Optimality hypotheses for river network development

Earth's surface is full of fascinating patterns and many of them are created by the flow of water. For example, natural stream reaches are composed of regular repetition of riffles and pools. Streams meander like snakes. In a greater scale, rivers show gradual patterns in the distribution of grain size of bed material and slope (often finer material and milder slope in the downstream direction). As streams flow downward, they merge with other streams creating overall a 'network' of streams. This river network looks very similar to a tree, originating from its downstream end (outlet) and bifurcates as it goes upstream (Figure 1).

These interesting patterns have inspired scientists to think about a kind of governing principle behind the evolution of fluvial systems. This viewpoint has been formed over a century, and yet there are many open questions in this metaphysical topic. Some studies tried to find the governing principle from a statistical viewpoint while efforts to describe the basic system's behavior from thermodynamic principles have often been regarded as more fundamental.

For the case of treelike river network formation, a group of scientists hypothesized that the evolution of stream networks is toward a certain kind of an optimized state. Such a state is assumed to be quantifiable as an extreme (minimum or maximum) value of an ideal quantity. As an example, let us take a look at one of the hypotheses, called the minimum total energy expenditure. In this hypothesis, river network organization has been regarded as the problem of connecting points on a plane. Imagine a 2-D space composed of a fixed number of points. As shown in Figure 2, there are very different ways of connecting all points on the plane, such as spiral, explosive, and tree patterns. Interestingly, if we calculated the value of the total energy expenditure (the equation shown in Figure 3), the tree pattern shows the smallest value. Therefore, it is plausible that the tree pattern corresponds to the state of the minimum total energy expenditure. Maybe this provides some insights on why we do not see streams in a spiral or explosive pattern in nature.

Nevertheless, there have been other kinds of optimality conditions suggested to describe various features of landscapes (Figure 3). With our current scientific knowledge, we are not sure which of the optimality criteria is a more realistic one. One way of studying this problem is to search for a theoretical river network that satisfies an optimality criterion for each optimality hypothesis and examine which of the obtained 'solution' river networks is closest to the natural river network. In fact, some of these criteria can result in treelike organizations of river networks, as shown in the example of the minimum total energy expenditure. But natural river networks also show concave longitudinal profiles (slopes become milder in the downstream) in most cases. A recent study showed that none of the criteria listed in Figure 3 captures both patterns of treelike organization and concavity.

There are more fundamental issues in this subject. For example, whether the landscape is destined to follow certain optimality conditions (if any) or the seemingly optimality criterion is merely a consequent signature of spontaneous landscape evolution has been a topic for serious thought. Some optimality criteria exhibit conflicts with others and there are research studies that criticize present forms of optimality criteria. Optimality criteria suggested thus far have been formulated for very ideal settings and a more general principle should incorporate complex feedback mechanisms between hydrological, geological, atmospheric, and ecological processes. Obviously, this will be very challenging. In this light, new ideas on a more fundamental physical basis such as the maximum entropy production are highly sought for guiding further research direction. Or the entire idea of optimality hypotheses may be fundamentally wrong and simply illusive. In any case, the depth of scientific knowledge to be gained through this ongoing discovery procedure will significantly improve our understanding of the dynamics of Earth's surface processes. Besides this great scientific importance, the overall idea of optimality criteria for landscape formation is simply fascinating without a doubt. Today, many scientists are continuously investigating this research topic.

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