On the finite-time singularity for an inviscid vortex ring model

In Moffat and Kimura [1], a low degree-of-freedom model for describing the approach to finite-time singularity of the incompressible Euler fluid equations was introduced. The model is obtained by assuming an initial finite-energy configuration of two vortex rings placed symmetrically on two tilted planes and then approximating the subsequent evolution. In [2] we obtained the Hamiltonian structure of the inviscid limit of the model. The associated noncanonical Poisson bracket [3] and two invariants, one that serves as the Hamiltonian and the other a Casimir invariant were discovered. It is shown that the system is integrable with a solution that lies on the intersection of the two invariants, just as for the free rigid body of mechanics whose solution lies on the intersection of the kinetic energy and angular momentum surfaces. Also, a direct quadrature is given and used to demonstrate the Leray form for finite-time singularity in the model. To the extent the Moffat and Kimura model accurately represents Euler's ideal fluid equations of motion, we showed the existence of finite-time singularity. This talk is a continuation of [3], with an emphasis on the physical interpretation of the invariants and subsequent evolution.

[1] H. K. Moffatt and Y. Kimura, J. Fluid. Mech. 86, 930, (2019); J. Fluid. Mech. 870, R1 (2019).

[2] P. J. Morrison and Y. Kimura, arXiv:2011.10864v1 [physics.fludyn] 21 Nov 2020

[3] P. J. Morrison, Rev. Mod. Phys. 70, 467 (1998).