scholarly journals Scaling laws and warning signs for bifurcations of SPDEs

2018 ◽  
Vol 30 (5) ◽  
pp. 853-868
Author(s):  
CHRISTIAN KUEHN ◽  
FRANCESCO ROMANO
Keyword(s):  
Differential Equations ◽  
Scaling Laws ◽  
Scaling Law ◽  
Warning Signs ◽  
Covariance Operator ◽  
Large Classes ◽  
Practical Applications ◽  
Critical Transitions ◽  
Degenerate Eigenvalues ◽  
Adjoint Operators

Critical transitions (or tipping points) are drastic sudden changes observed in many dynamical systems. Large classes of critical transitions are associated with systems, which drift slowly towards a bifurcation point. In the context of stochastic ordinary differential equations, there are results on growth of variance and autocorrelation before a transition, which can be used as possible warning signs in applications. A similar theory has recently been developed in the simplest setting for stochastic partial differential equations (SPDEs) for self-adjoint operators in the drift term. This setting leads to real discrete spectrum and growth of the covariance operator via a certain scaling law. In this paper, we develop this theory substantially further. We cover the cases of complex eigenvalues, degenerate eigenvalues as well as continuous spectrum. This provides a fairly comprehensive theory for most practical applications of warning signs for SPDE bifurcations.

scholarly journals Asymmetric Unimodal Maps with Non-universal Period-Doubling Scaling Laws

2020 ◽  
Vol 379 (1) ◽  
pp. 103-143
Author(s):  
Oleg Kozlovski ◽  
Sebastian van Strien
Keyword(s):  
Scaling Laws ◽  
Scaling Law ◽  
Period Doubling ◽  
Cantor Sets ◽  
Unimodal Maps ◽  
Renormalization Theory

Abstract We consider a family of strongly-asymmetric unimodal maps $$\{f_t\}_{t\in [0,1]}$$ { f t } t ∈ [ 0 , 1 ] of the form $$f_t=t\cdot f$$ f t = t · f where $$f:[0,1]\rightarrow [0,1]$$ f : [ 0 , 1 ] → [ 0 , 1 ] is unimodal, $$f(0)=f(1)=0$$ f ( 0 ) = f ( 1 ) = 0 , $$f(c)=1$$ f ( c ) = 1 is of the form and $$\begin{aligned} f(x)=\left\{ \begin{array}{ll} 1-K_-|x-c|+o(|x-c|)&{} \text{ for } x<c, \\ 1-K_+|x-c|^\beta + o(|x-c|^\beta ) &{} \text{ for } x>c, \end{array}\right. \end{aligned}$$ f ( x ) = 1 - K - | x - c | + o ( | x - c | ) for x < c , 1 - K + | x - c | β + o ( | x - c | β ) for x > c , where we assume that $$\beta >1$$ β > 1 . We show that such a family contains a Feigenbaum–Coullet–Tresser $$2^\infty $$ 2 ∞ map, and develop a renormalization theory for these maps. The scalings of the renormalization intervals of the $$2^\infty $$ 2 ∞ map turn out to be super-exponential and non-universal (i.e. to depend on the map) and the scaling-law is different for odd and even steps of the renormalization. The conjugacy between the attracting Cantor sets of two such maps is smooth if and only if some invariant is satisfied. We also show that the Feigenbaum–Coullet–Tresser map does not have wandering intervals, but surprisingly we were only able to prove this using our rather detailed scaling results.

scholarly journals Scaling and Similarity of the Anisotropic Coherent Eddies in Near-Surface Atmospheric Turbulence

2018 ◽  
Vol 75 (3) ◽  
pp. 943-964 ◽  
Author(s):  
Khaled Ghannam ◽  
Gabriel G. Katul ◽  
Elie Bou-Zeid ◽  
Tobias Gerken ◽  
Marcelo Chamecki
Keyword(s):  
Scaling Laws ◽  
Velocity Structure ◽  
Scaling Law ◽  
Longitudinal Velocity ◽  
Structure Functions ◽  
Length Scale ◽  
Near Surface ◽  
Vegetation Canopies ◽  
Velocity Spectra ◽  
Attached Eddies

