Scaling Laws in Geophysics: Application to Potential Fields of Methods Based on the Laws of Self-similarity and Homogeneity

2015 ◽  
pp. 1-18 ◽  
Author(s):  
Maurizio Fedi
Keyword(s):  
Scaling Laws ◽  
Potential Fields ◽  
Self Similarity

scholarly journals Solar wind low-frequency magnetohydrodynamic turbulence: extended self-similarity and scaling laws

1996 ◽  
Vol 3 (4) ◽  
pp. 247-261 ◽  
Author(s):  
V. Carbone ◽  
P. Veltri ◽  
R. Bruno
Keyword(s):  
Solar Wind ◽  
Scaling Laws ◽  
Velocity Structure ◽  
Fluid Flows ◽  
Structure Functions ◽  
Point Of View ◽  
Self Similarity ◽  
Extended Self ◽  
Scaling Exponents ◽  
Magnetohydrodynamic Turbulence

Abstract. In this paper we review some of the work done in investigating the scaling properties of Magnetohydrodynamic turbulence, by using velocity fluctuations measurements performed in the interplanetary space plasma by the Helios spacecraft. The set of scaling exponents ξq for the q-th order velocity structure functions, have been determined by using the Extended Self-Similarity hypothesis. We have found that the q-th order velocity structure function, when plotted vs. the 4-th order structure function, displays a range of self-similarity which extends over all the lengths covered by measurements, thus allowing for a very good determination of ξq. Moreover the results seem to show that the scaling exponents are the same regardless the various observation periods considered. The obtained scaling exponents have been compared with the results of some intermittency models for Kraichnan's turbulence, derived in the framework of infinitely divisible fragmentation processes, showing the good agreement between these models and our observations. Finally, on the basis of the actually available data sets, we show that scaling laws in Solar Wind turbulence seem to be different from turbulent scaling laws in the ordinary fluid flows. This is true for high-order velocity structure functions, while low-order velocity structure functions show the same scaling laws. Since our measurements involve length scales which extend over many order of magnitude where dissipation is practically absent, our results show that Solar Wind turbulence can be regarded as a testing bench for the investigation of general scaling behaviour in turbulent flows. In particular our results strongly support the point of view which attributes a key role to the inertial range dynamics in determining the intermittency characteristics in fluid flows, in contrast with the point of view which attributes intermittency to a finite Reynolds number effect.

scholarly journals Eulerian and Lagrangian Structure Function’s Scaling Exponents in Turbulent Channel Flow

2006 ◽  
Vol 61 (12) ◽  
pp. 624-628
Author(s):  
Jian-Ping Luo ◽  
Tatsuo Ushijima ◽  
Osami Kitoh ◽  
Zhi-Ming Lu ◽  
Yu-Lu Liu ◽  
...  
Keyword(s):  
Channel Flow ◽  
Scaling Laws ◽  
Turbulent Channel Flow ◽  
Self Similarity ◽  
Extended Self ◽  
Scaling Exponents ◽  
Turbulent Channel Flows ◽  
Frame Work ◽  
Simulation Results ◽  
Similarity Method

The relation between Eulerian structure function’s scaling exponents and Lagrangian ones in turbulent channel flows is explored both theoretically and numerically. A nonlinear parametric transformation between Eulerian structure function’s scaling exponents and Lagrangian ones is derived, following Landau and Novikov’s frame work. This relation is then compared to some known experimental and numerical results, but mainly to our DNS (direct numerical simulation) results of a fully developed channel flow with Reτ = 100. The scaling exponents are evaluated in terms of the ESS (extended self-similarity) method, since the Reynolds number is too low to make the standard scaling laws applicable. The agreement between theory and simulation is satisfactory

scholarly journals Limits of Predictability of a Global Self-Similar Routing Model in a Local Self-Similar Environment

Atmosphere ◽  
2020 ◽  
Vol 11 (8) ◽  
pp. 791
Author(s):  
Nicolas Velasquez ◽  
Ricardo Mantilla
Keyword(s):  
Scaling Laws ◽  
Hydraulic Geometry ◽  
Distributed Hydrological Model ◽  
Self Similarity ◽  
Cross Sectional ◽  
Actual Observation ◽  
Scaling Parameters ◽  
Self Similar ◽  
Distributed Hydrological Models ◽  
Similarity Laws

Regional Distributed Hydrological models are being adopted around the world for prediction of streamflow fluctuations and floods. However, the details of the hydraulic geometry of the channels in the river network (cross sectional geometry, slope, drag coefficients, etc.) are not always known, which imposes the need for simplifications based on scaling laws and their prescription. We use a distributed hydrological model forced with radar-derived rainfall fields to test the effect of spatial variations in the scaling parameters of Hydraulic Geometric (HG) relationships used to simplify routing equations. For our experimental setup, we create a virtual watershed that obeys local self-similarity laws for HG and attempt to predict the resulting hydrographs using a global self-similar HG parameterization. We find that the errors in the peak flow value and timing are consistent with the errors that are observed when trying to replicate actual observation of streamflow. Our results provide evidence that local self-similarity can be a more appropriate simplification of HG scaling laws than global self-similarity.

