  G.Luo and T.Y.Hou (Proceedings of the National Academy of Sciences 111.36
(2014): 12968-12973) have investigated a potentially singular solution of the
radially bounded, 3D, axisymmetric Euler equations via a hybrid, 6th-order
Galerkin and 6th-order finite-difference method on an adaptive (moving) mesh.
Given the importance of this problem, we have developed a pseudospectral,
Fourier-Chebyshev scheme to study this solution by an independent numerical
technique, which leads to new insights into this (potentially) singular
solution: (a) the time of singularity is preceded, in any, practical,
spectrally truncated direct numerical simulation (DNS), by the formation of
structures called tygers, first investigated in the one-dimensional (1D)
Burgers and two-dimensional (2D) Euler equations; (b) these tygers herald the
thermalization of our spectrally truncated system of equations. We demonstrate
how we can analyse this (potential) singularity by using singularity-detection
criteria based on (a) the Beale-Kato-Majda theorem and (b) a generalization of
the analyticity-strip method. We find that the width of the analyticity strip,
in the axial direction, decays faster than an exponential and, in the radial
direction, it decays linearly; this linear decay can be extrapolated to obtain
an estimate for the time at which the (potential) singularity occurs; our
estimate is consistent with that of G.Luo and T.Y.Hou. We examine the
DNS-resolution dependence of (a) the tyger-formation time, (b) thermalization,
(c) energy and helicity conservation, and (d) our estimate for the singularity
time.
