44 research outputs found
Power laws statistics of cliff failures, scaling and percolation
The size of large cliff failures may be described in several ways, for
instance considering the horizontal eroded area at the cliff top and the
maximum local retreat of the coastline. Field studies suggest that, for large
failures, the frequencies of these two quantities decrease as power laws of the
respective magnitudes, defining two different decay exponents. Moreover, the
horizontal area increases as a power law of the maximum local retreat,
identifying a third exponent. Such observation suggests that the geometry of
cliff failures are statistically similar for different magnitudes. Power laws
are familiar in the physics of critical systems. The corresponding exponents
satisfy precise relations and are proven to be universal features, common to
very different systems. Following the approach typical of statistical physics,
we propose a "scaling hypothesis" resulting in a relation between the three
above exponents: there is a precise, mathematical relation between the
distributions of magnitudes of erosion events and their geometry. Beyond its
theoretical value, such relation could be useful for the validation of field
catalogs analysis. Pushing the statistical physics approach further, we develop
a numerical model of marine erosion that reproduces the observed failure
statistics. Despite the minimality of the model, the exponents resulting from
extensive numerical simulations fairly agree with those measured on the field.
These results suggest that the mathematical theory of percolation, which lies
behind our simple model, can possibly be used as a guide to decipher the
physics of rocky coast erosion and could provide precise predictions to the
statistics of cliff collapses.Comment: 20 pages, 13 figures, 1 table. To appear in Earth Surface Processes
and Lanforms (Rocky Coast special issue
Chemical etching of a disordered solid: from experiments to field theory
We present a two-dimensional theoretical model for the slow chemical
corrosion of a thin film of a disordered solid by suitable etching solutions.
This model explain different experimental results showing that the corrosion
stops spontaneously in a situation in which the concentration of the etchant is
still finite while the corrosion surface develops clear fractal features. We
show that these properties are strictly related to the percolation theory, and
in particular to its behavior around the critical point. This task is
accomplished both by a direct analysis in terms of a self-organized version of
the Gradient Percolation model and by field theoretical arguments.Comment: 7 pages, 3 figure
Field theory of self-organized fractal etching
We propose a phenomenological field theoretical approach to the chemical
etching of a disordered-solid. The theory is based on a recently proposed
dynamical etching model. Through the introduction of a set of Langevin
equations for the model evolution, we are able to map the problem into a field
theory related to isotropic percolation. To the best of the authors knowledge,
it constitutes the first application of field theory to a problem of chemical
dynamics. By using this mapping, many of the etching process critical
properties are seen to be describable in terms of the percolation
renormalization group fixed point. The emerging field theory has the
peculiarity of being ``{\it self-organized}'', in the sense that without any
parameter fine-tuning, the system develops fractal properties up to certain
scale controlled solely by the volume, , of the etching solution.
In the limit the upper cut-off goes to infinity and the system
becomes scale invariant. We present also a finite size scaling analysis and
discuss the relation of this particular etching mechanism with Gradient
Percolation.
Finally, the possibility of considering this mechanism as a new generic path
to self-organized criticality is analyzed, with the characteristics of being
closely related to a real physical system and therefore more directly
accessible to experiments.Comment: 9 pages, 3 figures. Submitted to Phys. Rev.
Optimal branching asymmetry of hydrodynamic pulsatile trees
Most of the studies on optimal transport are done for steady state regime
conditions. Yet, there exists numerous examples in living systems where supply
tree networks have to deliver products in a limited time due to the pulsatile
character of the flow. This is the case for mammals respiration for which air
has to reach the gas exchange units before the start of expiration. We report
here that introducing a systematic branching asymmetry allows to reduce the
average delivery time of the products. It simultaneously increases its
robustness against the unevitable variability of sizes related to
morphogenesis. We then apply this approach to the human tracheobronchial tree.
We show that in this case all extremities are supplied with fresh air, provided
that the asymmetry is smaller than a critical threshold which happens to fit
with the asymmetry measured in the human lung. This could indicate that the
structure is adjusted at the maximum asymmetry level that allows to feed all
terminal units with fresh air.Comment: 4 pages, 4 figure
The fractal nature of a diffusion front and the relation to percolation
International audienceUsing a two dimensional simulation, a diffusion front is shown to have a fractal geometry in a range increasing with the diffusion length. The number of particles on the front, and the width measuring its spread, follow power laws as a function of the diffusion length. The associated exponents and the fractal dimension can be expressed as simple functions of the critical exponents of the two dimensional percolation problem
Diffusion-Reorganized Aggregates: Attractors in Diffusion Processes?
A process based on particle evaporation, diffusion and redeposition is
applied iteratively to a two-dimensional object of arbitrary shape. The
evolution spontaneously transforms the object morphology, converging to
branched structures. Independently of initial geometry, the structures found
after long time present fractal geometry with a fractal dimension around 1.75.
The final morphology, which constantly evolves in time, can be considered as
the dynamic attractor of this evaporation-diffusion-redeposition operator. The
ensemble of these fractal shapes can be considered to be the {\em dynamical
equilibrium} geometry of a diffusion controlled self-transformation process.Comment: 4 pages, 5 figure
How winding is the coast of Britain ? Conformal invariance of rocky shorelines
We show that rocky shorelines with fractal dimension 4/3 are conformally
invariant curves by measuring the statistics of their winding angles from
global high-resolution data. Such coastlines are thus statistically equivalent
to the outer boundary of the random walk and of percolation clusters. A simple
model of coastal erosion gives an explanation for these results. Conformal
invariance allows also to predict the highly intermittent spatial distribution
of the flux of pollutant diffusing ashore
Artificial boundaries and formulations for the incompressible Navier-Stokes equations. Applications to air and blood flows.
International audienceWe deal with numerical simulations of incompressible Navier-Stokes equations in truncated domain. In this context, the formulation of these equations has to be selected carefully in order to guarantee that their associated artificial boundary conditions are relevant for the considered problem. In this paper, we review some of the formulations proposed in the literature, and their associated boundary conditions. Some numerical results linked to each formulation are also presented. We compare different schemes, giving successful computations as well as problematic ones, in order to better understand the difference between these schemes and their behaviours dealing with systems involving Neumann boundary conditions. We also review two stabilization methods which aim at suppressing the instabilities linked to these natural boundary conditions