Preliminary Evidence for a Theory of the Fractal City

1996 ◽  
Vol 28 (10) ◽  
pp. 1745-1762 ◽  
Author(s):  
M Batty ◽  
Y Xie
Keyword(s):  
Fractal Dimension ◽  
Urban Form ◽  
Scaling Laws ◽  
Fractal Dimensions ◽  
Power Laws ◽  
Preliminary Evidence ◽  
Residential Development ◽  
Self Similarity ◽  
Estimation Techniques ◽  
Data Source

In this paper, we argue that the geometry of urban residential development is fractal. Both the degree to which space is filled and the rate at which it is filled follow scaling laws which imply invariance of function, and self-similarity of urban form across scale. These characteristics are captured in population density functions based on inverse power laws whose parameters are fractal dimensions. First we outline the relevant elements of the theory in terms of scaling relations and then we introduce two methods for estimating fractal dimension based on varying the size of cities and the scale at which their form is detected. Exact and statistical estimation techniques are applied to each method respectively generating dimensions which measure the extent and the rate of space filling. These methods are then applied to residential development patterns in six industrial cities in the northeastern United States, with an innovative data source from the TIGER/Line files. The results support the theory of the fractal city outlined in books by Batty and Longley and Frankhauser, but with the clear conclusion that different scale and estimation techniques generate different types of fractal dimension.

scholarly journals Shapes of river networks

2018 ◽  
Vol 474 (2215) ◽  
pp. 20180081 ◽  
Author(s):  
Robert S. Yi ◽  
Álvaro Arredondo ◽  
Eric Stansifer ◽  
Hansjörg Seybold ◽  
Daniel H. Rothman
Keyword(s):  
Aspect Ratio ◽  
Scaling Laws ◽  
Aridity Index ◽  
River Network ◽  
River Networks ◽  
Self Similarity ◽  
Scaling Exponents ◽  
Structural Consequence ◽  
Climatic Aridity ◽  
Morphological Expression

River network scaling laws describe how their shape varies with their size. However, the regional variation of this size-dependence remains poorly understood. Here we show that river network scaling laws vary systematically with the climatic aridity index. We find that arid basins do not change their proportions with size, while humid basins do. To explore why, we study an aspect ratio L ⊥ / L ∥ between basin width L ⊥ and basin length L ∥ . We find that the aspect ratio exhibits a dependence on climate and argue that this can be understood as a structural consequence of the confluence angle. We then find that, in humid basins, the aspect ratio decreases with basin size, which we attribute to a common hydrogeological hierarchy. Our results offer an explanation of the variability in network scaling exponents and suggest that the absence of self-similarity in humid basins can be understood as a morphological expression of subsurface processes.

SCALING LAWS IN A TURBULENT BAROCLINIC INSTABILITY

Fractals ◽  
1995 ◽  
Vol 03 (02) ◽  
pp. 297-314 ◽  
Author(s):  
C. SERIO ◽  
V. TRAMUTOLI
Keyword(s):  
Power Law ◽  
Scaling Laws ◽  
Spatial Scales ◽  
Baroclinic Instability ◽  
Satellite Image ◽  
Horizontal Resolution ◽  
Turbulent Motion ◽  
Self Similarity ◽  
Wavelet Transform Analysis ◽  
Self Similar

This work provides an empirical investigation of scaling laws in a cloud system generated and advected by a strong baroclinic instability. An infrared satellite image with a spatial (horizontal) resolution of about 1 km has been analyzed. The presence of two sizeable and unmistakable scaling regions, one extending from 1 to 15 km and characterized by a power law with an exponent close to 1, the other stretching from 20 km up to 100 km and characterized by a power law with exponent close to 1/3, have been revealed by variogram analysis. These two scaling laws are in agreement with the idea of scale invariance of the turbulent motion and also suggest the presence of a self-similar structure. To explore this possibility, wavelet transform analysis at different spatial scales has been used. Our findings are that self-similarity is present at the smallest scales, but this universal characteristic may be masked by non-universal effects which influence the homogeneity of the underlying turbulent motion. The implications of the two scaling exponents, 1 and 1/3, are also discussed.

Self-focusing of thin liquid jets

2007 ◽  
Vol 464 (2089) ◽  
pp. 197-206 ◽  
Author(s):  
Laurent Duchemin
Keyword(s):  
Free Surface ◽  
Time Evolution ◽  
Scaling Laws ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Liquid Jets ◽  
Self Similarity ◽  
Long Time ◽  
Self Similar ◽  
Long Time Evolution

The nonlinear evolution of an initially perturbed free surface perpendicularly accelerated, or of an initially flat free surface subject to a perturbed velocity profile, gives rise to the emergence of thin spikes of fluid. We are investigating the long-time evolution of a thin inviscid jet of this kind, subject or not to a body force acting in the direction of the jet itself. A fully nonlinear theory for the long-time evolution of the jet is given. In two dimensions, the curvature of the tip scales like t 3 , where t is time, and the peak undergoes an overshoot in acceleration which evolves like t −5 . In three dimensions, the jet evolves towards an axisymmetric shape, and the curvature and the overshoot in acceleration obey asymptotic laws in t 2 and t −4 , respectively. The asymptotic self-similar shape of the spike is found to be a hyperbola in two dimensions, a hyperboloid in three dimensions. Scaling laws and self-similarity are confronted with two-dimensional computations of the Richtmyer–Meshkov instability.

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