Abstract
The quantitative analysis of watershed geomorphology started with the study of Horton (1945) and thereafter many results have been reported. It was suggested that there is a definite relationship between stream lengths and drainage areas. Hack (1957) has found that relationship between mainstream lengths and drainage areas can be expressed by the equation L=cAr, where constants c and r are 1.4 and 0.6 respectively. This relationship is called Hack's law.
In this paper, the Strahler's and the link magnitude ordering systems are applied to measure the drainage basin. The link magnitude system is a new, simple and rational one advocated by Shreve (1967). The mainstream length is defined by this link magnitude system. The author attempted to examine the Hack's law, which is a relationship between mainstream lengths and drainage areas, for 19 stream networks in Hokkaido where the Norton's laws hold good (Table 1). The results are summarized as follows :
1. The Hack's law holds good to the investigated 155 drainage basins larger than 5th order basins (Fig. 4) and to each from the 1st to the 7th order basins respectively (Fig. 5, Table2). It is also applicable to each of 19 networks, that is, an individual network (Table 3). The exponent r exceeds 0.5 and approaches to about 0.6 in both cases, and this result agrees with Hack et al. but not with Sakaguchi and Mueller.
2. The exponent r can be derived from the Horton's laws (ratio of stream length and ratio of drainage area), but it becomes somewhat smaller when the stream length ratio by the Strahler's ordering system is used (Table 4).
3. The effect of the map scale used to Hack's law is not significant, but it affects the coefficient more than the exponent (Fig. 6). This is because the length is more affects by the map scale than the area is (Fig. 7). Accordingly, we must be careful to use maps of different scales.
4. The Hack's law between basin lengths and drainage areas holds good, too (Fig. 8). But, both the coefficient and the exponent become somewhat smaller than in the relationship between mainstream lengths and drainage areas.
5. The deviation of r from 0.5 depends upon both mainstream sinuosity and shape change of drainge basin. Among the roughly equal drainage areas, it is more affected by shape change than mainstream sinuosity (Fig. 9, Table 2). But, both effects become nearly equal in case of wider range of drainage area (Table 2). In case of relationship within an identical network both effects become nearly equal on average in 19 networks, but their effects are reflected in geomorphic characteristics of individual networks (Table 3).
6. The Hack's law also holds good for the simulated drainage networks by a random walk model. The effects of mainstream sinuosity and shape change of drainage basin, mentioned in 5, are also found for these simulated drainage networks (Table 6).
7. Relative contribution of the mainstream sinuosity to the shape change of a drain- age basin relates with the entropy calculated from probabilities deciding stream flow direc- tions. The shape change is large when the entropy is small, but the sinuosity is large when the entropy is large (Fig. 10). The relative contribution could be a new index to indicate geomorphic characteristics of a drainage basin.=
