Dipole Dynamics in the Point Vortex Model

Abstract

Presented here is a thorough theoretical and numerical investigation into the dynamics of vortex dipoles in point vortex and point vortex adjacent systems. We begin by reviewing the basic laws of fluid flow and thus standard hydrodynamic turbulence after which, motivated by the vortex filament description of fluid flow, we introduce the intriguing phenomenon of quantum turbulence. From this quantum turbulence we motivate study of the point vortex model, including a description of the basic theory of point vortex systems. Using Kirchoff’s Hamiltonian formulation of the 2D point vortex model in the infinite domain, we consider work done into dipole-vortex collisions, as well as extending this to consider the dipole-dipole collisions in both the integrable and non-integrable cases. Here we solve for important dynamical quantities such as the scattering angle, of great importance to the mixing of turbulent systems, and also the vortex separations at the dipole periapsis, as well as the extremum values of dipole separation, and we discuss the possibility of dipole creation in such systems. We consider this as a numerical analysis but also analyse each interaction regime theoretically where possible, finding perfect agreement between theoretical predictions and numerical results where this is done. The study then continues by considering the collisions of dipoles with same-signed rotating vortex clusters, and also analysing the possibility of approximating the dynamics of a dipole colliding with large vortex clusters by the dynamics of a dipole colliding with equivalently strong single point vortices, we find such an approximation is even more effective than may first be considered, with the dipole-vortex simulations often faithfully reproducing the dynamical quantities of the dipole-clusters. We also consider the scattering angles and dipole creation possibility as before, noting the highly chaotic behaviour found at impact parameters close to zero. We then move from the 2D infinite domain point vortex model in the strict sense in order to consider how point vortex systems may capture behaviour of standard turbulent systems; by considering large N periodic point vortex systems we realise the 2D turbulent phenomenon of the inverse energy cascade through additional forcing and dampening mechanisms. Finally, we give a study on the effect of the interaction of sound waves with a vortex dipole. Investigating the notion that sound will lead to an eventual vortex collapse, we find this is highly dependent upon the initial conditions present, more specifically the initial sound distribution, as this behaviour although realised in Gaussian distributed sound is not realised in Rayleigh-Jean distributed sound. Therefore, it is speculated that the shrinkage and annihilation processes occur not due to the presence of sound alone but instead through the out-of-equilibrium motion as the system relaxes towards the eventual statistical equilibrium of vortex annihilation.