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 rb 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 δa, or the quantity of water accumulated on the soil surface, and the rate q8 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 sedimentation 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 overland 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 xc.

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 competitive 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 generally 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, including 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.

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