Yaglom's Law Research Articles

Uniturbulence statistics and analysis of factors influencing the energy spectrum

This paper investigates the dynamics of unidirectionally propagating surface Alfvén waves, employing magnetohydrodynamic numerical simulations and statistical methodologies. The primary goal of this work is to enhance our understanding of the nonlinear self-cascade of surface Alfvén waves, which we term as uniturbulence, by unraveling the complex relationships between various length scales and their interplay with turbulent energy transfer mechanisms. To achieve this, we extensively analyze the phenomenon of uniturbulence using methods such as power spectrum analysis, radially averaged Fourier transform, and kurtosis. We employ these techniques to investigate the spatiotemporal distributions of kinetic and magnetic energy in uniturbulent flows. We also reveal the crucial role of the density contrast's variations and the role of Yaglom's law in characterizing energy transfer mechanisms. Our findings reveal that the inertial range of the perpendicular kinetic energy and magnetic energy along the z-axis depicts a progressive change in slope values, ultimately approaching the often-observed values of −5/3 and −3/2, respectively. Furthermore, our kurtosis analysis highlights the non-Gaussian behavior of the flow field at different length scales and over time, offering a perspective on uniturbulence dynamics. The correlations observed among diverse statistical approaches emphasize the complex interplay between different length scales in the context of uniturbulence. Our findings contribute to understanding this phenomenon, establishing a basis for future investigations to clarify the connections regulating these turbulent dynamics.

The Yaglom Scaling of the Third-order Structure Functions in the Inner Heliosphere Observed by Helios 1 and 2

Abstract The third-order scaling law, Yaglom law, of Elsässer fluctuations in the solar wind is believed to reflect the inertial range energy cascade of the MHD turbulence and provides an approach to evaluate the cascade rate. However, the occurrence ratio with the Yaglom scaling law, the fraction of the intervals where the Yaglom linear scaling is observed, is reported to be low (0.05–0.30) in the high-latitude solar wind observed by the Ulysses spacecraft. Whether the occurrence ratio could be higher in other conditions remains unknown. Here, we analyze the occurrence of the third-order scaling in the inner heliosphere with the first 100 days of observation of the Helios 1 and Helios 2 spacecraft. We investigate 162 intervals in the leading edges and 323 intervals in the trailing edges of the high-speed streams, respectively. All of these intervals have a time duration of 9 hr. We find that in the inner heliosphere the occurrence ratio is relatively high in the leading edges (0.58) and moderate in the trailing edges (0.45). Among the data intervals with the Yaglom scaling in the leading edges, 94.7% of intervals give positive rates, while in the trailing edges 78.6% give negative rates. The variations of the occurrence ratio with various turbulence parameters are shown. The cascade rate is found to be higher than the proton heating rate calculated from the data, which have third-order scaling. These new results raise several questions related to the nature and origin of the third-order scaling law and may initiate new studies on solar wind turbulence.

Reaction analogy based forcing for incompressible scalar turbulence

We present a novel reaction analogy (RA) based forcing method for generating stationary scalar fields in incompressible turbulence. The new method can produce more general scalar PDFs (e.g. double-delta) than current methods, while ensuring that scalar fields remain bounded. Such features are useful for generating initial fields in non-premixed combustion or for studying non-Gaussian scalar turbulence. The RA method mathematically models hypothetical chemical reactions that convert reactants in a mixed state back into its pure unmixed components. Various types of chemical reactions are formulated and the corresponding mathematical expressions derived such that the reaction term is smooth in the scalar space and is consistent with mass conservation. For large values of the scalar dissipation rate, the method produces statistically steady double-delta scalar PDFs. Quasi-uniform, Gaussian, and stretched exponential scalar statistics are recovered for smaller values of the scalar dissipation rate. The shape of the scalar PDF can be further controlled by changing the stoichiometric coefficients of the reaction. The ability of the new method to produce fully developed passive scalar fields with quasi-Gaussian PDFs is also investigated, by exploring the convergence of the third order mixed structure function to the "four-thirds" Yaglom's law.

THE YAGLOM LAW IN THE EXPANDING SOLAR WIND

We study the Yaglom law, which relates the mixed third order structure function to the average dissipation rate of turbulence, in a uniformly expanding solar wind by using the two scales expansion model of magnetohydrodynamic (MHD) turbulence. We show that due to the expansion of the solar wind two new terms appear in the Yaglom law. The first term is related to the decay of the turbulent energy by nonlinear interactions, whereas the second term is related to the non-zero cross-correlation of the Els\"asser fields. Using magnetic field and plasma data from WIND and Helios 2 spacecrafts, we show that at lower frequencies in the inertial range of MHD turbulence the new terms become comparable to Yaglom's third order mixed moment, and therefore they cannot be neglected in the evaluation of the energy cascade rate in the solar wind.

ON THE OCCURRENCE OF THE THIRD-ORDER SCALING IN HIGH LATITUDE SOLAR WIND

The occurrence and nature of a nonlinear energy cascade within the intermediate scales of solar wind Alfvenic turbulence represents an important open issue. Using in situ measurements of fast, high latitude solar wind taken by the Ulysses spacecraft at solar minima, it is possible to show that a nonlinear energy cascade of imbalanced turbulence is only observed when the solar wind owns peculiar properties. These are the reduction of the local correlation between velocity and magnetic field (weak cross-helicity); the presence of large-scale velocity shears; and the steepening and extension down to low frequencies of the turbulent spectra. Our observations suggest the important role of both large-scale velocity and Alfvenicity of the field fluctuations for the validation of the Yaglom law in solar wind turbulence.

Anisotropic Third-Moment Estimates of the Energy Cascade in Solar Wind Turbulence Using Multispacecraft Data

The first direct determination of the inertial range energy cascade rate, using an anisotropic form of Yaglom's law for magnetohydrodynamic turbulence, is obtained in the solar wind with multispacecraft measurements. The two-point mixed third-order structure functions of Elsässer fluctuations are integrated over a sphere in magnetic field-aligned coordinates, and the result is consistent with a linear scaling. Therefore, volume integrated heating and cascade rates are obtained that, unlike previous studies, make only limited assumptions about the underlying spectral geometry of solar wind turbulence. These results confirm the turbulent nature of magnetic and velocity field fluctuations in the low frequency limit, and could supply the energy necessary to account for the nonadiabatic heating of the solar wind.

Observation of Inertial Energy Cascade in Interplanetary Space Plasma

Direct evidence for the presence of an inertial energy cascade, the most characteristic signature of hydromagnetic turbulence (MHD), is observed in the solar wind by the Ulysses spacecraft. After a brief rederivation of the equivalent of Yaglom's law for MHD turbulence, a linear relation is indeed observed for the scaling of mixed third-order structure functions involving Elsässer variables. This experimental result firmly establishes the turbulent character of low-frequency velocity and magnetic field fluctuations in the solar wind plasma.

Normalisation d'un processus de branchement critique dans un environnement aléatoire

Let ( Z n ) be a critical branching process in an independent and identically distributed (i.i.d.) random environment. For each fixed environment ω, let C n = E ω [ Z n ∣ Z n >0] be the conditional expectation of Z n given Z n >0. We prove an analogue of Yaglom's law: as n→∞, the conditional law of Z n / C n , conditional on Z n >0, converges to a non-degenerate law on [0,∞). We give also an analogue of Kolmogorov's law, as well as a local limit theorem for the semi-group of probability generating functions. To cite this article: Y. Guivarc'h et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).