Convergence Rates in L^2 for Elliptic Homogenization Problems

We study rates of convergence of solutions in L^2 and H^{1/2} for a family of elliptic systems {L_\epsilon} with rapidly oscillating oscillating coefficients in Lipschitz domains with Dirichlet or Neumann boundary conditions. As a consequence, we obtain convergence rates for Dirichlet, Neumann, and Steklov eigenvalues of {L_\epsilon}. Most of our results, which rely on the recently established uniform estimates for the L^2 Dirichlet and Neumann problems in \cite{12,13}, are new even for smooth domains.Comment: 25 page

Large-scale impact of Saharan dust on the North Atlantic Ocean circulation

The potential for a dynamical impact of Saharan mineral dust on the North Atlantic Ocean large-scale circulation is investigated. To this end, an ocean general circulation model forced by atmospheric fluxes is perturbed by an idealized, seasonally varying, net shortwave flux anomaly, as it results from remote sensing observations of aerosol optical thickness representing Saharan dust load in the atmosphere. The dust dynamical impact on the circulation is assessed through a comparison between perturbed and an unperturbed run. Results suggest that, following the dust-induced shortwave irradiance anomaly, a buoyancy anomaly is created in the Atlantic offshore the African coast, which over the course of the time propagates westward into the interior Atlantic while progressively subducting. Changes in the large-scale barotropic and overturning circulations are significant after 3 years, which coincides with the elapsed time required by the bulk of the buoyancy perturbation to reach the western boundary of the North Atlantic. Although small in amplitude, the changes in the meridional overturning are of the same order as interannual-to-decadal variability. Variations in the amplitude of the forcing lead to changes in the amplitude of the response, which is almost linear during the first 3 years. In addition, a fast, but dynamically insignificant, response can be observed to propagate poleward along the eastern boundary of the North Atlantic, which contributes to a nonlinear response in the subpolar region north of 40°N

Reactive Turbulent Flow in Low-Dimensional, Disordered Media

We analyze the reactions $A+A \to \emptyset$ and $A + B \to \emptyset$ occurring in a model of turbulent flow in two dimensions. We find the reactant concentrations at long times, using a field-theoretic renormalization group analysis. We find a variety of interesting behavior, including, in the presence of potential disorder, decay rates faster than that for well-mixed reactions.Comment: 6 pages, 4 figures. To appear in Phys. Rev.

Diffusive transport and self-consistent dynamics in coupled maps

The study of diffusion in Hamiltonian systems has been a problem of interest for a number of years. In this paper we explore the influence of self-consistency on the diffusion properties of systems described by coupled symplectic maps. Self-consistency, i.e. the back-influence of the transported quantity on the velocity field of the driving flow, despite of its critical importance, is usually overlooked in the description of realistic systems, for example in plasma physics. We propose a class of self-consistent models consisting of an ensemble of maps globally coupled through a mean field. Depending on the kind of coupling, two different general types of self-consistent maps are considered: maps coupled to the field only through the phase, and fully coupled maps, i.e. through the phase and the amplitude of the external field. The analogies and differences of the diffusion properties of these two kinds of maps are discussed in detail.Comment: 13 pages, 14 figure

Dispersion Coefficients by a Field-Theoretic Renormalization of Fluid Mechanics

We consider subtle correlations in the scattering of fluid by randomly placed obstacles, which have been suggested to lead to a diverging dispersion coefficient at long times for high Peclet numbers, in contrast to finite mean-field predictions. We develop a new master equation description of the fluid mechanics that incorporates the physically relevant fluctuations, and we treat those fluctuations by a renormalization group procedure. We find a finite dispersion coefficient at low volume fraction of disorder and high Peclet numbers.Comment: 4 pages, 1 figure; to appear in Phys. Rev. Let

