Carbon price and optimal extraction of a polluting fossil fuel with restricted carbon capture

Among technological options to mitigate greenhouse gas (GHG) emissions, Carbon Capture and Storage technology (CCS) seems particularly promising. This technology allows to keep on extracting polluting fossil fuels without drastically increasing CO2 atmospheric concentration. We examine here a two-sector model with two primary energy resources, a polluting exhaustible resource and an expensive carbon-free renewable resource, in which an environmental regulation is imposed through a cap on the atmospheric carbon stock. We assume that only the emissions from one sector can be captured. Previous literature, based on one-sector models in which all emissions are capturable, finds that CCS technology should not be used before the threshold has been reached. We find that, when technical constraints make it impossible to capture emissions from both sectors, this result does not always hold. CCS technology should be used before the ceiling is reached if non capturable emissions are large enough. In that case, we find that energy prices paths must differ between sectors reflecting the difference of social cost of the resource according to its use. Numerical exercise shows that the initial carbon tax should equal 52$/t CO2 and that using CCS before the ceiling is optimal.Nonrenewable Resources, Externalities, Carbon Capture.

Multiscale Analysis of the Stress State in a Granular Slope in Transition to Failure

By means of contact dynamics simulations, we analyze the stress state in a granular bed slowly tilted towards its angle of repose. An increasingly large number of grains are overloaded in the sense that they are found to carry a stress ratio above the Coulomb yield threshold of the whole packing. Using this property, we introduce a coarse-graining length scale at which all stress ratios are below the packing yield threshold. We show that this length increases with the slope angle and jumps to a length comparable to the depth of the granular bed at an angle below the angle of repose. This transition coincides with the onset of dilatation in the packing. We map this transition into a percolation transition of the overloaded grains, and we argue that in the presence of long-range correlations above the transition angle, the granular slope is metastable.Comment: 11 pages, 14 Fig, submitted to PR

Diphasic non-local model for granular surface flows

Considering recent results revealing the existence of multi-scale rigid clusters of grains embedded in granular surface flows, i.e. flows down an erodible bed, we describe here the surface flows rheology through a non-local constitutive law. The predictions of the resulting model are compared quantitatively to experimental results: The model succeeds to account for the counter-intuitive shape of the velocity profile observed in experiments, i.e. a velocity profile decreasing exponentially with depth in the static phase and remaining linear in the flowing layer with a velocity gradient independent of both the flowing layer thickness, the angle between the flow and the horizontal, and the coefficient of restitution of the grains. Moreover, the scalings observed in rotating drums are recovered, at least for small rotating speed.Comment: 7 pages, submitted to Europhys. Let

Logarithmic rate dependence in deforming granular materials

Rate-independence for stresses within a granular material is a basic tenet of many models for slow dense granular flows. By contrast, logarithmic rate dependence of stresses is found in solid-on-solid friction, in geological settings, and elsewhere. In this work, we show that logarithmic rate-dependence occurs in granular materials for plastic (irreversible) deformations that occur during shearing but not for elastic (reversible) deformations, such as those that occur under moderate repetitive compression. Increasing the shearing rate, \Omega, leads to an increase in the stress and the stress fluctuations that at least qualitatively resemble what occurs due to an increase in the density. Increases in \Omega also lead to qualitative changes in the distributions of stress build-up and relaxation events. If shearing is stopped at t=0, stress relaxations occur with \sigma(t)/ \sigma(t=0) \simeq A \log(t/t_0). This collective relaxation of the stress network over logarithmically long times provides a mechanism for rate-dependent strengthening.Comment: 4 pages, 5 figures. RevTeX

Stress in frictionless granular material: Adaptive Network Simulations

We present a minimalistic approach to simulations of force transmission through granular systems. We start from a configuration containing cohesive (tensile) contact forces and use an adaptive procedure to find the stable configuration with no tensile contact forces. The procedure works by sequentially removing and adding individual contacts between adjacent beads, while the bead positions are not modified. In a series of two-dimensional realizations, the resulting force networks are shown to satisfy a linear constraint among the three components of average stress, as anticipated by recent theories. The coefficients in the linear constraint remain nearly constant for a range of shear loadings up to about .6 of the normal loading. The spatial distribution of contact forces shows strong concentration along ``force chains". The probability of contact forces of magnitude f shows an exponential falloff with f. The response to a local perturbing force is concentrated along two characteristic rays directed downward and laterally.Comment: 8 pages, 8 figure

Stochastic Games: Existence of the Minmax

The existence of the value for stochastic games with finitely many states and actions, as well as for a class of stochastic games with infinitely many states and actions, is proved in [2]. Here we use essentially the same tools to derive the existence of the minmax and maxmin for n-player stochasti

Dissipation of vibration in rough contact

The relationship which links the normal vibration occurring during the sliding of rough surfaces and the nominal contact area is investigated. Two regimes are found. In the first one, the vibrational level does not depend on the contact area, while in the second one, it is propor- tional to the contact area. A theoretical model is proposed. It is based on the assumption that the vibrational level results from a competition between two processes of vibration damping, the internal damping of the material and the contact damping occurring at the interface

A model for collisions in granular gases

We propose a model for collisions between particles of a granular material and calculate the restitution coefficients for the normal and tangential motion as functions of the impact velocity from considerations of dissipative viscoelastic collisions. Existing models of impact with dissipation as well as the classical Hertz impact theory are included in the present model as special cases. We find that the type of collision (smooth, reflecting or sticky) is determined by the impact velocity and by the surface properties of the colliding grains. We observe a rather nontrivial dependence of the tangential restitution coefficient on the impact velocity.Comment: 11 pages, 2 figure