Inverse cascades and primordial magnetic fields

Olesen, P.

doi:10.1016/s0370-2693(97)00235-9

Cited by 111 publications

(151 citation statements)

References 14 publications

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“…This has been done by one of the authors in the case where viscosity was ignored [5]. Here we shall generalize these results.…”

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confidence: 54%

Scaling and the prediction of energy spectra in decaying hydrodynamic turbulence

Ditlevsen¹,

Jensen²,

Olesen³

2004

Physica A: Statistical Mechanics and its Applications

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Few rigorous results are derived for fully developed turbulence. By applying the scaling properties of the Navier-Stokes equation we have derived a relation for the energy spectrum valid for unforced or decaying isotropic turbulence. We find the existence of a scaling function ψ. The energy spectrum can at any time by a suitable rescaling be mapped onto this function. This indicates that the initial (primordial) energy spectrum is in principle retained in the energy spectrum observed at any later time, and the principle of permanence of large eddies is derived. The result can be seen as a restoration of the determinism of the Navier-Stokes equation in the mean. We compare our results with a windtunnel experiment and find good agreement.The Navier-Stokes equation (NSE) for hydrodynamic flow has been known for many years, and several interesting numerical results have been found. However, very few analytic statements have been derived from these equations [1,2]. In this Letter we consider decaying isotropic turbulence, i.e. hydrodynamics without external forcing.One well known feature of the hydrodynamic equations is their invariance under re-scaling when the coordinates are scaled by an arbitrary quantity l. Any solution to the NSE can then be mapped onto another solution with a corresponding change in the velocity, the time and the diffusion coefficient ν. The same scaling argument applies to the energy density. In the following we shall show that this leads to a general scaling behavior of the energy density considered as a function of k, t, ν.The result resembles the k 4 backscattering at integral scales obtained in EDQNM closure calculations [3]. In studies of the decay of homogeneous and isotropic turbulence self-similarity in the shape of the energy spectra are often assumed, expressed as the principle of permanence of large eddies (PLE) [2]. In a recent interesting phenomenological study it was proposed that for initial spectra not as steep as k 4 there will be three ranges of self-similarity where PLE only applies to the very large scales [4].We shall now derive the scaling properties of the energy density directly from the scaling properties of the NSE. This has been done by one of the authors in the case where viscosity was ignored [5]. Here we shall generalize these results. Consider the energy density in D dimensions, given bywhere L and K are infrared and ultraviolet cutoffs, respectively, and Ω D is the solid angle. We assume isotropy such that the energy density only depends on the modulus k = |k| of the wave vector. The total energy density is given byUsing the invariance of the unforced NSE under the scalingwhere h is an arbitrary parameter, the relationtrivially follows.In the following we assume the cutoffs such that for the relevant physical range we have 1/L ≪ k ≪ K. We will therefore suppress the explicit dependence of the energy density on the cutoffs. The viscosity provides a physical cutoff for large k while the cutoff 1/L is determined by the boundary conditions at the integral scale.Let...

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“…This has been done by one of the authors in the case where viscosity was ignored [5]. Here we shall generalize these results.…”

supporting

confidence: 54%

Scaling and the prediction of energy spectra in decaying hydrodynamic turbulence

Ditlevsen¹,

Jensen²,

Olesen³

2004

Physica A: Statistical Mechanics and its Applications

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Section: Introductionmentioning

confidence: 99%

“…[7,8] and derive, for unforced, incompressible, homogeneous and isotropic [10] three-dimensional MHD (3DMHD) turbulence, expressions for the decay of the kinetic and magnetic energies and the growth of the Taylor and integral scales, when we start with power-law initial conditions E 0 a (k) ∼ k q (q > −1), where the subscript a is v for the kinetic energy and b for the magnetic energy. Such initial conditions are of interest in the astrophysical context of the decay of power-law 'primordial' energy spectra [8]. We also derive bounds for the decay of the cross-and magnetic helicity.We then show by systematic numerical studies of a shell model for MHD turbulence [11,12] and an 80 3 pseudospectral DNS of the 3DMHD equations that, given power-law initial conditions, the kinetic and magnetic energies and the Taylor and integral scales follow the decay expressions mentioned above within a regime governed by the temporal evolution of the integral scales.…”

Section: Introductionmentioning

confidence: 99%

Decay of magnetohydrodynamic turbulence from power-law initial conditions

Kalelkar

Pandit

2004

Phys. Rev. E

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We derive relations for the decay of the kinetic and magnetic energies and the growth of the Taylor and integral scales in unforced, incompressible, homogeneous and isotropic three-dimensional magnetohydrodynamic (3DMHD) turbulence with power-law initial energy spectra. We also derive bounds for the decay of the cross-and magnetic helicities. We then present results from systematic numerical studies of such decay both within the context of an MHD shell model and direct numerical simulations (DNS) of 3DMHD. We show explicitly that our results about the power-law decay of the energies hold for times t < t * , where t * is the time at which the integral scales become comparable to the system size. For t < t * , our numerical results are consistent with those predicted by the principle of 'permanence of large eddies'.

