On the von Karman-Howarth equations for Hall MHD flows

The von Karman-Howarth equations are derived for three-dimensional (3D) Hall magnetohydrodynamics (MHD) in the case of an homogeneous and isotropic turbulence. From these equations, we derive exact scaling laws for the third-order correlation tensors. We show how these relations are compatible with previous heuristic and numerical results. These multi-scale laws provide a relevant tool to investigate the non-linear nature of the high frequency magnetic field fluctuations in the solar wind or, more generally, in any plasma where the Hall effect is important.Comment: 11 page

Passive Scalar Structures in Supersonic Turbulence

We conduct a systematic numerical study of passive scalar structures in supersonic turbulent flows. We find that the degree of intermittency in the scalar structures increases only slightly as the flow changes from transonic to highly supersonic, while the velocity structures become significantly more intermittent. This difference is due to the absence of shock-like discontinuities in the scalar field. The structure functions of the scalar field are well described by the intermittency model of She and L\'{e}v\^{e}que [Phys. Rev. Lett. 72, 336 (1994)], and the most intense scalar structures are found to be sheet-like at all Mach numbers.Comment: 4 pages, 3 figures, to appear in PR

Statistics of mixing in three-dimensional Rayleigh--Taylor turbulence at low Atwood number and Prandtl number one

Three-dimensional miscible Rayleigh--Taylor (RT) turbulence at small Atwood number and at Prandtl number one is investigated by means of high resolution direct numerical simulations of the Boussinesq equations. RT turbulence is a paradigmatic time-dependent turbulent system in which the integral scale grows in time following the evolution of the mixing region. In order to fully characterize the statistical properties of the flow, both temporal and spatial behavior of relevant statistical indicators have been analyzed. Scaling of both global quantities ({\it e.g.}, Rayleigh, Nusselt and Reynolds numbers) and scale dependent observables built in terms of velocity and temperature fluctuations are considered. We extend the mean-field analysis for velocity and temperature fluctuations to take into account intermittency, both in time and space domains. We show that the resulting scaling exponents are compatible with those of classical Navier--Stokes turbulence advecting a passive scalar at comparable Reynolds number. Our results support the scenario of universality of turbulence with respect to both the injection mechanism and the geometry of the flow

One-Parameter Homothetic Motion in the Hyperbolic Plane and Euler-Savary Formula

In \cite{Mul} one-parameter planar motion was first introduced and the relations between absolute, relative, sliding velocities (and accelerations) in the Euclidean plane $\mathbb{E}^2$ were obtained. Moreover, the relations between the Complex velocities one-parameter motion in the Complex plane were provided by \cite{Mul}. One-parameter planar homothetic motion was defined in the Complex plane, \cite{Kur}. In this paper, analogous to homothetic motion in the Complex plane given by \cite{Kur}, one-parameter planar homothetic motion is defined in the Hyperbolic plane. Some characteristic properties about the velocity vectors, the acceleration vectors and the pole curves are given. Moreover, in the case of homothetic scale $h$ identically equal to 1, the results given in \cite{Yuc} are obtained as a special case. In addition, three hyperbolic planes, of which two are moving and the other one is fixed, are taken into consideration and a canonical relative system for one-parameter planar hyperbolic homothetic motion is defined. Euler-Savary formula, which gives the relationship between the curvatures of trajectory curves, is obtained with the help of this relative system

Spartan Random Processes in Time Series Modeling

A Spartan random process (SRP) is used to estimate the correlation structure of time series and to predict (extrapolate) the data values. SRP's are motivated from statistical physics, and they can be viewed as Ginzburg-Landau models. The temporal correlations of the SRP are modeled in terms of `interactions' between the field values. Model parameter inference employs the computationally fast modified method of moments, which is based on matching sample energy moments with the respective stochastic constraints. The parameters thus inferred are then compared with those obtained by means of the maximum likelihood method. The performance of the Spartan predictor (SP) is investigated using real time series of the quarterly S&P 500 index. SP prediction errors are compared with those of the Kolmogorov-Wiener predictor. Two predictors, one of which explicit, are derived and used for extrapolation. The performance of the predictors is similarly evaluated.Comment: 10 pages, 3 figures, Proceedings of APFA

