146 research outputs found
Thermodynamic Formalism of the Harmonic Measure of Diffusion Limited Aggregates: Phase Transition and Converged
We study the nature of the phase transition in the multifractal formalism of
the harmonic measure of Diffusion Limited Aggregates (DLA). Contrary to
previous work that relied on random walk simulations or ad-hoc models to
estimate the low probability events of deep fjord penetration, we employ the
method of iterated conformal maps to obtain an accurate computation of the
probability of the rarest events. We resolve probabilities as small as
. We show that the generalized dimensions are infinite for
, where . In the language of this means
that is finite. We present a converged curve.Comment: accepted for Physical Review Letter
Carbon choices determine US cities committed to futures below sea level
Anthropogenic carbon emissions lock in long-term sea-level rise that greatly exceeds projections for this century, posing profound challenges for coastal development and cultural legacies. Analysis based on previously published relationships linking emissions to warming and warming to rise indicates that unabated carbon emissions up to the year 2100 would commit an eventual global sea-level rise of 4.3–9.9 m. Based on detailed topographic and population data, local high tide lines, and regional long-term sea-level commitment for different carbon emissions and ice sheet stability scenarios, we compute the current population living on endangered land at municipal, state, and national levels within the United States. For unabated climate change, we find that land that is home to more than 20 million people is implicated and is widely distributed among different states and coasts. The total area includes 1,185–1,825 municipalities where land that is home to more than half of the current population would be affected, among them at least 21 cities exceeding 100,000 residents. Under aggressive carbon cuts, more than half of these municipalities would avoid this commitment if the West Antarctic Ice Sheet remains stable. Similarly, more than half of the US population-weighted area under threat could be spared. We provide lists of implicated cities and state populations for different emissions scenarios and with and without a certain collapse of the West Antarctic Ice Sheet. Although past anthropogenic emissions already have caused sea-level commitment that will force coastal cities to adapt, future emissions will determine which areas we can continue to occupy or may have to abandon
Bi-Laplacian Growth Patterns in Disordered Media
Experiments in quasi 2-dimensional geometry (Hele Shaw cells) in which a
fluid is injected into a visco-elastic medium (foam, clay or
associating-polymers) show patterns akin to fracture in brittle materials, very
different from standard Laplacian growth patterns of viscous fingering. An
analytic theory is lacking since a pre-requisite to describing the fracture of
elastic material is the solution of the bi-Laplace rather than the Laplace
equation. In this Letter we close this gap, offering a theory of bi-Laplacian
growth patterns based on the method of iterated conformal maps.Comment: Submitted to PRL. For further information see
http://www.weizmann.ac.il/chemphys/ander
Stress field around arbitrarily shaped cracks in two-dimensional elastic materials
The calculation of the stress field around an arbitrarily shaped crack in an
infinite two-dimensional elastic medium is a mathematically daunting problem.
With the exception of few exactly soluble crack shapes the available results
are based on either perturbative approaches or on combinations of analytic and
numerical techniques. We present here a general solution of this problem for
any arbitrary crack. Along the way we develop a method to compute the conformal
map from the exterior of a circle to the exterior of a line of arbitrary shape,
offering it as a superior alternative to the classical Schwartz-Cristoffel
transformation. Our calculation results in an accurate estimate of the full
stress field and in particular of the stress intensity factors K_I and K_{II}
and the T-stress which are essential in the theory of fracture.Comment: 7 pages, 4 figures, submitted for PR
Quasi-Static Fractures in Disordered Media and Iterated Conformal Maps
We study the geometrical characteristic of quasi-static fractures in
disordered media, using iterated conformal maps to determine the evolution of
the fracture pattern. This method allows an efficient and accurate solution of
the Lam\'e equations without resorting to lattice models. Typical fracture
patterns exhibit increased ramification due to the increase of the stress at
the tips. We find the roughness exponent of the experimentally relevant
backbone of the fracture pattern; it crosses over from about 0.5 for small
scales to about 0.75 for large scales, in excellent agreement with experiments.
We propose that this cross-over reflects the increased ramification of the
fracture pattern.Comment: submitted to Physical Review Letter
Potential climatic transitions with profound impact on Europe
We discuss potential transitions of six climatic subsystems with large-scale impact on Europe, sometimes denoted as tipping elements. These are the ice sheets on Greenland and West Antarctica, the Atlantic thermohaline circulation, Arctic sea ice, Alpine glaciers and northern hemisphere stratospheric ozone. Each system is represented by co-authors actively publishing in the corresponding field. For each subsystem we summarize the mechanism of a potential transition in a warmer climate along with its impact on Europe and assess the likelihood for such a transition based on published scientific literature. As a summary, the ‘tipping’ potential for each system is provided as a function of global mean temperature increase which required some subjective interpretation of scientific facts by the authors and should be considered as a snapshot of our current understanding. <br/
Conformal mapping methods for interfacial dynamics
The article provides a pedagogical review aimed at graduate students in
materials science, physics, and applied mathematics, focusing on recent
developments in the subject. Following a brief summary of concepts from complex
analysis, the article begins with an overview of continuous conformal-map
dynamics. This includes problems of interfacial motion driven by harmonic
fields (such as viscous fingering and void electromigration), bi-harmonic
fields (such as viscous sintering and elastic pore evolution), and
non-harmonic, conformally invariant fields (such as growth by
advection-diffusion and electro-deposition). The second part of the article is
devoted to iterated conformal maps for analogous problems in stochastic
interfacial dynamics (such as diffusion-limited aggregation, dielectric
breakdown, brittle fracture, and advection-diffusion-limited aggregation). The
third part notes that all of these models can be extended to curved surfaces by
an auxilliary conformal mapping from the complex plane, such as stereographic
projection to a sphere. The article concludes with an outlook for further
research.Comment: 37 pages, 12 (mostly color) figure
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A High-End Estimate of Sea Level Rise for Practitioners
Sea level rise (SLR) is a long-lasting consequence of climate change because global anthropogenic warming takes centuries to millennia to equilibrate for the deep ocean and ice sheets. SLR projections based on climate models support policy analysis, risk assessment and adaptation planning today, despite their large uncertainties. The central range of the SLR distribution is estimated by process-based models. However, risk-averse practitioners often require information about plausible future conditions that lie in the tails of the SLR distribution, which are poorly defined by existing models. Here, a community effort combining scientists and practitioners builds on a framework of discussing physical evidence to quantify high-end global SLR for practitioners. The approach is complementary to the IPCC AR6 report and provides further physically plausible high-end scenarios. High-end estimates for the different SLR components are developed for two climate scenarios at two timescales. For global warming of +2°C in 2100 (RCP2.6/SSP1-2.6) relative to pre-industrial values our high-end global SLR estimates are up to 0.9 m in 2100 and 2.5 m in 2300. Similarly, for a (RCP8.5/SSP5-8.5), we estimate up to 1.6 m in 2100 and up to 10.4 m in 2300. The large and growing differences between the scenarios beyond 2100 emphasize the long-term benefits of mitigation. However, even a modest 2°C warming may cause multi-meter SLR on centennial time scales with profound consequences for coastal areas. Earlier high-end assessments focused on instability mechanisms in Antarctica, while here we emphasize the importance of the timing of ice shelf collapse around Antarctica. This is highly uncertain due to low understanding of the driving processes. Hence both process understanding and emission scenario control high-end SLR
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