Dynamics of Nearly Circular Vortex Filaments
Nelson, Sybil Prince
We derive a model for the dynamics of a pair of nearly-circular vortex filaments, and obtain a system of coupled partial differential equations that incorporate both the self-induced dynamics due to the vorticity of each filament and the interaction of the two filaments. Our derivation follows the one by Klein, Majda, and Damodaran  for the case of nearly-parallel vortex filaments. The evolution equations for nearly circular vortex filaments contain a curvature dependent term in the self-induction component of the vector field. Such component is obtained by linearizing the Vortex Filament Equation around a circle. We discuss special time-independent solutions using perturbation methods and Fourier-series methods for the associated partial differential equations.
Vortex-motion; Differential equations, Partial.