Vortex-Driven Sensitivity in Deformation Flow

Abstract

A sensitivity mechanism for the interaction of two vortices in a two-dimensional deformation background flow is explored. A nonlinear model describing the vortex interaction up to a critical merging distance is developed. This model shows that in a confluent or diffluent background flow, two vortices can obtain a global minimum distance by counterclockwise turning and a local minimum distance by clockwise turning about each other. In a major portion of the phase space, the global minimum distance is smaller than the local minimum distance. Therefore, vortex merger is more likely to occur during the process in which the two vortices approach their global smallest distance. Analysis of the governing equations shows that there exists a ?most sensitive line? of initial vortex positions. When an initial state is on one side of the most sensitive line, the two vortices can obtain their global smallest distance and thus have a higher chance for merger. Conversely, when an initial state is on the other side relative to the most sensitive line, the two vortices can only obtain their local smallest distance. Consequently, they have a lower chance for merger. When an initial state is close to the most sensitive line, a very small initialization error can put this initial state on the wrong side of the most sensitive line, and the forecast evolution of the pair of vortices may be very different from the observed evolution. Numerical simulations that support these theoretical results are shown.