Basic Concepts and Management Issues of Rivers - River Dynamics and Integrated River Management - page 4

where Ȧ = stream order; N Z = number of Ȧ -th order streams; L Z = length of Ȧ -th order streams; s Z =
slope of Ȧ -th order streams; and A Z = drainage area of Ȧ -th order stream; and A , B , C , D , E , F , G , H =
constants. These empirical laws were not derived from basic theories in physics or other fundamental
theories. However, the validity of these equations have been independently verified and accepted as the
basic laws in river morphology. An example of Horton's laws is shown in Fig. 1.4 for the Rogue River
basin in Oregon (Yang, 1971).
Fig. 1.4 An example of Horton's laws for the Rogue River basin (after Yang, 1971)
Model of network growth —Horton (1945) not only pioneered the quantitative description and analysis
of channel networks and established "laws" of network composition, but he also proposed a model of
network growth by overland flow. Horton suggested that on a steep, newly exposed surface a series of parallel
rills develop and that, with time, cross-grading and micro-piracy among these rills produce an integrated
dendritic network (Fig. 1.5 Model A). A second model of network growth is the headward growth type
(Howard, 1971), in which the drainage network develops fully at the edge of an undissected area (Fig. 1.5
Model B). As growing headward and bifurcate, the channels fill the space available and form a fully
developed dendritic network. In the third model suggested by Glock (1931), the drainage area is rapidly
subdivided by channels and the addition of tributaries then fills the available space (Fig. 1.5 Model C).
Among the three models of network growth, one extreme is the Horton model (Fig. 1.5 Model A), in
which parallel channels develop almost instantaneously over the surface, and the final pattern of the
network progressively emerges by internal changes (capture) and replacement of the initial pattern of rills.
This model occurs only in very small drainage area, generally smaller than 1 km 2 . At the other extreme is
the headward-growth model in which a "wave of dissection'' (Howard, 1971; Schumm, 1956) can be
envisioned at the tips of first-order channels, which is the active zone of network headward growth
(Fig. 1.5 Model B). As this wave progresses into the undissected basin, the fingertip channels lengthen
and bifurcate, leaving behind a channel system that is almost fully developed. In this developed portion
of the network few additions or losses of channels occur during the continued extension of the network.
The significant feature of this model is that the network is almost fully developed as the wave of dissection
passes a particular point. This model may occur in large drainage areas, for instance, 1000 km 2 . The third
model (Fig. 1.5 Model C) lies between the two extremes and occurs in drainage areas smaller than 100 km 2 .