A Fractal Framework for Channel-Hillslope Coupling

Abstract. Questions of landscape scale in coupled channel-hillslope landscape evolution have been a significant focus of geomorphological research for decades. Studies to date have suggested a characteristic landscape length that marks the shift from fluvial channels to hillslopes, limiting fluvial incision and setting the length of hillslopes. The representation of real-world landscapes in slope-area plots, however, makes it challenging to identify the exact transition from hillslopes to channels, owing to the existence of an intermediary colluvial valley region. Without a rigorous explanation for the scaling of the channel hillslope transition, the use of computational models, which are forced to implement a finite grid resolution, is limited by the scaling of the physical parameters of the model relative to the grid resolution. Grid resolution is also tied to the width of channels, which is undetermined without a rigorous explanation of where channels begin.

Building on existing work, we demonstrate the existence and implications of the characteristic landscape length and its relationship to grid resolution. We derive the characteristic landscape length as the horizontal length in a one-dimensional landscape evolution framework required to form an inflection point. On a two-dimensional domain, channel heads form in steady state at the characteristic area, the square of the characteristic length, independent of grid resolution. We present a box-counting fractal definition using the grid resolution, revealing that the dimension of the contributing drainage region on steady-state hillslopes is expressed as a multifractal system. In sum, channels have contributing drainage areas, therefore a dimension of two, whereas, by definition, unchannelized locations or nodes have a dimension between zero and two, so not a well-defined area. This conceptualization aligns with the observed scaling of channel width. It also importantly suggests that real-world landscapes have something analogous to the concept of a grid resolution, as this paper demonstrates. In doing so, our works clarifies several unresolved properties of channel-hillslope coupling, with potential for substantially improving the accuracy of coupled landscape evolution models in replicating landscape forms.