scholarly journals Fractal structures in freezing brine

2017 ◽  
Vol 826 ◽  
pp. 975-995
Author(s):  
Sergey Alyaev ◽  
Eirik Keilegavlen ◽  
Jan Martin Nordbotten ◽  
Iuliu Sorin Pop
Keyword(s):  
Characteristic Length ◽  
Complex Problem ◽  
Rate Of Change ◽  
Critical Threshold ◽  
Fractal Structures ◽  
Primary Interest ◽  
External Temperature ◽  
Self Similar ◽  
Salt Diffusion ◽  
Heuristic Analysis

The process of initial ice formation in brine is a highly complex problem. In this paper, we propose a mathematical model that captures the dynamics of nucleation and development of ice inclusions in brine. The primary emphasis is on the interaction between ice growth and salt diffusion, subject to external forcing provided by temperature. Within this setting two freezing regimes are identified, depending on the rate of change of the temperature: a slow freezing regime where a continuous ice domain is formed; and a fast freezing regime where recurrent nucleation appears within the fluid domain. The second regime is of primary interest, as it leads to fractal-like ice structures. We analyse the critical threshold between the slow and fast regimes by identifying the explicit rates of external temperature control that lead to self-similar salt-concentration profiles in the fluid domains. Subsequent heuristic analysis provides estimates of the characteristic length scales of the fluid domains depending on the time-variation of the temperature. The analysis is confirmed by numerical simulations.

scholarly journals Wavelet introscopy of human organism bionets

2021 ◽  
pp. 63-72
Author(s):  
Gennady M. Aldonin ◽  
◽  
Vasily V. Cherepanov ◽  
Keyword(s):  
Functional State ◽  
Human Body ◽  
Dynamic Models ◽  
Golden Ratio ◽  
The State ◽  
The Body ◽  
Human Organism ◽  
Fractal Structures ◽  
Self Similar ◽  
The Creation

In domestic and foreign practice, a great deal of experience has been accumulated in the creation of means for monitoring the functional state of the human body. The existing complexes mainly analyze the electrocardiogram, blood pressure and a number of other physiological parameters. Diagnostics is often based on formal statistical data which are not always correct due to the nonstationarity of bioprocesses and without taking into account their physical nature. An urgent task of monitoring the state of the cardiovascular system is the creation of effective algorithms for computer technologies to process biosignals based on nonlinear dynamic models of body systems since biosystems and bioprocesses have a nonlinear nature and fractal structure. The nervous and muscular systems of the heart, the vascular and bronchial systems of the human body are examples of such structures. The connection of body systems with their organization in the form of self-similar fractal structures with scaling close to the “golden ratio” makes it possible to diagnose them topically. It is possible to obtain detailed information about the state of the human body’s bio-networks for topical diagnostics on the basis of the wavelet analysis of biosignals (the so-called wavelet-introscopy). With the help of wavelet transform, it is possible to reveal the structure of biosystems and bioprocesses, as a picture of the lines of local extrema of wavelet diagrams of biosignals. Mathematical models and software for wavelet introscopy make it possible to extract additional information from biosignals about the state of biosystems. Early detection of latent forms of diseases using wavelet introscopy can shorten the cure time and reduce the consequences of disorders of the functional state of the body (FSO), and reduce the risk of disability. Taking into account the factors of organizing the body’s biosystems in the form of self-similar fractal structures with a scaling close to the “golden ratio” makes it possible to create a technique for topical diagnostics of the most important biosystems of the human body.

Shadow of a disformal Kerr black hole in quadratic degenerate higher-order scalar–tensor theories

2020 ◽  
Vol 80 (12) ◽  
Author(s):  
Fen Long ◽  
Songbai Chen ◽  
Mingzhi Wang ◽  
Jiliang Jing
Keyword(s):  
Black Hole ◽  
Deformation Parameter ◽  
Kerr Black Hole ◽  
Arbitrary Spin ◽  
Higher Order ◽  
Fractal Structures ◽  
Extreme Black Hole ◽  
Spin Parameter ◽  
Result Show ◽  
Self Similar

AbstractWe have studied the shadow of a disformal Kerr black hole with an extra deformation parameter, which belongs to non-stealth rotating solutions in quadratic degenerate higher-order scalar–tensor (DHOST) theory. Our result show that the size of the shadow increases with the deformation parameter for the black hole with arbitrary spin parameter. However, the effect of the deformation parameter on the shadow shape depends heavily on the spin parameter of black hole and the sign of the deformation parameter. The change of the shadow shape becomes more distinct for the black hole with the more quickly rotation and the more negative deformation parameter. Especially, for the near-extreme black hole with negative deformation parameter, there exist a “pedicel”-like structure appeared in the shadow, which increases with the absolute value of deformation parameter. The eyebrow-like shadow and the self-similar fractal structures also appear in the shadow for the disformal Kerr black hole in DHOST theory. These features in the black hole shadow originating from the scalar field could help us to understand the non-stealth disformal Kerr black hole and quadratic DHOST theory.

