Stability of the Leray scaling solution for vortex reconnection in Euler flows

As a model for studying the evolution towards finite-time singularity of the Navier-Stokes equation, a dynamical system was proposed for describing the behavior of vortex reconnection of two vortex rings placed symmetrically on two tilted planes [1][2]. For the Euler limit, it was shown that this dynamical system can be written in noncanonical Hamiltonian form with Hamiltonian, H, and a Casimir invariant, C [3]. The special case where both H and C vanish gives a singular solution, obtained by quadrature, which has the exact scaling proposed by Leray for studying the self-similar solutions of the Navier-Stokes equations. We investigate the stability of this Leray solution by employing time dependent stretched coordinates that transform the singular solution into a fixed point, and then examining a linear perturbation about the fixed point. It will be shown that there is one unstable mode among the three eigen solutions. The physical meaning of the stability analysis will be considered.

[1] Moffatt, H.K. & Kimura, Y., Towards a finite-time singularity of the Navier-Stokes equations. Part 1. Derivation and analysis of dynamical system. J. Fluid Mech. (2019) 861 930-967.

[2] Moffatt, H.K. & Kimura, Y., Towards a finite-time singularity of the Navier-Stokes equations. Part 2. Vortex reconnection and singularity evasion. J. Fluid Mech. (2019) 870 R1.

[3] Morrison, P.J. & Kimura, Y., A Hamiltonian description of finite-time singularity in Euler's fluid equations. arXiv: 2011.10864v1 [physics.flu-dyn].