ON THE CYCLICITY AND ALLOMETRIC GROWTH OF DRAINAGE NETWORKS

Abstract

The average number _??_ελ of streams of order A entering a stream of order ic from the sides provides two parameters: ε₁=_??_ and K=(_??_)
A model of drainage networks is built on the assumption that each parameter is constant in a network. The model is a cyclic system because it not only satisfies the condition that each cycle is entirely similar to the previous and following cycles but also includes structurally Hortoniann networks as a special case (K=O). The law of allometric growth of drainage networks is. formulated by using E1 and K on the assumption that a basin can be devided into infinitesimally small basins and interbasin areas according to the above mentioned cycle. The order m (t) of a subnetwork at time t is expressed by the following equation.
_??_
where l is the lowest order of streams on topographic maps or aerial photos of a given scale, δ is constant,
_??_
and Q=_??_
The above equation holds exactly for networks of infinitely large value of (m(t)-l) and to a fairly good approximation for networks of comparatively large value of it. Setting ε₁=1 and K=2 in the equation leads to the equation which expresses allometric growth of a random graph model (the average or the most probable state of subnetworks in infinite topologically random channel networks). The model corresponds to drainage networks in a stationary state and includes the random graph model as a special case.