Extreme-value statistics from Lagrangian convex hull analysis for homogeneous turbulent Boussinesq convection and MHD convection

We investigate the utility of the convex hull of many Lagrangian tracers to analyze transport properties of turbulent flows with different anisotropy. In direct numerical simulations of statistically homogeneous and stationary Navier-Stokes turbulence, neutral fluid Boussinesq convection, and MHD Boussinesq convection a comparison with Lagrangian pair dispersion shows that convex hull statistics capture the asymptotic dispersive behavior of a large group of passive tracer particles. Moreover, convex hull analysis provides additional information on the sub-ensemble of tracers that on average disperse most efficiently in the form of extreme value statistics and flow anisotropy via the geometric properties of the convex hulls. We use the convex hull surface geometry to examine the anisotropy that occurs in turbulent convection. Applying extreme value theory, we show that the maximal square extensions of convex hull vertices are well described by a classic extreme value distribution, the Gumbel distribution. During turbulent convection, intermittent convective plumes grow and accelerate the dispersion of Lagrangian tracers. Convex hull analysis yields information that supplements standard Lagrangian analysis of coherent turbulent structures and their influence on the global statistics of the flow.Comment: 18 pages, 10 figures, preprin

Shape Theorems for Poisson Hail on a Bivariate Ground

We consider the extension of the Euclidean stochastic geometry Poisson Hail model to the case where the service speed is zero in some subset of the Euclidean space and infinity in the complement. We use and develop tools pertaining to sub-additive ergodic theory in order to establish shape theorems for the growth of the ice-heap under light tail assumptions on the hailstone characteristics. The asymptotic shape depends on the statistics of the hailstones, the intensity of the underlying Poisson point process and on the geometrical properties of the zero speed set.Comment: Final version accepted in Advances in Applied Probabilit

Burgers Turbulence

The last decades witnessed a renewal of interest in the Burgers equation. Much activities focused on extensions of the original one-dimensional pressureless model introduced in the thirties by the Dutch scientist J.M. Burgers, and more precisely on the problem of Burgers turbulence, that is the study of the solutions to the one- or multi-dimensional Burgers equation with random initial conditions or random forcing. Such work was frequently motivated by new emerging applications of Burgers model to statistical physics, cosmology, and fluid dynamics. Also Burgers turbulence appeared as one of the simplest instances of a nonlinear system out of equilibrium. The study of random Lagrangian systems, of stochastic partial differential equations and their invariant measures, the theory of dynamical systems, the applications of field theory to the understanding of dissipative anomalies and of multiscaling in hydrodynamic turbulence have benefited significantly from progress in Burgers turbulence. The aim of this review is to give a unified view of selected work stemming from these rather diverse disciplines.Comment: Review Article, 49 pages, 43 figure

Abrasion of flat rotating shapes

We report on the erosion of flat linoleum "pebbles" under steady rotation in a slurry of abrasive grit. To quantify shape as a function of time, we develop a general method in which the pebble is photographed from multiple angles with respect to the grid of pixels in a digital camera. This reduces digitization noise, and allows the local curvature of the contour to be computed with a controllable degree of uncertainty. Several shape descriptors are then employed to follow the evolution of different initial shapes toward a circle, where abrasion halts. The results are in good quantitative agreement with a simple model, where we propose that points along the contour move radially inward in proportion to the product of the radius and the derivative of radius with respect to angle

Distribution of Aligned Letter Pairs in Optimal Alignments of Random Sequences

Considering the optimal alignment of two i.i.d. random sequences of length $n$, we show that when the scoring function is chosen randomly, almost surely the empirical distribution of aligned letter pairs in all optimal alignments converges to a unique limiting distribution as $n$ tends to infinity. This result is interesting because it helps understanding the microscopic path structure of a special type of last passage percolation problem with correlated weights, an area of long-standing open problems. Characterizing the microscopic path structure yields furthermore a robust alternative to optimal alignment scores for testing the relatedness of genetic sequences

Singularities and the distribution of density in the Burgers/adhesion model

We are interested in the tail behavior of the pdf of mass density within the one and $d$-dimensional Burgers/adhesion model used, e.g., to model the formation of large-scale structures in the Universe after baryon-photon decoupling. We show that large densities are localized near ``kurtoparabolic'' singularities residing on space-time manifolds of codimension two ($d \le 2$) or higher ($d \ge 3$). For smooth initial conditions, such singularities are obtained from the convex hull of the Lagrangian potential (the initial velocity potential minus a parabolic term). The singularities contribute {\em \hbox{universal} power-law tails} to the density pdf when the initial conditions are random. In one dimension the singularities are preshocks (nascent shocks), whereas in two and three dimensions they persist in time and correspond to boundaries of shocks; in all cases the corresponding density pdf has the exponent -7/2, originally proposed by E, Khanin, Mazel and Sinai (1997 Phys. Rev. Lett. 78, 1904) for the pdf of velocity gradients in one-dimensional forced Burgers turbulence. We also briefly consider models permitting particle crossings and thus multi-stream solutions, such as the Zel'dovich approximation and the (Jeans)--Vlasov--Poisson equation with single-stream initial data: they have singularities of codimension one, yielding power-law tails with exponent -3.Comment: LATEX 11 pages, 6 figures, revised; Physica D, in pres