Ideal evolution of magnetohydrodynamic turbulence when imposing Taylor-Green symmetries

Brachet, M. E.; Bustamante, M. D.; Krstulovic, G.; Mininni, P. D.; Pouquet, A.; Rosenberg, D.

doi:10.1103/physreve.87.013110

Title: Ideal evolution of magnetohydrodynamic turbulence when imposing Taylor-Green symmetries

Journal Article · Wed Jan 30 00:00:00 EST 2013 · Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

DOI:https://doi.org/10.1103/physreve.87.013110· OSTI ID:1565059

Brachet, M. E. ^[1]; Bustamante, M. D. ^[2]; Krstulovic, G. ^[3]; Mininni, P. D. ^[4]; Pouquet, A. ^[5]; Rosenberg, D. ^[5]

Centre National de la Recherche Scientifique (CNRS), Paris (France). Lab. de Physique Statistique de l'École Normale Supérieure
Univ. College Dublin, Dublin (Ireland). School of Mathematical Sciences
Centre National de la Recherche Scientifique (CNRS), Paris (France)
National Center for Atmospheric Research, Boulder, CO (United States). Computational and Information Systems Lab.; Univ. of Buenos Aires (Argentina). Dept. de Física
National Center for Atmospheric Research, Boulder, CO (United States). Computational and Information Systems Lab.

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 fourfold 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 regridding 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 $t = 2.70$. More robust conclusions will require higher resolution runs and grid-point interpolation measurements of maximum current and vorticity.

View Accepted Manuscript (DOE)

Cite

Export

Save

Research Organization:: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States). Oak Ridge Leadership Computing Facility (OLCF)

Sponsoring Organization:: USDOE Office of Science (SC)

Grant/Contract Number:: AC05-00OR22725

OSTI ID:: 1565059

Alternate ID(s):: OSTI ID: 1101982

Journal Information:: Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics, Vol. 87, Issue 1; ISSN 1539-3755

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

Citation Metrics:

Cited by: 22 works

Citation information provided by
Web of Science

References (67)

