A Hierarchical Cluster System Based on HortonStrahler Rules for River Networks

We consider a cluster system in which each cluster is characterized by two parameters: an "order" i , following HortonStrahler rules, and a "mass" j following the usual additive rule. Denoting by c _i,j ( t ) the concentration of clusters of order i and mass j at time t , we derive a coagulation-like ordinary differential system for the time dynamics of these clusters. Results about the existence and the behavior of solutions as   t   are obtained; in particular, we prove that   c _i,j ( t ) 0  and   N _i ( c ( t )) 0  as   t ,  where the functional   N _i (·)  measures the total amount of clusters of a given fixed order i . Exact and approximate equations for the time evolution of these functionals are derived. We also present numerical results that suggest the existence of self-similar solutions to these approximate equations and discuss their possible relevance for an interpretation of Horton's law of river numbers.