Abstract
A new diffusion model was introduced to simulate the profile evolution of the river consisting of gravelly bed materials and receiving no water from tributaries. In this model, diffusion coefficient a (x) is expressed as an exponential function of position x;
a(x)=k exp (rx)
where k and r are constants. Sediment flux J per unit width is proposed to be proportional to the gradient ux of river bed;
J=- a(x)ux Therefore, change of river bed height with time is expressed by the following equation;ut=-Jx=∂/∂x{a(x)ux=}
This equation is wellposed if the upper boundary condition is given by the flux and grain size of sediment supplied at the top of alluvial fan, and the lower one by the sea level as a free boundary problem concerning the passively moving river mouth and the material excha-nge ratio to the sea.
The steady solution of the differential quation shows an exponential curve:u=(Je/kr)exp(-rx)+C
for the longitudinal profile of the graded river, where Je is equilibrium flux of sediment and C is an integral constant that will be determined by the sea level.
The author examined the model as a Markov process. Sternberg's abrasion law was newly interpreted on a basis of stochastic concept as follows: the stay-time of grains at a given position of river bed is randomly changed after progressive decreasing of grain weight/diameter, and the exponential diffusion coefficient is derived from this random time-change for the basic diffusion process (Wiener process) and Shulits's empirical equation;ux∝p (or d)
where p (d) is weight (diameter) of grains.
In conclusion, a clear explanation of the empirical equation for the graded rivers;u=C1exp(-rx)+C2(C1, C2; const.)
was given under the above-mentioned theoretical basis.
