Superstrings,Cantorian-fractal Spacetime and Quantum-like Chaos in Atmospheric Flows

Atmospheric flows exhibit long-range spatiotemporal correlations manifested as the fractal geometry to the global cloud cover pattern concomitant with the inverse power law form for spectra of temporal fluctuations. Such non-local connections are ubiquitous to dynamical systems in nature and are identified as signatures of self-organized criticality. A recently developed cell dynamical system model for atmospheric flows predicts the observed self-organized criticality as a natural consequence of quantum-like mechanics governing flow dynamics. The model is based on the concept that spatial integration of enclosed small scale fluctuations results in the formation of large eddy circulations. The model predicts the following: (a) The flow structure consists of an overall logarithmic spiral trajectory with the quasiperiodic Penrose tiling pattern for the internal structure. (b) Conventional power spectrum analysis will resolve such spiral trajectories as a continuum of eddies with progressive increase in phase. (c) Increments in phase are concomitant with increases in period length and also represent the variance, a characteristic of quantum systems identified as Berrys phase. (d) The universal algorithm for self-organized criticality is expressed in terms of the universal Feigenbaum constants, a and d, as 2a²=πd, where the fractional volume intermittency of occurrence πd contributes to the total variance 2a² of fractal structures. (e) The Feigenbaum constants are expressed as functions of the golden mean. ( f) The quantum mechanical constants fine structure constant and ratio of proton mass to electron mass, which are pure numbers and are obtained by experimental observations only, are now derived in terms of the Feigenbaum constant, a. (g) Atmospheric flow structure follows Keplers third law of planetary motion. Therefore, Newtons inverse square law for gravitation also applies to eddy masses. The centripetal acceleration representing the inertial masses (of eddies) are equivalent to gravitational masses. The fractal-Cantorian structure of spacetime can also be visualized as a nested continuum of vortex (eddy) circulations, whose inertial masses obey Newtons inverse square law of gravitation. The model concept resembles a superstring model for subatomic dynamics which incorporates gravitational forces.     

Velocity fields with power-law spectra for modeling turbulent flows

We consider a generalization of homogeneous and isotropic Çinlar velocity fields to capture power-law spectra. The random velocity field is non-Gaussian with a representation motivated by Lagrangian and Eulerian observations. A wide range of turbulent flows can be generated by varying the stochastic parameters of the model. The velocity field being a functional version of Poisson shot-noise is constructed as the superposition of eddies randomized through their types and arrival times. We introduce a dependence between the eddy types which are spatial parameters and the decay parameter which is temporal. As a result, long-range correlation in space and a power-law spectrum previously used with Ornstein–Uhlenbeck velocity fields are achieved. We show that a corresponding power-law form for the probability distribution of the eddy diameter is sufficient for this result. The parameters of the probability distribution are further specified in view of Kolmogorov theory of the inertial scales. In particular, ∣k∣^−5/3 scaling of the spectrum is obtained. In the diffusive limit, we show that the parameters governing the decay and the arrival rate, and the speed of rotation of an eddy increase while its diameter decreases. That is, the eddies arrive fast, decay fast, and rotate fast with a small radius for a Brownian limit.     

The LANS-alpha turbulence model in primitive-equation ocean modeling

The Lagrangian-Averaged Navier-Stokes alpha (LANS-α) and Leray turbulence parameterizations are demonstrated in a primitive-equation ocean model using an idealized channel domain. For LANS-α, turbulence statistics such as kinetic energy, eddy kinetic energy, and temperature profiles resemble doubled-resolution statistics with the standard model. In a North Atlantic domain with realistic topography, the Leray model increases eddy activity. The LANS-α and Leray models show great promise to improve heat transport and temperature distributions in global ocean-climate simulations, as these processes depend on better resolution of eddies near the grid-scale. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

表示湍流场的一种新设想

本文仿照量子场论中描述基本粒子产生湮灭的方法来描述湍流中涡旋的产生和消灭.因为当某一基本粒子存在的时候,我们可以认为它是一个不变实体,而湍流中涡旋则在时间过程中不断变化和耗散,所以在类比应用量子场论方法时首先要解决怎样的湍流涡旋可认为是同一个涡旋.根据线性化理论的特点,我们认为在时间过程中按相似性规律变化时湍流涡旋才算是同一个涡旋,而把不具有相似性的涡旋出现或消失,看成是方程(2.6)中相互作用项φ_i所引起的湮火和产生的结果.然后,我们采用和量子场论相类似的产生算符和消灭算符来描述湍流涡旋系统所处的状态.最后,我们利用原N-S方程中相互作用项来构成涡旋相互作用的“Schr?dinger”方程以描述其状态的变化.这样就得类似于量子场论的湍流涡旋相互作用理论.     

