Multiple scales in river basin morphology

Abstract

Existing data reveal that the steepness S of many river basins with mountainous headwaters, defined as the ratio of total relief to total length of a basin, is related to the logarithm of basin size. The relief of these same river basins is observed to be non-linearly related to basin steepness, with maximal values of relief occurring in basins for which the value of S is about 0.03 or, equivalently, in basins in which the average slope is about 2°. These systematic relationships between size, relief, and steepness of some basins, which span nine orders of magnitude of area, are inter- preted in terms of an intermediate limit process bounded by well separated spatial scales. The lower-limiting scale corresponds to a threshold between channel formation and significant mass wastage, and the upper-limiting scale corresponds to the size of the largest landmass. If the observed relationships between basin size, relief, and steepness indeed reflect a limit process, then they are independent of the details of the physics contributing to landscape development and instead result from the constraints imposed by the bounding, spatial scales. Exceptions to the limiting relationships explored here are readily noted, however, and are interpreted to reflect regional geological and temporal controls. The delineation of the importance of multiple scales provides a potentially powerful basis for the examination of morphological relation- ships in river basins. introduction River basins dominate the landscape, and the study of their morphology reveals important information about their formation, sediment yield, and fill. I examine here the quantitative relationship between the area and steepness of river basins which drain mountainous terrain in terms of the bounding and intermediate length scales imposed by geological process and form. The steepness of a river basin is considered in terms of the ratio of its total relief to its total length. This ratio is a dimensionless number that has found wide use in the comparison of basins (Schumm 1956, 1963) and other landscape features, such as alluvial fans (Melton 1965), which vary greatly in size. Its value represents the average slope of the longitudinal profile of a basin and thus represents the basin-wide magnitude of the gravitational component of landscape- forming processes. In the next section I review ideas regarding the existence and separation of spatial scales in fluvial landscapes. This is followed by a simple analysis of physical systems bounded exclusively by two well-separated spatial scales. The results of the analysis are then used to examine the relationships observed between river basin area, steepness, and relief. Additional points of interest are explored in the discussion. These include the situation when an intermediate spatial scale is imposed by regional geological structure and the evaluation of a denudational time scale and its relationship to basin size. I conclude with a summary of important findings. separation of multiple, bounding scales in river basin morphology Montgomery and Dietrich (1992) found a simple relationship between the length L of a drainage basin and its area A given by