Power law wave number spectra of fractal particle distributions advected by flowing fluids

1996 ◽  
Vol 8 (9) ◽  
pp. 2426-2434 ◽  
Author(s):  
Arthur Namenson ◽  
Thomas M. Antonsen ◽  
Edward Ott
Keyword(s):  
Power Law ◽  
Wave Number ◽  
Particle Distributions ◽  
Fractal Particle

scholarly journals HELICAL MAGNETIC FIELDS FROM INFLATION

2009 ◽  
Vol 18 (09) ◽  
pp. 1395-1411 ◽  
Author(s):  
LEONARDO CAMPANELLI
Keyword(s):  
Magnetic Fields ◽  
Power Law ◽  
Background Field ◽  
Wave Number ◽  
Conformal Time ◽  
De Sitter ◽  
Simple Power ◽  
Dimensionless Constant ◽  
Real Positive Number

We analyze the generation of seed magnetic fields during de Sitter inflation considering a noninvariant conformal term in the electromagnetic Lagrangian of the form [Formula: see text], where I(ϕ) is a pseudoscalar function of a nontrivial background field ϕ. In particular, we consider a toy model that could be realized owing to the coupling between the photon and either a (tachyonic) massive pseudoscalar field or a massless pseudoscalar field nonminimally coupled to gravity, where I follows a simple power law behavior I(k,η) = g/(-kη)β during inflation, while it is negligibly small subsequently. Here, g is a positive dimensionless constant, k the wave number, η the conformal time, and β a real positive number. We find that only when β = 1 and 0.1 ≲ g ≲ 2 can astrophysically interesting fields be produced as excitation of the vacuum, and that they are maximally helical.

Multi-dimensional complex phase transition model by catastrophe theory developing Kolmogorov turbulence theory

2020 ◽  
Vol 34 (15) ◽  
pp. 2050159
Author(s):  
Zhuo Zhou ◽  
Jiu Hui Wu ◽  
Xiao Liang ◽  
Mei Lin ◽  
Xiao Yang Yuan ◽  
...  
Keyword(s):  
Phase Transition ◽  
Power Law ◽  
Catastrophe Theory ◽  
Equilibrium Phase ◽  
Turbulence Theory ◽  
Wave Number ◽  
Transition Model ◽  
Kolmogorov Turbulence ◽  
Equilibrium Phase Transition ◽  
Non Equilibrium

In this paper, a novel multi-dimensional complex non-equilibrium phase transition model is put forward to describe quantitatively the physical development process of turbulence and develop the Kolmogorov turbulence theory from the catastrophe theory, in which the well-known −5/3 power law is only a special case in this paper proving the accuracy of our methods. Catastrophe theory is a highly generalized mathematical tool that summarizes the laws of non-equilibrium phase transition. Every control variable in catastrophe theory could be skillfully expanded into multi-parameter multiplication with different indices and the relationship among these characteristic indices can be determined by dimensionless analysis. Thus, the state variables can be expressed quantitatively with multiple parameters, and the multi-dimensional non-equilibrium phase transition theory is established. As an example, by adopting the folding catastrophe model, we strictly derive out the quantitative relationship between energy and wave number with respect to a new scale index [Formula: see text] to quantitative study the whole process of the laminar flow to turbulence, in which [Formula: see text] varies from [Formula: see text] to [Formula: see text] corresponding to energy containing range and [Formula: see text] to energy containing scale where [Formula: see text] power law is deduced, and at [Formula: see text] the [Formula: see text] law of Kolmogorov turbulence theory is obtained, and fully developed turbulence phase starts at [Formula: see text] giving [Formula: see text] law. Furthermore, this theory presented is verified by our wind tunnel experiments. This novel non-equilibrium phase transition methods cannot only provide a new insight into the turbulence model, but also be applied to other non-equilibrium phase transitions.

scholarly journals On the spectral distribution of kinetic energy in large-scale atmospheric flow

