Vortex intensification and collapse of the Lissajous-elliptic ring: single- and multi-filament Biot-Savart simulations and visiometrics

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Abstract
The collapsing ‘Lissajous-elliptic’ (LE) vortex ring is examined via quantifications of Single- and multi-filament Biot-Savart numerical simulations. In the single-filament simulations, parametric studies show simple relationships between the collapse boundary and the impulse and energy invariants. Collapse becomes non-monotonic in time, for a sufficiently small initial core ‘radius’. Self-similar, singular-like behaviour of the off-filament strain-rate growth has been observed in a small interval, just prior to core overlapping. The computation of the strain-rate eigenvalues and vortex stretching in a diagnostics box surrounding the collapse region yields patterns observed previously in continuum simulations. New diagnostics are presented, including line densities of the energy and the linear and angular momentum, all of which approach zero in the collapse region of the ring. These diagnostics may provide critical parameters for initiating surgery in a topology-changing algorithm. Our multi-filament simulations exhibit layer-like vortex regions and a ‘torus’-shaped vortex stretching pattern observed previously in continuum periodic-domain simulations of vortex reconnection. Quantifications in a cross-section of the collapse region indicate that the circulation tends to concentrate in the head or frontside of the convecting dipolar structure. This is also the location of the incipient ‘bridge’ which is evolving from the weak filaments that have been convected from the initially outer-vortex regions. The formation of this smaller scale vortex structure exhibits the largest vorticity amplification in the variable-core model simulations.