1
|
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] [What about the content of this article? (0)] [Affiliation(s)] [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.
Collapse
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
| |
Collapse
|
2
|
Baerenzung J, Politano H, Ponty Y, Pouquet A. Spectral modeling of magnetohydrodynamic turbulent flows. Phys Rev E Stat Nonlin Soft Matter Phys 2008; 78:026310. [PMID: 18850939 DOI: 10.1103/physreve.78.026310] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/24/2008] [Indexed: 05/26/2023]
Abstract
We present a dynamical spectral model for large-eddy simulation of the incompressible magnetohydrodynamic (MHD) equations based on the eddy damped quasinormal Markovian approximation. This model extends classical spectral large-eddy simulations for the Navier-Stokes equations to incorporate general (non-Kolmogorovian) spectra as well as eddy noise. We derive the model for MHD flows and show that the introduction of an eddy damping time for the dynamics of spectral tensors, in the absence of equipartition between the velocity and magnetic fields, leads to better agreement with direct numerical simulations, an important point for dynamo computations.
Collapse
Affiliation(s)
- J Baerenzung
- TNT/NCAR, P.O. Box 3000, Boulder, Colorado 80307-3000, USA
| | | | | | | |
Collapse
|
3
|
Baerenzung J, Politano H, Ponty Y, Pouquet A. Spectral modeling of turbulent flows and the role of helicity. Phys Rev E Stat Nonlin Soft Matter Phys 2008; 77:046303. [PMID: 18517728 DOI: 10.1103/physreve.77.046303] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/04/2007] [Revised: 11/06/2007] [Indexed: 05/26/2023]
Abstract
We present a version of a dynamical spectral model for large eddy simulation based on the eddy damped quasinormal Markovian approximation [S. A. Orszag, in, edited by R. Balian, Proceedings of Les Houches Summer School, 1973 (Gordon and Breach, New York, 1977), p. 237; J. P. Chollet and M. Lesievr, J. Atmos. Sci. 38, 2747 (1981)]. Three distinct modifications are implemented and tested. On the one hand, whereas in current approaches, a Kolmogorov-like energy spectrum is usually assumed in order to evaluate the non-local transfer, in our method the energy spectrum of the subgrid scales adapts itself dynamically to the large-scale resolved spectrum; this first modification allows in particular for a better treatment of transient phases and instabilities, as shown on one specific example. Moreover, the model takes into account the phase relationships of the small scales, embodied, for example, in strongly localized structures such as vortex filaments. To that effect, phase information is implemented in the treatment of the so-called eddy noise in the closure model. Finally, we also consider the role that helical small scales may play in the evaluation of the transfer of energy and helicity, the two invariants of the primitive equations in the inviscid case; this leads as well to intrinsic variations in the development of helicity spectra. Therefore, our model allows for simulations of flows for a variety of circumstances and a priori at any given Reynolds number. Comparisons with direct numerical simulations of the three-dimensional Navier-Stokes equation are performed on fluids driven by an Arnold-Beltrami-Childress (ABC) flow which is a prototype of fully helical flows (velocity and vorticity fields are parallel). Good agreements are obtained for physical and spectral behavior of the large scales.
Collapse
Affiliation(s)
- J Baerenzung
- Laboratoire Cassiopée, UMR 6202, Observatoire de la Côte d'Azur, B.P. 4229, 06304 Nice Cedex 4, France
| | | | | | | |
Collapse
|
4
|
Alexakis A, Bigot B, Politano H, Galtier S. Anisotropic fluxes and nonlocal interactions in magnetohydrodynamic turbulence. Phys Rev E Stat Nonlin Soft Matter Phys 2007; 76:056313. [PMID: 18233762 DOI: 10.1103/physreve.76.056313] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/06/2007] [Indexed: 05/25/2023]
Abstract
We investigate the locality or nonlocality of the energy transfer and the spectral interactions involved in the cascade for decaying magnetohydrodynamic (MHD) flows in the presence of a uniform magnetic field B at various intensities. The results are based on a detailed analysis of three-dimensional numerical flows at moderate Reynolds numbers. The energy transfer functions, as well as the global and partial fluxes, are examined by means of different geometrical wave number shells. On the one hand, the transfer functions of the two conserved Elsässer energies E+ and E- are found local in both the directions parallel (k|| direction) and perpendicular (kperpendicular direction) to the magnetic guide field, whatever the B strength. On the other hand, from the flux analysis, the interactions between the two counterpropagating Elsässer waves become nonlocal. Indeed, as the B intensity is increased, local interactions are strongly decreased and the interactions with small k|| modes dominate the cascade. Most of the energy flux in the kperpendicular direction is due to modes in the plane at k||=0, while the weaker cascade in the k|| direction is due to the modes with k||=1. The stronger magnetized flows tend thus to get closer to the weak turbulence limit, where three-wave resonant interactions are dominant. Hence, the transition from the strong to the weak turbulence regime occurs by reducing the number of effective modes in the energy cascade.