1998 ◽  
Vol 5 (3) ◽  
pp. 187-192
Author(s):  
A. Wiin-Nielsen
Keyword(s):  
Steady State ◽  
Kinetic Energy ◽  
Power Law ◽  
Spectral Distribution ◽  
Large Scale ◽  
Wave Number ◽  
Initial State ◽  
Atmospheric Flow ◽  
Linear Interaction ◽  
Spectral Components

Abstract. A one-dimensional form of the equation of motion with forcing and dissipation is formulated in the spectral domain and used to make long term integrations from which the spectral distribution of the kinetic energy is determined The forcing in the wave number domain is determined in advance and kept constant for the duration of the time integrations. The dissipation is proportional to the second derivative of the velocity. The applied equation is made non-dimensional by selecting a length scale from which the time scale and the velocity scale may be determined. The resulting equation contains no parameters apart from the forcing. The integrations use a large number of spectral components and no approximation is made with respect to the non-linear interaction among the spectral components. Starting from an initial state in which all the velocity components are set to zero the equation is integrated for a long time to see if it reaches a steady state. The spectral distribution of the kinetic energy is determined in the steady state, and it is found that the distribution, in agreement with observational studies, may be approximated by a power law of the form n-3 within certain wave number regions. The wave numbers for which the -3 power law applies is found between the region of maximum forcing and the dissipation range. The intensity of the maximum forcing is varied to see how the resulting steady state varies. In addition, the maximum number of spectral components is varied. However, the available computing power sets an upper limit to the number of components.

scholarly journals Formation of quasi-power law weak Langmuir turbulence spectrum by harmonic generation

2004 ◽  
Vol 11 (2) ◽  
pp. 267-274
Author(s):  
P. H. Yoon
Keyword(s):  
Power Law ◽  
Langmuir Wave ◽  
Wave Number ◽  
Wave Turbulence ◽  
Langmuir Turbulence ◽  
Turbulence Spectrum ◽  
Langmuir Waves ◽  
Multi Scale ◽  
History Of ◽  
Simulation Results

Abstract. Langmuir wave turbulence generated by a beam-plasma interaction has been studied since the early days of plasma physics research. Despite a long history of investigation on this subject, among the outstanding issues is the generation of harmonic Langmuir waves observed in both laboratory and computer-simulated experiments. However, the phenomenon has not been adequately explained in terms of theory, nor has it been fully characterized by means of numerical simulations. In this paper, a theory of harmonic Langmuir wave generation is put forth and tested against the Vlasov simulation results. It is found that the harmonic Langmuir mode spectra exhibit quasi power-law feature implying a multi-scale structure in both frequency and wave number space spanning several orders of magnitude.

scholarly journals A Note on the Ocean Surface Roughness Spectrum*

2011 ◽  
Vol 28 (3) ◽  
pp. 436-443 ◽  
Author(s):  
Paul A. Hwang
Keyword(s):  
Surface Roughness ◽  
Wind Speed ◽  
Power Law ◽  
Field Data ◽  
Wave Spectrum ◽  
Phase Speed ◽  
Short Wave ◽  
Wave Number ◽  
Power Law Function ◽  
Good Agreement

Abstract In a recent study, the dimensionless surface roughness spectrum has been empirically parameterized as a power-law function of the dimensionless wind speed expressed as the ratio of wind friction velocity and phase speed of the surface roughness wave component. The wave-number-dependent proportionality coefficient, A, and exponent, a, of the power-law function are derived from field measurements of the short-wave spectrum. To extend the roughness spectrum model beyond the wavenumber range of field data, analytical functions are formulated such that A and a approach their asymptotic limits: A0 and a0 toward the lowest wavenumber, and A∞ and a∞ toward the highest wavenumber. Of the four asymptotic values, A∞ is considered most questionable for the lack of reference information. When applied to the normalized radar cross-section (NRCS) computation, the results are in good agreement (within about 2 dB) with field data or geophysical model functions (GMFs) for incidence angles between 20° and 40° but significant underestimation occurs for higher incidence angles. The comparison study of NRCS computation offers helpful guidelines for adjusting the asymptotic factors, especially the numerical value of A∞. Improved agreement between the computed NRCS (vertical polarization) using the new roughness spectrum with GMF is expanded to incidence angles between 20° and 60°. The wind speed range of good agreement between calculation and GMF is below about 15 m s−1 for Ku band and about 30 m s−1 for C band.

