Growth laws for channel networks incised by groundwater flow

The re-emergence of groundwater at the surface shapes the Earth's topography through a process known as seepage erosion(1-5). In combination with flowover land(6), seepage erosion contributes to the initiation and growth of channel networks(1-5). Seepage processes have also been invoked in the formation of enigmatic amphitheatre-headed channel networks on both Earth(7-11) and Mars(12-14). However, the role of seepage in producing such channels remains controversial(11,15,16). One proposed growth law for channel development suggests that the velocity at which channel heads advance is proportional to the flux of groundwater to the heads(17). Here we use field observations and physical theory to show that this simple model, combined with a second linear response that relates channel branching to the total groundwater flux to the network, is sufficient to characterize key aspects of the growth and form of a kilometre-scale seepage-driven channel network in Florida(18). We find that the dynamics for the advance of channel heads are reversible, which allows us to estimate the age of the channel network and reconstruct the history of its growth. Our theory also predicts the evolution of the characteristic length scale between channels(19), thereby linking network growth dynamics to geometric form.