Tracking Point Vortices and Circulations via Advected Passive Particles: An Estimation Approach

We present a novel method for estimating the circulations and positions of point vortices in a two-dimensional (2D) environment using trajectory data of passive particles in the presence of Gaussian noise. The method comprises two algorithms: the first one calculates the vortex circulations, while the second one reconstructs the vortex trajectories. This reconstruction is done thanks to a hierarchy of optimization problems, involving the integration of systems of differential equations, over time sub-intervals all with the same amplitude defined by the autocorrelation function for the advected passive particles' trajectories. Our findings indicate that accurately tracking the position of vortices and determining their circulations is achievable, even when passive particle trajectories are affected by noise.

Automated summary

We introduce a novel method to estimate the circulations and positions of point vortices in a 2D environment using trajectory data of passive particles affected by Gaussian noise. The method consists of two algorithms: one for calculating the vortex circulations, and the other for reconstructing the vortex trajectories. Through a hierarchy of optimization problems, involving the integration of systems of differential equations over time sub-intervals with the same amplitude defined by the autocorrelation function of the advected passive particles' trajectories, we find that accurately tracking the vortex position and determining its circulation is achievable even in the presence of noise.