SCIENTIFIC PROGRAMS AND ACTIVITIES

2008-09 talks held at the Fields Institute

March 4, 2009
3:10 p.m.

Joint Fields/Physics Colloquium
Ehud Meron, Ben-Gurion University of the Negev
Periodic vs. scale-free patterns: reconciling the dichotomy of dryland vegetation
Field observations of vegetation patchiness in drylands reveal periodic patterns with characteristic length scales along with scale-free patterns characterized by broad patch-size distributions, often reported to obey power-laws. Despite the numerous theoretical and experimental studies that have been devoted to vegetation patchiness, this dichotomy of patterns is still poorly understood. Using a mathematical model that captures basic feedbacks between biomass and water and between above-ground and below-ground biomass, we elucidate mechanisms that control patch-size distributions in water-limited systems, and identify physical and ecological circumstances that lead to periodic patterns and to scale-free patterns.

Jan. 14, 2009
3:10pm

Consider the problem of optimally matching two measures on the circle, or equivalently two periodic measures on R, where the cost c(x, y) of
matching two points x, y satisfies the Monge condition: c(p, q) + c(r, s) < c(p, s) + c(r, q) whenever p < r and q < s. Motivated by the weak KAM theory, we introduce a notion of locally optimal transport plan and show that all locally optimal transport plans are conjugate to shifts. The theory is applied to a transportation problem arising in image processing: for two sets of point masses, both of which have the same total mass, find an optimal transport plan with respect to a given cost function that satisfies the Monge condition. For the case of N real-valued point masses we present an O(N log epsilon) algorithm that approximates the optimal cost within epsilon; when all masses are integer multiples of 1/M, the algorithm gives an exact solution in O(N log M) operations.
Julie Delon (Telecom Paris), Julien Salomon (U. Paris-Dauphine/CEREMADE), and Andrei Sobolevskii (U. Moscow)

Dust, droplets and other ?nite-size impurities with a large mass density suspended in incompressible turbulent flows are commonly encountered in many natural phenomena and industrial processes, such as the growth of raindrops in subtropical clouds, the formation of planetesimals in the early Solar system, and the combustion in Diesel engines. The most salient feature of such suspensions is the presence of strong inhomogeneities in the spatial distribution of particles, a phenomenon dubbed preferential concentration. We show that, depending on the spatial scale at which it is observed, the particle distribution can be of very different natures.

At dissipative scales, where the fluid flow is differentiable, the phase-space density of particles is supported on a dynamically evolving fractal set. This attractor is characterized by non-trivial multiscaling properties, which imply multiscaling of the coarse-grained spatial distribution of the mass
of particles. For larger length scales inside the inertial range of turbulence, the particle distribution is characterized by large voids where the mass is orders of magnitude below its average. Such regions are typically correlated with the vortical structures of the ?ow; this con?rms the classical phenomenological pictures that in turbulent ?ows, eddies act as small centrifuges and eject heavy particles leading to their concentration in the strain-dominated regions. The signature of this voids in the coarse-grained mass probability distribution is an algebraic behavior at small densities. We present a simple model for mass transport that reproduces the same distribution.

References:
J. Bec, L. Biferale, M. Cencini, A. Lanotte, S. Musacchio & F. Toschi, Heavy particle concentration in turbulence at dissipative and inertial scales, Phys. Rev. Lett. 98, 084502, 2007, [arXiv:nlin/0608045] J. Bec & R. Chétrite, Toward a phenomenological approach to the clustering of heavy particles in turbulent flows, New J. Phys. 9, 77 (1-16), 2007, [arXiv:nlin/0701033] J. Bec, M. Cencini, R. Hillerbrand & K. Turitsyn, Stochastic suspensions of heavy particles, Physica D 237, 2037-2050, 2008, [arXiv:0710.2507]