ABSTRACT
By applying the law of the numbers, we know that there are nZ, streams of order
(3.31)
Fig. 3.17 : Theoretical watersheds with different confluence ratios and their hydrographs (based on Chow, 1988).
By definition, n: 1. Therefore, the total length of the streams of order Z is expressed as:
(3.32)
This makes it easy to determine the total length of the drainage network Ltot from
the sum of the lengths of the streams of orders 1 through ::
(3.33)
This last total again represents a geometric series of ratio (RB /RL) -1, with a first
term order (RBRL) :1 and the nth term equal to one. The sum of this series is thus:
(3.34)
If RBRL1 and : tends to infinity, then the series converges and the total length of the network tends towards:
(3.35)
However, in most cases, this series diverges, and the sum we are trying to find tends towards infinity because : tends to infinity. This gives us:
(3.36)
We also know that the average length of 1st order streams is equal to:
(3.37)
With some manipulation of these last two equations, we obtain the following relationship:
(3.38)
This relationship fits the definition of a law of scale as defined by Mandelbrot (Mandelbrot, 1995; Mandelbrot, 1997). Basically, two quantities are linked by a law
of scale if there is an exponent H such as A BH, or more generally AaBH4. In the case of Equation 3.30, this gives us
(3.39)
This result underscores the fractal nature of the drainage network.
