Priyanka Shukla

The problem of finding initial conditions that lead to self-similar motion of point vortices is formulated as a linear system. The linearity in the equations is used to check for the existence of similarity solutions with a given shape and, in particular, to numerically find self-similar vortex configurations with or without any prior knowledge of circulations. Algorithms for computing the one-parameter family of collapse and expansion configurations and the finitely many rigid-vortex configurations present in the family are also discussed. Typical families are shown to have vortices parametrized along closed curves, and the conditions for which they are not closed are investigated via several numerical examples.

Point vortex systems that include vortices with constant coordinate functions are largely unexplored, even though they have reasonable physical interpretations in the geophysical context. Here, we investigate the dynamical aspects of the restricted three-vortex problem when one of the point vortices is assumed to be fixed at a location in the plane. The motion of the passive tracer is explored from a rotating frame of reference within which the free vortex with non-zero circulation remains stationary. By using basic dynamical system theory, it is shown that the vortex motion is always bounded, and any configuration of the three vortices must go through at least one collinear state. The present analysis reveals that any non-relative equilibrium solution of the vortex system either has periodic inter-vortex distances or it will asymptotically converge to a relative equilibrium configuration. The initial conditions required for different types of motion are explained in detail by exploiting the Hamiltonian structure of the problem. The underlying effects of a fixed vortex on the motion of vortices are also explored.

The dynamics of a constrained three-vortex system, a pair of point vortices of arbitrary non-zero circulations in the velocity field of a fixed point vortex, is investigated. The underlying dynamical system is simplified using a coordinate transformation and categorized into two cases based on the zero and non-zero values of the constant of angular impulse. For each case, dynamical features of the vortex motion are studied analytically in the transformed plane to completely classify the vortex motions and understand the boundedness and periodicity of the inter-vortex distances. The theoretical predictions are also verified numerically and illustrated for various sets of initial conditions and circulations.