scholarly journals First-Order Structure Function Analysis of Statistical Scale Invariance in the AIRS-Observed Water Vapor Field

2012 ◽  
Vol 25 (16) ◽  
pp. 5538-5555 ◽  
Author(s):  
Kyle G. Pressel ◽  
William D. Collins
Keyword(s):  
Water Vapor ◽  
Structure Function ◽  
Power Law ◽  
Scale Invariance ◽  
Structure Functions ◽  
Scale Dependence ◽  
Global Maximum ◽  
Order Structure ◽  
Scaling Exponents ◽  
First Order

Abstract The power-law scale dependence, or scaling, of first-order structure functions of the tropospheric water vapor field between 58°S and 58°N is investigated using observations from the Atmospheric Infrared Sounder (AIRS). Power-law scale dependence of the first-order structure function would indicate that the water vapor field exhibits statistical scale invariance. Directional and directionally independent first-order structure functions are computed to assess the directional dependence of derived first-order structure function scaling exponents (H) for a range of scales from 50 to 500 km. In comparison to other methods of assessing statistical scale invariance, the methodology used here requires minimal assumptions regarding the homogeneity of the spatial distribution of data within regions of analysis. Additionally, the methodology facilitates the evaluation of anisotropy and quantifies the extent to which the structure functions exhibit scale invariance. The spatial and seasonal dependence of the computed scaling exponents are explored. Minimum scaling exponents at all levels are shown to occur proximate to the equator, while the global maximum is shown to occur in the middle troposphere near the tropical–subtropical margin of the winter hemisphere. From a detailed analysis of AIRS maritime scaling exponents, it is concluded that the AIRS observations suggest the existence of two scaling regimes in the extratropics. One of these regimes characterizes the statistical scale invariance the free troposphere with H approximately = 0.55 and a second that characterizes the statistical scale invariance of the boundary layer with H approximately = ⅓.

scholarly journals Extended power-law scaling of air permeabilities measured on a block of tuff

2012 ◽  
Vol 16 (1) ◽  
pp. 29-42 ◽  
Author(s):  
M. Siena ◽  
A. Guadagnini ◽  
M. Riva ◽  
S. P. Neuman
Keyword(s):  
Power Law ◽  
Structure Functions ◽  
Self Similarity ◽  
Scaling Exponents ◽  
Multi Scale ◽  
Sample Structure ◽  
Heavy Tailed ◽  
Non Gaussian ◽  
Nonlinear Fashion ◽  
Extended Power

Abstract. We use three methods to identify power-law scaling of multi-scale log air permeability data collected by Tidwell and Wilson on the faces of a laboratory-scale block of Topopah Spring tuff: method of moments (M), Extended Self-Similarity (ESS) and a generalized version thereof (G-ESS). All three methods focus on q-th-order sample structure functions of absolute increments. Most such functions exhibit power-law scaling at best over a limited midrange of experimental separation scales, or lags, which are sometimes difficult to identify unambiguously by means of M. ESS and G-ESS extend this range in a way that renders power-law scaling easier to characterize. Our analysis confirms the superiority of ESS and G-ESS over M in identifying the scaling exponents, ξ(q), of corresponding structure functions of orders q, suggesting further that ESS is more reliable than G-ESS. The exponents vary in a nonlinear fashion with q as is typical of real or apparent multifractals. Our estimates of the Hurst scaling coefficient increase with support scale, implying a reduction in roughness (anti-persistence) of the log permeability field with measurement volume. The finding by Tidwell and Wilson that log permeabilities associated with all tip sizes can be characterized by stationary variogram models, coupled with our findings that log permeability increments associated with the smallest tip size are approximately Gaussian and those associated with all tip sizes scale show nonlinear variations in ξ(q) with q, are consistent with a view of these data as a sample from a truncated version (tfBm) of self-affine fractional Brownian motion (fBm). Since in theory the scaling exponents, ξ(q), of tfBm vary linearly with q we conclude that nonlinear scaling in our case is not an indication of multifractality but an artifact of sampling from tfBm. This allows us to explain theoretically how power-law scaling of our data, as well as of non-Gaussian heavy-tailed signals subordinated to tfBm, are extended by ESS. It further allows us to identify the functional form and estimate all parameters of the corresponding tfBm based on sample structure functions of first and second orders.

