Note on the Rate of Vortex Stretching for Axisymmetric Euler Flows Without Swirl

TL;DR

This paper investigates the growth rate of vorticity in axisymmetric Euler flows without swirl, providing new lower bounds under certain initial decay conditions, and advances understanding of vortex dynamics.

Contribution

It introduces the generalized vertical moment and proves its monotonicity to establish improved lower bounds on vorticity growth for specific initial conditions.

Findings

Lower bound of vorticity growth is approximately t^{1/2-} for certain initial decay rates.

Monotonicity of the generalized vertical moment is established.

Results support Childress's conjecture in specific flow configurations.

Abstract

In this paper, we investigate Childress's conjecture proposed in [Phys.D 237(14-17):1921-1925, 2008] on the growth rate of the vorticity maximum for axisymmetric swirl-free Euler flows in three and higher dimensions. We consider the setting that the axial vorticity is non-positive in the upper half space and odd in the last coordinate, which corresponds to the flow setup for head-on collision of anti-parallel vortex rings. By introducing the \emph{generalized vertical moment} and proving its monotonicity, we obtain a lower bound for the growth of the vorticity maximum, contingent on the initial decay rate in the $z$-direction. Specifically, for three-dimensional flows with initial vorticity sufficiently fast decay in $z$, we obtain a lower bound of $t^{\frac{1}{2}-}$, thereby improving upon existing results.