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A physical explanation for an observed area-slope-elevation relationship for catchments with declining relief

Authors

  • Garry Willgoose


Abstract

A general relationship between the contributing area, slope, and mean elevation of catchments with relief declining after a tectonic uplift event is presented. This relationship is based on the continuity equation for runoff and erosion processes in the catchment. The key hypothesis underlying this relationship is that as a catchment declines, the nondimensionalized catchment approaches a constant form. This hypothesis is verified for computer simulated catchments. The area-slope-elevation relationship covers several cases: catchments declining toward a peneplain; catchments declining from a high slope dynamic equilibrium (resulting from a high rate of tectonic uplift) to a low slope one (resulting from a lower rate of tectonic uplift); and catchments declining from an elevated initial condition, as, for example, in the decline of a mine spoil heap. A previously published relationship between slope and area for catchments in dynamic equilibrium and based on runoff and erosion physics is shown to be a special case of this general relationship. The new area-slope-elevation relationship is compared with data from simulated catchments and a field catchment. It is thus shown that for declining catchments the area-slope-elevation relationship is a good predictor of catchment form for catchments with declining relief. It is argued that the slope-area-elevation relationship is sufficient, with the planiform drainage pattern, to completely define the elevation properties of the catchment such as, for instance, the hypsometric curve.

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