Predictability of tipping points with rate-dependent effects

2021 ◽  
Author(s):  
Johannes Lohmann ◽  
Peter Ditlevsen
Keyword(s):  
Early Warning ◽  
Control Parameter ◽  
Initial Conditions ◽  
Ocean Model ◽  
Rate Of Change ◽  
Global Ocean ◽  
Critical Threshold ◽  
Tipping Points ◽  
Operating Space ◽  
Safe Operating

<p>Due to non-linearities in the dynamics of crucial elements in the climate system, Earth’s safe operating space is limited. Beyond a certain level of a control parameter, such as the atmospheric Greenhouse gas concentration, qualitative regime shifts in one or more sub-systems may take place. Additionally, theoretical studies suggest that abrupt, irreversible change can happen already prior to the crossing of a critical threshold in a control parameter.</p><p>In these so-called rate-induced transitions, the effective parameter level to induce tipping depends on the rate of change, or more generally the precise trajectory of the changing control parameter. Here we show rate-induced tipping points of the overturning circulation in a global ocean model. Due to the chaotic dynamics of the system, whether there will be tipping or not depends both on the rate and initial conditions in a very sensitive, non-smooth way. This raises questions about whether the safe operating space is still well-defined, and whether an approach of its boundary can be predicted.</p><p>For tipping points associated with slow passages across a bifurcation, generic early-warning signals have been developed for these purposes. Due to the necessarily fast parameter changes involved in rate-induced tipping, early-warning is more challenging. In many cases the tipping involves a saddle escape, which results in a delay of the actual transition and can be exploited for early-warning. Here this is demonstrated in the context of low-dimensional models. While due to the sensitivity of the dynamics around the saddle one might not be able to predict with certainty whether and when the system will tip, the indicators presented here may allow issuing a warning as the system gets close to tipping.</p>

Projected Measures: A Simple Way to Characterize Fractal Structures and Interfaces

Fractals ◽  
1997 ◽  
Vol 05 (02) ◽  
pp. 295-308
Author(s):  
Massimiliano Giona ◽  
Manuela Giustiniani ◽  
Oreste Patierno
Keyword(s):  
Closed Form ◽  
Fourier Transforms ◽  
Fractal Structures ◽  
Multifractal Spectra ◽  
Self Similar ◽  
The Moment ◽  
Dynamic Phenomena

The properties of projected measures of fractal objects are investigated in detail. In general, projected measures display multifractal features which play a role in the evolution of dynamic phenomena on/through fractal structures. Closed-form results are obtained for the moment hierarchy of model fractal interfaces. The distinction between self-similar and self-affine interfaces is discussed by considering the properties of multifractal spectra, the orientational effects in the behavior of the moment hierarchies, and the scaling of the corresponding Fourier transforms. The implications of the properties of projected measures in the characterization of transfer phenomena across fractal interfaces are briefly analyzed.

scholarly journals Non-Differentiable Structure in the Generalized Mandelbrot Set

1988 ◽  
Vol 43 (3) ◽  
pp. 287-288
Author(s):  
J. Peinke ◽  
J. Parisi ◽  
B. Röhricht ◽  
O. E. Rössler ◽  
W. Metzler
Keyword(s):  
Vortex Structure ◽  
Julia Set ◽  
Mandelbrot Set ◽  
Fractal Structures ◽  
Differentiable Structure ◽  
Self Similar

Abstract The generalized Mandelbrot set, described previously, contains - like the original Mandelbrot set - non-differentiable self-similar fractal structures. An example looking like a vortex structure is presented along with a corresponding generalized Julia set.

Thresholds of change in a multi-use conservation landscape of South Africa: historical land-cover, future transformation and consequences for environmental decision-making

2016 ◽  
Vol 43 (3) ◽  
pp. 253-262 ◽  
Author(s):  
KAERA L. COETZER-HANACK ◽  
E.T.F. WITKOWSKI ◽  
BAREND F.N. ERASMUS
Keyword(s):  
South Africa ◽  
Decision Making ◽  
Land Cover ◽  
Recent Observation ◽  
Rate Of Change ◽  
Critical Threshold ◽  
Rapid Decline ◽  
Remotely Sensed Data ◽  
Environmental Decision Making ◽  
Environmental Decision

