Change in the Mathematical Function Type Describing the Longitudinal Profile of a River through an Evolutionary Process

Abstract

Japanese rivers where a large amount of sediment accumulates along the middle reaches are today not at grade. Nevertheless, most of their longitudinal profiles can be described by one of the mathematical functions proposed for graded rivers. For these rivers, the mathematical function should show an aspect of fluvial processes different from the explanations of grade theories. The geomorphological significance of differences in the mathematical functions describing the longitudinal profiles of rivers is discussed here. A dynamic change in the shape of longitudinal profile of a river accompanying the change in mathematical function type is proposed in relation to changes in fluvial processes and stage of evolution of a river. Most Japanese rivers can be described by either exponential or power functions. The fluvial processes of the rivers described by exponential functions are in the depositional state. The front of depositional area of gravels increases in altitude and migrates downstream. The rivers expressed by power functions are in the transportational state. The front of depositional area of gravels decreases in altitude and migrates downstream. The rivers matched by linear functions, indicating that their longitudinal profiles are almost straight lines, are in mass equilibrium: the sediment load is balanced between inflow and outflow. Through aggradational processes, the shape of longitudinal profile of a river changes with the change in mathematical function type, from exponential, to power, and finally to linear functions. The difference in the type of function best fitting the longitudinal profile of a river reflects variations in fluvial processes and the evolutionary stage of the river.