scholarly journals Towards mapping turbulence in the intra-cluster medium

2019 ◽  
Vol 629 ◽  
pp. A144 ◽  
Author(s):  
E. Cucchetti ◽  
N. Clerc ◽  
E. Pointecouteau ◽  
P. Peille ◽  
F. Pajot
Keyword(s):  
Structure Function ◽  
Spatial Scales ◽  
Companion Paper ◽  
Structure Functions ◽  
Order Structure ◽  
Sampling Variance ◽  
Function Estimation ◽  
X Ray ◽  
Spatially Resolved Spectroscopy ◽  
Field Unit

X-ray observations of the hot gas filling the intra-cluster medium (ICM) provide a wealth of information on the dynamics of clusters of galaxies. The global equilibrium of the ICM is believed to be ensured by non-thermal and thermal pressure support sources, among which gas movements and the dissipation of energy through turbulent motions. Accurate mapping of turbulence using X-ray emission lines is challenging due to the lack of spatially resolved spectroscopy. Only future instruments such as the X-ray Integral Field Unit (X-IFU) on Athena will have the spatial and spectral resolution to quantitatively investigate the ICM turbulence over a broad range of spatial scales. Powerful diagnostics for these studies are line shift and the line broadening maps, and the second-order structure function. When estimating these quantities, instruments will be limited by uncertainties of their measurements, and by the sampling variance (also known as cosmic variance) of the observation. Here, we extend the formalism started in our companion Paper I to include the effect of statistical uncertainties of measurements in the estimation of these line diagnostics, in particular for structure functions. We demonstrate that statistics contribute to the total variance through different terms, which depend on the geometry of the detector, the spatial binning and the nature of the turbulent field. These terms are particularly important when probing the small scales of the turbulence. An application of these equations is performed for the X-IFU, using synthetic turbulent velocity maps of a Coma-like cluster. Results are in excellent agreement with the formulas both for the structure function estimation (≤3%) and its variance (≤10%). The expressions derived here and in Paper I are generic, and ensure an estimation of the total errors in any X-ray measurement of turbulent structure functions. They also open the way for optimisations in the upcoming instrumentation and in observational strategies.

scholarly journals Shallow water wave turbulence

2019 ◽  
Vol 874 ◽  
pp. 1169-1196 ◽  
Author(s):  
Pierre Augier ◽  
Ashwin Vishnu Mohanan ◽  
Erik Lindborg
Keyword(s):  
Reynolds Number ◽  
Structure Function ◽  
Shallow Water ◽  
Water Wave ◽  
Structure Functions ◽  
Wave Turbulence ◽  
Order Structure ◽  
Third Order ◽  
Shallow Water Wave ◽  
The Mean