Lagrangian coherent structures,transport and chaotic mixing in simple kinematic ocean models

Methods of dynamical system’s theory are used for numerical study of transport and mixing of passive particles (water masses, temperature, salinity, pollutants, etc.) in simple kinematic ocean models composed with the main Eulerian coherent structures in a randomly fluctuating ocean—a jet-like current and an eddy. Advection of passive tracers in a periodically-driven flow consisting of a background stream and an eddy (the model inspired by the phenomenon of topographic eddies over mountains in the ocean and atmosphere) is analyzed as an example of chaotic particle’s scattering and transport. A numerical analysis reveals a non-attracting chaotic invariant set Λ that determines scattering and trapping of particles from the incoming flow. It is shown that both the trapping time for particles in the mixing region and the number of times their trajectories wind around the vortex have hierarchical fractal structure as functions of the initial particle’s coordinates. Scattering functions are singular on a Cantor set of initial conditions, and this property should manifest itself by strong fluctuations of quantities measured in experiments. The Lagrangian structures in our numerical experiments are shown to be similar to those found in a recent laboratory dye experiment at Woods Hole. Transport and mixing of passive particles is studied in the kinematic model inspired by the interaction of a current (like the Gulf Stream or the Kuroshio) with an eddy in a noisy environment. We demonstrate a non-trivial phenomenon of noise-induced clustering of passive particles and propose a method to find such clusters in numerical experiments. These clusters are patches of advected particles which can move together in a random velocity field for comparatively long time. The clusters appear due to existence of regions of stability in the phase space which is the physical space in the advection problem.     

Cantorian Fractal Spacetime and Information in Neural Network of the Human Brain

The neural networks of the human brain act as very efficient parallel processing computers co-ordinating memory related responses to a multitude of input signals from sensory organs. Information storage, update and appropriate retrieval are controlled at the molecular level by the neuronal cytoskeleton which serves as the internal communication network within neurons. Information flow in the highly ordered parallel networks of the filamentous protein polymers which make up the cytoskeleton may be compared to atmospheric flows which exhibit long-range spatiotemporal correlations, i.e. long-term memory. Such long-range spatiotemporal correlations are ubiquitous to real world dynamical systems and is recently identified as signature of self-organized criticality or chaos. The signatures of self-organized criticality i.e. long-range temporal correlations have recently been identified in the electrical activity of the brain. The physics of self-organized criticality or chaos is not yet identified. A recently developed non-deterministic cell dynamical system model for atmospheric flows predicts the observed long-range spatiotemporal correlations as intrinsic to quantum-like mechanics governing flow dynamics. The model visualises large scale circulations to form as the result of spatial integration of enclosed small scale perturbations with intrinsic two-way ordered energy flow between the scales. Such a concept maybe applied for the collection and integration of a multitude of signals at the cytoskeletal level and manifested in activation of neurons in the macroscale. The cytoskeleton networks inside neurons may be the elementary units of a unified dynamic memory circulation network with intrinsic global response to local stimuli. A cell dynamical system model for human memory circulation network analogous to atmospheric circulations network is presented in this paper. The model like the analysis of Koruga et al. make use of certain connections to the concept of Cantorian-Fractal spacetime.     

A dynamical approach to large eddy simulation of turbulent flows: existence of weak solutions

We consider a system of equations coming from turbulence models using a large eddy simulation (LES) technique. The idea of this approach bases on decomposing the velocity into a part containing large flow structures and a part consisting of small scales. The equations for large‐scale quantities are derived from the Navier–Stokes equations with an additional constitutive relation for the contribution of small eddies. The mathematical difficulties in this paper focus on the non‐linear and non‐local turbulent term. Copyright © 2005 John Wiley & Sons, Ltd.     

