Anisotropic Homogeneous Turbulence: hierarchy and intermittency of scaling exponents in the anisotropic sectors

We present the first measurements of anisotropic statistical fluctuations in perfectly homogeneous turbulent flows. We address both problems of intermittency in anisotropic sectors and hierarchical ordering of anisotropies on a direct numerical simulation of a three dimensional random Kolmogorov flow. We achieved an homogeneous and anisotropic statistical ensemble by randomly shifting the forcing phases. We observe high intermittency as a function of the order of the velocity correlation within each fixed anisotropic sector and a hierarchical organization of scaling exponents at fixed order of the velocity correlation at changing the anisotropic sector.Comment: 6 pages, 3 eps figure

The Scaling Structure of the Velocity Statistics in Atmospheric Boundary Layer

The statistical objects characterizing turbulence in real turbulent flows differ from those of the ideal homogeneous isotropic model.They containcontributions from various 2d and 3d aspects, and from the superposition ofinhomogeneous and anisotropic contributions. We employ the recently introduceddecomposition of statistical tensor objects into irreducible representations of theSO(3) symmetry group (characterized by $j$ and $m$ indices), to disentangle someof these contributions, separating the universal and the asymptotic from the specific aspects of the flow. The different $j$ contributions transform differently under rotations and so form a complete basis in which to represent the tensor objects under study. The experimental data arerecorded with hot-wire probes placed at various heights in the atmospheric surfacelayer. Time series data from single probes and from pairs of probes are analyzed to compute the amplitudes and exponents of different contributions to the second order statistical objects characterized by $j=0$, $j=1$ and $j=2$. The analysis shows the need to make a careful distinction between long-lived quasi 2d turbulent motions (close to the ground) and relatively short-lived 3d motions. We demonstrate that the leading scaling exponents in the three leading sectors ($j = 0, 1, 2$) appear to be different butuniversal, independent of the positions of the probe, and the large scaleproperties. The measured values of the exponent are $\zeta^{(j=0)}_2=0.68 \pm 0.01$, $\zeta^{(j=1)}_2=1.0\pm 0.15$ and $\zeta^{(j=2)}_2=1.38 \pm 0.10$. We present theoretical arguments for the values of these exponents usingthe Clebsch representation of the Euler equations; neglecting anomalous corrections, the values obtained are 2/3, 1 and 4/3 respectively.Comment: PRE, submitted. RevTex, 38 pages, 8 figures included . Online (HTML) version of this paper is avaliable at http://lvov.weizmann.ac.il

A numerical comparison of theories of violent relaxation

Using N-body simulations with a large set of massless test particles we compare the predictions of two theories of violent relaxation, the well known Lynden-Bell theory and the more recent theory by Nakamura. We derive ``weaken'' versions of both theories in which we use the whole equilibrium coarse-grained distribution function as a constraint instead of the total energy constraint. We use these weaken theories to construct expressions for the conditional probability $K_i(\tau)$ that a test particle initially at the phase-space coordinate $\tau$ would end-up in the $i$'th macro-cell at equilibrium. We show that the logarithm of the ratio $R_{ij}(\tau) \equiv K_i(\tau)/K_j(\tau)$ is directly proportional to the initial phase-space density $f_0(\tau)$ for the Lynden-Bell theory and inversely proportional to $f_0(\tau)$ for the Nakamura theory. We then measure $R_{ij}(\tau)$ using a set of N-body simulations of a system undergoing a gravitational collapse to check the validity of the two theories of violent relaxation. We find that both theories are at odds with the numerical results, qualitatively and quantitatively.Comment: Replaced with a revised version, which is now accepted to MNRAS. LaTeX, 12 pages, 6 figure

Universality and saturation of intermittency in passive scalar turbulence

The statistical properties of a scalar field advected by the non-intermittent Navier-Stokes flow arising from a two-dimensional inverse energy cascade are investigated. The universality properties of the scalar field are directly probed by comparing the results obtained with two different types of injection mechanisms. Scaling properties are shown to be universal, even though anisotropies injected at large scales persist down to the smallest scales and local isotropy is not fully restored. Scalar statistics is strongly intermittent and scaling exponents saturate to a constant for sufficiently high orders. This is observed also for the advection by a velocity field rapidly changing in time, pointing to the genericity of the phenomenon. The persistence of anisotropies and the saturation are both statistical signatures of the ramp-and-cliff structures observed in the scalar field.Comment: 4 pages, 8 figure

Statistical conservation laws in turbulent transport

We address the statistical theory of fields that are transported by a turbulent velocity field, both in forced and in unforced (decaying) experiments. We propose that with very few provisos on the transporting velocity field, correlation functions of the transported field in the forced case are dominated by statistically preserved structures. In decaying experiments (without forcing the transported fields) we identify infinitely many statistical constants of the motion, which are obtained by projecting the decaying correlation functions on the statistically preserved functions. We exemplify these ideas and provide numerical evidence using a simple model of turbulent transport. This example is chosen for its lack of Lagrangian structure, to stress the generality of the ideas

Computing Topology Preservation of RBF Transformations for Landmark-Based Image Registration

In image registration, a proper transformation should be topology preserving. Especially for landmark-based image registration, if the displacement of one landmark is larger enough than those of neighbourhood landmarks, topology violation will be occurred. This paper aim to analyse the topology preservation of some Radial Basis Functions (RBFs) which are used to model deformations in image registration. Mat\'{e}rn functions are quite common in the statistic literature (see, e.g. \cite{Matern86,Stein99}). In this paper, we use them to solve the landmark-based image registration problem. We present the topology preservation properties of RBFs in one landmark and four landmarks model respectively. Numerical results of three kinds of Mat\'{e}rn transformations are compared with results of Gaussian, Wendland's, and Wu's functions

Inhomogeneous Anisotropic Passive Scalars

We investigate the behaviour of the two-point correlation function in the context of passive scalars for non homogeneous, non isotropic forcing ensembles. Exact analytical computations can be carried out in the framework of the Kraichnan model for each anisotropic sector. It is shown how the homogeneous solution is recovered at separations smaller than an intrinsic typical lengthscale induced by inhomogeneities, and how the different Fourier modes in the centre-of-mass variable recombine themselves to give a ``beating'' (superposition of power laws) described by Bessel functions. The pure power-law behaviour is restored even if the inhomogeneous excitation takes place at very small scales.Comment: 14 pages, 5 figure

Dark matter density profiles from the Jeans equation

We make a simple analytical study of radial profiles of dark matter structures, with special attention to the question of the central radial density profile. We let our theoretical assumptions be guided by results from numerical simulations, and show that at any radius where both the radial density profile, rho, and the phase-space-like density profile, rho/sigma^epsilon, are exact power laws, the only allowed density slopes in agreement with the spherical symmetric and isotropic Jeans equation are in the range 1< beta <3, where beta = - dln(rho)/dln(r). We also allow for a radial variation of these power laws, as well as anisotropy, and show how this allows for more shallow central slopes.Comment: 4 pages, no figures, minor typos correcte

Statistical geometry in scalar turbulence

A general link between geometry and intermittency in passive scalar turbulence is established. Intermittency is qualitatively traced back to events where tracer particles stay for anomalousy long times in degenerate geometries characterized by strong clustering. The quantitative counterpart is the existence of special functions of particle configurations which are statistically invariant under the flow. These are the statistical integrals of motion controlling the scalar statistics at small scales and responsible for the breaking of scale invariance associated to intermittency.Comment: 4 pages, 5 figure