Global chaotization of fluid particle trajectories in a sheared two-layer two-vortex flow.

Abstract

In a two-layer quasi-geostrophic approximation, we study the irregular dynamics of fluid particles arising due to two interacting point vortices embedded in a deformation flow consisting of shear and rotational components. The two vortices are arranged within the bottom layer, but an emphasis is on the upper-layer fluid particle motion. Vortices moving in one layer induce stirring of passive scalars in the other layer. This is of interest since point vortices induce singular velocity fields in the layer they belong to; however, in the other layer, they induce regular velocity fields that generally result in a change in passive particle stirring. If the vortices are located at stagnation points, there are three different types of the fluid flow. We examine how properties of each flow configuration are modified if the vortices are displaced from the stagnation points and thus circulate in the immediate vicinity of these points. To that end, an analysis of the steady-state configurations is presented with an emphasis on the frequencies of fluid particle oscillations about the elliptic stagnation points. Asymptotic relations for the vortex and fluid particle zero-oscillation frequencies are derived in the vicinity of the corresponding elliptic points. By comparing the frequencies of fluid particles with the ones of the vortices, relations between the parameters that lead to enhanced stirring of fluid particles are established. It is also demonstrated that, if the central critical point is elliptic, then the fluid particle trajectories in its immediate vicinity are mostly stable making it harder for the vortex perturbation to induce stirring. Change in the type of the central point to a hyperbolic one enhances drastically the size of the chaotic dynamics region. Conditions on the type of the central critical point also ensue from the derived asymptotic relations.