On taming Moffatt-Kimura vortices of doom in the viscous case

TL;DR

This paper introduces a two-layer viscous mechanism that prevents finite-time singularities in vortex collision models by analyzing turbulent dissipation and vortex-stretching properties, offering new insights into fluid stability.

Contribution

It proposes a novel two-layer viscous mechanism combining turbulent dissipation analysis and vortex-stretching cancellation properties to prevent singularities in vortex models.

Findings

The problem is at worst critical with bounds on sparseness and analyticity.

A subtle mechanism can drive sparseness into the dissipation range.

Analytic cancellation properties help prevent singularity formation.

Abstract

In this note we propose a two-layer viscous mechanism for preventing finite time singularity formation in the Moffatt-Kimura model of two counter-rotating vortex rings colliding at a nontrivial angle. In the first layer the scenario is recast within the framework of the study of turbulent dissipation based on a suitably defined `scale of sparseness' of the regions of intense fluid activity. Here it is found that the problem is (at worst) critical, i.e., the upper bound on the scale of sparseness of the vorticity super-level sets is comparable to the lower bound on the radius of spatial analyticity. In the second layer, an additional more subtle mechanism is identified, potentially capable of driving the scale of sparseness into the dissipation range and preventing the formation of a singularity. The mechanism originates in certain analytic cancellation properties of the vortex-stretching term in the sense of compensated compactness in Hardy spaces which then convert information on local mean oscillations of the vorticity direction (boundedness in certain log-composite weighted local bmo spaces) into log-composite faster decay of the vorticity super-level sets.