Collapse and backward motion of axisymmetric toroidal vortices in an accretion flow
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TICAL, NONLINEAR, AND SOFT MATTER PHYSICS
Collapse and Backward Motion of Axisymmetric Toroidal Vortices in an Accretion Flow E. Yu. Bannikovaa,b, V. M. Kontorovicha,b,*, and S. A. Poslavskyb a
Institute of Radio Astronomy, National Academy of Sciences of Ukraine, Chervonopraporna ul. 4, Kharkov, 61002 Ukraine b Karazin Kharkov National University, pl. Svobody 4, Kharkov, 61022 Ukraine *email: [email protected] Received March 21, 2013
Abstract—The problem of the interaction of two coaxial, counterrotating ring vortices in the presence of a convergent (accretion) flow with a sink at the center of symmetry has been solved. The vortices that would recede from each other in the absence of a flow (the problem inverse to the Helmholtz problem) are shown to be brought closer together by the flow and then ejected with acceleration along the axis of symmetry. The ejection velocity increases with sink strength. However, if the sink strength exceeds some critical value that depends on the initial conditions, then no ejection occurs and the vortices are captured by the flow and col lapse. A similar capture and collapse are also possible during the motion of a single vortex in a flow. The dif ference from the planar case, where no collapse occurs, is significant. The detected phenomenon can be applied when studying nonlinear processes in atmospheric vortices as well as in active galactic nuclei and planetary atmospheres. DOI: 10.1134/S1063776113100117
1. INTRODUCTION The interaction of ring vortices with flows is of con siderable interest and has numerous applications (see the jubilee issue of the journal [1] devoted to the 150th anniversary of Helmholtz’s classical work [2] and ref erences therein). In twodimensional hydrodynamics, a ring vortex is known to be simulated by a pair of point vortices with opposite signs of circulation located symmetrically relative to the axis of motion. In the absence of a flow, the pair moves uniformly and recti linearly along the axis of symmetry [3]. When a radial flow is taken into account, the character of motion becomes distinctly different [4]. In a divergent flow the distance between the pair components increases and the velocity slows down, while in a convergent flow the situation is reversed: a decrease in the distance between the pair components causes their velocity to increase. In a certain domain of parameters, the vortex can execute a backward motion. A system of two mirrorsymmetric ring vortices (a dipole toroidal vortex) is also of considerable inter est in connection with astrophysical applications [5– 10]. The dynamics of a dipole toroidal vortex in a radial flow was investigated in a 2D description in [11]. In this approximation, a dipole toroidal vortex can be represented by four point vortices. Starting from the paper by Grobli [12], it is well known that the compo nents of the pairs switch places as a result of their headon collision, while the new pairs fly apart at a right angle to the direction of motion with the original (in magnitude) velocities. The p
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