KPZ in one dimensional random geometry of multiplicative cascades

We prove a formula relating the Hausdorff dimension of a subset of the unit interval and the Hausdorff dimension of the same set with respect to a random path matric on the interval, which is generated using a multiplicative cascade. When the random variables generating the cascade are exponentials of Gaussians, the well known KPZ formula of Knizhnik, Polyakov and Zamolodchikov from quantum gravity appears. This note was inspired by the recent work of Duplantier and Sheffield proving a somewhat different version of the KPZ formula for Liouville gravity. In contrast with the Liouville gravity setting, the one dimensional multiplicative cascade framework facilitates the determination of the Hausdorff dimension, rather than some expected box count dimension.Comment: 14 page

The Microstructure of Turbulent Flow

In 1941 a general theory of locally isotropic turbulence was proposed by Kolmogoroff which permitted the prediction of a number of laws of turbulent flow for large Reynolds numbers. The most important of these laws, the dependence of the mean square of the difference in velocities at two points on their distance and the dependence of the coefficient of turbulence diffusion on the scale of the phenomenon, were obtained by both Kolmogoroff and Obukhoff in the same year. At the present time these laws have been experimentally confirmed by direct measurements carried out in aerodynamic wind tunnels in the laboratory, in the atmosphere, and also on the ocean. In recent years in the Laboratory of Atmospheric Turbulence of the Geophysics Institute of the Soviet Academy of Sciences, a number of investigations have been conducted in which this theory was further developed. The results of several of these investigations are presented

On the noise-induced passage through an unstable periodic orbit II: General case

Consider a dynamical system given by a planar differential equation, which exhibits an unstable periodic orbit surrounding a stable periodic orbit. It is known that under random perturbations, the distribution of locations where the system's first exit from the interior of the unstable orbit occurs, typically displays the phenomenon of cycling: The distribution of first-exit locations is translated along the unstable periodic orbit proportionally to the logarithm of the noise intensity as the noise intensity goes to zero. We show that for a large class of such systems, the cycling profile is given, up to a model-dependent change of coordinates, by a universal function given by a periodicised Gumbel distribution. Our techniques combine action-functional or large-deviation results with properties of random Poincar\'e maps described by continuous-space discrete-time Markov chains.Comment: 44 pages, 4 figure

Intermittency of Magnetohydrodynamic Turbulence: Astrophysical Perspective

Intermittency is an essential property of astrophysical fluids, which demonstrate an extended inertial range. As intermittency violates self-similarity of motions, it gets impossible to naively extrapolate the properties of fluid obtained computationally with relatively low resolution to the actual astrophysical situations. In terms of Astrophysics, intermittency affects turbulent heating, momentum transfer, interaction with cosmic rays, radio waves and many more essential processes. Because of its significance, studies of intermittency call for coordinated efforts from both theorists and observers. In terms of theoretical understanding we are still just scratching a surface of a very rich subject. We have some theoretically well justified models that are poorly supported by experiments, we also have She-Leveque model, which could be vulnerable on theoretical grounds, but, nevertheless, is well supported by experimental and laboratory data. I briefly discuss a rather mysterious property of turbulence called ``extended self-similarity'' and the possibilities that it opens for the intermittency research. Then I analyze simulations of MHD intermittency performed by different groups and show that their results do not contradict to each other. Finally, I discuss the intermittency of density, intermittency of turbulence in the viscosity-dominated regime as well as the intermittency of polarization of Alfvenic modes. The latter provides an attractive solution to account for a slower cascading rate that is observed in some of the numerical experiments. I conclude by claiming that a substantial progress in the field may be achieved by studies of the turbulence intermittency via observations.Comment: 14 pages, 7 figures, invited lecture at Trieste, published International Journal of Modern Physics D, July 2

PDF of Velocity Fluctuation in Turbulence by a Statistics based on Generalized Entropy

An analytical formula for the probability density function (PDF) of the velocity fluctuation in fully-developed turbulence is derived, non-perturbatively, by assuming that its underlying statistics is the one based on the generalized measures of entropy, the R\'{e}nyi entropy or the Tsallis-Havrda-Charvat (THC) entropy. The parameters appeared in the PDF, including the index $q$ which appears in the measures of the R\'{e}nyi entropy or of the THC entropy are determined self-consistently with the help of the observed value $\mu$ of the intermittency exponent. The derived PDF explains quite well the experimentally observed density functions.Comment: 10 pages, 2 figure