Effective Dissipation and Turbulence in Spectrally Truncated Euler Flows Cichowlas, Cyril; Bonaïti, Pauline; Debbasch, Fabrice Physical Review Letters, Vol. 95, Issue 26 https://doi.org/10.1103/PhysRevLett.95.264502	journal	December 2005
Long-time properties of magnetohydrodynamic turbulence and the role of symmetries Stawarz, Julia E.; Pouquet, Annick; Brachet, Marc-Etienne Physical Review E, Vol. 86, Issue 3 https://doi.org/10.1103/PhysRevE.86.036307	journal	September 2012
Inertial ranges and resistive instabilities in two‐dimensional magnetohydrodynamic turbulence Politano, H.; Pouquet, A.; Sulem, P. L. Physics of Fluids B: Plasma Physics, Vol. 1, Issue 12 https://doi.org/10.1063/1.859051	journal	December 1989
Transport of solar wind into Earth's magnetosphere through rolled-up Kelvin–Helmholtz vortices Hasegawa, H.; Fujimoto, M.; Phan, T. -D. Nature, Vol. 430, Issue 7001 https://doi.org/10.1038/nature02799	journal	August 2004
Local interaction between vorticity and shear in a perfect incompressible fluid Vieillefosse, P. Journal de Physique, Vol. 43, Issue 6 https://doi.org/10.1051/jphys:01982004306083700	journal	January 1982
Evidence for a singularity of the three‐dimensional, incompressible Euler equations Kerr, Robert M. Physics of Fluids A: Fluid Dynamics, Vol. 5, Issue 7 https://doi.org/10.1063/1.858849	journal	July 1993
Singular, weak and absent: Solutions of the Euler equations Constantin, Peter Physica D: Nonlinear Phenomena, Vol. 237, Issue 14-17 https://doi.org/10.1016/j.physd.2008.01.006	journal	August 2008
A Sparse-Mode Spectral Method for the Simulation of Turbulent Flows Meneguzzi, M.; Politano, H.; Pouquet, A. Journal of Computational Physics, Vol. 123, Issue 1 https://doi.org/10.1006/jcph.1996.0003	journal	January 1996
Flow visualization and field line advection in computational fluid dynamics: application to magnetic fields and turbulent flows Mininni, Pablo; Lee, Ed; Norton, Alan New Journal of Physics, Vol. 10, Issue 12 https://doi.org/10.1088/1367-2630/10/12/125007	journal	December 2008
Three-Dimensional Periodic Flows with High-Symmetry Kida, Shigeo Journal of the Physical Society of Japan, Vol. 54, Issue 6 https://doi.org/10.1143/JPSJ.54.2132	journal	June 1985
Alfvén waves and ideal two-dimensional Galerkin truncated magnetohydrodynamics Krstulovic, Giorgio; Brachet, Marc-Etienne; Pouquet, Annick Physical Review E, Vol. 84, Issue 1 https://doi.org/10.1103/PhysRevE.84.016410	journal	July 2011
Finite dissipation and intermittency in magnetohydrodynamics Mininni, P. D.; Pouquet, A. Physical Review E, Vol. 80, Issue 2 https://doi.org/10.1103/PhysRevE.80.025401	journal	August 2009
A hybrid MPI–OpenMP scheme for scalable parallel pseudospectral computations for fluid turbulence Mininni, Pablo D.; Rosenberg, Duane; Reddy, Raghu Parallel Computing, Vol. 37, Issue 6-7 https://doi.org/10.1016/j.parco.2011.05.004	journal	June 2011
Euler equations for incompressible ideal fluids Bardos, Claude; Titi, Edriss Russian Mathematical Surveys, Vol. 62, Issue 3 https://doi.org/10.1070/RM2007v062n03ABEH004410	journal	June 2007
High-Reynolds-number simulation of turbulent mixing Yeung, P. K.; Donzis, D. A.; Sreenivasan, K. R. Physics of Fluids, Vol. 17, Issue 8 https://doi.org/10.1063/1.2001690	journal	August 2005
Possibility of an inverse cascade of magnetic helicity in magnetohydrodynamic turbulence Frisch, U.; Pouquet, A.; LÉOrat, J. Journal of Fluid Mechanics, Vol. 68, Issue 4 https://doi.org/10.1017/S002211207500122X	journal	April 1975
Direct numerical simulation of transition to turbulence from a high‐symmetry initial condition Boratav, Oluş N.; Pelz, Richard B. Physics of Fluids, Vol. 6, Issue 8 https://doi.org/10.1063/1.868166	journal	August 1994
