Analogies in constitution of natural erosive structures and the Koch's Curve: preconditionsof qualitative comparison

The ontogenesis of a classical fractal set - the Koch's Curve and classicalgeomorphological object (the erosive form) - is studied in this article at the qualitative level. Modeling natural objects - valleys of theNozhnitsy Ravine and of the Zhilaya Rel'nya River - have been chosen for the qualitative comparison in the Medvedica River basin,Saratov region, Lysogorskiy district. As a result of detailed studying of valleys of the ravine and of the river in field conditions and bymethods of remote sounding, their cross-section profiles are constructed, which reflect the basic stages of development ofgeomorphological object modelling. The Koch's (snowflake) Curve is a reference example of fractals. The construction of the Koch'sCurve begins with an equipotential triangle, then on each party the average third is replaced with a broken line consisting of two piecesof the same length. This procedure (iteration) repeats indefinitely, therefore, the Koch's Curve becomes a fractal object. The growth ofthe erosive valley is qualitatively similar to the consistent increase in length of the Koch's Curve. The surface deprived of erosiveelements (repeating the primary relief and ideally non-existent in nature) is qualitatively similar to the initiator. A prefractal of the firstorder complicated by the V-shaped form is similar to an erosive furrow arising on a slope at which weak depth is present and there is nolateral erosion. A prefractal of the second order is similar to a gully, which increases depth and extends in sizes. Each following step ofthe erosive cycle corresponds to the increase of the prefractal order. The analysis of the fractal properties of a series of cross-sectionprofiles allows solving important problems of structural geology and geomorphology by means of the device of nonlinear dynamics: bymeans of working out and introduction of original techniques of search, revealing spatial features of geological structures and theirseparate elements, elements of fault and crack network, etc. At the present stage of development within urbanized territories manyravines and beams, which qualitatively represented high usages of prefractals, test return changes. The filling in of ravines returns thesurface to the position of the initiator, and the high order of a prefractal remains "buried". In the geoecological relation the real surfaceof a high order of a prefractal represents the steadiest system and anthropogenic changes are inevitably reflected in ecological safety ofdwellings of the population and condition of engineering constructions. On the basis of the comparative analysis a law is observed: thequalitative change of the erosive form in transition from one stage to another corresponds to the quantitative change of prefractal orderand, finally, to complication of fractal borders of the Koch's Curve.

Keywords

Authors

References