Abstract
The purpose of this paper is to test theoretically one of mathematical models for longitudinal river profiles and to evaluate it by a numerical simulation of terrace development of the Tama River. Significance of mathematical model describing the change of height with time in landform profiles developed by Hirano (1976),
Ut=aUxx+bUx
was tested with the view of clarifying flow rate of sediment,
J=(aUx+bU) where Uxx is curvature, Ux gradient and U height of river bed. The first term of the equation expresses deposition and the second does erosion in the concave river profile. This second order differential equation is the same one that describes the diffusion process having drift.
The analogy of river profile development to diffusion process is basically justified either by the existence of Brownian motion of particles in the river system or by a presumption apriori that the flow rate of sediment is proportional to the gradient of river bed. But it is difficult to identify what is the drift under constraint in the river system, because the flow by the forced drift is opposite in direction of the gradient of river bed and the diffusion flow. Although this model has these weak points in physical basis, it has a big advantage that its steady solution;
U=Clexp(-rx)+C2 (r=b/a)
shows an exponential curve which is accepted empirically to be the most suitable one for the longitudinal river profile.
This dynamic model was applied to simulate the terrace development of the Tama River. The Tama River is originated from Kanto Mountains, enters the depositional hanto Plain and ends at Tokyo Bay. Its ca. 1000km2 drainage basin is roughly estimated to discharge ca. 1.5×10-4km3/yr of sediment. And well developed terraces has been studied so in detail that the area has been a type area of Japanese late Pleistocene and Holocene.
The values of coefficient a and b in the equation are estimated from the flow rate of sediment of present River in grade. A shape index r=b/a determines the shape of graded profile, and the flow rate gives a rate of asymptotic convergence to the graded form. The profile (Juen, 1965b) of the Musashino Terrace (60×103 yr PP) was given as an initial value of simulation. The gradient at the upper boundary was given as a function of time. Another boundary condition was given by the sea level as a function of time. But it is assumed that the river mouth (the lower boundary) has been so conservative there in gradient that the position of the river mouth was floating horizontally in response both to the change of sea level and the profile itself (as a free boundary problem).
The results of simulation well represent the actual development of river profile, which is stated by the early Wi rm fill-strathing in the upper-most reach of the River, the latest Pleistocene over-flowing on the older surfaces in the upper-middle reach, the forming of valley-topography in the lower reach due to the maximum sea level lowering and of drowned valley-topography due to the postglacial sea level rising, and then river mouth advancing toward sea caused by fluvial accretion during the last seven thousand years.