Interactive desktop analysis of high resolution simulations: application to turbulent plume dynamics and current sheet formation Clyne, John; Mininni, Pablo; Norton, Alan New Journal of Physics, Vol. 9, Issue 8 https://doi.org/10.1088/1367-2630/9/8/301	journal	August 2007
Cascades, thermalization, and eddy viscosity in helical Galerkin truncated Euler flows Krstulovic, G.; Mininni, P. D.; Brachet, M. E. Physical Review E, Vol. 79, Issue 5 https://doi.org/10.1103/PhysRevE.79.056304	journal	May 2009
Large-scale flow effects, energy transfer, and self-similarity on turbulence Mininni, P. D.; Alexakis, A.; Pouquet, A. Physical Review E, Vol. 74, Issue 1 https://doi.org/10.1103/PhysRevE.74.016303	journal	July 2006
3D Euler equations and ideal MHD mapped to regular systems: Probing the finite-time blowup hypothesis Bustamante, Miguel D. Physica D: Nonlinear Phenomena, Vol. 240, Issue 13 https://doi.org/10.1016/j.physd.2011.03.006	journal	June 2011
Numerical simulations of possible finite time singularities in the incompressible Euler equations: Comparison of numerical methods Grafke, Tobias; Homann, Holger; Dreher, Jürgen Physica D: Nonlinear Phenomena, Vol. 237, Issue 14-17 https://doi.org/10.1016/j.physd.2007.11.006	journal	August 2008
Remarks on the breakdown of smooth solutions for the 3-D Euler equations Beale, J. T.; Kato, T.; Majda, A. Communications in Mathematical Physics, Vol. 94, Issue 1 https://doi.org/10.1007/BF01212349	journal	March 1984
Measurement of the rugged invariants of magnetohydrodynamic turbulence in the solar wind Matthaeus, William H.; Goldstein, Melvyn L. Journal of Geophysical Research, Vol. 87, Issue A8 https://doi.org/10.1029/JA087iA08p06011	journal	January 1982
A weak turbulence theory for incompressible magnetohydrodynamics Galtier, S.; Nazarenko, S. V.; Newell, A. C. Journal of Plasma Physics, Vol. 63, Issue 5 https://doi.org/10.1017/S0022377899008284	journal	June 2000
Turbulence Regimes and Turbulence Intermittency in the Stable Boundary Layer during CASES-99 Sun, Jielun; Mahrt, Larry; Banta, Robert M. Journal of the Atmospheric Sciences, Vol. 69, Issue 1 https://doi.org/10.1175/JAS-D-11-082.1	journal	January 2012
Anisotropic Turbulence of Shear-Alfvén Waves Galtier, S.; Nazarenko, S. V.; Newell, A. C. The Astrophysical Journal, Vol. 564, Issue 1 https://doi.org/10.1086/338791	journal	January 2002
Geophysical–astrophysical spectral-element adaptive refinement (GASpAR): Object-oriented h-adaptive fluid dynamics simulation Rosenberg, Duane; Fournier, Aimé; Fischer, Paul Journal of Computational Physics, Vol. 215, Issue 1 https://doi.org/10.1016/j.jcp.2005.10.031	journal	June 2006
3D Euler about a 2D symmetry plane Bustamante, Miguel D.; Kerr, Robert M. Physica D: Nonlinear Phenomena, Vol. 237, Issue 14-17 https://doi.org/10.1016/j.physd.2008.02.007	journal	August 2008
Adaptive mesh refinement with spectral accuracy for magnetohydrodynamics in two space dimensions Rosenberg, D.; Pouquet, A.; Mininni, P. D. New Journal of Physics, Vol. 9, Issue 8 https://doi.org/10.1088/1367-2630/9/8/304	journal	August 2007
Blowup or no blowup? The interplay between theory and numerics Hou, Thomas Y.; Li, Ruo Physica D: Nonlinear Phenomena, Vol. 237, Issue 14-17 https://doi.org/10.1016/j.physd.2008.01.018	journal	August 2008
Interplay between the Beale-Kato-Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem Bustamante, Miguel D.; Brachet, Marc Physical Review E, Vol. 86, Issue 6 https://doi.org/10.1103/PhysRevE.86.066302	journal	December 2012
Evidence for a Singularity in Ideal Magnetohydrodynamics: Implications for Fast Reconnection Kerr, Robert M.; Brandenburg, Axel Physical Review Letters, Vol. 83, Issue 6 https://doi.org/10.1103/PhysRevLett.83.1155	journal	August 1999
