Consideration on the composition of drainage networks and their evolution
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概要
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An extended interpretation of the term of the "cyclic net" defined by Scheidegger leads to a better understanding of the laws of drainage composition and the evolution of drainage basins. The average number _κε_λ of streams of order λ entering a stream of order κ from the sides provides two parameters: ε_1=_κε_＜κ-1＞ and K=(_κε_λ/_κε_＜κ-1＞)^＜1/(κ-λ-1)＞ . A model of drainage basins is built on the assumption that each of the parameters is constant for various values of κ and λ in a network. The law of stream numbers of the model is formulated as κμλ=Q(Q^＜κ-λ-1＞-P^＜κ-λ-1＞)(2+ε_1-P)/(Q-P)+P^＜κ-λ-1＞(2+ε_1), where _κμ_λ is the average number of streams of order λ in a basin order κ, P=[2+ε_1+K-√(2+ε_1+K)^2-8K]/2 and Q=[2+ε_1+K+√(2+ε_1+K)^2-8K]/2. On some reasonable assumptions, the law of basin areas and the law of stream lengths are also formalized by using ε_1 and K, viz., A_λ=Q^＜λ-l＞A_l and L_λ=Q^＜(λ-l)/2＞L_l where l is the lowest order of streams or basins, A_λ is the average area of basins of order λ, A_l is the average area of basins of the lowest order, L_λ is the average length of streams of order λ and L_l is the average length of stream of the lowest order. The condition of the "cyclic net" is satisfied basically in the model, because the relation of the streams of order λ to the streams of order (λ+η) is the same as the relation of the streams of order (λ+1) to the streams of order (λ+1+η). The equation which describes the law of stream numbers gives graphs on the Horton diagram which tend to be concave upward, except the case of K=0, and seems to be more adequate to describe the relationship between stream orders and numbers of actual drainage networks than Horton's formula. The average values of ε_1 and K in infinite topologically random channel networks are 1 and 2 respectively for various values of κ and λ. The most probable networks in the set of infinite topologically random channel networks also satisfy ε_1=1 and K=2. The law of allometric growth of drainage basins is formulated by using ε_1 and K as m(t)=[δt+ln{(Q-P)/(2+ε_1-P)}]/lnQ+l, where m(t) is the order of a basin at time t and δ is constant. This equation holds exactly for basins of infinitely large value of [m(t)-l] and to a fairly good approximation for basins of a comparatively large value of it. It can be said that the model corresponds to basins in an equilibrium state and encompasses basins of the maximum entropy as a special case. The model seems to be very advantageous not only to investigate the composition of drainage networks but also to explain their development.
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