Abstract
Channel networks increase in complexity as the importance of erosion
grows compared to diffusion by soil creep, giving rise to a
channelization cascade. In this cascade, smaller channels join to form
progressively larger ones with an alternation of ridges and valleys
involving a multitude of wavelengths. Simulations of landscape evolution
models and laboratory experiments are used to uncover the signature of
such a cascade in the wavenumber spectrum of elevation fluctuations.
Power spectra at intermediate distances from the boundaries are
characterized by a peak wavenumber (the most energetic mode) that is
related to the quasi-cyclic valleys superimposed on power-law scaling
with exponent ($\alpha$) across a wide range of smaller
scales. Dimensional analysis and self-similarity arguments are used to
reveal the controlling factors on $\alpha$, showing
that $\alpha$ is uniquely linked to the power-law
relation (with exponent $m$) between erosion potential and the
specific drainage area via $\alpha = 2m -3$.