Numerical evidence of smooth self‐similar dynamics and possibility of subsequent collapse for three‐dimensional ideal flows Brachet, M. E.; Meneguzzi, M.; Vincent, A. Physics of Fluids A: Fluid Dynamics, Vol. 4, Issue 12 https://doi.org/10.1063/1.858513	journal	December 1992
Remarks on Singularities, Dimension and Energy Dissipation for Ideal Hydrodynamics and MHD Caflisch, Russel E.; Klapper, Isaac; Steele, Gregory Communications in Mathematical Physics, Vol. 184, Issue 2 https://doi.org/10.1007/s002200050067	journal	April 1997
A magnetic reconnection X-line extending more than 390 Earth radii in the solar wind Phan, T. D.; Gosling, J. T.; Davis, M. S. Nature, Vol. 439, Issue 7073 https://doi.org/10.1038/nature04393	journal	January 2006
Small-Scale Structures in Three-Dimensional Magnetohydrodynamic Turbulence Mininni, P. D.; Pouquet, A. G.; Montgomery, D. C. Physical Review Letters, Vol. 97, Issue 24 https://doi.org/10.1103/PhysRevLett.97.244503	journal	December 2006
The three-dimensional Euler equations: Where do we stand? Gibbon, J. D. Physica D: Nonlinear Phenomena, Vol. 237, Issue 14-17 https://doi.org/10.1016/j.physd.2007.10.014	journal	August 2008
Geometric Statistics in Turbulence Constantin, Peter SIAM Review, Vol. 36, Issue 1 https://doi.org/10.1137/1036004	journal	March 1994
Intermittency in MHD turbulence and coronal nanoflares modelling Veltri, P.; Nigro, G.; Malara, F. Nonlinear Processes in Geophysics, Vol. 12, Issue 2 https://doi.org/10.5194/npg-12-245-2005	journal	January 2005
Paradigmatic flow for small-scale magnetohydrodynamics: Properties of the ideal case and the collision of current sheets Lee, E.; Brachet, M. E.; Pouquet, A. Physical Review E, Vol. 78, Issue 6 https://doi.org/10.1103/PhysRevE.78.066401	journal	December 2008
A Lagrangian study of dynamics and singularity formation at magnetic null points in ideal three‐dimensional magnetohydrodynamics Klapper, I.; Rado, A.; Tabor, M. Physics of Plasmas, Vol. 3, Issue 11 https://doi.org/10.1063/1.871559	journal	November 1996
Evolution of complex singularities in Kida–Pelz and Taylor–Green inviscid flows Cichowlas, C.; Brachet, M-E Fluid Dynamics Research, Vol. 36, Issue 4-6 https://doi.org/10.1016/j.fluiddyn.2004.09.005	journal	April 2005
Small-scale structure of the Taylor–Green vortex Brachet, Marc E.; Meiron, Daniel I.; Orszag, Steven A. Journal of Fluid Mechanics, Vol. 130, Issue -1 https://doi.org/10.1017/S0022112083001159	journal	May 1983
Depression of Nonlinearity in Decaying Isotropic MHD Turbulence Servidio, S.; Matthaeus, W. H.; Dmitruk, P. Physical Review Letters, Vol. 100, Issue 9 https://doi.org/10.1103/PhysRevLett.100.095005	journal	March 2008
The size and dynamics of magnetic flux structures in magnetohydrodynamic turbulence Brandenburg, Axel; Procaccia, Itamar; Segel, Daniel Physics of Plasmas, Vol. 2, Issue 4 https://doi.org/10.1063/1.871393	journal	April 1995
Rapid Alignment of Velocity and Magnetic Field in Magnetohydrodynamic Turbulence Matthaeus, W. H.; Pouquet, A.; Mininni, P. D. Physical Review Letters, Vol. 100, Issue 8 https://doi.org/10.1103/PhysRevLett.100.085003	journal	February 2008
On Hydromagnetic Equilibrium Woltjer, L. Proceedings of the National Academy of Sciences, Vol. 44, Issue 9 https://doi.org/10.1073/pnas.44.9.833	journal	September 1958
Shell-to-shell energy transfer in magnetohydrodynamics. I. Steady state turbulence Alexakis, Alexandros; Mininni, Pablo D.; Pouquet, Annick Physical Review E, Vol. 72, Issue 4 https://doi.org/10.1103/PhysRevE.72.046301	journal	October 2005
Current sheet formation in two-dimensional incompressible MHD turbulence from a lagrangian point of view Klapper, I.; Tabor, M. Geophysical & Astrophysical Fluid Dynamics, Vol. 73, Issue 1-4 https://doi.org/10.1080/03091929308203623	journal	December 1993
