Compression and Decay of Hillslope Topographic Variance in Fourier Wavenumber Domain

Abstract

Three mathematical models of hillslope sediment transport are common: linear diffusion, nonlinear diffusion, and nonlocal transport. Each of these is supported by a different theory, but each contains land-surface slope as a central ingredient. As such, land-surface evolution by all three of these models is largely similar in that topographic highs degrade and lows fill in. However, details of land-surface form reveal diagnostic clues to linear or nonlinear behavior of the land surface. We cast land-surface evolution into wavenumber (Fourier) domain, which effectively separates signals into coarse- and fine-scale elements of land-surface form, such as hillslope-valley sequences and pit-mound features, respectively. In wavenumber domain linear diffusion results in vertical spectral decay, which is associated with landform straightening and smoothing of sharp concavities. Nonlinear diffusion results in spectral compression toward low wavenumbers, which is associated with landform lengthening and is similar to slope replacement. Nonlocal processes share elements of linearity or nonlinearity but are modified by the particular form of the distribution of particle travel distance. Ultimately, all processes tend toward zero topographic variance, but by distinctly different styles as revealed in wavenumber domain. Spectral compression by nonlinear processes can result in temporary spectral growth over certain spectral bands and is interpreted as a signature of nonlinear processes for certain landforms. The signatures come from the evolution of topographic details and landforms with sharp concavities highlight this behavior, whereas landforms with low concavities obscure these diagnostic behaviors.