Channelized drainage networks transport water and sediment over length scales ranging from centimeters to thousands of kilometers. Their dendritic structure typically results from the interaction of fluid-mechanical processes and topographic roughness. Relevant flows occur not only on the surface but also in the subsurface. In the latter case, groundwater "seepage" flows that exit the subsurface at channel heads can drive channel growth. Seepage flows have been suggested as an explanation of enigmatic channelization patterns on Earth and Mars and are thought to play a significant role in the erosion of levees. Here we show how networks of seepage-driven channels grow. We study a natural channel network cut through several kilometers of ancient sands on the Florida Panhandle. Using ground-penetrating radar, we obtain a three-dimensional reconstruction of the subsurface water table which suggests that flow to each channel head originates primarily from regions closer to that channel head than to any other. We then show that the size of this area, and thus the flux of groundwater to channel heads, is proportional to the velocity at which channel heads advance. The reversibility of this linear growth law allows us to reconstruct the network's growth.
Analysis of the growth history provides evidence of a second linear
response: new tributaries are generated at a rate proportional to the area drained by the network. At long times, the ratio of the coefficients that define the two linear response relations determines the characteristic length scale between channels. Consequently the network's static geometry derives from its time-dependent growth in a manner that is independent of topographic roughness and material heterogeneity.
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