Constraints on finite-time current sheet formation at null points in two-dimensional ideal incompressible magnetohydrodynamics Klapper, I. Physics of Plasmas, Vol. 5, Issue 4 https://doi.org/10.1063/1.872659	journal	April 1998
Lack of universality in decaying magnetohydrodynamic turbulence Lee, E.; Brachet, M. E.; Pouquet, A. Physical Review E, Vol. 81, Issue 1 https://doi.org/10.1103/PhysRevE.81.016318	journal	January 2010
General introduction Eyink, Gregory; Frisch, Uriel; Moreau, René Physica D: Nonlinear Phenomena, Vol. 237, Issue 14-17 https://doi.org/10.1016/j.physd.2008.05.006	journal	August 2008
Adaptive Mesh Refinement for Singular Solutions of the Incompressible Euler Equations Grauer, Rainer; Marliani, Christiane; Germaschewski, Kai Physical Review Letters, Vol. 80, Issue 19 https://doi.org/10.1103/PhysRevLett.80.4177	journal	May 1998
Shell-to-shell energy transfer in magnetohydrodynamics. II. Kinematic dynamo Mininni, Pablo; Alexakis, Alexandros; Pouquet, Annick Physical Review E, Vol. 72, Issue 4 https://doi.org/10.1103/PhysRevE.72.046302	journal	October 2005
Internal motion of a small element of fluid in an inviscid flow Vieillefosse, P. Physica A: Statistical Mechanics and its Applications, Vol. 125, Issue 1 https://doi.org/10.1016/0378-4371(84)90008-6	journal	April 1984
Geometry of singular structures in magnetohydrodynamic flows Grauer, Rainer; Marliani, Christiane Physics of Plasmas, Vol. 5, Issue 7 https://doi.org/10.1063/1.872939	journal	July 1998
Velocity and scaling of collapsing Euler vortices Kerr, Robert M. Physics of Fluids, Vol. 17, Issue 7 https://doi.org/10.1063/1.1905183	journal	July 2005
Simulation of vortex reconnection Kerr, Robert M.; Hussain, Fazle Physica D: Nonlinear Phenomena, Vol. 37, Issue 1-3 https://doi.org/10.1016/0167-2789(89)90151-6	journal	July 1989
Creation and dynamics of vortex tubes in three-dimensional turbulence Constantin, Peter; Procaccia, Itamar; Segel, Daniel Physical Review E, Vol. 51, Issue 4 https://doi.org/10.1103/PhysRevE.51.3207	journal	April 1995
Current-Sheet Formation in 3D Ideal Incompressible Magnetohydrodynamics Grauer, Rainer; Marliani, Christiane Physical Review Letters, Vol. 84, Issue 21 https://doi.org/10.1103/PhysRevLett.84.4850	journal	May 2000
On the validity of a nonlocal approach for MHD turbulence Gomez, T.; Politano, H.; Pouquet, A. Physics of Fluids, Vol. 11, Issue 8 https://doi.org/10.1063/1.870092	journal	August 1999
The dynamics of unforced turbulence at high Reynolds number for Taylor–Green vortices generalized to MHD Pouquet, A.; Lee, E.; Brachet, M. E. Geophysical & Astrophysical Fluid Dynamics, Vol. 104, Issue 2-3 https://doi.org/10.1080/03091920903304080	journal	April 2010
A possible generation mechanism of interplanetary rotational discontinuities: INTERPLANETARY ROTATIONAL DISCONTINUITY Lin, C. C.; Tsai, C. L.; Chen, H. J. Journal of Geophysical Research: Space Physics, Vol. 114, Issue A8 https://doi.org/10.1029/2008JA014008	journal	August 2009
Symmetry and the hydrodynamic blow-up problem Pelz, Richard B. Journal of Fluid Mechanics, Vol. 444 https://doi.org/10.1017/S0022112001005298	journal	September 2001
Dynamic Depletion of Vortex Stretching and Non-Blowup of the 3-D Incompressible Euler Equations Hou, Thomas Y.; Li, Ruo Journal of Nonlinear Science, Vol. 16, Issue 6 https://doi.org/10.1007/s00332-006-0800-3	journal	August 2006
MHD turbulence in the solar wind: self-similarity, intermittency and coherent structures Veltri, Pierluigi Plasma Physics and Controlled Fusion, Vol. 41, Issue 3A https://doi.org/10.1088/0741-3335/41/3A/071	journal	January 1999

