On the non-local geometry of turbulence

A multi-scale methodology for the study of the non-local geometry of eddy structures in turbulence is developed. Starting from a given three-dimensional field, this consists of three main steps: extraction, characterization and classification of structures. The extraction step is done in two stages. First, a multi-scale decomposition based on the curvelet transform is applied to the full three-dimensional field, resulting in a finite set of component three-dimensional fields, one per scale. Second, by iso-contouring each component field at one or more iso-contour levels, a set of closed iso-surfaces is obtained that represents the structures at that scale. The characterization stage is based on the joint probability density function (p.d.f.), in terms of area coverage on each individual iso-surface, of two differential-geometry properties, the shape index and curvedness, plus the stretching parameter, a dimensionless global invariant of the surface. Taken together, this defines the geometrical signature of the iso-surface. The classification step is based on the construction of a finite set of parameters, obtained from algebraic functions of moments of the joint p.d.f. of each structure, that specify its location as a point in a multi-dimensional ‘feature space’. At each scale the set of points in feature space represents all structures at that scale, for the specified iso-contour value. This then allows the application, to the set, of clustering techniques that search for groups of structures with a common geometry. Results are presented of a first application of this technique to a passive scalar field obtained from 5123 direct numerical simulation of scalar mixing by forced, isotropic turbulence (Reλ = 265). These show transition, with decreasing scale, from blob-like structures in the larger scales to blob- and tube-like structures with small or moderate stretching in the inertial range of scales, and then toward tube and, predominantly, sheet-like structures with high level of stretching in the dissipation range of scales. Implications of these results for the dynamical behaviour of passive scalar stirring and mixing by turbulence are discussed

Cluster Growth in two- and three-dimensional Granular Gases

Dissipation in granular media leads to interesting phenomena as there are cluster formation and crystallization in non-equilibrium dynamical states. The freely cooling system is examined concerning the energy decay and the cluster evolution in time, both in two and three dimensions. Interesting parallels to percolation theory are obtained in three dimensions.Comment: 9 pages, 12 figure

Towards a Landau-Ginzburg-type Theory for Granular Fluids

In this paper we show how, under certain restrictions, the hydrodynamic equations for the freely evolving granular fluid fit within the framework of the time dependent Landau-Ginzburg (LG) models for critical and unstable fluids (e.g. spinodal decomposition). The granular fluid, which is usually modeled as a fluid of inelastic hard spheres (IHS), exhibits two instabilities: the spontaneous formation of vortices and of high density clusters. We suppress the clustering instability by imposing constraints on the system sizes, in order to illustrate how LG-equations can be derived for the order parameter, being the rate of deformation or shear rate tensor, which controls the formation of vortex patterns. From the shape of the energy functional we obtain the stationary patterns in the flow field. Quantitative predictions of this theory for the stationary states agree well with molecular dynamics simulations of a fluid of inelastic hard disks.Comment: 19 pages, LaTeX, 8 figure

Multichannel Approach to Clustering Matter

An approach is developed, combining the ideas of quantum statistical mechanics and multichannel theory of scattering, for treating statistical systems whose constituents can possess different bound states realized as compact clusters. The main principles for constructing multichannel cluster Hamiltonians are formulated: principle of statistical correctness, principle of cluster coexistence, and principle of potential scaling. The importance of the principle of statistical correctness is emphasized by showing that when it does not hold the behaviour of thermodynamic functions becomes essentially distorted. And moreover, unphysical instabilities can appear. The ideas are carefully illustrated by a statistical model of hot nuclear matter.Comment: 1 file, LaTex, no figure

Clustering in the Phase Space of Dark Matter Haloes. II. Stable Clustering and Dark Matter Annihilation

We present a model for the structure of the particle phase space average density ($P^2SAD$) in galactic haloes, introduced recently as a novel measure of the clustering of dark matter. Our model is based on the stable clustering hypothesis in phase space, the spherical collapse model, and tidal disruption of substructures, which is calibrated against the Aquarius simulations. Using this model, we can predict the behaviour of $P^2SAD$ in the numerically unresolved regime, down to the decoupling mass limit of generic WIMP models. This prediction can be used to estimate signals sensitive to the small scale structure of dark matter. For example, the dark matter annihilation rate can be estimated for arbitrary velocity-dependent cross sections in a convenient way using a limit of $P^2SAD$ to zero separation in physical space. We illustrate our method by computing the global and local subhalo annihilation boost to that of the smooth dark matter distribution in a Milky-Way-size halo. Two cases are considered, one where the cross section is velocity independent and one that approximates Sommerfeld-enhanced models. We find that the global boost is $\sim10-30$, which is at the low end of current estimates (weakening expectations of large extragalactic signals), while the boost at the solar radius is below the percent level. We make our code to compute $P^2SAD$ publicly available, which can be used to estimate various observables that probe the nanostructure of dark matter haloes.Comment: 12 pages, 7 figures, version published in MNRAS (minor corrections), publicly available code in IDL at http://spaces.perimeterinstitute.ca/p2sad