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Gibbon JD, Gupta A, Krstulovic G, Pandit R, Politano H, Ponty Y, Pouquet A, Sahoo G, Stawarz J. Depletion of nonlinearity in magnetohydrodynamic turbulence: Insights from analysis and simulations. Phys Rev E 2016; 93:043104. [PMID: 27176387 DOI: 10.1103/physreve.93.043104] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/18/2015] [Indexed: 11/07/2022]
Abstract
It is shown how suitably scaled, order-m moments, D_{m}^{±}, of the Elsässer vorticity fields in three-dimensional magnetohydrodynamics (MHD) can be used to identify three possible regimes for solutions of the MHD equations with magnetic Prandtl number P_{M}=1. These vorticity fields are defined by ω^{±}=curlz^{±}=ω±j, where z^{±} are Elsässer variables, and where ω and j are, respectively, the fluid vorticity and current density. This study follows recent developments in the study of three-dimensional Navier-Stokes fluid turbulence [Gibbon et al., Nonlinearity 27, 2605 (2014)NONLE50951-771510.1088/0951-7715/27/10/2605]. Our mathematical results are then compared with those from a variety of direct numerical simulations, which demonstrate that all solutions that have been investigated remain in only one of these regimes which has depleted nonlinearity. The exponents q^{±} that characterize the inertial range power-law dependencies of the z^{±} energy spectra, E^{±}(k), are then examined, and bounds are obtained. Comments are also made on (a) the generalization of our results to the case P_{M}≠1 and (b) the relation between D_{m}^{±} and the order-m moments of gradients of magnetohydrodynamic fields, which are used to characterize intermittency in turbulent flows.
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Affiliation(s)
- J D Gibbon
- Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
| | - A Gupta
- Department of Physics, University of Rome "Tor Vergata," 00133 Rome, Italy
| | - G Krstulovic
- Laboratoire Lagrange, Université Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, Blvd de l'Observatoire, CS 34229, 06304 Nice cedex 4, France
| | - R Pandit
- Centre for Condensed Matter Theory, Indian Institute of Science, Bangalore, 560 012, India
| | - H Politano
- Laboratoire Dieudonné, Université de Nice Sophia-Antipolis, France
| | - Y Ponty
- Laboratoire Lagrange, Université Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, Blvd de l'Observatoire, CS 34229, 06304 Nice cedex 4, France
| | - A Pouquet
- National Center for Atmospheric Research, P.O. Box 3000, Boulder, Colorado 80307, USA.,Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, Colorado 80303, USA
| | - G Sahoo
- Department of Physics, University of Rome "Tor Vergata," 00133 Rome, Italy
| | - J Stawarz
- Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, Colorado 80303, USA
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Stawarz JE, Pouquet A. Small-scale behavior of Hall magnetohydrodynamic turbulence. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:063102. [PMID: 26764833 DOI: 10.1103/physreve.92.063102] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/17/2015] [Indexed: 06/05/2023]
Abstract
Decaying Hall magnetohydrodynamic (HMHD) turbulence is studied using three-dimensional (3D) direct numerical simulations with grids up to 768(3) points and two different types of initial conditions. Results are compared to analogous magnetohydrodynamic (MHD) runs and both Laplacian and Laplacian-squared dissipative operators are examined. At scales below the ion inertial length, the ratio of magnetic to kinetic energy as a function of wave number transitions to a magnetically dominated state. The transition in behavior is associated with the advection term in the momentum equation becoming subdominant to dissipation. Examination of autocorrelation functions reveals that, while current and vorticity structures are similarly sized in MHD, HMHD current structures are narrower and vorticity structures are wider. The electric field autocorrelation function is significantly narrower in HMHD than in MHD and is similar to the HMHD current autocorrelation function at small separations. HMHD current structures are found to be significantly more intense than in MHD and appear to have an enhanced association with strong alignment between the current and magnetic field, which may be important in collisionless plasmas where field-aligned currents can be unstable. When hyperdiffusivity is used, a longer region consistent with a k(-7/3) scaling is present for right-polarized fluctuations when compared to Laplacian dissipation runs.
