Exact scaling relations in relativistic hydrodynamic turbulence (original) (raw)
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Scaling Relations of Compressible MHD Turbulence
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We study scaling relations of compressible strongly magnetized turbulence. We find a good correspondence of our results with the Fleck (1996) model of compressible hydrodynamic turbulence. In particular, we find that the density-weighted velocity, i.e. u ≡ ρ 1/3 v, proposed in Kritsuk et al. (2007) obeys the Kolmogorov scaling, i.e. E u (k) ∼ k −5/3 for the high Mach number turbulence. Similarly, we find that the exponents of the third order structure functions for u stay equal to unity for the all the Mach numbers studied. The scaling of higher order correlations obeys the She-Lévêque (1994) scalings corresponding to the two-dimensional dissipative structures, and this result does not change with the Mach number either. In contrast to v which exhibits different scaling parallel and perpendicular to the local magnetic field, the scaling of u is similar in both directions. In addition, we find that the peaks of density create a hierarchy in which both physical and column densities decrease with the scale in accordance to the Fleck (1996) predictions. This hierarchy can be related ubiquitous small ionized and neutral structures (SINS) in the interstellar gas. We believe that studies of statistics of the column density peaks can provide both consistency check for the turbulence velocity studies and insight into supersonic turbulence, when the velocity information is not available.
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Turbulent scaling phenomena are studied in an ultracold Bose gas away from thermal equilibrium. Fixed points of the dynamical evolution are characterized in terms of universal scaling exponents of correlation functions. The scaling behavior is determined analytically in the framework of quantum field theory, using a nonperturbative approximation of the two-particle irreducible effective action. While perturbative Kolmogorov scaling is recovered at higher energies, scaling solutions with anomalously large exponents arise in the infrared regime of the turbulence spectrum. The extraordinary enhancement in the momentum dependence of long-range correlations could be experimentally accessible in dilute ultracold atomic gases. Such experiments have the potential to provide insight into dynamical phenomena directly relevant also in other present-day focus areas like heavy-ion collisions and early-universe cosmology.
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Magnetohydrodynamic (MHD) turbulence in the majority of natural systems, including the interstellar medium, the solar corona, and the solar wind, has Reynolds numbers far exceeding the Reynolds numbers achievable in numerical experiments. Much attention is therefore drawn to the universal scaling properties of small-scale fluctuations, which can be reliably measured in the simulations and then extrapolated to astrophysical scales. However, in contrast with hydrodynamic turbulence, where the universal structure of the inertial and dissipation intervals is described by the Kolmogorov self-similarity, the scaling for MHD turbulence cannot be established based solely on dimensional arguments due to the presence of an intrinsic velocity scale-the Alfvén velocity. In this Letter, we demonstrate that the Kolmogorov first self-similarity hypothesis cannot be formulated for MHD turbulence in the same way it is formulated for the hydrodynamic case. Besides profound consequences for the analytical consideration, this also imposes stringent conditions on numerical studies of MHD turbulence. In contrast with the hydrodynamic case, the discretization scale in numerical simulations of MHD turbulence should decrease faster than the dissipation scale, in order for the simulations to remain resolved as the Reynolds number increases.
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A scale-invariant model of statistical mechanics is applied to present a modified statistical theory of turbulence. It is shown that a homogeneous isotropic turbulent fluid is composed of a spectrum of eddies (energy levels) each composed of a spectrum of molecular clusters with energy spectra governed by the invariant Planck distribution law and harmonious with the Kolmogorov κ − 5/3 law. The stability of clusters, de Broglie wave packets, is due to a potential that acts as Poincaré stress in the Schrödinger equation.
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We develop a new formalism for the study of turbulence using the scale relativity framework (applied in v-space according to de Montera’s proposal). We first review some of the various ingredients which are at the heart of the scale relativity approach (scale dependence and fractality, chaotic paths, irreversibility) and recall that they indeed characterize fully developped turbulent flows. Then we show that, in this framework, the time derivative of the Navier-Stokes equation can be transformed into a macroscopic Schrödinger-like equation. The local velocity PDF is given by the squared modulus of a solution of this equation. This implies the presence of null minima Pv(vi) ≈ 0 in this PDF. We also predict a new acceleration component in Lagrangian representation, Aq = ±Dv ∂v lnPv, which is therefore expected to diverge in these minima. Then we check these theoretical predictions by data analysis of available turbulence experiments: (1) Empty zones are in effect detected in observed ...
1996
The present note contains the text of lectures discussing the problem of universality in fully developed turbulence. After a brief description of Kolmogorov’s 1941 scaling theory of turbulence and a comparison between the statistical approach to turbulence and field theory, we discuss a simple model of turbulent advection which is exactly soluble but whose exact solution is still difficult to analyze. The model exhibits a restricted universality. Its correlation functions contain terms with universal but anomalous scaling but with non-universal amplitudes typically diverging with the growing size of the system. Strict universality applies only after such terms have been removed leaving renormalized correlators with normal scaling. We expect that the necessity of such an infrared renormalization is a characteristic feature of universality in turbulence. 1