The composition of the stream system of a drainage basin can be expressed quantitatively in terms of stream order, drainage density, bifurcation ratio, and stream-length ratio. Stream orders are so chosen.that the fingertip or unbranched tributaries are of the 1st order;
streams which receive 1st order tributaries, but these only, are of the 2d order; third order streams receive 2d or 1st and 2d order tributaries, and so on, until, finally, the main stream is of the highest order and characterizes the order of the drainage basin. Two fundamental laws connect the numbers and lengths of streams of different orders in a drainage
basin:
(1) The law of stream numbers. This expresses the relation between the number of streams of a
given order and the stream order in terms of an inverse geometric series, of which the bifurcation ratio
ft is the base. (2) The law of stream lengths expresses the average length of streams of a given order in terms
of stream order, average length of streams of the 1st order, and the stream-length ratio. This law
takes the form of a direct geometric series. These two laws extend Playfair''s law and give it a
quantitative meaning. The infiltration theory of surface runoff is based on two fundamental concepts:
(1) There is a maximum or limiting rate at which the soil, when in a given condition, can absorb
rain as it falls. This is the infiltration-capacity. It is a volume per unit of time. (2) When runoff takes place from any soil surface, there is a definite functional relation between
the depth of surface detention Sa, or the quantity of water accumulated on the soil surface, and the
rate q, of surface runoff or channel inflow. For a given terrain there is a minimum length xc of overland flow required to produce sufficient
runoff volume to initiate erosion. The critical length xc depends on surface slope, runoff intensity, infiltration-capacity, and resistivity of the soil to erosion. This is the most important single factor
involved in erosion phenomena and, in particular, in connection with the development of stream
systems and their drainage basins by aqueous erosion. The erosive force and the rate at which erosion can take place at a distance x from the watershed
line is directly proportional to the runoff intensity, in inches-per hour, the distance x, a function of the
slope angle, and a proportionality factor Ke, which represents the quantity of material which can be
torn loose and eroded per unit of time and surface area, with unit runoff intensity, slope, and terrain. The rate of erosion is the quantity of material actually removed from the soil surface per unit of
time and area, and this may be governed by either the transporting power of overland flow or the
actual rate of erosion, whichever is smaller. If the quantity of material torn loose and carried in
suspension in overland flow exceeds the quantity which can be transported, deposition or sedimenta- tion on the soil surface will take place. On newly exposed terrain, resulting, for example, from the recession of a coast line, sheet erosion
occurs first where the distance from the watershed line to the coast line first exceeds the critical length
xc, and sheet erosion spreads laterally as the width of the exposed terrain increases. Erosion of such
a newly exposed plane surface initially develops a series of shallow, close-spaced, shoestring gullies or
rill channels. The rills flow parallel with or are consequent on the original slope. As a result of
various causes, the divides between adjacent rill channels are broken down locally, and the flow in
the shallower rill channels more remote from the initial rill is diverted into deeper rills more closely
adjacent thereto, and a new system of rill channels is developed having a direction of flow at an angle
to the initial rill channels and producing a resultant slope toward the initial rill. This is called cross- grading. With progressive exposure of new terrain, streams develop first at points where the length of over- land flow first exceeds the critical length xc, and streams starting at these points generally become the
primary or highest-order streams of the ultimate drainage basins. The development of a rilled
surface on each side of the main stream, followed by cross-grading, creates lateral slopes toward the
main stream, and on these slopes tributary streams develop, usually one on either side, at points
where the length of overland flow in the new resultant slope direction first exceeds the critical length
OCc. Cross-grading and recross-grading of a given portion of the area will continue, accompanied in
each case by the development of a new order of tributary streams, until finally the length of overland
flow within the remaining areas is everywhere less than the critical length xc. These processes fully
account for the geometric-series laws of stream numbers and stream lengths. A belt of no erosion exists around the margin of each drainage basin and interior subarea while the
development of the stream system is in progress, and this belt of no erosion finally covers the entire
area when the stream development becomes complete. The development of interior divides between subordinate streams takes place as the result of com- petitive erosion, and such divides, as well as the exterior divide surrounding the drainage basin, are
generally sinuous in plan and profile as a result of competitive erosion on the two sides of the divide, with the general result that isolated hills commonly occur along divides, particularly on cross divides, at their junctions with longitudinal divides. These interfluve hills are not uneroded areas, as their
summits had been subjected to more or less repeated cross-grading previous to the development of
the divide on which they are located. With increased exposure of terrain weaker streams may be absorbed by the stronger, larger streams
by competitive erosion, and the drainage basin grows in width at the same time that it increases in
length. There is, however, always a triangular area of direct drainage to the coast line intermediate
between any two major streams, with the result that the final form of a drainage basin is usually ovoid
or pear-shaped. The drainage basins of the first-order tributaries are the last developed on a given area, and such
streams often have steep-sided, V-shaped, incised channels adjoined by belts of no erosion. The end point of stream development occurs when the tributary subareas have been so completely subdivided by successive orders of stream development that there nowhere remains a length of
overland flow exceeding the critical length xc. Stream channels may, however, continue to develop
to some extent through headward erosion, but stream channels do not, in general, extend to the
watershed line. Valley and stream development occur together and are closely related. At a given cross section
the valley cannot grade below the stream, and the valley supplies the runoff and sediment which
together determine the valley and stream profiles. As a result of cross-grading antecedent to the
development of new tributaries, the tributaries and their valleys are concordant with the parent
stream and valley at the time the new streams are formed and remain concordant thereafter. Valley cross sections, when grading is complete, and except for first-order tributaries, are gen- erally S-shaped on each side of the stream, with a point of contraflexure on the upper portion of the
slope, and downslope from this point the final form is determined by a combination of factors, includ- ing erosion rate, transporting power, and the relative frequencies of occurrence of storms and runoff
of different intensities. The longitudinal profile of a valley along the stream bank and the cross
section of the valley are closely related, and both are related to the resultant slope at a given location. Many areas on which meager stream development has taken place, and which are commonly classified as youthful, are really mature, because the end point of stream development and erosion
for existing conditions has already been reached. When the end point of stream and valley gradation has arrived in a given drainage basin, the
remaining surface is usually concave upward, more or less remembling a segment of a parabaloid, ribbed by cross and longitudinal divides and containing interfluve hills and plateaus. This is called
a ""graded"" surface, and it is suggested that the term ""peneplain"" is not appropriate, since this surface
is neither a plane nor nearly a plane, nor does it approach a plane as an ultimate limiting form. The hydrophysical concepts applied to stream and valley development account for observed
phenomena from the time of exposure of the terrain. Details of these phenomena of stream and
valley development on a given area may be modified by geologic structures and subsequent geologic
changes, as well as local variations of infiltration-capacity and resistance to erosion. In this paper stream development and drainage-basin topography are considered wholly from the
viewpoint of the operation of hydrophysical processes. In connection with the Davis erosion cycle
the same subject is treated largely with reference to the effects of antecedent geologic conditions and
subsequent geologic changes. The two views bear much the same relation as two pictures of the same
object taken in different lights, and one supplements the other. The Davis erosion cycle is, in effect, usually assumed to begin after the development of at least a partial stream system; the hydrophysical
concept carries stream development back to the original newly exposed surface.