The dynamics of irrotational shallow water wave turbulence forced at large scales and dissipated at small scales is investigated. First, we derive the shallow water analogue of the ‘four-fifths law’ of Kolmogorov turbulence for a third-order structure function involving velocity and displacement increments. Using this relation and assuming that the flow is dominated by shocks, we develop a simple model predicting that the shock amplitude scales as $(\unicode[STIX]{x1D716}d)^{1/3}$, where $\unicode[STIX]{x1D716}$ is the mean dissipation rate and $d$ the mean distance between the shocks, and that the $p$th-order displacement and velocity structure functions scale as $(\unicode[STIX]{x1D716}d)^{p/3}r/d$, where $r$ is the separation. Then we carry out a series of forced simulations with resolutions up to $7680^{2}$, varying the Froude number, $F_{f}=(\unicode[STIX]{x1D716}L_{f})^{1/3}/c$, where $L_{f}$ is the forcing length scale and $c$ is the wave speed. In all simulations a stationary state is reached in which there is a constant spectral energy flux and equipartition between kinetic and potential energy in the constant flux range. The third-order structure function relation is satisfied with a high degree of accuracy. Mean energy is found to scale approximately as $E\sim \sqrt{\unicode[STIX]{x1D716}L_{f}c}$, and is also dependent on resolution, indicating that shallow water wave turbulence does not fit into the paradigm of a Richardson–Kolmogorov cascade. In all simulations shocks develop, displayed as long thin bands of negative divergence in flow visualisations. The mean distance between the shocks is found to scale as $d\sim F_{f}^{1/2}L_{f}$. Structure functions of second and higher order are found to scale in good agreement with the model. We conclude that in the weak limit, $F_{f}\rightarrow 0$, shocks will become denser and weaker and finally disappear for a finite Reynolds number. On the other hand, for a given $F_{f}$, no matter how small, shocks will prevail if the Reynolds number is sufficiently large.

Applicability of Kolmogorov's and Monin's equations of turbulence

1997 ◽  
Vol 353 ◽  
pp. 67-81 ◽  
Author(s):  
REGINALD J. HILL
Keyword(s):  
Structure Function ◽  
Structure Functions ◽  
Inertial Range ◽  
Homogeneous Turbulence ◽  
Order Structure ◽  
Third Order ◽  
Local Isotropy ◽  
The Third ◽  
Moderate Reynolds Number ◽  
Wide Range

The equation relating second- and third-order velocity structure functions was presented by Kolmogorov; Monin attempted to derive that equation on the basis of local isotropy. Recently, concerns have been raised to the effect that Kolmogorov's equation and an ancillary incompressibility condition governing the third-order structure function were proven only on the restrictive basis of isotropy and that the statistic involving pressure that appears in the derivation of Kolmogorov's equation might not vanish on the basis of local isotropy. These concerns are resolved. In so doing, results are obtained for the second- and third-order statistics on the basis of local homogeneity without use of local isotropy. These results are applicable to future studies of the approach toward local isotropy. Accuracy of Kolmogorov's equation is shown to be more sensitive to anisotropy of the third-order structure function than to anisotropy of the second-order structure function. Kolmogorov's 4/5 law for the inertial range of the third-order structure function is obtained without use of the incompressibility conditions on the second- and third-order structure functions. A generalization of Kolmogorov's 4/5 law, which applies to the inertial range of locally homogeneous turbulence at very large Reynolds numbers, is shown to also apply to the energy-containing range for the more restrictive case of stationary, homogeneous turbulence. The variety of derivations of Kolmogorov's and Monin's equations leads to a wide range of applicability to experimental conditions, including, in some cases, turbulence of moderate Reynolds number.

Percolation

2017 ◽  
Author(s):  
Paul Charbonneau
Keyword(s):  
Phase Transitions ◽  
Water Vapor ◽  
Power Law ◽  
Critical Behavior ◽  
Scale Invariance ◽  
Granular Medium ◽  
Statistical Physics ◽  
Two Dimensions ◽  
Natural Systems ◽  
Fractal Clusters

This chapter explores a lattice-based system where complex structures can arise from pure randomness: percolation, typically described as the passage of liquid through a porous or granular medium. In its more abstract form, percolation is an exemplar of criticality, a concept in statistical physics related to phase transitions. A classic example of criticality is liquid water boiling into water vapor, or freezing into ice. The chapter first provides an overview of percolation in one and two dimensions before discussing the use of a tagging algorithm for identifying and sizing clusters. It then considers fractal clusters on a lattice at the percolation threshold, scale invariance of power-law behavior, and critical behavior of natural systems. The chapter includes exercises and further computational explorations, along with a suggested list of materials for further reading.

