Ideal evolution of MHD turbulence when imposing Taylor-Green symmetries

Abstract

We investigate the ideal and incompressible magnetohydrodynamic (MHD) equations in three space dimensions for the development of potentially singular structures. The methodology consists in implementing the four-fold symmetries of the Taylor-Green vortex generalized to MHD, leading to substantial computer time and memory savings at a given resolution; we also use a re-gridding method that allows for lower-resolution runs at early times, with no loss of spectral accuracy. One magnetic configuration is examined at an equivalent resolution of $6144^3$ points, and three different configurations on grids of $4096^3$ points. At the highest resolution, two different current and vorticity sheet systems are found to collide, producing two successive accelerations in the development of small scales. At the latest time, a convergence of magnetic field lines to the location of maximum current is probably leading locally to a strong bending and directional variability of such lines. A novel analytical method, based on sharp analysis inequalities, is used to assess the validity of the finite-time singularity scenario. This method allows one to rule out spurious singularities by evaluating the rate at which the logarithmic decrement of the analyticity-strip method goes to zero. The result is that the finite-time singularity scenario cannot be ruled out, and the singularity time could be somewhere between $t=2.33$ and $t=\mathrm{2.70.}$ More robust conclusions will require higher resolution runs and grid-point interpolation measurements of maximum current and vorticity.

18 pages, 13 figures, 2 tables; submitted to Physical Review E

References in corpus (13)

• Dynamic Depletion of Vortex Stretching and Non-Blowup of the 3-D Incompressible Euler Equations
• Large scale flow effects, energy transfer, and self-similarity on turbulence
• Rapid directional alignment of velocity and magnetic field in magnetohydrodynamic turbulence
• Thermodynamic ground states of platinum metal nitrides
• Small scale structures in three-dimensional magnetohydrodynamic turbulence
• Numerical simulations of possible finite time singularities in the incompressible Euler equations: comparison of numerical methods
• Interplay between the Beale-Kato-Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem
• Alfvén waves and ideal two-dimensional Galerkin truncated magnetohydrodynamics
• A paradigmatic flow for small-scale magnetohydrodynamics: properties of the ideal case and the collision of current sheets
• Long-time properties of MHD turbulence and the role of symmetries
• Adaptive mesh refinement with spectral accuracy for magnetohydrodynamics in two space dimensions
• 3D Euler equations and ideal MHD mapped to regular systems: probing the finite-time blowup hypothesis
• Forced MHD turbulence in three dimensions using Taylor-Green symmetries

Cited by in corpus (12)

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• Evidence for Bolgiano-Obukhov scaling in rotating stratified turbulence using high-resolution direct numerical simulations
• Magnetic reconnection: Self-similar current sheet collapse triggered by "ideal" tearing
• Turbulent channel without boundaries: The periodic Kolmogorov flow
• Turbulent cascade, bottleneck and thermalized spectrum in hyperviscous flows
• Effects of electron inertia in collisionless magnetic reconnection
• Coronal Heating Topology: the Interplay of Current Sheets and Magnetic Field Lines
• Equilibria, Dynamics and Current Sheets Formation in Magnetically Confined Coronae
• Velocity and acceleration statistics in particle-laden turbulent swirling flows
• Self-truncation and scaling in Euler-Voigt-$\alpha$ and related fluid models
• Symmetry-plane model of 3D Euler flows and mapping to regular systems to improve blowup assessment using numerical and analytical solutions
• Magnetized Decaying Turbulence in the Weakly Compressible Taylor-Green Vortex