Bounds for Euler from vorticity moments and line divergence

The inviscid growth of a range of vorticity moments is compared using Euler calculations of anti-parallel vortices with a new initial condition. The primary goal is to understand the role of nonlinearity in the generation of a new hierarchy of rescaled vorticity moments in Navier–Stokes calculations where the rescaled moments obey Dm ≥ Dm+1, the reverse of the usual Ωm+1 ≥ Ωm Hölder ordering of the original moments. Two temporal phases have been identified for the Euler calculations. In the first phase the 1 < m < ∞ vorticity moments are ordered in a manner consistent with the new Navier–Stokes hierarchy and grow in a manner that skirts the lower edge of possible singular growth with D2 m → � sup ӏωӏ ~ Am(Tc-t)-1 where the Am are nearly independent of m. In the second phase, the new Dm ordering breaks down as the Ωm converge towards the same super-exponential growth for all m. The transition is identified using new inequalities for the upper bounds for the -dD-2m/dt that are based solely upon the ratios Dm+1/Dm, and the convergent super-exponential growth is shown by plotting log(d log Ωm/dt). Three-dimensional graphics show significant divergence of the vortex lines during the second phase, which could be what inhibits the initial power-law growth