A pseudospectral study of a potentially singular solution of the three-dimensional axisymmetric incompressible Euler equation: tygers and thermalization

Luo and Hou [1] 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 [2, 3] and (b) a generalization of the analyticity-strip method [4]. 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 Ref. [1]. 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. Two hundred and sixty five years ago, Euler introduced the equations for an inviscid, incompressible, three-dimensional (3D) fluid in Principes généraux du mouvement des fluides [5–7]. The incompressible Euler partial differential equation (PDE) and its viscous descendant, namely, the incompressible Navier-Stokes PDE [8, 9] govern, respectively, ideal and viscous fluid flows at low Mach numbers. They are, therefore, among the most prominent equations in theoretical physics; and their solutions are of importance in a variety of engineering, fluid-dynamical, astrophysical, geophysical, and laboratory settings. Furthermore, these equations pose challenges for mathematicians: In particular, we do not know with certainty if any solutions of these 3D equations develop a singularity in a finite time, if we start with analytic initial data (for non-analytic initial data, see Ref. [10]). The answer to this grand-challenge problem has important implications for turbulence in fluids, even if we use the 3D Euler PDE, as conjectured by Onsager [11] (see, e.g., Ref. [12, 13]); for the Navier-Stokes case, this is one of the Clay mathematics problems [14]. Explorations of this problem (for the Euler case see, e.g., Refs. [7, 15]) often use direct numerical simulations (DNSs), which have not, so far, yielded unambiguous results for or against a finite-time singularity in the 3D Euler PDE with any analytic initial velocity field. A study by Luo and Hou [1] has explored a potentially singular solution of the radially bounded, 3D, axisymmetric Euler equations via a hybrid, 6th-order Galerkin and 6thorder finite-difference method on an adaptive (moving) mesh. Given the importance of this problem, it behooves us to study this potentially singular solution by a completely different numerical scheme. Therefore, we have developed a pseudospectral, Fourier-