Abstract The low-wavenumber regime of the spectrum of turbulence commensurate with Townsend’s “attached” eddies is investigated here for the near-neutral atmospheric surface layer (ASL) and the roughness sublayer (RSL) above vegetation canopies. The central thesis corroborates the significance of the imbalance between local production and dissipation of turbulence kinetic energy (TKE) and canopy shear in challenging the classical distance-from-the-wall scaling of canonical turbulent boundary layers. Using five experimental datasets (two vegetation canopy RSL flows, two ASL flows, and one open-channel experiment), this paper explores (i) the existence of a low-wavenumber k−1 scaling law in the (wind) velocity spectra or, equivalently, a logarithmic scaling ln(r) in the velocity structure functions; (ii) phenomenological aspects of these anisotropic scales as a departure from homogeneous and isotropic scales; and (iii) the collapse of experimental data when plotted with different similarity coordinates. The results show that the extent of the k−1 and/or ln(r) scaling for the longitudinal velocity is shorter in the RSL above canopies than in the ASL because of smaller scale separation in the former. Conversely, these scaling laws are absent in the vertical velocity spectra except at large distances from the wall. The analysis reveals that the statistics of the velocity differences Δu and Δw approach a Gaussian-like behavior at large scales and that these eddies are responsible for momentum/energy production corroborated by large positive (negative) excursions in Δu accompanied by negative (positive) ones in Δw. A length scale based on TKE dissipation collapses the velocity structure functions at different heights better than the inertial length scale.

scholarly journals On the integrability problem for systems of partial differential equations in one unknown function, I

2019 ◽  
Vol 11 (4) ◽  
pp. 35-71 ◽  
Author(s):  
Antonio Kumpera
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Unknown Function ◽  
Contact Structures ◽  
Partial Differential ◽  
Practical Applications ◽  
First Order ◽  
Integrability Problem ◽  
Integration Problem ◽  
In The Beginning

We discuss the integration problem for systems of partial differential equations in one unknown function and special attention is given to the first order systems. The Grassmannian contact structures are the basic setting for our discussion and the major part of our considerations inquires on the nature of the Cauchy characteristics in view of obtaining the necessary criteria that assure the existence of solutions. In all the practical applications of partial differential equations, what is mostly needed and what in fact is hardest to obtains are the solutions of the system or, occasionally, some specific solutions. This work is based on four most enlightening Mémoires written by Élie Cartan in the beginning of the last century.

Adhesive and normal stress-dependent dynamic friction of a gelatin hydrogel

2021 ◽  
pp. 135065012110446
Author(s):  
Nitish Sinha ◽  
Arun Kumar Singh ◽  
Vinit Gupta ◽  
Jitendra Kumar Katiyar
Keyword(s):  
Experimental Data ◽  
Normal Stress ◽  
Scaling Law ◽  
Shear Velocity ◽  
Friction Model ◽  
Dynamic Friction ◽  
Practical Applications ◽  
Soft Solids ◽  
Gelatin Hydrogel ◽  
Stress Dependent

Adhesion and friction of soft solids on hard surfaces are the important properties for a variety of practical applications. In the present study, Coulomb's law of friction is used for characterizing adhesive friction as well as normal stress-dependent dynamic friction of a gelatin hydrogel on a fixed glass surface. The experimental data, concerning normal stress-dependent dynamic friction of different shear velocity, are obtained from literature. It is observed that both components of friction increase with shear velocity. More importantly, the scaling law shows that adhesive stress varies almost linearly with corresponding coefficient of friction of the hydrogel. A dynamic friction model is also used to analyze the same experimental data to predict a negative normal stress at which dynamic friction reduces to zero, and this result matches closely with the experimental value.

scholarly journals Origin of the scaling laws of sediment transport

2017 ◽  
Vol 473 (2197) ◽  
pp. 20160785 ◽  
Author(s):  
Sk Zeeshan Ali ◽  
Subhasish Dey
Keyword(s):  
Sediment Transport ◽  
Froude Number ◽  
Scaling Laws ◽  
Scaling Law ◽  
Bedload Transport ◽  
Inertial Range ◽  
Channel Width ◽  
Densimetric Froude Number ◽  
Relative Roughness ◽  
Contraction Ratio

In this paper, we discover the origin of the scaling laws of sediment transport under turbulent flow over a sediment bed, for the first time, from the perspective of the phenomenological theory of turbulence. The results reveal that for the incipient motion of sediment particles, the densimetric Froude number obeys the ‘(1 +  σ )/4’ scaling law with the relative roughness (ratio of particle diameter to approach flow depth), where σ is the spectral exponent of turbulent energy spectrum. However, for the bedforms, the densimetric Froude number obeys a ‘(1 +  σ )/6’ scaling law with the relative roughness in the enstrophy inertial range and the energy inertial range. For the bedload flux, the bedload transport intensity obeys the ‘3/2’ and ‘(1 +  σ )/4’ scaling laws with the transport stage parameter and the relative roughness, respectively. For the suspended load flux, the non-dimensional suspended sediment concentration obeys the ‘ − Z ’ scaling law with the non-dimensional vertical distance within the wall shear layer, where Z is the Rouse number. For the scour in contracted streams, the non-dimensional scour depth obeys the ‘4/(3 −  σ )’, ‘−4/(3 −  σ )’ and ‘−(1 +  σ )/(3 −  σ )’ scaling laws with the densimetric Froude number, the channel contraction ratio (ratio of contracted channel width to approach channel width) and the relative roughness, respectively.

