On the Topology of the Atmosphere Advected by a Periodic Array of Axisymmetric Thin-cored Vortex Rings

The fluid motion produced by a spatially periodic array of identical, axisymmetric, thin-cored vortex rings is investigated. It is well known that such an array moves uniformly without change of shape or form in the direction of the central axis of symmetry, and is therefore an equilibrium solution of Euler’s equations. In a frame of reference moving with the system of vortex rings, the motion of passive fluid particles is investigated as a function of the two nondimensional parameters that define this system: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon=a/R$$\end{document}, the ratio of minor radius to major radius of the torus-shaped vortex rings, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda=L/R$$\end{document}, the separation of the vortex rings normalized by their radii. Two bifurcations in the streamline topology are found that depend significantly on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}; these bifurcations delineate three distinct shapes of the “atmosphere” of fluid particles that move together with the vortex ring for all time. Analogous to the case of an isolated vortex ring, the atmospheres can be “thin-bodied” or “thick-bodied”. Additionally, we find the occurrence of a “connected” system, in which the atmospheres of neighboring rings touch at an invariant circle of fluid particles that is stationary in a frame of reference moving with the vortex rings.