Scaling Analysis of the China France Oceanography Satellite Along Track Wave and Wind Data

2020 ◽  
Author(s):  
Yang Gao ◽  
Francois G Schmitt ◽  
Jianyu Hu ◽  
Yongxiang Huang
Keyword(s):  
Power Spectrum ◽  
Power Law ◽  
Function Analysis ◽  
Second Order ◽  
Order Structure ◽  
Scaling Analysis ◽  
Scaling Exponents ◽  
Fourier Power Spectrum ◽  
Along Track ◽  
Dual Power

<p>Turbulence or turbulence-like phenomena are ubiquitous in nature, often showing a power-law behavior of the Fourier power spectrum in either spatial or temporal domains. This power-law behavior is due to interactions among different scales of motion, and to the absence of characteristic scale among several scale ranges. It can be further interpreted in the framework of turbulent cascade with movements on continuous range of scales. The power-law feature and the associate cascade picture are vitally important to our understanding of the ocean and atmosphere dynamics. In this work, we consider the China France Oceanography SATellite (CFOSAT) data in the general framework of ocean and atmosphere multi-scale dynamics. We apply both Fourier power spectrum analysis and second-order structure-function analysis, used in the fields of turbulence, to extract multiscale information from the wind speed (WS) and significant wave-height (Hs) data provided by CFOSAT project. The data analyzed here are along track data spatially collected from 29<sup>th</sup> July to 31<sup>th</sup> December 2019. The measured Fourier power spectrums for both WS and Hs illustrate a dual power-law behavior respectively from 5 to 25 km, and 30 to 500 km with measured scaling exponents β close to 2 and 5/3. The measured second-order structure-functions confirm the existence of the dual power-law behavior. The corresponding measured scaling exponents  ζ(2) close to 1 and 2/3 for the spatial scales mentioned above. Our preliminary results confirm the relevance of using multiscale statistical tools and turbulent theory to characterize the large-scale movements of both ocean and atmosphere.</p>

Equilibrium similarity solution of the turbulent transport equation along the centreline of a round jet

2015 ◽  
Vol 772 ◽  
pp. 740-755 ◽  
Author(s):  
H. Sadeghi ◽  
P. Lavoie ◽  
A. Pollard
Keyword(s):  
Experimental Data ◽  
Kinetic Energy ◽  
Structure Function ◽  
Transport Equation ◽  
Power Law ◽  
Turbulent Jet ◽  
Order Structure ◽  
Current Analysis ◽  
Power Law Decay ◽  
The Mean

A novel similarity-based form is derived of the transport equation for the second-order velocity structure function of$\langle ({\it\delta}q)^{2}\rangle$along the centreline of a round turbulent jet using an equilibrium similarity analysis. This self-similar equation has the advantage of requiring less extensive measurements to calculate the inhomogeneous (decay and production) terms of the transport equation. It is suggested that the normalised third-order structure function can be uniquely determined when the normalised second-order structure function, the power-law exponent of$\langle q^{2}\rangle$and the decay rate constants of$\langle u^{2}\rangle$and$\langle v^{2}\rangle$are available. In addition, the current analysis demonstrates that the assumption of similarity, combined with an inverse relation between the mean velocity$U$and the streamwise distance$x-x_{0}$from the virtual origin (i.e. $U\propto (x-x_{0})^{-1}$), is sufficient to predict a power-law decay for the turbulence kinetic energy ($\langle q^{2}\rangle \propto (x-x_{0})^{m}$), rather than requiring a power-law decay ($m=-2$) as an additionalad hocassumption. On the basis of the current analysis, it is suggested that the mean kinetic energy dissipation rate,$\langle {\it\epsilon}\rangle$, varies as$(x-x_{0})^{m-2}$. These theoretical results are tested against new experimental data obtained along the centreline of a round turbulent jet as well as previously published data on round jets for$11\,000\leqslant \mathit{Re}_{D}\leqslant 184\,000$over the range$10\leqslant x/D\leqslant 90$. For the present experiments,$\langle q^{2}\rangle$exhibits power-law behaviour with$m=-1.83$. The validity of this solution is confirmed using other experimental data where$\langle q^{2}\rangle$follows a power law with$-1.89\leqslant m\leqslant -1.78$. The present similarity form of the transport equation for$\langle ({\it\delta}q)^{2}\rangle$is also shown to be closely satisfied by the experimental data.

