Shell models of magnetohydrodynamic turbulence

Plunian, Franck; Stepanov, Rodion; Frick, Peter

doi:10.1016/j.physrep.2012.09.001

Cited by 122 publications

(88 citation statements)

References 193 publications

(316 reference statements)

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“…The energy flux from inner usphere to outer u-sphere (Π implies that the growth of magnetic energy at larger scales (leftward shift of the E b peak) is not due to any inverse cascade of magnetic energy, as conjectured by some of the earlier work [18,25]. The leftward shift of the peak of the E b (k) however is due to the shift of dominant flux Π u< b> to smaller wavenumbers, as exhibited in Figure 5.…”

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confidence: 62%

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Energy transfers and magnetic energy growth in small-scale dynamo

Kumar¹,

Verma²,

Samtaney³

2013

EPL

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-In this letter we investigate the dynamics of magnetic energy growth in small-scale dynamo by studying energy transfers, mainly energy fluxes and shell-to-shell energy transfers. We perform dynamo simulations for magnetic Prandtl number Pm = 20 on 1024 3 grid using pseudospectral method. We demonstrate that the magnetic energy growth is caused by nonlocal energy transfers from the large-scale or forcing-scale velocity field to small-scale magnetic field. The peak of these energy transfers move towards lower wavenumbers as dynamo evolves, which is the reason why the integral scale of the magnetic field increases with time. The energy transfers U 2U (velocity to velocity) and B2B (magnetic to magnetic) are forward and local.Spontaneous generation of magnetic fields in magnetohydrodynamics (MHD), particularly in stars, planets, and galaxies, is known as "dynamo effect" [1]. A small seed magnetic field is amplified by self-induced currents. It has been argued that the swirling and twisting of the magnetic field lines lead to this growth [2,3]. In this letter we attempt to quantify the energy transfers during the dynamo process.The magnetic energy growth is called "small-scale dynamo" (SSD) or "large-scale dynamo" (LSD) depending on the scale at which magnetic field grows maximally. Following Cattaneo et al. [4], in LSD (SSD), the magnetic field is generated at characteristic lengths larger (smaller) than those of the velocity field. The nature of dynamo generally depends on the magnetic Prandtl number Pm, which is the ratio of the kinematic viscosity (ν) and the magnetic diffusivity (η) of the fluid. Typically, SSD is observed for larger Pm [5], while LSD is for smaller Pm [13]. Yet, there are many exceptions. For example, both SSD and LSD coexist in solar dynamo for which Pm ≪ 1 [4].Scale-dependent energy transfers in MHD have been computed using numerical and analytical tools. Primarily this is done by computing various energy fluxes and (a) E-mail: rohitkr@iitk.ac.in (b) E-mail: mkv@iitk.ac.in (c) E-mail: ravi.samtaney@kaust.edu.sa shell-to-shell energy transfers of MHD. Several simulations [6-8] employed logarithmic-binned shells, while Alexakis et al.[9] used linearly-binned shells. For unit Prandtl number, these diagnostics demonstrate that under steady state, the large-scale magnetic field is fed by the large-scale velocity field. The magnetic energy thus enhanced at large scales cascades forward to small scales. The magnetic energy at large scales is maintained by this mechanism.Moll et al.[10] employed Alexakis et al.'s method [9] to compute energy transfers for small-scale dynamo with high Pm. They showed that during dynamo action, kinetic energy at large scales is transferred to the magnetic energy at smaller scales. In our letter we compute energy transfers using Dar et al.'s method [6] for a small-scale dynamo on a high-resolution grid. We observe that the magnetic energy growth is due to nonlocal energy transfers from large-scale velocity fields to small-scale magnetic fields. The peak of the...