Scaling Laws From Statistical Data and Dimensional Analysis

2004 ◽  
Vol 72 (5) ◽  
pp. 648-657 ◽  
Author(s):  
Patricio F. Mendez ◽  
Fernando Ordóñez
Keyword(s):  
Dimensional Analysis ◽  
Scaling Laws ◽  
Ad Hoc ◽  
Scaling Law ◽  
Engineering Systems ◽  
User Input ◽  
Backward Elimination ◽  
Dimensionless Groups ◽  
Complex Engineering ◽  
Complex Engineering Systems

Scaling laws provide a simple yet meaningful representation of the dominant factors of complex engineering systems, and thus are well suited to guide engineering design. Current methods to obtain useful models of complex engineering systems are typically ad hoc, tedious, and time consuming. Here, we present an algorithm that obtains a scaling law in the form of a power law from experimental data (including simulated experiments). The proposed algorithm integrates dimensional analysis into the backward elimination procedure of multivariate linear regressions. In addition to the scaling laws, the algorithm returns a set of dimensionless groups ranked by relevance. We apply the algorithm to three examples, in each obtaining the scaling law that describes the system with minimal user input.

Determination method of dynamic distorted scaling laws and applicable structure size intervals of a rotating thin-wall short cylindrical shell

2014 ◽  
Vol 229 (5) ◽  
pp. 806-817 ◽  
Author(s):  
Z Luo ◽  
YP Zhu ◽  
XY Zhao ◽  
DY Wang
Keyword(s):  
Cylindrical Shell ◽  
Scaling Laws ◽  
Good Accuracy ◽  
Scaling Law ◽  
Thin Wall ◽  
Determination Method ◽  
Governing Equations ◽  
7050 Aluminum Alloy ◽  
Boundary Functions

This study investigates the applicability of distortion models for predicting dynamic characteristics of a rotating thin-wall short cylindrical shell. The significance of this study is that it provides a necessary scaling law, applicable structure size intervals, and its boundary functions, which can guide the design of distortion models. Sensitivity analysis and governing equations are employed to establish the scaling law between the model and the prototype. Then a commonly used 7050 aluminum alloy cylindrical shell is analyzed as a prototype. The determination of applicable structure size intervals is discussed, and the boundary functions of the applicable structure size intervals are investigated. The applicability of the scaling law and the applicable intervals of rotating thin-wall short cylindrical shell are verified numerically. The results indicate that distortion models that satisfy the structure size applicable intervals can predict the characteristics of the prototype with good accuracy.

Effect of finite sampling time on estimation of Brownian fluctuation

2015 ◽  
Vol 767 ◽  
pp. 65-84 ◽  
Author(s):  
Shahram Pouya ◽  
Di Liu ◽  
Manoochehr M. Koochesfahani
Keyword(s):  
Diffusion Coefficient ◽  
Scaling Laws ◽  
Solid Wall ◽  
Exact Analytical Solution ◽  
Scaling Law ◽  
Time Interval ◽  
Free Diffusion ◽  
Mean Velocity ◽  
Hindered Diffusion ◽  
Near Wall

AbstractWe present a study of the effect of finite detector integration/exposure time $E$, in relation to interrogation time interval ${\rm\Delta}t$, on analysis of Brownian motion of small particles using numerical simulation of the Langevin equation for both free diffusion and hindered diffusion near a solid wall. The simulation result for free diffusion recovers the known scaling law for the dependence of estimated diffusion coefficient on $E/{\rm\Delta}t$, i.e. for $0\leqslant E/{\rm\Delta}t\leqslant 1$ the estimated diffusion coefficient scales linearly as $1-(E/{\rm\Delta}t)/3$. Extending the analysis to the parameter range $E/{\rm\Delta}t\geqslant 1$, we find a new nonlinear scaling behaviour given by $(E/{\rm\Delta}t)^{-1}[1-((E/{\rm\Delta}t)^{-1})/3]$, for which we also provide an exact analytical solution. The simulation of near-wall diffusion shows that hindered diffusion of particles parallel to a solid wall, when normalized appropriately, follows with a high degree of accuracy the same form of scaling laws given above for free diffusion. Specifically, the scaling laws in this case are well represented by $1-((1+{\it\epsilon})(E/{\rm\Delta}t))/3$, for $E/{\rm\Delta}t\leqslant 1$, and $(E/{\rm\Delta}t)^{-1}[1-((1+{\it\epsilon})(E/{\rm\Delta}t)^{-1})/3]$, for $E/{\rm\Delta}t\geqslant 1$, where the small parameter ${\it\epsilon}$ depends on the size of the near-wall domain used in the estimation of the diffusion coefficient and value of $E$. For the range of parameters reported in the literature, we estimate ${\it\epsilon}<0.03$. The near-wall simulations also show a bias in the estimated diffusion coefficient parallel to the wall even in the limit $E=0$, indicating an overestimation which increases with increasing time delay ${\rm\Delta}t$. This diffusion-induced overestimation is caused by the same underlying mechanism responsible for the previously reported overestimation of mean velocity in near-wall velocimetry.