Third-order structure function relations for quasi-geostrophic turbulence

2007 ◽  
Vol 572 ◽  
pp. 255-260 ◽  
Author(s):  
ERIK LINDBORG
Keyword(s):  
Structure Function ◽  
Three Dimensional ◽  
Structure Functions ◽  
Order Structure ◽  
Third Order ◽  
Longitudinal Structure ◽  
The Third ◽  
Inverse Cascade ◽  
Geostrophic Turbulence ◽  
Potential Enstrophy

We derive two third-order structure function relations for quasi-geostrophic turbulence, one for the forward cascade of potential enstrophy and one for the inverse cascade of energy. These relations are the counterparts of Kolmovorov's (1941) four-fifths law for the third-order longitudinal structure functions of three-dimensional turbulence.

Triplet and Higher Correlations in the Long Wavelength Limit

1975 ◽  
Vol 53 (4) ◽  
pp. 372-376 ◽  
Author(s):  
L. E. Ballentine ◽  
A. Lakshmi
Keyword(s):  
Structure Function ◽  
Electronic Properties ◽  
Lower Order ◽  
Liquid Metals ◽  
Structure Functions ◽  
Order Structure ◽  
Long Wavelength ◽  
Wavelength Limit ◽  
Ionic Potential ◽  
Superposition Approximation

The nth order structure function of a liquid, defined as the ensemble average of a product of n Fourier components of the atomic density, is studied in the limit of long wavelength for one or more Fourier components. It is shown by means of fluctuation theory that these limits are simply related to lower order structure functions and their derivatives with respect to pressure, and that they will be small in magnitude for normal liquids. This conclusion is important for the study of electronic properties of liquid metals because the screened ionic potential is always large in the long wavelength limit. Thus large spurious contributions may be obtained from the use of approximate structure functions that do not satisfy the correct long wavelength limits. In this respect the Kirkwood superposition approximation is very unsatisfactory.

scholarly journals Exact second-order structure-function relationships

2002 ◽  
Vol 468 ◽  
pp. 317-326 ◽  
Author(s):  
REGINALD J. HILL
Keyword(s):  
Structure Function ◽  
Scaling Laws ◽  
Velocity Structure ◽  
Structure Functions ◽  
Energy Dissipation Rate ◽  
Second Order ◽  
Order Structure ◽  
Third Order ◽  
Navier Stokes Equation ◽  
Local Isotropy

Equations that follow from the Navier–Stokes equation and incompressibility but with no other approximations are ‘exact’. Exact equations relating second- and third- order structure functions are studied, as is an exact incompressibility condition on the second-order velocity structure function. Opportunities for investigations using these equations are discussed. Precisely defined averaging operations are required to obtain exact averaged equations. Ensemble, temporal and spatial averages are all considered because they produce different statistical equations and because they apply to theoretical purposes, experiment and numerical simulation of turbulence. Particularly simple exact equations are obtained for the following cases: (i) the trace of the structure functions, (ii) DNS that has periodic boundary conditions, and (iii) an average over a sphere in r-space. Case (iii) introduces the average over orientations of r into the structure-function equations. The energy dissipation rate ε appears in the exact trace equation without averaging, whereas in previous formulations ε appears after averaging and use of local isotropy. The trace mitigates the effect of anisotropy in the equations, thereby revealing that the trace of the third-order structure function is expected to be superior for quantifying asymptotic scaling laws. The orientation average has the same property.

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