scholarly journals Density/area power-law models for separating multi-scale anomalies of ore and toxic elements in stream sediments in Gejiu mineral district, Yunnan Province, China

2010 ◽  
Vol 7 (3) ◽  
pp. 4273-4293 ◽  
Author(s):  
Q. Cheng ◽  
Q. Xia ◽  
W. Li ◽  
S. Zhang ◽  
Z. Chen ◽  
...  
Keyword(s):  
Energy Density ◽  
Power Law ◽  
Stream Sediments ◽  
Landscape Patterns ◽  
Wave Number ◽  
Copper Production ◽  
Spectral Energy ◽  
Spectral Energy Density ◽  
Mineral District ◽  
Fractal Filtering

Abstract. This contribution introduces a fractal filtering technique newly developed on the basis of a spectral energy density vs. area power-law model in the context of multifractal theory. It can be used to map anisotropic singularities of geochemical landscapes created from geochemical concentration values in various surface media such as soils, stream sediments, tills and water. A geochemical landscape can be converted into a Fourier domain in which the spectral energy density is plotted against the area (in wave number units), and the relationship between the spectrum energy density (S) and the area (A) enclosed by the above-threshold spectrum energy density can be fitted by power-law models. Mixed geochemical landscape patterns can be fitted with different S-A power-law models in the frequency domain. Fractal filters can be defined according to these different S-A models and used to decompose the geochemical patterns into components with different self-similarities. The fractal filtering method was applied to a geochemical dataset from 7349 stream sediment samples collected from Gejiu mineral district, which is famous for its word-class tin and copper production. Anomalies in three different scales were decomposed from total values of the trace elements As, Sn, Cu, Zn, Pb, and Cd. These anomalies generally correspond to various geological features and geological processes such as sedimentary rocks, intrusions, fault intersections and mineralization.

Time-dependent scaling relations and a cascade model of turbulence

1978 ◽  
Vol 88 (2) ◽  
pp. 369-391 ◽  
Author(s):  
Thomas L. Bell ◽  
Mark Nelkin
Keyword(s):  
Energy Spectrum ◽  
Power Law ◽  
Isotropic Turbulence ◽  
Cascade Model ◽  
Time Dependent ◽  
Wave Number ◽  
Grid Turbulence ◽  
A Value ◽  
Model Equations ◽  
Spatial Dimensionality

We study the time-dependent solutions of a nonlinear cascade model for homogeneous isotropic turbulence first introduced by Novikov & Desnyansky. The dynamical variables of the model are the turbulent kinetic energies in discrete wave-number shells of thickness one octave. The model equations contain a parameter C whose size governs the amount of energy cascaded to small wavenumbers relative to the amount cascaded to large wavenumbers. We show that the equations permit scale-similar evolution of the energy spectrum. For 0 ≤ C ≤ 1 and no external force, the freely evolving energy spectrum displays the Kolmogorov k power law, and the total energy decreases in time as a power t−w, where the exponent w depends on the value of C. Grid-turbulence experiments seem to favour a value of C in the range 0·3-0·6. In the presence of an external stirring force acting near a wavenumber k0, the model predicts, in addition to the Kolmogorov k spectrum for k > k0, a scale-similar flow of energy to wavenumbers k < k0. This backward energy flow falls off as a power law in time, and establishes a stationary energy spectrum for k < k0 which is a power law in k less steep than k. We discuss the similarity of the behaviour of the model for C > 1 to the behaviour of turbulent fluid for a spatial dimensionality near 2. The model is shown to approach the Kovasznay and the Leith diffusion approximation equations in the limit in which the thickness of the wavenumber shells approaches zero. However, the cascade model with finite shell thicknesses appears to behave in a more physically reasonable way than the limiting differential equations.