Cited By (9)

Evidence for Bolgiano-Obukhov scaling in rotating stratified turbulence using high-resolution direct numerical simulations Rosenberg, D.; Pouquet, A.; Marino, R. Physics of Fluids, Vol. 27, Issue 5 https://doi.org/10.1063/1.4921076	journal	May 2015
Generation of turbulence through frontogenesis in sheared stratified flows Sujovolsky, N. E.; Mininni, P. D.; Pouquet, A. Physics of Fluids, Vol. 30, Issue 8 https://doi.org/10.1063/1.5043293	journal	August 2018
Equilibria, Dynamics, and Current Sheet Formation in Magnetically Confined Coronae Rappazzo, A. F. The Astrophysical Journal, Vol. 815, Issue 1 https://doi.org/10.1088/0004-637x/815/1/8	journal	December 2015
Finite-time singularities and flow regularization in a hydromagnetic shell model at extreme magnetic Prandtl numbers Nigro, G.; Carbone, V. New Journal of Physics, Vol. 17, Issue 7 https://doi.org/10.1088/1367-2630/17/7/073038	journal	July 2015
Turbulent cascade, bottleneck, and thermalized spectrum in hyperviscous flows Agrawal, Rahul; Alexakis, Alexandros; Brachet, Marc E. Physical Review Fluids, Vol. 5, Issue 2 https://doi.org/10.1103/physrevfluids.5.024601	journal	February 2020
Coronal Heating Topology: The Interplay of Current Sheets and Magnetic Field Lines Rappazzo, A. F.; Matthaeus, W. H.; Ruffolo, D. The Astrophysical Journal, Vol. 844, Issue 1 https://doi.org/10.3847/1538-4357/aa79f2	journal	July 2017
Equilibria, Dynamics and Current Sheets Formation in Magnetically Confined Coronae Rappazzo, A. F. arXiv https://doi.org/10.48550/arxiv.1505.04370	text	January 2015
Coronal Heating Topology: the Interplay of Current Sheets and Magnetic Field Lines Rappazzo, A. F.; Matthaeus, W. H.; Ruffolo, D. arXiv https://doi.org/10.48550/arxiv.1706.08983	text	January 2017
Generation of turbulence through frontogenesis in sheared stratified flows Sujovolsky, N. E.; Mininni, P. D.; Pouquet, A. arXiv https://doi.org/10.48550/arxiv.1706.10287	preprint	January 2017

Similar Records

An adaptive finite element method for magnetohydrodynamics

Journal Article · Thu Dec 10 00:00:00 EST 1998 · Journal of Computational Physics · OSTI ID:1565059

Strauss, H R; Longcope, D W

Variational Integration for Ideal Magnetohydrodynamics and Formation of Current Singularities

Thesis/Dissertation · Fri Sep 01 00:00:00 EDT 2017 · OSTI ID:1565059

Zhou, Yao

Converging cylindrical shocks in ideal magnetohydrodynamics

Journal Article · Mon Sep 15 00:00:00 EDT 2014 · Physics of Fluids (1994) · OSTI ID:1565059

Pullin, D. I.; Mostert, W.; Wheatley, V.; +1 more

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
Physics

Title: Ideal evolution of magnetohydrodynamic turbulence when imposing Taylor-Green symmetries

Citation Formats

References (67)

Cited By (9)

Similar Records

Related Subjects