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

Zoran Grujic

TL;DR

The paper tackles the question of whether the viscous Moffatt-Kimura model for two counter-rotating vortex rings can develop a finite-time singularity. It develops a two-layer dissipation mechanism: first, a critical sparseness analysis tied to the radius of spatial analyticity to assess a potential blow-up; second, a log-composite, compensated-compactness pathway in Hardy spaces that translates local vorticity-direction oscillations into accelerated decay of vorticity-level sets, potentially driving the system subcritical. A dissipation argument via the harmonic-measure maximum principle shows that, under these hypotheses, the $L^\infty$ norm of the vorticity remains bounded, arguing against blow-up and highlighting the crucial role of viscosity and local coherence of the vorticity direction. The work clarifies why such dynamics are difficult to detect in simulations and emphasizes the interplay between analytic smoothing and viscous reconnection in regulating singular behavior in 3D incompressible flows.

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.

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

TL;DR

The paper tackles the question of whether the viscous Moffatt-Kimura model for two counter-rotating vortex rings can develop a finite-time singularity. It develops a two-layer dissipation mechanism: first, a critical sparseness analysis tied to the radius of spatial analyticity to assess a potential blow-up; second, a log-composite, compensated-compactness pathway in Hardy spaces that translates local vorticity-direction oscillations into accelerated decay of vorticity-level sets, potentially driving the system subcritical. A dissipation argument via the harmonic-measure maximum principle shows that, under these hypotheses, the norm of the vorticity remains bounded, arguing against blow-up and highlighting the crucial role of viscosity and local coherence of the vorticity direction. The work clarifies why such dynamics are difficult to detect in simulations and emphasizes the interplay between analytic smoothing and viscous reconnection in regulating singular behavior in 3D incompressible flows.

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.

Paper Structure

This paper contains 6 sections, 4 theorems, 28 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Criticality
  3. Beyond criticality?
  4. Dissipation via the harmonic measure maximum principle
  5. Conclusion
  6. Acknowledgments

Key Result

Proposition 4.3

Let $\Omega$ be an open, connected set in $\mathbb{C}$ such that its boundary has nonzero Hausdorff dimension, and let $K$ be a Borel subset of the boundary. Suppose that $u$ is a subharmonic function on $\Omega$ satisfying Then (Here, $h(z,\Omega,K)$ denotes the harmonic measure of $K$ with respect to $\Omega$, evaluated at $z$.)

Figures (3)

  • Figure 1: initial configuration Moffatt2023
  • Figure 2: viscous reconnection, bridges and threads at $\emph{Re} = 4 000$Yao2020
  • Figure 3: viscous reconnection, avalanche at $\emph{Re} = 40 000$Yao2020A

Theorems & Definitions (7)

  • Definition 4.1
  • Definition 4.2
  • Proposition 4.3: Ransford1995
  • Proposition 4.4: Solynin1997
  • Theorem 4.5: Bradshaw2019
  • Theorem 4.6
  • proof