A quantitative nonequilibrium phase transition theory for analyzing the turbulence development process

2019 ◽  
Vol 33 (22) ◽  
pp. 1950243
Author(s):  
Zhuo Zhou ◽  
Jiu Hui Wu ◽  
Xiao Liang ◽  
Xiaoyang Yuan ◽  
Mei Lin ◽  
...  
Keyword(s):  
Phase Transition ◽  
Power Law ◽  
Development Process ◽  
Transition Theory ◽  
Wave Number ◽  
Index Formula ◽  
Nonequilibrium Phase Transition ◽  
Nonequilibrium Phase ◽  
Catastrophe Model ◽  
Scaling Index

In this paper, a quantitative nonequilibrium multi-dimensional phase transition theory is proposed for describing the turbulence spectrum (energy E with wave number k and scaling index [Formula: see text]) of the turbulence development process by a fold catastrophe model. Each of the control variables in this catastrophe model is subtly expressed into a relative multi-parameter multiplication, and then the state variable can be quantitatively described by these parameters. By using this nonequilibrium phase transition theory, the quantitative relationship in the process of turbulence formation can be strictly derived through dimensionless analysis. Therefore, the turbulence development process can be described with respect to a scaling index [Formula: see text], in which there exists an energy containing range with −1.12 power law (E [Formula: see text] k[Formula: see text]) when [Formula: see text] varies from −2 to −1.2, and an inertial subrange with −1.69 power law (E [Formula: see text] k[Formula: see text]) that is almost identical with the famous Kolmogorov’s −5/3 power law when [Formula: see text] varies from −1.2 to −0.8, and then the dissipation range with −2.52 power law (E [Formula: see text] k[Formula: see text]) when [Formula: see text] varies from −0.8 to 0. Furthermore, this quantitative nonequilibrium phase transition theory has been verified by the corresponding theoretical comparison and experiment. This theory provides not only a new understanding of turbulence, but also a new perspective for other complex nonequilibrium phase transitions.

scholarly journals Density/area power-law models for separating multi-scale anomalies of ore and toxic elements in stream sediments in Gejiu mineral district, Yunnan Province, China

2010 ◽  
Vol 7 (10) ◽  
pp. 3019-3025 ◽  
Author(s):  
Q. Cheng ◽  
Q. Xia ◽  
W. Li ◽  
S. Zhang ◽  
Z. Chen ◽  
...  
Keyword(s):  
Energy Density ◽  
Power Law ◽  
Stream Sediments ◽  
Landscape Patterns ◽  
Wave Number ◽  
Copper Production ◽  
Spectral Energy ◽  
Spectral Energy Density ◽  
Mineral District ◽  
Fractal Filtering

Abstract. This contribution introduces a fractal filtering technique newly developed on the basis of a spectral energy density vs. area power-law model in the context of multifractal theory. It can be used to map anisotropic singularities of geochemical landscapes created from geochemical concentration values in various surface media such as soils, stream sediments, tills and water. A geochemical landscape can be converted into a Fourier domain in which the spectral energy density is plotted against the area (in wave number units), and the relationship between the spectrum energy density (S) and the area (A) enclosed by the above-threshold spectrum energy density can be fitted by power-law models. Mixed geochemical landscape patterns can be fitted with different S-A power-law models in the frequency domain. Fractal filters can be defined according to these different S-A models and used to decompose the geochemical patterns into components with different self-similarities. The fractal filtering method was applied to a geochemical dataset from 7,349 stream sediment samples collected from Gejiu mineral district, which is famous for its word-class tin and copper production. Anomalies in three different scales were decomposed from total values of the trace elements As, Sn, Cu, Zn, Pb, and Cd. These anomalies generally correspond to various geological features and geological processes such as sedimentary rocks, intrusions, fault intersections and mineralization.

Power Law Wave-Number Spectra of Scum on the Surface of a Flowing Fluid

1995 ◽  
Vol 75 (19) ◽  
pp. 3438-3441 ◽  
Author(s):  
Thomas M. Antonsen, Jr. ◽  
Arthur Namenson ◽  
Edward Ott ◽  
John C. Sommerer
Keyword(s):  
Power Law ◽  
Wave Number ◽  
Flowing Fluid
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