LAWS OF THE COMPOSITION OF DIVIDE-SEGMENT SYSTEM AND DRAINAGE BASIN MORPHOLOGE

Abstract

In order to investigate three dimensional form of drainage basin statistically, the laws to describe the composition of drainage divide system are necessary in addition to the laws of drainage composition. The law of divide-segment numbers has been already proposed together with a classification method of divide segments bordering a basin and in the basin (Fig. 1: Tokunaga, 1984). The relation between number of divide segments and order is formalized by using the parameter _κε_λ that is the mean number of streams of order λ in a basin, entering into a stream of order κ from the sides. Namely, let ε₁=_κε_κ-1 and K=(_κε_λ/_κε_κ-1)^1/(κ-λ-1), then the mean number _κD_λ of divide segments of order λ in a basin of order κ is given by the following equation when ε₁ and K are constant for various values of κ and λ.
_κD_λ=[(_λ+2D_λ-P_λ+1D_λ)/(Q-P)]Q^κ-λ-1+[(_λ+2D_λ-Q_λ+1D_λ)/(P-Q)]P^κ-λ-1 (1)
where P=[2+ε₁+K-_??_(2+ε₁+K)²-8K]/2, Q=[2+ε₁+K+_??_(2+ε₁+K)²-8K]/2, _λ+2D_λ=(2+ε₁)_λ+1D_λ+ε₁K_λD_λ and _λ+1D_λ=(2+ε₁)_λD_λ-1. Then _λD_λ is the number of λ-th order segments in the divide surrounding a basin of order λ and _λD_λ=2.
Equation (1) is named the law of divide-segment numbers (Tokunaga, 1984). The data derived from basins in the Tama Hills and the Tanzawa Mountains in the Kanto district, Japan, support sufficiently this law (Fig. 4, Table 1 and Table 2). Following three laws are obtained by analyzing the data on lengths, slopes and falls of divide segments.
The law of divide-segment lengths states that
L_{D, Κ}=L_{D, λ}D_ι^(κ-λ) (2)
where L_{D, Κ} is the mean length of divide segments of order κ and D_ι is the divide-segment length ratio.
The law of divide-segment slopes is
S_{D, Κ}=S_{D, λ}D_s-(κ-λ) (3)
where S_{D, Κ} is the mean slope of divide segments of order κ and D_S is the divide-segment slope ratio.
The law of divide-segment falls is
F_{D, Κ}=F_{D, λ}D_ƒ^(κ-λ) (4) where F_{D, Κ} is the mean fall of divide segments of order κ and D_ƒ is the divide-segment fall ratio.
The data show that D_ι, D_S and D_ƒ are constant for each basin (Fig. 5, Fig. 7, Fig. 9). The data also show that D_S is smaller than the stream slope ratio and that D_ƒ is larger than 1.0.