Relation of Fractal Dimension and Sinuosity coefficient in Meandering River

Meandering rivers, as prime examples of nature tendency to reach a regular form, have been the focus of many researchers. These rivers contain a series of alternating bends and curves, joined by short straight intervals across their plan and flow over gently sloping channels in which sedimentary load settles as point loads on the inner wall of the bend. River morphology studies the geometrical form of rivers in the plan, longitudinal profile (channel slope), cross section geometry and topography. Morphological analysis of meandering rivers is performed in two stages: determination of independent variables (flow and sedimentary discharge), calculation of geometrical parameters of river morphology through physical or experimental relationships. Such parameters are mostly investigated using Euclidean geometry. Sinuosity, for example, has been calculated with Euclidean attitude in Cartesian coordinates. Quantifying geometrical parameters of meandering rivers morphology in a Euclidean approach arises problems such as inaccuracy or complexity in calculation. Instead, Fractal geometry is widely used in river engineering in recent literature, due to its more detailed perspective of an object and its non-Euclidean properties. In Fractal Geometry, the mathematical space classified into one-, two-, and three-dimensional spaces on the basis of Euclidean geometry, is expressed as is fractal spaces in which the irregularities of the shapes are expressed in terms of fractal dimension (a real dimension and not necessarily a natural number). Single-fractal analyses are mainly carried out using methods such as box counting, variation, scale change, and Brownian motion methods, while multi-fractal analyses include methods such as spectral or wavelet analysis. Box counting is one of the fractal dimension calculation methods, widely used in rivers and shorelines. In this method, the set of points is meshed on a curve or a surface with squares (boxes) and the number of squares covering each part of the curve is calculated. Variation method also is one of the most accurate and popular method that can be used to calculate fractal dimension in various fields, however it is rarly used in river engineering up to now. In the present study, the fractal dimension in the Mond River was calculated over a 15-year period from 2000 to 2015. Mond river, with 685 km length is one of the most important rivers in southern Iran, originating in Fars province and flowing into the Persian Gulf through Bushehr province. Two fractal methods namely, box counting and variational methods were applied to calculate fractal dimensions in I) the whole river II) 3 longest bends III) 13 meanders. The results were then compared with those of sinusoidal coefficient. To calculate the fractal dimension by changes method, the area covered by different characteristic lengths is calculated in fixed intervals. Then, for different characteristic lengths the area covered by meander curve is calculated using code written in Matlab. The correlation coefficient values for the river coordinate data at each of the river intervals are obtained and compared in the bends. In the box counting method, different dimensions of the box and therefore different grids were considered. Then, in order to calculate the fractal dimension, the number of boxes involved was calculated for different widths using codes written in Matlab. Variations in the box width with the number of boxes in logarithmic scale are used to calculate the fractal dimension in the box counting method. The values for fractal dimension ranged between 1.01 to 1.09 and 1.0027 to 1.991 using box counting method and changes method, respectively. Additionally, the calculated fractal dimension values were compared with sinusoidal coefficients in three long meanders and fourteen bends of the river. Results indicated high correlations (R2 = 0.94-0.99) between fractal and sinusoidal coefficients in the meanders. The fractal dimension obtained in 2005 (1.05) was larger than those in other years. The largest fractal dimension was met in the second meander, with a value of 1.06. Highest sinusoidal coefficient was also found in the second meander indicating a direct relationship between these two parameters. There was a high correlation coefficient (close to 1) between the fractal dimension and the sinusoidal coefficient in the long meanders. A considerably high correlation coefficient of 0.96 was obtained between the parameters of the sinusoidal coefficient and the central angle calculated from the morphological analysis, which indicates a direct relationship between these parameters. The correlation coefficient of 0.85 between the fractal dimension parameters and the sinusoidal coefficient as well as the correlation coefficient of 0.86 between the fractal dimension parameters with the central angle indicates that the fractal dimension parameter is an appropriate indicator for expressing the changes and complexity of the meandering rivers.