scholarly journals Scaling laws for pressure, temperature, and ionization with two-temperature equation-of-state effects in laser-produced plasmas

1992 ◽  
Vol 10 (1) ◽  
pp. 23-40 ◽  
Author(s):  
H. Szichman ◽  
S. Eliezer
Keyword(s):  
Equation Of State ◽  
Scaling Laws ◽  
Interpolation Method ◽  
Scaling Law ◽  
Plasma Medium ◽  
Degree Of Ionization ◽  
Ablation Pressure ◽  
Two Temperature ◽  
Temperature Equation ◽  
Thermal Conductivities

A two-temperature equation of state (EOS) for a plasma medium was developed. The cold-electron temperature is taken from semiempirical calculations, while the thermal contribution of the electrons is calculated from the Thomas–Fermi–Dirac model. The ion EOS is obtained by the Gruneisen–Debye solid–gas interpolation method. These EOS are well behaved and are smoothed over the whole temperature and density regions, so that the thermodynamical derivatives are also well behaved. Moreover, these EOS were used to calculate the average ionization 〈Z〉 and 〈Z2〉 of the plasma medium. Furthermore, the thermal conductivities have been calculated with the use of an extrapolation between the conductivities in the solid and plasma (Spitzer) states. The two-temperature EOS, the average ionization 〈Z〉 and 〈Z2〉, and the thermal conductivities for electrons and ions were introduced into a two-fluid hydrodynamic code to calculate the laser-plasma interaction in carbon, aluminum, copper, and gold slab targets. It was found that the two EOS are important mainly from the ablation surface outward (toward the laser). In particular, the creation of cavitons in the distribution of the electrons is predicted here, especially for light materials such as aluminum. These studies enable us also to establish that the commonly used exponential scaling laws of the type Pa = A[I/(1014 W/cm2)]α for the ablation pressure and similar laws for the temperature are valid only for absorbed laser intensities in the range 3 × 1012-3 × 1014 W/cm2, while the degree of ionization (at the corona) follows a quite different scaling law. We also found that the parameters A and α. in the above expression are dependent on material, laser wavelength, and pulse shape. Thus we determined for the ablation pressure, using a trapezoidal Nd laser pulse, that α varies between 0.78 and 0.84 and that A varies between 14 and 8 Mbar for 6 ≤ Z ≤ 79. Beyond the range of validity the scaling laws may give values at least twice as large as those obtained by the simulation.

Scaling Effect on Dynamic Impact Response of Structures in Fluid Fields

2013 ◽  
Author(s):  
Zhongheng Guo ◽  
Lingyu Sun ◽  
Taikun Wang ◽  
Junmin Du ◽  
Han Li ◽  
...  
Keyword(s):  
Numerical Simulation ◽  
Dimensional Analysis ◽  
Large Scale ◽  
Scaling Laws ◽  
Scaling Law ◽  
Fluid Pressure ◽  
Structural Safety ◽  
Scale Model ◽  
Small Scale ◽  
Water Tank

At the conceptual design phase of a large-scale underwater structure, a small-scale model in a water tank is often used for the experimental verification of kinematic principles and structural safety. However, a general scaling law for structure-fluid interaction (FSI) problems has not been established. In the present paper, the scaling laws for three typical FSI problems under the water, rigid body moves at a given kinematic equation or is driven by time-dependent fluids with given initial condition, as well as elastic-plastic body moves and then deforms subject to underwater impact loads, are investigated, respectively. First, the power laws for these three types of FSI problems were derived by dimensional analysis method. Then, the laws for the first two types were verified by numerical simulation. In addition, a multipurpose small-scale water sink test device was developed for numerical model updating. For the third type of problem, the dimensional analysis is no longer suitable due to its limitation on identifying the fluid pressure and structural stress, a simulation-based procedure for dynamics evaluation of large-scale structure was provided. The results show that, for some complex FSI problems, if small-scale prototype is tested safely, it doesn’t mean the full-scale product is also safe if both their pressure and stress are the main concerns, it needs further demonstration, at least by numerical simulation.

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