Strong MHD helical turbulence and the nonlinear dynamo effect
TL;DR: In this paper, a nonlinear theory is developed for three-dimensional, homogeneous, isotropic, incompressible MHD turbulence with helicity, i.e. not statistically invariant under plane reflexions.
Abstract: To understand the turbulent generation of large-scale magnetic fields and to advance beyond purely kinematic approaches to the dynamo effect like that introduced by Steenbeck, Krause & Radler (1966)’ a new nonlinear theory is developed for three-dimensional, homogeneous, isotropic, incompressible MHD turbulence with helicity, i.e. not statistically invariant under plane reflexions. For this, techniques introduced for ordinary turbulence in recent years by Kraichnan (1971~~)’ Orszag (1970, 1976) and others are generalized to MHD; in particular we make use of the eddy-damped quasi-normal Markovian approximation. The resulting closed equations for the evolution of the kinetic and magnetic energy and helicity spectra are studied both theoretically and numerically in situations with high Reynolds number and unit magnetic Prandtl number. Interactions between widely separated scales are much more important than for non-magnetic turbulence. Large-scale magnetic energy brings to equipartition small-scale kinetic and magnetic excitation (energy or helicity) by the ‘AlfvBn effect ’; the small-scale ‘residual’ helicity, which is the difference between a purely kinetic and a purely magnetic helical term, induces growth of largescale magnetic energy and helicity by the ‘helicity effect’. In the absence of helicity an inertial range occurs with a cascade of energy to small scales; to lowest order it is a - power law with equipartition of kinetic and magnetic energy spectra as in Kraichnan (1965) but there are - 2 corrections (and possibly higher ones) leading to a slight excess of magnetic energy. When kinetic energy is continuously injected, an initial seed of magnetic field willgrow to approximate equipartition, at least in the small scales. If in addition kinetic helicity is injected, an inverse cascade of magnetic helicity is obtained leading to the appearance of magnetic energy and helicity in ever-increasing scales (in fact, limited by the size of the system). This inverse cascade, predicted by Frisch et aZ. (1975), results from a competition between the helicity and Alfvh effects and yields an inertial range with approximately - 1 and - 2 power laws for magnetic energy and helicity. When kinetic helicity is injected at the scale Zinj and the rate k (per unit mass), the time of build-up of magnetic energy with scale L 9 Zinl is t % L( prp;nj)-k 21 FLM 77
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TL;DR: The current understanding of astrophysical magnetic fields is reviewed in this paper, focusing on their generation and maintenance by turbulence, where analytical and numerical results are discussed both for small scale dynamos, which are completely isotropic, and for large scale dynamo, where some form of parity breaking is crucial.
1,548 citations
TL;DR: In this paper, the authors focus on a topic of fundamental importance for both plasma physics and astrophysics, namely the occurrence of large-amplitude low-frequency fluctuations of the fields that describe the plasma state.
Abstract: In this review we will focus on a topic of fundamental importance for both plasma physics and astrophysics, 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, Ulysses’ high latitude observations 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.
1,212 citations
TL;DR: In this article, a two-part review summarizes the observations, theory, and simulations of interstellar turbulence and their implications for many fields of astrophysics, including basic fluid equations, solenoidal and compressible modes, global inviscid quadratic invariants, scaling arguments for the power spectrum, phenomenological models for the scaling of higher-order structu...
Abstract: ▪ Abstract Turbulence affects the structure and motions of nearly all temperature and density regimes in the interstellar gas. This two-part review summarizes the observations, theory, and simulations of interstellar turbulence and their implications for many fields of astrophysics. The first part begins with diagnostics for turbulence that have been applied to the cool interstellar medium and highlights their main results. The energy sources for interstellar turbulence are then summarized along with numerical estimates for their power input. Supernovae and superbubbles dominate the total power, but many other sources spanning a large range of scales, from swing-amplified gravitational instabilities to cosmic ray streaming, all contribute in some way. Turbulence theory is considered in detail, including the basic fluid equations, solenoidal and compressible modes, global inviscid quadratic invariants, scaling arguments for the power spectrum, phenomenological models for the scaling of higher-order structu...
1,195 citations
Cites methods from "Strong MHD helical turbulence and t..."
...Pouquet et al. (1976) addressed the MHD energy cascade using EDQNM closure and predicted an inverse cascade for large magnetic helicities....
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...Pouquet et al. (1976) were among the first to address the locality and direction of the MHD energy cascade using a closure method....
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...…cross-energy transfer between kinetic and magnetic energies are direct, meaning to larger wavenumbers, whereas the magnetic and kinetic helical contributions give an inverse cascade for the four sets of possible magnetic and kinetic energy exchanges, confirming the result of Pouquet et al. (1976)....
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TL;DR: The theory of two-dimensional turbulence is reviewed and unified, and some hydrodynamic and plasma applications are considered in this paper, where some equations of incompressible hydrodynamics, absolute statistical equilibrium, spectral transport of energy and enstrophy, turbulence on the surface of a rotating sphere, turbulent diffusion, MHD turbulence, and two dimensional superflow are discussed.
