Chaotic advection of fluid particles in a field of point vortices and vortex rings

Abstract

Processes of formation of vortex rings and mushroom structures in the ocean, in geological structures in Earth’s mantle, and in various applications in chemical technology,1 provide conditions for advanced interest in the phenomena of internal mixing. In order to model such processes, an approach based on a direct study of the trajectories of motion of Lagrangian particles in a Eulerian velocity field currently attracts special attention. Although the ideas of such an approach may be found in the works of Maxwell2 and Reynolds,3 comparatively little has been done toward obtaining a clear view of what goes on, except in the simplest cases. To the more complicated cases belong the processes of chaotic advection of particles in bounded domains.4 This paper presents the results of a study of properties of chaotic particle advection of an ideal fluid in unbounded and bounded domains under free motion of point vortices or vortex rings. Data on the process of particle motion for particles that initially form the ‘‘atmosphere’’ of a vortex pair when it interacts with a single vortex, or with other noncollinear vortex pairs, are presented for a two-dimensional unbounded domain. Depending on the type of interaction—either ‘‘direct’’ or ‘‘exchange’’ collision or mutual trapping5–7—an estimate of the effectiveness of stirring of atmosphere particles and chaotization of their trajectories is given. The problem of advection of particles in a two-dimensional rectangular domain with a free vortex is solved. Although the locked trajectory of the vortex and the period of its revolution are explicitly calculated by means of Jacobi and Weierstrass elliptic functions, a complicated dependence of the streamfunction on time leads to the formation of ‘‘whorls’’ and ‘‘tendrils,’’ and to effective stirring of some domains. Poincaré sections of particles for characteristic values of the geometrical parameters of the domain and initial positions of the vortex are given. Various types of interaction such as ‘‘leap-frogging,’’ passing and mutual trapping,8 are studied for the axisymmetric motion of two coaxial vortex rings with the intensities of vorticity of the same or different signs. An analysis of chaotization of the motion for particles initially forming the atmosphere of the rings has been carried out for some cases of interaction.