Abstract
Combinatorial analysis showed that the average number mμl of streams of order l merging into streams of varoius orders higher than l in a subnetwork of order m which is a portion of infinite topologically random channel networks is (Tokunaga, 1974)**:
_??_(1)
This equation provides two prerequisite conceptions for understanding some laws of drainage composition of an idealized basin. One of them is the conception of an “ideal drainage basin”. It means a supposed drainage basin in which the law of stream numbers given by Eq. (1) holds for every subbasins as well as itself and which can be finally divided into infinitesimal basins and interbasin areas. The other is the conception of a “unit drainage basin”. It means the basin regarded as a basic basin in considering the laws of drainage composition and its order is determined arbitrarily. Consequently orders of basins have only relative meanig in the ideal drainage basin, and may be determined depending upon a scale of topographic maps or aerial photos in actual drainage basins. Thus both the order 1 of the unit drainage basin and the order m of given drainage basin are treated as variables in this paper.
According to the above mentioned rules, some equations to describe the laws of drain-age composition of the ideal drainage basin can be deduced from Eq. (1) after algebraic procedures.
1) The law of basin areas is derived from Eq. (1), on a reasonable assumption that the interbasin area which is formed among basins of given orders and a stream of order higher than them is in any case smaller than the basins. It is written by Eq. (2).
am=4m-lα, (2)
where am is the average area of basins of order m and al is the average area of the unit drainage basins.
2) It is easily deduced from Eq. (1) that the number of streams of order l and orders higher than l which merge directly into a stream of order m is
Ni, m, l=2m-l+ 1, and the number of interbasin areas contacting with a stream of order m is equal to that number. Conse-quently the number Ni, m, l of such interbasin areas is
Ni, m, l=2m-l+1. (3)
Then Eq. (3) is named a “law of numbers of interbasin areas” in this paper. 3) A “law of interbasin areas” is derived from Eq. (1), (2) and (3). The average area βm, l of interbasin areas contacting with a stream of order m is written by Eq. (4)
βm, l _??_ (4)
4) Assume the next relation between the average length L2 of streams of order A and average area a2 of basins of order A,
_??_
where e is constant independent from orders of basins and streams, and substitute the above relation into Eq. (2). Then the law of s tram lengths is given by Eq. (5).
Lm=2m-1L1, (5)
where Lm is the average length of streams of order m and Ll is the average length of streams of order L.
5) Combining Eq. (1), (2) and (5) leads to the equation, which expresses the drainage density Dm, 1, after some algebraic procedures.
_??_
Introduction of a parameter msa which is the average number of streams of order A entering into a stream of order m from sides and assumptions that ε1=λελ-1 and K=mελ-1/mελ take same values respectively for various values of A in a basin provide the laws of drainage composition of the basin in a mathematically generalized form. Corresponding equations to Eq. (2), (3), (4), (5) and (6) are written by Eq. (7), (S), (9), (10) and (11) respectively.
6) The law of basin areas.
_??_
where _??_.
7) The law of numbers of interbasin areas.
_??_
8) The law of interbasin areas.
_??_
9) The law of stream lengths.
_??_
10) The drainage density.
_??_
Hack's law is checked in the basin given in the mathematically generalized form as well as the ideal drainage basin.
