She-Leveque Model Research Articles

Interactions of velocity structures between large and small scales in microelectrokinetic turbulence

We investigated the high intermittency of electrokinetic turbulence with experiments and the She-Leveque model. The intermittency factor 𝛽 can be larger than 1 in electrokinetic turbulence. 𝛽 > 1 indicates that the probability of velocity structures on small scales are determined by all the probabilities of velocity structures on large scales. The investigation finds a route to turbulence through a different but tighter relationship between large and small scale velocity structures.

SPECTRAL AND INTERMITTENCY PROPERTIES OF RELATIVISTIC TURBULENCE

High-resolution numerical simulations are utilized to examine isotropic turbulence in a compressible fluid when long wavelength velocity fluctuations approach light speed. Spectral analysis reveals an inertial sub-range of relativistic motions with a broadly 5/3 index. The use of generalized Lorentz-covariant structure functions based on the four-velocity is proposed. These structure functions extend the She-Leveque model for intermittency into the relativistic regime.

Hierarchical structure of the optical path length of the supersonic turbulent boundary layer

The optical path length (OPL) of supersonic turbulent boundary layer of Mach number 3.0 is obtained with the nanoparticle-based planar laser scattering technique, and its structure is analyzed within the framework of hierarchical symmetry assumption. Our result offers reasonable evidence for that the OPL obeys this assumption with parameter β depending on q. The scaling exponent ζ(q) of structure function is computed and compared with the theoretical prediction of She-Leveque model. The curve ζ(q) we obtained is convex and smaller than the theoretical value for small q, which is attributed to the large scale structure of the OPL.

Universal hierarchical symmetry for turbulence and general multi-scale fluctuation systems

Scaling is an important measure of multi-scale fluctuation systems. Turbulence as the most remarkable multi-scale system possesses scaling over a wide range of scales. She-Leveque (SL) hierarchical symmetry, since its publication in 1994, has received wide attention. A number of experimental, numerical and theoretical work have been devoted to its verification, extension, and modification. Application to the understanding of magnetohydrodynamic turbulence, motions of cosmic baryon fluids, cosmological supersonic turbulence, natural image, spiral turbulent patterns, DNA anomalous composition, human heart variability are just a few among the most successful examples. A number of modified scaling laws have been derived in the framework of the hierarchical symmetry, and the SL model parameters are found to reveal both the organizational order of the whole system and the properties of the most significant fluctuation structures. A partial set of work related to these studies are reviewed. Particular emphasis is placed on the nature of the hierarchical symmetry. It is suggested that the SL hierarchical symmetry is a new form of the self-organization principle for multi-scale fluctuation systems, and can be employed as a standard analysis tool in the general multi-scale methodology. It is further suggested that the SL hierarchical symmetry implies the existence of a turbulence ensemble. It is speculated that the search for defining the turbulence ensemble might open a new way for deriving statistical closure equations for turbulence and other multi-scale fluctuation systems.

Local turbulence simulations for the multiphase ISM

In this paper we show results of numerical simulations for the turbulence in the interstellar medium. These results were obtained using a Riemann solver-free numerical scheme for high-Mach number hyperbolic equations. Here we especially concentrate on the physical properties of the ISM. That is, we do not present turbulence simulations trimmed to be applicable to the interstellar medium. The simulations are rather based on physical estimates for the relevant parameters of the interstellar gas. Applying our code to simulate the turbulent plasma motion within a typical interstellar molecular cloud, we investigate the influence of different equations of state (isothermal and adiabatic) on the statistical properties of the resulting turbulent structures. We find slightly different density power spectra and dispersion maps, while both cases yield qualitatively similar dissipative structures, and exhibit a departure from the classical Kolmogorov case towards a scaling described by the She-Leveque model. Solving the full energy equation with realistic heating/cooling terms appropriate for the diffuse interstellar gas, we are able to reproduce a realistic two-phase distribution of cold and warm plasma. When extracting maps of polarised intensity from our simulation data, we find encouraging similarity to actual observations. Finally, we compare the actual magnetic field strength of our simulations to its value inferred from the rotation measure. We find these to be systematically different by a factor of about 1.5, thus highlighting the often underestimated influence of varying line-of-sight particle densities on the magnetic field strength derived from observed rotation measures.

Multiscaling Dynamics of Impurity Transport in Drift-Wave Turbulence

Intermittency effects and the associated multiscaling spectrum of exponents are investigated for impurities advection in tokamak edge plasmas. The two-dimensional Hasagawa-Wakatani model of resistive drift-wave turbulence is used as a paradigm to describe edge tokamak turbulence. Impurities are considered as a passive scalar advected by the plasma turbulent flow. The use of the extended self-similarity technique shows that the structure function relative scaling exponent of impurity density and vorticity follows the She-Leveque model. This confirms the intermittent character of the impurities advection in the turbulent plasma flow and suggests that impurities are advected by vorticity filaments.

An interpretation of the She-Lévêque model based on order statistics

We present an interpretation of the She-Leveque model in fully developed turbulence based on order statistics. Turbulent behavior at large values of the Reynolds number is often studied through the scaling behavior of moments of the distribution of the velocity differences and of the energy dissipation. The present interpretation leads to a derivation of the scaling exponents ζp and τp of these moments, without any postulate about a universal relation over the fluctuation structures such as the one used by She and Leveque. The interpretation is based on the fact that the hierarchy of fluctuation structures imposes statistical constraints, whereupon the order p itself is seen as a random variable. As proposed by She and Leveque, the hierarchy of the structures is such that the structures of larger order affect locally lower order structures through an entrainment process. This phenomenon leads to the Fisher-Tippett law, one of three asymptotic distributions for the extreme value of a random sample as the size of the sample grows to infinity.

Inertial range scaling in numerical turbulence with hyperviscosity.

Numerical turbulence with hyperviscosity is studied and compared with direct simulations using ordinary viscosity and data from wind tunnel experiments. It is shown that the inertial range scaling is similar in all three cases. Furthermore, the bottleneck effect is approximately equally broad (about one order of magnitude) in these cases and only its height is increased in the hyperviscous case-presumably as a consequence of the steeper decent of the spectrum in the hyperviscous subrange. The mean normalized dissipation rate is found to be in agreement with both wind tunnel experiments and direct simulations. The structure function exponents agree with the She-Leveque model. Decaying turbulence with hyperviscosity still gives the usual t(-1.25) decay law for the kinetic energy, and also the bottleneck effect is still present and about equally strong.

Scaling properties of three-dimensional magnetohydrodynamic turbulence

The scaling properties of three-dimensional magnetohydrodynamic turbulence with finite magnetic helicity are obtained from direct numerical simulations using 512(3) modes. The results indicate that the turbulence does not follow the Iroshnikov-Kraichnan phenomenology. The scaling exponents of the structure functions can be described by a modified She-Leveque model zeta(p) = p/9+1-(1/3)(p/3), corresponding to basic Kolmogorov scaling and sheetlike dissipative structures. In particular, we find zeta(2) approximately 0.7, consistent with the energy spectrum E(k) approximately k(-5/3) as observed in the solar wind, and zeta(3) approximately 1, confirming a recent analytical result.