Fractal behavior in space and time in a simplified model of fluvial landform evolution

Two general approaches have been applied to understanding the fractal structure of fluvial topography: (1) deterministic, process-based models, and (2) stochastic partial differential equations (PDE). Deterministic models reproduce the fractal behavior of fluvial topography but have two limitations: they often underestimate the amount of lateral valley and ridge migration that occurs in nature, and the complexity has made it difficult to identify the precise origin of fractal behavior in fluvial landscapes. The simplicity of stochastic PDE models has made them useful for investigating fractal behavior, but they incorrectly suggest that fractal behavior is only possible with stochastic forcing. In this paper I investigate whether simplified, deterministic PDE models of landform evolution also exhibit fractal behavior and other features of complexity (i.e. deterministic chaos). These models are based on the KPZ equation, well known in the physics literature. This equation combines diffusion (i.e. hillslope processes) and nonlinear advection (i.e. bedrock or alluvial channel incision). Two models are considered: (1) a deterministic model with uniform erodibility and random initial topography, and (2) a deterministic model with random erodibility and uniform initial topography. Results illustrate that both of these deterministic models exhibit fractal behavior and deterministic chaos. In this context, chaotic behavior means that valley and ridge migration and nonlinear amplification of small perturbations in these models prevent an ideal steady state landscape from ever developing in the large-system limit. These results suggest that fractal structure and deterministic chaos are intrinsic features of the evolution of fluvial landforms, and that these features result from an inverse cascade of energy from small to large wavelengths in drainage basins. This inverse cascade differs from the direct cascade of three-dimensional turbulence in which energy flows from large to small wavelengths.