Channel concavity controls drainage network complexity

The first-order morphology of mountain ranges is controlled by the topologic complexity of the channel networks that drain them. Some networks are characterized by simple flow paths that follow the regional topographic gradient. Other networks are more complex, showing tortuous flow paths and asymmetric distribution of drainage area with respect to the main trunks. The degree of network complexity controls the distribution of slope magnitude and aspect, as well as the local relief of mountainous terrains, placing a strong control over their geomorphic, hydrologic, and ecologic functionality. 

Some of the variability in network complexity could be attributed to the level of heterogeneity in the environmental and boundary conditions. Spatial gradients in tectonics, climate, and lithology are likely linked to more complex network topology. However, previous numerical studies of landscape evolution showed that variability in complexity appears even when the environmental and boundary conditions are uniform. This means that drainage complexity could emerge from autogenic network dynamics.

To explore the controls over network complexity, we adopt a new metric that quantifies complexity as the distribution of differences in flow length between pairs of flow paths that diverge from a common divide and merge downstream. Symmetric flow lengths indicate low complexity, and increased flow-length asymmetry is indicative of a complex network. Consistent with previous numerical studies, we show, for the first time for natural mountain ranges, that plan-view network complexity, as expressed by lengthwise asymmetry, is a strong function of the concavity index that characterizes channel long profiles.

An analytic model of channel pairs that diverge from a stable drainage divide and obeys Hack’s law predicts that low concavity channels can sustain a stable divide only if they are lengthwise symmetric. In contrast, high concavity channels can sustain stable divides under a range of lengthwise symmetry conditions. The analytic model explains the increase in asymmetry (complexity) median and variance with increased channel concavity documented in both natural and numerical mountain ranges.

An optimal channel network perspective provides further intuition. Starting from a random network, the energy gain of reducing network complexity is high only when the concavity is low. Therefore, high-concavity, complex networks have a lower energetic incentive to reduce their complexity via changes in network topology. In contrast, complex networks of medium and low concavity tend to change their topology via drainage divide migration to achieve a less complicated and lower energy configuration.

Our findings provide a way to quantify channel concavity by evaluating the plan-form network complexity. Our results further imply that reduction in channel concavity, due to, for example, a transition to a drier climate, is expected to induce a phase of drainage reorganization that reduces the network complexity. In contrast, increased concavity is likely to cause minor or no changes in network topology and complexity.