Effects of Turbulent Mixing on the Critical Behavior

Effects of strongly anisotropic turbulent mixing on the critical behavior are studied by means of the renormalization group. Two models are considered: the equilibrium model A, which describes purely relaxational dynamics of a nonconserved scalar order parameter, and the Gribov model, which describes the nonequilibrium phase transition between the absorbing and fluctuating states in a reaction-diffusion system. The velocity is modelled by the d-dimensional generalization of the random shear flow introduced by Avellaneda and Majda within the context of passive scalar advection. Existence of new nonequilibrium types of critical regimes (universality classes) is established.Comment: Talk given in the International Bogolyubov Conference "Problems of Theoretical and Mathematical Physics" (Moscow-Dubna, 21-27 August 2009

The Navier wall law at a boundary with random roughness

We consider the Navier-Stokes equation in a domain with irregular boundaries. The irregularity is modeled by a spatially homogeneous random process, with typical size \eps \ll 1. In a parent paper, we derived a homogenized boundary condition of Navier type as \eps \to 0. We show here that for a large class of boundaries, this Navier condition provides a O(\eps^{3/2} |\ln \eps|^{1/2}) approximation in $L^2$, instead of O(\eps^{3/2}) for periodic irregularities. Our result relies on the study of an auxiliary boundary layer system. Decay properties of this boundary layer are deduced from a central limit theorem for dependent variables

Spectrum of the Fokker-Planck operator representing diffusion in a random velocity field

We study spectral properties of the Fokker-Planck operator that represents particles moving via a combination of diffusion and advection in a time-independent random velocity field, presenting in detail work outlined elsewhere [J. T. Chalker and Z. J. Wang, Phys. Rev. Lett. {\bf 79}, 1797 (1997)]. We calculate analytically the ensemble-averaged one-particle Green function and the eigenvalue density for this Fokker-Planck operator, using a diagrammatic expansion developed for resolvents of non-Hermitian random operators, together with a mean-field approximation (the self-consistent Born approximation) which is well-controlled in the weak-disorder regime for dimension d>2. The eigenvalue density in the complex plane is non-zero within a wedge that encloses the negative real axis. Particle motion is diffusive at long times, but for short times we find a novel time-dependence of the mean-square displacement, $\sim t^{2/d}$ in dimension d>2, associated with the imaginary parts of eigenvalues.Comment: 8 pages, submitted to Phys Rev

Stochastic reconstruction of sandstones

A simulated annealing algorithm is employed to generate a stochastic model for a Berea and a Fontainebleau sandstone with prescribed two-point probability function, lineal path function, and ``pore size'' distribution function, respectively. We find that the temperature decrease of the annealing has to be rather quick to yield isotropic and percolating configurations. A comparison of simple morphological quantities indicates good agreement between the reconstructions and the original sandstones. Also, the mean survival time of a random walker in the pore space is reproduced with good accuracy. However, a more detailed investigation by means of local porosity theory shows that there may be significant differences of the geometrical connectivity between the reconstructed and the experimental samples.Comment: 12 pages, 5 figure

Malvinas-slope water intrusions on the northern Patagonia continental shelf

The Patagonia continental shelf located off southeastern South America is bounded offshore by the Malvinas Current, which extends northward from northern Drake Passage (~55° S) to nearly 38° S. The transition between relatively warm-fresh shelf waters and Subantarctic Waters from the western boundary current is characterized by a thermohaline front extending nearly 2500 km. We use satellite derived sea surface temperature, and chlorophyll-<I>a</I> data combined with hydrographic and surface drifter data to document the intrusions of slope waters onto the continental shelf near 41° S. These intrusions create vertically coherent localized negative temperature and positive salinity anomalies extending onshore about 150 km from the shelf break. The region is associated with a center of action of the first mode of non-seasonal sea surface temperature variability and also relatively high chlorophyll-<I>a</I> variability, suggesting that the intrusions are important in promoting the local development of phytoplankton. The generation of slope water penetrations at this location may be triggered by the inshore excursion of the 100 m isobath, which appears to steer the Malvinas Current waters over the outer shelf