The density of the transversal homoclinic points in the Henon-like strange attractors

We consider the Henon-like strange attractors Λ in a family which is a nonsingular perturbation of a d-modal family. The existence of the Henon-like strange attractors in this family was proved by Diaz et al. [Inventions Math. 125 (1996) 37]. We prove that the transversal homoclinic points are dense in Λ, and that hyperbolic periodic points are dense in Λ. Moreover, the hyperbolic periodic points that are heteroclinically related to the primary periodic point (transversal intersection of stable and unstable manifolds) are dense in Λ.     

Towards backward perturbation bounds for approximate dual Krylov subspaces

Given a matrix A,n by n, and two subspaces K and L of dimension m, we consider how to determine a backward perturbation E whose norm is as small as possible, such that k and L are Krylov subspaces of A+E and its adjoint, respectively. We first focus on determining a perturbation matrix for a given pair of biorthonormal bases, and then take into account how to choose an appropriate biorthonormal pair and express the Krylov residuals as a perturbation of the matrix A. Specifically, the perturbation matrix is globally optimal when A is Hermitian and K=L. The results show that the norm of the perturbation matrix can be assessed by using the norms of the Krylov residuals and those of the biorthonormal bases. Numerical experiments illustrate the efficiency of our strategy.     

On the eigenvalues of specially low-rank perturbed matrices

We study the eigenvalues of a matrix A perturbed by a few special low-rank matrices. The perturbation is constructed from certain basis vectors of an invariant subspace of A, such as eigenvectors, Jordan vectors, or Schur vectors. We show that most of the eigenvalues of the low-rank perturbed matrix stayed unchanged from the eigenvalues of A; the perturbation can only change the eigenvalues of A that are related to the invariant subspace. Existing results mostly studied using eigenvectors with full column rank for perturbations, we generalize the results to more general settings. Applications of our results to a few interesting problems including the Google’s second eigenvalue problem are presented.     

Numerical investigation of vortical evolution in a backward-facing step expansion flow

A numerical investigation of laminar flow over a backward-facing step is presented for the Reynolds number in the range of 50Re2500. The objective of this numerical investigation is to add to the existing knowledge of the backward-facing step flow to deepen our understanding of the expansion flow structure. We proceed with the analysis by verifying the computer code through the Pearson vortex problem. We then perform a parametric study by varying the Reynolds number, with the aim of determining whether or not there exists a critical Reynolds number, above which reattachment length on the channel floor decreases. We also concentrate on subjects that have been little explored in the flow, examples of which are the onset of a single vortex in the primary eddy and how the recirculating bubble containing flow reversals is torn into smaller eddies. Eddy distortion, leading to mobile saddle points, and the merging of eddies are also discussed in this study.     

Existence of KAM tori in the phase-space of lattice vortex systems

Under very mild conditions on the circulations, and for arbitrary vortex configurations, the existence of quasi-periodic solutions for a lattice vortex model is shown.Control over the size of the perturbation in the KAM-theory is achieved by uniform scalings of the circulations, the vortex separations, and time. Thus, additional restrictions on the circulations and the ratios of vortex separations are not required; this makes the result physically meaningful.     

Steady flows of Jeffrey–Hamel type from the half-plane into an infinite channel. 2. Linearization on a symmetric solution

The steady flow of a Navier–Stokes fluid is analysed in a two-dimensional asymmetric unbounded domain $\text{Ω}$, having two outlets to infinity, namely a half-plane K and a semi-infinite channel Π₋. Assuming that $\text{Ω}$ differs from a symmetric domain $\text{Ω}$ only by a small perturbation, we show the existence of a unique solution to the Navier–Stokes system in $\text{Ω}$. The solution is obtained as a perturbation of the symmetric solution and, at large distances in K, it takes the Jeffrey–Hamel form. Curiously, our results are valid only if the flux Φ, besides being small, is directed from the half-plane towards the semi-infinite channel, i.e. Φ is negative.The main ingredients in our proofs are estimates in weighted spaces with detached asymptotics and the study of a model problem resulting from the linearization around the symmetric solution which, for non-zero flux, leads, in contrast to the linearization around the zero solution, to the absence of compatibility conditions for the convective term and, for Φ<0, to the domination of nonlinear terms by the linear ones. We also provide some explicit examples of the domain perturbation.     

Absolute and relative Weyl theorems for generalized eigenvalue problems

Weyl-type eigenvalue perturbation theories are derived for Hermitian definite pencils A-λB, in which B is positive definite. The results provide a one-to-one correspondence between the original and perturbed eigenvalues, and give a uniform perturbation bound. We give both absolute and relative perturbation results, defined in the standard Euclidean metric instead of the chordal metric that is often used.     

