The spectrum of random magnetic fields in the mean field dynamo theory of the Galactic magnetic field

Kulsrud, Russell M.; Anderson, Stephen W.

doi:10.1086/171743

Cited by 427 publications

(593 citation statements)

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“…It is instructive to express its growth rate in terms of the growth rate of the magnetic energy B 2 . In three dimensions, this gives γ K = (16/5)γ 2 /2, which agrees with the result Malyshkin [35] obtained by a direct calculation of K 2 in the spirit of the Kulsrud-Anderson theory [23]. We also see that the mean-square mirror force [Eq.…”

Section: Statistics Of Lorentz Tension and Magnetic-field-line Cusupporting

confidence: 89%

Structure of small-scale magnetic fields in the kinematic dynamo theory

et al. 2001

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A weak fluctuating magnetic field embedded into a turbulent conducting medium grows exponentially while its characteristic scale decays. In the interstellar medium and protogalactic plasmas, the magnetic Prandtl number is very large, so a broad spectrum of growing magnetic fluctuations is excited at small (subviscous) scales. The condition for the onset of nonlinear back reaction depends on the structure of the field lines. We study the statistical correlations that are set up in the field pattern and show that the magnetic-field lines possess a folding structure, where most of the scale decrease is due to the field variation across itself (rapid transverse direction reversals), while the scale of the field variation along itself stays approximately constant. Specifically, we find that, though both the magnetic energy and the mean-square curvature of the field lines grow exponentially, the field strength and the field-line curvature are anticorrelated, i.e., the curved field is relatively weak, while the growing field is relatively flat. The detailed analysis of the statistics of the curvature shows that it possesses a stationary limiting distribution with the bulk located at the values of curvature comparable to the characteristic wave number of the velocity field and a power tail extending to large values of curvature where it is eventually cut off by the resistive regularization. The regions of large curvature, therefore, occupy only a small fraction of the total volume of the system. Our theoretical results are corroborated by direct numerical simulations. The implication of the folding effect is that the advent of the Lorentz back reaction occurs when the magnetic energy approaches that of the smallest turbulent eddies. Our results also directly apply to the problem of statistical geometry of the material lines in a random flow. PACS number(s): 47.27. Gs, 98.35.Eg, 47.65.+a, 05.10.Gg

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Section: Statistics Of Lorentz Tension and Magnetic-field-line Cusupporting

confidence: 89%

Structure of small-scale magnetic fields in the kinematic dynamo theory

et al. 2001

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“…This implies that a dynamo can be excited by magnetic fluctuations, themselves presumably arising from a small-scale dynamo process, or perhaps an MHD instability of some sort. Since the small-scale dynamo is usually considered harmful to mean fields [5], this is an interesting possibility-a buildup of magnetic noise on small scales may cause a coherent large-scale dynamo to develop. Interestingly, the u fluctuations that allow for this magnetic shear-current effect arise solely through the pressure response of the fluid; i.e., the ∇p term in Eq.…”

Section: Specific Results For Unstratified Shear Dynamosmentioning

confidence: 99%

Electromotive force due to magnetohydrodynamic fluctuations in sheared rotating turbulence

Squire¹,

Bhattacharjee²

2015

Phys. Rev. E

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This article presents a calculation of the mean electromotive force arising from general small-scale magnetohydrodynamical turbulence, within the framework of the second-order correlation approximation. With the goal of improving understanding of the accretion disk dynamo, effects arising through small-scale magnetic fluctuations, velocity gradients, density and turbulence stratification, and rotation, are included. The primary result, which supplements numerical findings, is that an off-diagonal turbulent resistivity due to magnetic fluctuations can produce large-scale dynamo action-the magnetic analog of the "shear-current" effect. In addition, consideration of α effects in the stratified regions of disks gives the puzzling result that there is no strong prediction for a sign of α, since the effects due to kinetic and magnetic fluctuations, as well as those due to shear and rotation, are each of opposing signs and tend to cancel each other.

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“…This process occurs exponentially fast (Kulsrud & Anderson 1992;Schekochihin et al 2002a), so the corresponding time can be estimated as t $ À1 2 log Pr 1=2 m , which amounts to several eddy turnover times. The PDF of B with account taken of diffusion has not as yet been found, although Chertkov et al (1999), under certain additional assumptions, did derive the moments hB n i in the diffusive regime.…”

Section: Intermittencymentioning

confidence: 99%

“…The weak-field (kinematic) limit has been an attractive object of analytical study since the seminal work of Kazantsev (1967). The spectral theory of the kinematic dynamo driven by a random velocity field predicts exponential growth of the magnetic energy and its accumulation at the resistive scales (see Kazantsev 1967;Kulsrud & Anderson 1992;Gruzinov, Cowley, & Sudan 1996;Schekochihin, Boldyrev, & Kulsrud 2002a, and references therein). More recently, it was realized that small-scale magnetic fields generated by this '' stretch-and-fold '' dynamo possess a distinctive spatial folding structure ( Fig.…”

Section: Introductionmentioning

confidence: 99%

The Small‐Scale Structure of Magnetohydrodynamic Turbulence with Large Magnetic Prandtl Numbers

Schekochihin

Maron

Cowley

et al. 2002

ApJ

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We study the intermittency and field-line structure of the MHD turbulence in plasmas with very large magnetic Prandtl numbers. In this regime, which is realized in the interstellar medium, some accretion disks, protogalaxies, galaxy-cluster gas, the early universe, etc., magnetic fluctuations can be excited at scales below the viscous cutoff. The salient feature of the resulting small-scale magnetic turbulence is the folded structure of the fields. It is characterized by very rapid transverse spatial oscillation of the field direction, while the field lines remain largely unbent up to the scale of the flow. Quantitatively, the fluctuation level and the field-line geometry can be studied in terms of the statistics of the field strength and of the field-line curvature. In the kinematic limit, the distribution of the field strength is an expanding lognormal, while that of the field-line curvature K is stationary and has a power tail $K À13/7 . The field strength and curvature are anticorrelated, i.e., the growing fields are mostly flat, while the sharply curved fields remain relatively weak. The field, therefore, settles into a reduced-tension state. Numerical simulations demonstrate three essential features of the nonlinear regime. First, the total magnetic energy is equal to the total kinetic energy. Second, the intermittency is partially suppressed compared to the kinematic case, as the fields become more volume-filling and their distribution develops an exponential tail. Third, the folding structure of the field is unchanged from the kinematic case: the anticorrelation between the field strength and the curvature persists, and the distribution of the latter retains the same power tail. We propose a model of back-reaction based on the folding picture that reproduces all of the above numerical results.

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The spectrum of random magnetic fields in the mean field dynamo theory of the Galactic magnetic field

Cited by 427 publications

References 0 publications

Structure of small-scale magnetic fields in the kinematic dynamo theory

Structure of small-scale magnetic fields in the kinematic dynamo theory

Electromotive force due to magnetohydrodynamic fluctuations in sheared rotating turbulence

The Small‐Scale Structure of Magnetohydrodynamic Turbulence with Large Magnetic Prandtl Numbers

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