沙粒级配和沙丘分布的分形分析

研究了沙粒级配和沙丘分布的分形特征,并对其分形维数作了定量计算。结果表明:沙粒级配的粒径分布的分形维数在2.43~2.69之间,沙丘分布的盒分形维数为1.28,这对用数学模型进行风沙动力学的定量研究有重要的参考价值。

Fractal Analysis of Sand-grading and Dune Distribution

There are a great deal of scale invariant processes in nature, the concept of fractals provides a means of quantifying these processes. In this paper, the fractal analysis is applied to the problems of the sand grading and dune distribution. In many cases the frequency mass distribution of fragments satisfies the fractal condition. This is taken as evidence that the fragmentation mechanism is scale invariant. Fragments produced by weathering, explosions, and impacts often satisfy a fractal distribution condition over a wide range of scales. The number size distributions for sand grains also satisfy the fractal condition in many cases. In this paper, a power law relation is used to grain size distribution of sand, N=KR^-D_F, where K is a proportionality constant, N is the total number of particles with a characteristic linear dimension which is greater than a given comparative size R, thus the power distribution is equivalent to a fractal distribution with D_F as the fractal dimension. In order to apply the number based size relationship to the analysis of sands, assumption must be made regarding the unit weight of individual particles since particle size distribution of sand is determined by mass comparison using mesh screen analysis. The number based size relationship can be used for sand by modeling individual particles as uniform sphere and standard densities. By adjusting the size of the uniform particles to coincide with sieve screen dimensions, the number of particles bounded by each sieve can be determined by dividing the total weight of material retained on each sieve by the weight of an individual particle. The D_F for each grain size distribution was obtained by first developing a N depending r logarithmic plot and then determining the slope of the best fit line through the data points using least square linear regression. The data from the grain size distribution test results plotted as relatively straight lines on the N depending r logarithmic plot and therefore it was concluded that fractal theory was applicable. This paper presents an example of a typical grain size distribution curve and the corresponding fractal plot, similar plots were obtained for each of the nine grain size distributions of the desert dune sand in China. The box counting dimension can be used to quantify the dune distribution, in deriving the fractal dimension of the dune distribution, one fundamental requirement is that the size of the measurement step must be able to be varied. Although any random variance of measurement step size is valid theoretically in the dimensioning process, the box counting dimension involves a screen with the equal grids defined as the measurement step length in this paper. By knowing the projection of the dunes, the number of the grids including a portion of dune project can be calculated. Using consecutively smaller size of the grids results in different estimates of the sand distribution and provides for a varied level of scrutiny of the sand distribution. As the step length becomes smaller and smaller, an increasingly better portrayed of the actual sand dune distribution develops, and a series of sand values are obtained, the fractal dimension of sand dune distribution is determined by the slope k of the verse logarithmic plot, D=-k. As an example, a typical value of D of a desert dune is given to be 1.28 in this paper. Our primary conclusions are as follows: The grain size distribution of sands can often be considered fractal, and the typical fractal dimensions are between 2.43 and 2.69. As expected, the results of this investigation indicate that the higher the fragmentation fractal dimension, the higher the relative percentage of fine grained material within the distribution. The fragmentation of rock material is a consequence of the scale invariance of the fragmentation mechanism in that the zones of weakness along which fragmentation occurs can be found at all levels of scrutiny, and that the zones of weakness have a hierarchical structure where a large-scale fracture is a consequence of fractures at smaller scales. Additionally, fol lowing the physical process of weathering and erosion, rock fragments into smaller and smaller sizes in a complex and non-differentiale manner, and therefore appears random and irregular. It is therefore reasonable to assume that the grain size distributions of naturally sands are of ten fractal. The existence of a fractal dimension for number-size distribution is direct evidence for scale invariance. There are a lot of spatial self-organization in geomorphology:from ripples through dunes to megadunes of wind-blown sand, fractal analysis can be applied to dune distribution to provide quantifying values. On a more fundamental level, study of this particular natural systems can illuminate the role and character of feedback in the formation of patterns, especially, in understanding simple surfacial patterns emerge from complicated transport mechanisms and determining the general processes that influence the spatial scale of the patterns.