point vortices and their interaction in detail. It should be emphasized that at the present time, the properties of waves and separately of point vortices are well understood. However, in hydrodynamic media, as a rule, vortices and waves are present simultaneously. Therefore, to study the interaction between these objects is extremely important. Lighthill initiated the study of this effect in [1, 2]. He examined the generation of potential waves by vortex motion. In [3] using numerical simulation, the effect of periodic motion at point vortices was studied. The study of the inverse effect of potential waves on the evolution of a vortex began relatively recently [4, 5]. It has been found that under the influence of potential waves, the nature of the evolution of point vortices changes qualitatively. In this chapter we consider some unusual effects that appear when this interaction emerges.

Nontrivial Stationary Vortex Configurations

In the previous chapters, we have discussed nonstationary phenomena associated with the various types of point vortices. These vortices actually play a significant role in the formation of different stationary vortex flows. This chapter is focused mainly on the numerous stationary configurations that contain different types of singularities. Such configurations can include both the usual point vortices and more complicated singularities as well. All these solutions belong to two-dimensional hydrodynamics. In three-dimensional hydrodynamics, more complex configurations appear. In addition, a three-dimensional velocity field can form nontrivial topological vortex configurations, the so-called topological solitons. In these solitons, lines of force in the vortices are linked, which significantly increases their stability and their chances of survival in three-dimensional nonstationary fluid dynamics.

Vortices in Plasma Hydrodynamics

As we have seen, the point vortices play an important role in the two-dimensional hydrodynamics. Therefore, it is natural to want to generalize the concept of point vortices on the plasma hydrodynamics that would allow us to apply a number of ideas and methods from the fluid hydrodynamics to the plasma hydrodynamics. A lot of works deals with the approximation of point vortices for Charney–Hasegawa–Mima equation for the electrostatic drift waves in an inhomogeneous plasma or the large scale vortex motion in planetary atmospheres (e.g. [1–3]). The exact solutions which describe a point vortex model for the modon solution of the Charney–Hasegawa–Mima equation are found in [4]. The exact solutions in the form of point vortices for the current vortex filamentation of the nonlinear Alfven perturbations in high temperature magnetized plasma are found in [5–7]. This chapter describes the theory of point vortices in two-fluid plasma hydrodynamics very similar to the theory of point vortices in the ordinary hydrodynamics. These vortices form two classes: pure plasma vortices and vortices with a hydrodynamic envelope and plasma core. Such solutions are typical of the two-dimensional plasma hydrodynamics. In the case of the three-dimensional plasma hydrodynamics the nontrivial topological configurations of electron and ion fluids, so-called topological solitons, are possible. Such localized structures have the increased stability and can live a long time in the plasma.