Velocity and scaling of collapsing Euler vortices

Abstract

New analysis of the scaling structure of a numerical solution of the Euler equations finds that initially antiparallel vortex tubes collapse into two wings whose cross sections can be described using two length scales ρ and R. The first ρ∼(T−t) for the leading edge and the distance between the position of peak vorticity and the dividing plane. The second R∼(T−t)1∕2 describes the extent of the wings and the distance of the peak in vortical velocity supx∣v∣ from the peak in vorticity. All measures of singular growth within the inner region give the same singular time. This includes a blowup in the peak of vortical or axial velocity going as (T−t)−1∕2 at a distance R from the position of ∥ω∥∞. Outside this self-similar region, energy, enstrophy, circulation, and helicity accumulate. Twisting of vortex lines consistent with vortex line length growing to infinity is observed in the outer region. Vorticity in the intermediate zone between the inner and outer regions, while no longer growing at the singular rate, could be the major source of the strain interactions that drive the flow.