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 erosion1–5. In combination with flow over land6, seepage erosion contributes to the initiation and growth of channel networks1–5. Seepage processes have also been invoked in the formation of enigmatic amphitheatre-headed channel networks on both Earth7–11 and Mars12–14. However, the role of seepage in producing such channels remains controversial11,15,16. 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 heads17. 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 Florida18. 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 channels19, thereby linking network growth dynamics to geometric form. Networks of amphitheatre-headed channels known as ‘steephead streams’18 occur abundantly in Liberty County, Florida, east of the Apalachicola River on the Florida panhandle (Fig. 1). The steepheads are incised into 65m of laterally persistent, medium to coarse, fluviodeltaic and marine sands of Late Pliocene to Pleistocene origin20, deposited during progradation of the Apalachicola delta21. These sands unconformably overlie 15m of muddy Miocene marine carbonates and sands20. Steephead springs occur in the Late Pliocene to Pleistocene sands and examination of the deposit at spring sites reveals no obvious stratigraphic control on their vertical positions18. To investigate controls on the horizontal position of springs, we conducted a three-dimensional ground-penetrating radar survey of the water table near a highly bifurcated segment of the channel network (see Supplementary Information). Figure 2a shows that the water table descends as much as 6m from its highest point midway between channels before reaching the outer contour of the channel network. In general, the height of the water table is a complex function of the spatial distribution of sources (rainfall), sinks (the channel network) and subsurface heterogeneities22,23. As rainfall is uniform at this scale, we can test for the influence of heterogeneities by plotting water table height versus the distance to the nearest channel. The good correlation shown in Fig. 2b suggests