Small-Scale Structure of a Scalar Field Convected by Turbulence

Kraichnan, Robert H.

doi:10.1063/1.1692063

Cited by 829 publications

(888 citation statements)

References 13 publications

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“…An interesting laboratory in which to study this question is the Kazantsev-Kraichnan dynamo model [72,73], for the case of non-smooth advecting velocity field [74,75,76]. It is expected that advected curves and surfaces in this model will become fractal, just as in real turbulence [28].…”

Section: Turbulent Cascade Of Magnetic Fluxmentioning

confidence: 99%

The breakdown of Alfvén’s theorem in ideal plasma flows: Necessary conditions and physical conjectures

Eyink

Aluie

2006

Physica D: Nonlinear Phenomena

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This paper presents both rigorous results and physical theory on the breakdown of magnetic flux conservation for ideal plasmas, by nonlinear effects. Our analysis is based upon an effective equation for magnetohydrodynamic (MHD) modes at length-scales > ℓ, with smaller scales eliminated, as in renormalization-group methodology. We prove that flux-conservation can be violated for an arbitrarily small length-scale ℓ, and in the absence of any non-ideality, but only if singular current sheets and vortex sheets both exist and intersect in sets of large enough dimension. This result gives analytical support to and rigorous constraints on theories of fast turbulent reconnection. Mathematically, our theorem is analogous to

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Section: Turbulent Cascade Of Magnetic Fluxmentioning

confidence: 99%

The breakdown of Alfvén’s theorem in ideal plasma flows: Necessary conditions and physical conjectures

Eyink

Aluie

2006

Physica D: Nonlinear Phenomena

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“…These questions have been sharpened by recent work on the Kraichnan model of advection by a Gaussian random velocity field that is delta-correlated in time [22]. A novel phenomenon has been discovered there called spontaneous stochasticity: Lagrangian particle trajectories for a non-Lipschitz advecting velocity are non-unique and split to form a random process in pathspace for a fixed velocity realization [23,24,25,26,27,28,29].…”

mentioning

confidence: 99%

Turbulent diffusion of lines and circulations

Eyink

2007

Physics Letters A

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We study material lines and passive vectors in a model of turbulent flow at infiniteReynolds number, the Kraichnan-Kazantsev ensemble of velocities that are white-noise in time and rough (Hölder continuous) in space. It is argued that the phenomenon of "spontaneous stochasticity" generalizes to material lines and that conservation of circulations generalizes to a "martingale property" of the stochastic process of lines.

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“…Maybe the most known model of this type is a simple model of a passive scalar quantity advected by a random Gaussian velocity field, white in time and self-similar in space, the so-called Kraichnan's rapid-change model [4]. It was shown by both natural and numerical experimental investigations that the deviations from the predictions of the classical KO phenomenological theory is even more strongly displayed for a passively advected scalar field than for the velocity field itself (see, e.g., [5] and references cited therein).…”

Section: Introductionmentioning

confidence: 99%

Compressible advection of a passive scalar: two-loop scaling regimes

Hnatič¹,

Jurčišinová²,

Jurčišin³

et al. 2006

J. Phys. A: Math. Gen.

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Abstract. The influence of compressibility on the stability of the scaling regimes of the passive scalar advected by a Gaussian velocity field with finite correlation time is investigated by the field theoretic renormalization group within two-loop approximation. The influence of compressibility on the scaling regimes is discussed as a function of the exponents ε and η, where ε characterizes the energy spectrum of the velocity field in the inertial range E ∝ k 1−2ε , and η is related to the correlation time at the wave number k which is scaled as k −2+η . The restrictions given by nonzero compressibility on the regions with stable infrared fixed points which correspond to the stable infrared scaling regimes are discussed in detail. A special attention is paid to the case of so-called frozen velocity field, when the velocity correlator is time independent. In this case, explicit inequalities which must be fulfilled in the plane ε−η are determined within two-loop approximation. The existence of a "critical" value α c of the parameter of compressibility α at which one of the two-loop conditions is canceled as a result of the competition between compressible and incompressible terms is discussed. Brief general analysis of the stability of the scaling regime of the model with finite correlations in time of the velocity field within two-loop approximation is also given.

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Small-Scale Structure of a Scalar Field Convected by Turbulence

Cited by 829 publications

References 13 publications

The breakdown of Alfvén’s theorem in ideal plasma flows: Necessary conditions and physical conjectures

The breakdown of Alfvén’s theorem in ideal plasma flows: Necessary conditions and physical conjectures

Turbulent diffusion of lines and circulations

Compressible advection of a passive scalar: two-loop scaling regimes

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