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Affiliation(s)
- Julia E Stawarz
- Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, Colorado 80309, USA and Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, Colorado 80303, USA
| | - Annick Pouquet
- Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, Colorado 80303, USA
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Krstulovic G, Brachet ME, Pouquet A. Forced magnetohydrodynamic turbulence in three dimensions using Taylor-Green symmetries. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:043017. [PMID: 24827342 DOI: 10.1103/physreve.89.043017] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/07/2014] [Indexed: 06/03/2023]
Abstract
We examine the scaling laws of magnetohydrodynamic (MHD) turbulence for three different types of forcing functions and imposing at all times the fourfold symmetries of the Taylor-Green (TG) vortex generalized to MHD; no uniform magnetic field is present and the magnetic Prandtl number is equal to unity. We also include pumping in the induction equation, and we take the three configurations studied in the decaying case in Lee et al. [Phys. Rev. E 81, 016318 (2010)]. To that effect, we employ direct numerical simulations up to an equivalent resolution of 20483 grid points. We find that, similarly to the case when the forcing is absent, different spectral indices for the total energy spectrum emerge, corresponding to either a Kolmogorov law, an Iroshnikov-Kraichnan law that arises from the interactions of turbulent eddies and Alfvén waves, or to weak turbulence when the large-scale magnetic field is strong. We also examine the inertial range dynamics in terms of the ratios of kinetic to magnetic energy, and of the turnover time to the Alfvén time, and analyze the temporal variations of these quasiequilibria.
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Affiliation(s)
- G Krstulovic
- Laboratoire Lagrange, UMR7293, Université de Nice Sophia-Antipolis, CNRS, Observatoire de la Côte d'Azur, BP 4229, 06304 Nice Cedex 4, France
| | - M E Brachet
- Laboratoire de Physique Statistique de l'École Normale Supérieure, associé au CNRS et aux Universités ParisVI et VII, 24 Rue Lhomond, 75231 Paris, France
| | - A Pouquet
- Laboratory for Atmospheric and Space Sciences, and Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309, USA and Computational and Information Systems Laboratory, NCAR, Boulder, Colorado 80307, USA
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Dallas V, Alexakis A. Symmetry breaking of decaying magnetohydrodynamic Taylor-Green flows and consequences for universality. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:063017. [PMID: 24483567 DOI: 10.1103/physreve.88.063017] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/02/2013] [Indexed: 06/03/2023]
Abstract
We investigate the evolution and stability of a decaying magnetohydrodynamic Taylor-Green flow, using pseudospectral simulations with resolutions up to 2048(3). The chosen flow has been shown to result in a steep total energy spectrum with power law behavior k(-2). We study the symmetry breaking of this flow by exciting perturbations of different amplitudes. It is shown that for any finite amplitude perturbation there is a high enough Reynolds number for which the perturbation will grow enough at the peak of dissipation rate resulting in a nonlinear feedback into the flow and subsequently break the Taylor-Green symmetries. In particular, we show that symmetry breaking at large scales occurs if the amplitude of the perturbation is σ(crit)∼Re(-1) and at small scales occurs if σ(crit)∼Re(-3/2). This symmetry breaking modifies the scaling laws of the energy spectra at the peak of dissipation rate away from the k(-2) scaling and towards the classical k(-5/3) and k(-3/2) power laws.
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Affiliation(s)
- V Dallas
- Laboratoire de Physique Statistique, École Normale Supérieure, Université Pierre et Marié Curie, Université Paris Diderot, CNRS, 24 rue Lhomond, 75005 Paris, France
| | - A Alexakis
- Laboratoire de Physique Statistique, École Normale Supérieure, Université Pierre et Marié Curie, Université Paris Diderot, CNRS, 24 rue Lhomond, 75005 Paris, France
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Brachet ME, Bustamante MD, Krstulovic G, Mininni PD, Pouquet A, Rosenberg D. Ideal evolution of magnetohydrodynamic turbulence when imposing Taylor-Green symmetries. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:013110. [PMID: 23410449 DOI: 10.1103/physreve.87.013110] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/27/2012] [Indexed: 06/01/2023]
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 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.
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Affiliation(s)
- M E Brachet
- Laboratoire de Physique Statistique de l'École Normale Supérieure, associé au CNRS et aux Universités ParisVI et VII, 24 Rue Lhomond, 75231 Paris, France
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