A Concept Based on the Diffusion Equation Model of Concave River Profile Development

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  • NOGAMI Michio
    Department of Geography, College of Humanities and Sciences, Nihon University

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  • 河川縦断形発達の拡散モデルについて
  • カセン ジュウダンケイ ハッタツ ノ カクサン モデル ニ ツイテ

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Abstract

A diffusion equation developed by this author:<br>∂u/∂t=∂/∂x{dexp(rx)∂u/∂x} …Eq. 1<br>is able to simulate concave river profile development. The diffusion coefficient d exp(rx) is given by an exponential of distance x and its basis is attributable to a downstream exponential decrease in gravel size. The model gives an exponential curve as the equilibrium steady-state profile.<br>In this paper, the author develops a concept based on the model which gives the same equilibrium profile. The model is described in discrete quantities for simulation programs:<br>Δu=exp(-rx)Qin-Qout …Eq. 2<br>where Δu is the change in river bed height, and influx Qin is the product of the upstream gradient by the diffusion coefficient d exp(rx) as above, and Qout is outflux in the same manner. Here, the flux decrease of gravel is assumed to be<br>Qin(1-exp(-rx))<br>at distance x upstream and r is constant.<br>The model (Eq. 2) shows deviation from the conservation law, but the grain size of flux loss products is so fine that it can be moved directly to the river mouth or the delta as suspended load. Therefore the balance between erosion and deposition is conserved for the whole drainage basin. The model (Eq. 2) has two advantages: the model is described faithfully based on geomorphologic evidence; and in the operation of numerical simulation, the algorithm of the model is tolerant against running into divergent or runaway states.

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