Abstract: The theory of two-dimensional turbulence is reviewed and unified, and some hydrodynamic and plasma applications are considered. The topics covered include some equations of incompressible hydrodynamics, absolute statistical equilibrium, spectral transport of energy and enstrophy, turbulence on the surface of a rotating sphere, turbulent diffusion, MHD turbulence, and two-dimensional superflow. Finally, an attempt is made to assess the status and future of the principal research topics which have been discussed.
1,056 citations
TL;DR: In this article, it is argued that a turbulent hydromagnetic dynamo of some kind and an inverse cascade of magnetic energy gives the most plausible explanation for the regular galactic magnetic fields.
Abstract: ▪ Abstract We discuss current observational and theoretical knowledge of magnetic fields, especially the large-scale structure in the disks and halos of spiral galaxies. Among other topics, we consider the enhancement of global magnetic fields in the interarm regions, magnetic spiral arms, and representations as superpositions of azimuthal modes, emphasizing a number of unresolved questions. It is argued that a turbulent hydromagnetic dynamo of some kind and an inverse cascade of magnetic energy gives the most plausible explanation for the regular galactic magnetic fields. Primordial theory is found to be unsatisfactory, and fields of cosmological origin may not even be able to provide a seed field for a dynamo. Although dynamo theory has its own problems, the general form of the dynamo equations appears quite robust. Finally, detailed models of magnetic field generation in galaxies, allowing for factors such as spiral structure, starbursts, galactic winds, and fountains, are discussed and confronted with...
977 citations
Cites background from "Strong MHD helical turbulence and t..."
...Such motions also lead to an inverse cascade from the conservation properties of the magnetichelicity (Frisch et al 1975, Pouquet et al 1976), and to the cross-helicity e ect (Yoshizawa & Yokoi1993)....
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References
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01 Jan 1953
TL;DR: In this article, the kinematics of the field of homogeneous turbulence and the universal equilibrium theory of decay of the energy-containing eddies are discussed. But the authors focus on the dynamics of decay and not on the probability distribution of u(x).
Abstract: Preface 1 Introduction 2 Mathematics representation of the field of turbulence 3 The kinematics of homogeneous turbulence 4 Some linear problems 5 General dynamics of decay 6 The universal equilibrium theory 7 Decay of the energy-containing eddies 8 The probability distribution of u(x) Bibliography of research on homogeneous turbulence Index
3,121 citations
TL;DR: In this paper, it was shown that two-dimensional turbulence has both kinetic energy and mean square vorticity as inviscid constants of motion, and two formal inertial ranges, E(k)∼e2/3k−5/3/3, where e is the rate of cascade of kinetic energy per unit mass, η is the time taken to reach a cascade of mean square velocity, and k is the kinetic energy of the entire mass.
Abstract: Two‐dimensional turbulence has both kinetic energy and mean‐square vorticity as inviscid constants of motion. Consequently it admits two formal inertial ranges, E(k)∼e2/3k−5/3 and E(k)∼η2/3k−3, where e is the rate of cascade of kinetic energy per unit mass, η is the rate of cascade of mean‐square vorticity, and the kinetic energy per unit mass is ∫0∞E(k) dk. The −53 range is found to entail backward energy cascade, from higher to lower wavenumbers k, together with zero‐vorticity flow. The −3 range gives an upward vorticity flow and zero‐energy flow. The paradox in these results is resolved by the irreducibly triangular nature of the elementary wavenumber interactions. The formal −3 range gives a nonlocal cascade and consequently must be modified by logarithmic factors. If energy is fed in at a constant rate to a band of wavenumbers ∼ki and the Reynolds number is large, it is conjectured that a quasi‐steady‐state results with a −53 range for k « ki and a −3 range for k » ki, up to the viscous cutoff. The t...
2,950 citations
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1,572 citations
TL;DR: In this article, it was shown that certain formally deterministic fluid systems which possess many scales of motion are observationally indistinguishable from indeterministic systems; specifically, two states of the system differing initially by a small observable error will evolve into two states differing as greatly as randomly chosen states within a finite time interval, which cannot be lengthened by reducing the amplitude of the initial error.
Abstract: It is proposed that certain formally deterministic fluid systems which possess many scales of motion are observationally indistinguishable from indeterministic systems; specifically, that two states of the system differing initially by a small “observational error” will evolve into two states differing as greatly as randomly chosen states of the system within a finite time interval, which cannot be lengthened by reducing the amplitude of the initial error. The hypothesis is investigated with a simple mathematical model. An equation whose dependent variables are ensemble averages of the “error energy” in separate scales of motion is derived from the vorticity equation which governs two-dimensional incompressible flow. Solutions of the equation are determined by numerical integration, for cases where the horizontal extent and total energy of the system are comparable to those of the earth's atomsphere. It is found that each scale of motion possesses an intrinsic finite range of predictability, provided that the total energy of the system does not fall off too rapidly with decreasing wave length. With the chosen values of the constants, “cumulus-scale” motions can be predicted about one hour, “synoptic-scale” motions a few days, and the largest scales a few weeks in advance. The applicability of the model to real physical systems, including the earth's atmosphere, is considered. DOI: 10.1111/j.2153-3490.1969.tb00444.x
914 citations