Incompressible hydrodynamic turbulence from a chain reaction of vortex reconnection events

Abstract

From a new anti-parallel initial condition using long vortices, three-dimensional turbulence forms after two reconnection steps and the formation of at least one vortex ring. The long domain is needed in order to accommodate the multiple reconnections, which enhance vortex stretching rates and the generation of small-scale vortex structures within the vortex rings. In addition to making the initial vortices very long, new features introduced with this initial condition are an initial profile less likely to shed vortex sheets and an improved method for mapping the direction of the vorticity onto the three-dimensional mesh. To get to the turbulent state, the vortices progress through the following steps: First, until the first reconnection, the vortex dynamics is largely consistent with existing work on strong, possibly singular, growth of the vorticity in the Euler equations. Second, vortex reconnection at the junction of the primary symmetry planes meet. About half of the circulation from each vortex reconnects into two "bridges", leaving behind two "threads", with the bridges and threads arranged into 4 orthogonal pairs. In the third step, stretching is induced by new anti-parallel attractions due to the twisting that forms along the reconnected vortices. This extra stretching then pulls on the threads as they wind around the bridges, resulting in spirals over the entire vortices. In the fourth step, multiple vortex rings form through multiple reconnections, with each ring consisting of spiral vortices. A $k^{-5/3}$ energy spectra begins to form after the first ring separates and eventually covers one decade. It is argued that the spirals are the source of the $k^{-5/3}$ spectra that develops. Furthermore, a new hierarchy of rescaled vorticity moments is found where the lower-order moments bound the higher-order moments for all orders and all times.