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“…The pq diagram turns out to be a powerful diagnostic tool. Earlier work [8,14,15] has suggested that the decay behavior, and thus the positions of solutions in the pq diagram, depend on the exponent α for initial conditions of the form E ∼ k α e −k/k0 , where k 0 is a cutoff wavenumber. Motivated by earlier findings [2,11] of an inverse cascade in decaying MHD turbulence, Olesen considered the time-dependent energy spectra E(k, t) to be of the form [15] …”

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confidence: 99%

“…(2) that α = −3 + 2/q. He argues that for a given subinertial range spectral exponent α, the exponent q is given by [12,[15][16][17] q = 2/(3 + α)…”

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confidence: 99%

Classes of Hydrodynamic and Magnetohydrodynamic Turbulent Decay

2017

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We perform numerical simulations of decaying hydrodynamic and magnetohydrodynamic turbulence. We classify our time-dependent solutions by their evolutionary tracks in parametric plots between instantaneous scaling exponents. We find distinct classes of solutions evolving along specific trajectories toward points on a line of self-similar solutions. These trajectories are determined by the underlying physics governing individual cases, while the infrared slope of the initial conditions plays only a limited role. In the helical case, even for a scale-invariant initial spectrum (inversely proportional to wavenumber k), the solution evolves along the same trajectory as for a Batchelor spectrum (proportional to k 4 ). [7], galaxy clusters [8], and the early Universe [9,10]. In the latter case, cosmological magnetic fields generated in the early Universe provide the initial source of turbulence, which leads to a growth of the correlation length by an inverse cascade mechanism [11], in addition to the general cosmological expansion of the Universe. In the last two decades, this topic has gained significant attention [12]. The time span since the initial magnetic field generation is enormous, but it is still uncertain whether it is long enough to produce fields at sufficiently large length scales to explain the possibility of contemporary magnetic fields in the space between clusters of galaxies [13].In this Letter, we use direct numerical simulations (DNS) of both hydrodynamic (HD) and magnetohydrodynamic (MHD) decaying turbulence to classify different types by their decay behavior. The decay is characterized by the temporal change of the kinetic energy spec- * Electronic address: brandenb@nordita.org † Electronic address: tinatin@andrew.cmu.edu trum, E K (k, t), and, in MHD, also by the magnetic energy spectrum, E M (k, t). Here, k is the wavenumber and t is time. In addition to the decay laws of the energies E i (t) = E i (k, t) dk, with i = K or M for kinetic and magnetic energies, there are the kinetic and magnetic integral scales,We quantify the decay by the instantaneous scaling exponents p(t) ≡ d ln E/d ln t and q(t) ≡ d ln ξ/d ln t. Thus, we study the decay behaviors by plotting p(t) vs. q(t) in a parametric representation. The pq diagram turns out to be a powerful diagnostic tool. Earlier work [8,14,15] has suggested that the decay behavior, and thus the positions of solutions in the pq diagram, depend on the exponent α for initial conditions of the form E ∼ k α e −k/k0 , where k 0 is a cutoff wavenumber. Motivated by earlier findings [2,11] of an inverse cascade in decaying MHD turbulence, Olesen considered the time-dependent energy spectra E(k, t) to be of the form [15]where ξ(t) ∝ t q , with q being an as yet undetermined scaling exponent, and ψ is a function that depends on the dissipative and turbulent processes that lead to a departure from a powerlaw at large k. Moreover, the slope ψ ′ ≡ dψ/dκ with κ = kξ must vanish for κ → 0. This turns out to be a critical restriction. Olesen then makes use of the f...

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Inverse cascades and primordial magnetic fields

Cited by 111 publications

References 14 publications

Scaling and the prediction of energy spectra in decaying hydrodynamic turbulence

Scaling and the prediction of energy spectra in decaying hydrodynamic turbulence

Decay of magnetohydrodynamic turbulence from power-law initial conditions

Classes of Hydrodynamic and Magnetohydrodynamic Turbulent Decay

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