SUMMARYAs multi-use conservation landscapes, biosphere reserves (BRs) exemplify the landscape mosaic approach to environmental decision-making. In this study, time-series remotely-sensed data (1993–2006–2012) were used to monitor vegetation transformation in the Kruger to Canyons Biosphere Reserve (K2C) of South Africa, updating previous land-cover research. We identified changes in spatial extent, rate and intensity of land-cover change and extrapolated observed trends to 2018. The increased rate of change in the recent observation period (2.3 vs. 5.7%) was driven by more intensive gains in impacted vegetation and settlement since 2006 (>210 km2and >120 km2), with resultant transformation of intact habitat undermining regional connectivity. By 2012, intact vegetation had suffered losses of 6.3% (>350 km2) since 2006 and >14% (>750 km2) since 1993. A further 9.5% loss of intact habitat may represent a critical threshold, establishing K2C above the 50% threshold of landscape transformation, whereafter a rapid decline in landscape resilience is likely. Given the BR's spatial zonation, such a loss across the full extent of K2C is unlikely, at least in the short-term (i.e., by 2018). Yet, based on past trends of transformation in the unprotected transition zone, anticipating such losses in the longer term, is not unfounded (i.e., 2024).

AN INTRODUCTION TO FLOW AND TRANSPORT IN FRACTAL MODELS OF POROUS MEDIA: PART I

Fractals ◽  
2014 ◽  
Vol 22 (03) ◽  
pp. 1402001 ◽  
Author(s):  
JIANCHAO CAI ◽  
FERNANDO SAN JOSÉ MARTÍNEZ ◽  
MIGUEL ANGEL MARTÍN ◽  
EDMUND PERFECT
Keyword(s):  
Porous Media ◽  
Surface Erosion ◽  
Size Distributions ◽  
Particle Size Distributions ◽  
Fractal Structures ◽  
Flow And Transport ◽  
Fractal Models ◽  
Statics And Dynamics ◽  
Self Similar ◽  
Erosion Properties

This special issue gathers together a number of recent papers on fractal geometry and its applications to the modeling of flow and transport in porous media. The aim is to provide a systematic approach for analyzing the statics and dynamics of fluids in fractal porous media by means of theory, modeling and experimentation. The topics covered include lacunarity analyses of multifractal and natural grayscale patterns, random packing's of self-similar pore/particle size distributions, Darcian and non-Darcian hydraulic flows, diffusion within fractals, models for the permeability and thermal conductivity of fractal porous media and hydrophobicity and surface erosion properties of fractal structures.

scholarly journals Effect of gravitational wave on shadow of a Schwarzschild black hole

2021 ◽  
Vol 81 (6) ◽  
Author(s):  
Mingzhi Wang ◽  
Songbai Chen ◽  
Jiliang Jing
Keyword(s):  
Black Hole ◽  
Gravitational Wave ◽  
Legendre Polynomial ◽  
Vertical Direction ◽  
Einstein Equations ◽  
Schwarzschild Black Hole ◽  
Fractal Structures ◽  
Odd Order ◽  
Particular Solution ◽  
Self Similar

AbstractWe have studied the shadows of a Schwarzschild black hole under a special polar gravitational perturbation, which is a particular solution of Einstein equations expanded up to first order. It is shown that the black hole shadow changes periodically with time and the change of shadow depends on the Legendre polynomial order parameter l and the frequency $$\sigma $$ σ of gravitational wave. For the odd order of Legendre polynomial, the center of shadow oscillates along the direction which is vertical to equatorial plane. For even l, the center of shadow does not move, but the shadow alternately stretches and squeezes with time along the vertical direction. Moreover, the presence of the gravitational wave leads to the self-similar fractal structures appearing in the boundary of the black hole shadow. We also find that this special gravitational wave has a greater influence on the vertical direction of black hole shadow.

Cavity Expansion in Cohesive-Frictional Soils with Limited Dilation

Géotechnique ◽  
2021 ◽  
pp. 1-20
Author(s):  
John P. Carter ◽  
Hai-Sui Yu
Keyword(s):  
Analytical Solution ◽  
Characteristic Length ◽  
Cavity Expansion ◽  
Cavity Pressure ◽  
Initial Radius ◽  
Limit Pressure ◽  
Zero Radius ◽  
Self Similar ◽  
Limiting Pressure ◽  
Frictional Soils

The problem of cavity expansion from zero radius has no characteristic length and therefore possesses a similarity solution, in which the cavity pressure remains constant and the continuing deformation is geometrically self-similar. In this case, the incremental velocity approach first used by Hill (1950) to analyze cavity expansion in Tresca materials can be extended to derive a solution for limiting pressure of cavity expansion in other types of material. In this article, a rigorous semi-analytical solution is derived, following Hill's incremental velocity method, for the expansion of cavities from zero initial radius in cohesive-frictional soils with limited dilation. In particular, the radius of the elastic-plastic interface c is used in this article as the time scale and the solution for the limit pressure has been presented. Solutions are evaluated for a number of cases representative of a range of cohesive-frictional and dilatant soils. A comparison is also made between the solutions presented here and previous solutions for cohesive-frictional soils that have unlimited (on-going) plastic dilation. In particular, the influence of limited plastic dilation on the cavity limit pressure is identified and discussed.

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