A mechanistic understanding of self-formed channel shape and scale

Forming spontaneously when sediment laden water flows across an erodible material, terrestrial river channels possess a distinct shape, with robust relationships between channel width, depth and flow velocity that hold true over a million-fold change in water flux and for widths spanning less than a meter to more than a kilometer. These patterns have long since been described, however, a process-based understanding of what determines channel shape and scale is just beginning to emerge. This recent work has made clear the key elements to understanding terrestrial river channels which include: lateral momentum exchange within the flow, the frictional behavior of the channel boundary on the flow, and the dynamics of bedload sediment transport in the channel.  Bringing together these elements, a theoretical prediction for steady-state channel shape is derived directly from the Navier-Stokes equations of motion. 

The key result is an analytical description of channel geometry relating seven variables: flow width, depth, velocity, channel slope, and characteristic grain size, water flux and sediment flux. Using these equations, any four variables can be predicted if the other three are known. The theory was tested against 2500 terrestrial river reaches including both bedrock and alluvial rivers, where width varies by three orders of magnitude, and characteristic water flux varies by seven orders of magnitude. Using characteristic water flux, characteristic grain size, and a global average sediment transport rate, flow width, depth and velocity are predicted to within a factor of two for >80% of reaches.  

There are other long-lived geophysical and extraterrestrial flows over erodible materials that can be addressed with a general understanding of self-formed channels. With estimates of how fluid drag, turbulence generation and sediment transport might change on other planetary bodies, this model could be applied to extraterrestrial river networks such as those observed on Mars or Titan. More fundamentally, this work suggests a general approach to understand self-formed channels in an erodible medium generated by different kinds of flow, such as valley glaciers and subglacial river channels on both Earth and Mars.