Aerodynamic Sound Emission as a Singular Perturbation Problem

Sound emission from an eddy region involves three length scales: the eddy size I, wavelength λ of the sound, and a dimension L ofthe region. They are related by the Mach number M = l/λ, small for nearly incompressible eddies, and a parameter Λ = L/λ which plays no apparent role in current theories of aerodynamic sound. The theories of Lighthill and Ribner are examined in the case M ? 1, Λ ? 1. Ribner's result is found to contain an unacceptable improper integral. The utility of Lighthill's solution is found to depend on properties of the quadrupole moment T_ij that can be established only by studying the flow in more detail than Lighthill's theory allows. The general problem is posed in the form: given the body force f and vorticity ω find the density ρ and potential φ of the velocity u = ? × ψ{ ω } + ?φ The problem is solved for M ? 1, Λ ? 1 by matching a compressible eddy core scaled on I to a surrounding acoustic field scaled on λ. Lighthill's solution for ρ is shown to be adequate in both regions if T_ij is approximated by ρ₀υ_iυ_j, with v = ? × ψ. The situation M ? 1, Λ ? 1 is studied, and the conclusion is reached that sound emission from large bodies of turbulence is an open problem, Lighthill's theory notwithstanding.     

Series in a Lipschitz perturbation of the boundary for solving the Dirichlet problem

In a special Lipschitz domain treated as a perturbation of the upper half-space, we construct a perturbation theory series for a positive harmonic function with zero trace. The terms of the series are harmonic extensions to the half-space from its boundary of distributions defined by a recurrent formula and passage to the limit. The approximation error by a segment of the series is estimated via a power of the seminorm of the perturbation in the homogeneous Slobodestki? space b _N ^1?1/N . The series converges if the Lipschitz constant of the perturbation is small.     

Eigenvalue perturbation theory of classes of structured matrices under generic structured rank one perturbations

udy the perturbation theory of structured matrices under structured rank one perturbations, and then focus on several classes of complex matrices. Generic Jordan structures of perturbed matrices are identified. It is shown that the perturbation behavior of the Jordan structures in the case of singular J-Hamiltonian matrices is substantially different from the corresponding theory for unstructured generic rank one perturbation as it has been studied in [18, 28, 30, 31]. Thus a generic structured perturbation would not be generic if considered as an unstructured perturbation. In other settings of structured matrices, the generic perturbation behavior of the Jordan structures, within the confines imposed by the structure, follows the pattern of that of unstructured perturbations.     

Application of lattice Boltzmann method for incompressible viscous flows

Because of the presence of corner eddies that change in number and pattern the lid-driven cavity problem has been found suitable to study various aspects of the performance of solution algorithms for incompressible viscous flows. It retains all the difficult flow physics and is characterized by a large primary eddy at the centre and secondary eddies located near the cavity corners. In this work, lid-driven cavity flow is simulated by lattice Boltzmann method with single-relaxation-time and it is compared with those by lattice Boltzmann method with multi-relaxation-time and finite difference method. The effects of the Reynolds number on the size, centre position and number of vortices are studied in detail together with the flow pattern in the cavity. The close agreement of the results bears testimony to the validity of this relatively new approach. However lattice Boltzmann method with multi-relaxation-time model is seen to remove the difficulties faces by the lattice Boltzmann method with single-relaxation-time at higher Reynolds numbers.     

Robust exponential attractors for singularly perturbed Hodgkin-Huxley equations

Here we consider a singular perturbation of the Hodgkin-Huxley system which is derived from the Lieberstein's model. We study the associated dynamical system on a suitable bounded phase space, when the perturbation parameter ε (i.e., the axon specific inductance) is sufficiently small. We prove the existence of bounded absorbing sets as well as of smooth attracting sets. We deduce the existence of a smooth global attractor Aε. Finally we prove the main result, that is, the existence of a family of exponential attractors {Eε} which is Hölder continuous with respect to ε.     

Evaluating flame turbulence scale in diffusion combustion of diesel fuels

This paper presents results of mathematical simulations of dynamic flame behavior that occurs in burning diesel fuel, as well as experimental data on turbulent flame eddies provided by infrared thermography. The comparison of the simulated and experimental data documents a good match between the parameters of flame thermodynamics and the combustible gas flow in a turbulent eddy. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)