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confidence: 62%

“…There are a significant number of laboratory experiments [11][12][13], numerical simulations [14][15][16][17], and shell model computations [18,19] that address the dynamo process. It is important to contrast the energy transfer mechanisms for SSD and LSD.…”

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confidence: 99%

Energy transfers and magnetic energy growth in small-scale dynamo

Kumar¹,

Verma²,

Samtaney³

2013

EPL

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“…Starting from the old papers by Siggia (1977) different shell models have been introduced in literature for 3D fluid turbulence (Biferale, 2003). MHD shell models have been introduced to describe the MHD turbulent cascade (Plunian et al, 2012), starting from the paper by Gloaguen et al (1985).…”

Section: Shell Models For Turbulence Cascadementioning

confidence: 99%

The Solar Wind as a Turbulence Laboratory

Bruno

Carbone

2013

Living Rev. Solar Phys.

972

957

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In this review we will focus on a topic of fundamental importance for both astrophysics and plasma physics, namely the occurrence of large-amplitude low-frequency fluctuations of the fields that describe the plasma state. This subject will be treated within the context of the expanding solar wind and the most meaningful advances in this research field will be reported emphasizing the results obtained in the past decade or so. As a matter of fact, Helios inner heliosphere and Ulysses' high latitude observations, recent multi-spacecrafts measurements in the solar wind (Cluster four satellites) and new numerical approaches to the problem, based on the dynamics of complex systems, brought new important insights which helped to better understand how turbulent fluctuations behave in the solar wind. In particular, numerical simulations within the realm of magnetohydrodynamic (MHD) turbulence theory unraveled what kind of physical mechanisms are at the basis of turbulence generation and energy transfer across the spectral domain of the fluctuations. In other words, the advances reached in these past years in the investigation of solar wind turbulence now offer a rather complete picture of the phenomenological aspect of the problem to be tentatively presented in a rather organic way.

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“…It was firstly introduced by Parisi [176] in a hydrodynamical context and later extended to the case of MHD turbulence by Brandenburg, Enqvist, and Olesen [144]. Although many properties of turbulence, such as energy transfer, general spectral properties, etc., have been studied successfully using shell models, the physical validity of such toy models remains an open question (for a review on shell models of magnetohydrodynamic turbulence, see [177]). …”

Section: Freely Decaying Turbulent Magnetic Fieldsmentioning

confidence: 99%

Evolution of primordial magnetic fields in mean-field approximation

Campanelli

2014

Eur. Phys. J. C

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We study the evolution of phase-transitiongenerated cosmic magnetic fields coupled to the primeval cosmic plasma in the turbulent and viscous free-streaming regimes. The evolution laws for the magnetic energy density and the correlation length, both in the helical and the non-helical cases, are found by solving the autoinduction and Navier-Stokes equations in the mean-field approximation. Analytical results are derived in Minkowski spacetime and then extended to the case of a Friedmann universe with zero spatial curvature, both in the radiation-and the matterdominated era. The three possible viscous free-streaming phases are characterized by a drag term in the Navier-Stokes equation which depends on the free-streaming properties of neutrinos, photons, or hydrogen atoms, respectively. In the case of non-helical magnetic fields, the magnetic intensity B and the magnetic correlation length ξ B evolve asymptotically with the temperature, T , asHere, T i , N i , and v i are, respectively, the temperature, the number of magnetic domains per horizon length, and the bulk velocity at the onset of the particular regime. The coefficients κ B , κ ξ , 1 , 2 , 3 , and 4 , depend on the index of the assumed initial power-law magnetic spectrum, p, and on the particular regime, with the order-one constants κ B and κ ξ depending also on the cutoff adopted for the initial magnetic spectrum. In the helical case, the quasi-conservation of the magnetic helicity implies, apart from logarithmic corrections and a factor proportional to the initial fractional helicity, power-like evolution laws equal to those in the non-helical case, but with p equal to zero.

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Shell models of magnetohydrodynamic turbulence

Cited by 122 publications

References 193 publications

Energy transfers and magnetic energy growth in small-scale dynamo

Energy transfers and magnetic energy growth in small-scale dynamo

The Solar Wind as a Turbulence Laboratory

Evolution of primordial magnetic fields in mean-field approximation

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