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

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][3]. 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, and that a solution is obtained as the intersection of two surfaces, H=const. and C=const.  [4].  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. Among the three eigen solutions, there are two stable and one unstable modes. It is verified that the eigenvector of the unstable mode is parallel with the trajectory, and thus the Leray solution is linearly stable. 

[1] Moffatt, H.K. & Kimura, Y.,  J. Fluid Mech. (2019) 861  930-967. 

[2] Moffatt, H.K. & Kimura, Y.,  J. Fluid Mech. (2019) 870 R1.

[3] Moffatt, H.K. & Kimura, Y.,  J. Fluid Mech. (2023) 967 R1.

[4] Morrison, P.J. & Kimura, Y., Phys. Lett. A (2023) 484 129078.