Collapse
Affiliation(s)
- A Alexakis
- Laboratoire Cassiopée, UMR 6202, Observatoire de la Côte d'Azur, Boîte Postale 4229, Nice Cedex 4, France
| | | | | | | |
Collapse
|
5
|
Ponty Y, Mininni PD, Montgomery DC, Pinton JF, Politano H, Pouquet A. Numerical study of dynamo action at low magnetic Prandtl numbers. Phys Rev Lett 2005; 94:164502. [PMID: 15904232 DOI: 10.1103/physrevlett.94.164502] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/07/2004] [Indexed: 05/02/2023]
Abstract
We present a three-pronged numerical approach to the dynamo problem at low magnetic Prandtl numbers P(M). The difficulty of resolving a large range of scales is circumvented by combining direct numerical simulations, a Lagrangian-averaged model and large-eddy simulations. The flow is generated by the Taylor-Green forcing; it combines a well defined structure at large scales and turbulent fluctuations at small scales. Our main findings are (i) dynamos are observed from P(M)=1 down to P(M)=10(-2), (ii) the critical magnetic Reynolds number increases sharply with P(M)(-1) as turbulence sets in and then it saturates, and (iii) in the linear growth phase, unstable magnetic modes move to smaller scales as P(M) is decreased. Then the dynamo grows at large scales and modifies the turbulent velocity fluctuations.
Collapse
Affiliation(s)
- Y Ponty
- CNRS UMR6202, Laboratoire Cassiopée, Observatoire de la Côte d'Azur, BP 4229, Nice Cedex 04, France
| | | | | | | | | | | |
Collapse
|
6
|
Politano H, Gomez T, Pouquet A. von Kármán-Howarth relationship for helical magnetohydrodynamic flows. Phys Rev E Stat Nonlin Soft Matter Phys 2003; 68:026315. [PMID: 14525113 DOI: 10.1103/physreve.68.026315] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/01/2003] [Indexed: 05/20/2023]
Abstract
We derive an exact equation for homogeneous isotropic magnetohydrodynamic (MHD) turbulent flows with nonzero helicity; this result is of the same nature as the classical von Kármán-Howarth (VKH-HM) formulation for the kinetic energy of turbulent fluids. Helical MHD is relevant to the astrophysical flows such as in the solar corona, or the interstellar medium, and in the dynamo problem. The derivation involves the new writing of the general form of tensors for that case, for either vectors or (pseudo)axial vectors. It is shown that, for general third-order tensors, four generating functions are needed when taking into account the nonmirror invariance of helical fluids, instead of two as in the fully isotropic case. The new equation obtained, denoted by VKH-HM, links the dissipation of magnetic helicity to the third-order correlations involving combinations of the components of the velocity, the magnetic field, and the magnetic potential. Finally, in the long-time and nonresistive limit, this relationship leads to a linear scaling with separation of the third-order tensor, correlating the two normal components of the electromotive force and of the magnetic potential.
Collapse
Affiliation(s)
- H Politano
- Observatoire de la Côte d'Azur, Boîte Postale 4239, 06304 Nice, France
| | | | | |
Collapse
|
7
|
Gomez T, Politano H, Pouquet A. Exact relationship for third-order structure functions in helical flows. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 2000; 61:5321-5. [PMID: 11031579 DOI: 10.1103/physreve.61.5321] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/20/1999] [Indexed: 04/15/2023]
Abstract
An exact law for turbulent flows is written for third-order structure functions taking into account the invariance of helicity, a law akin to the so-called "4/5 law" of Kolmogorov. Here, the flow is assumed to be homogeneous, incompressible and isotropic but not invariant under reflectional symmetry. Our result is consistent with the derivation by O. Chkhetiani [JETP Lett. 10, 808, (1996)] of the von Karman-Howarth equation in the helical case, leading to a linear scaling relation for the third-order velocity correlation function. The alternative relation of the Kolmogorov type we derive here is written in terms of mixed structure functions involving combinations of differences of all components for both the velocity and vorticity fields. This relationship could prove to be a stringent test for the measuring of vorticity in the laboratory, and provide a supplementary tool for the study of the properties of helical flows.
Collapse
Affiliation(s)
- T Gomez
- Observatoire de la Cote d'Azur, CNRS UMR 6529, Nice, France
| | | | | |
Collapse
|
8
|
Politano H, Pouquet A. Model of intermittency in magnetohydrodynamic turbulence. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 1995; 52:636-641. [PMID: 9963465 DOI: 10.1103/physreve.52.636] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|
9
|
Brachet ME, Meneguzzi M, Vincent A, Politano H, Sulem PL. Numerical evidence of smooth self‐similar dynamics and possibility of subsequent collapse for three‐dimensional ideal flows. ACTA ACUST UNITED AC 1992. [DOI: 10.1063/1.858513] [Citation(s) in RCA: 100] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022]
|
10
|
Turlot E, Esteve D, Urbina C, Devoret M, Grauer R, Fernandez JC, Politano H, Reinisch G. Dynamical isoperimeter pattern in the square sine-Gordon system. Phys Rev B Condens Matter 1990; 42:8418-8425. [PMID: 9995016 DOI: 10.1103/physrevb.42.8418] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
|
11
|
Politano H, Pouquet A, Sulem PL. Inertial ranges and resistive instabilities in two‐dimensional magnetohydrodynamic turbulence. ACTA ACUST UNITED AC 1989. [DOI: 10.1063/1.859051] [Citation(s) in RCA: 95] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022]
|
12
|
Fernandez JC, Passot T, Politano H, Reinisch G, Taki M. Sturm-Liouville description of sine-Gordon soliton dynamics. Phys Rev B Condens Matter 1988; 37:7342-7347. [PMID: 9944022 DOI: 10.1103/physrevb.37.7342] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
|