Morphology, Materials, and Flows
2.1 The Watershed and the Stream Network .1 The Watershed
A watershed (also called drainage basin or catchment) is topographically defined as the area that contributes all the water that passes through a given cross section of a stream (figure 2.1a). The surface trace of the boundary that delimits a watershed is called a divide. The horizontal projection of the area of a watershed is the drainage area of the stream at (or above) the cross section. The stream cross section that defines the watershed is at the lowest elevation in the watershed and constitutes the watershed outlet; its location is determined by the purpose of the analysis. For geomorphological analyses, the watershed outlet is usually where the stream enters a larger stream, a lake, or the ocean. Water-resources analyses usually require quantitative analyses of streamflow data, so for this purpose the watershed outlet is usually at a gaging station where streamflow is monitored (see section 2.5.3).
The watershed is of fundamental importance because the water passing through the stream cross section at the watershed outlet originates as precipitation on the watershed, and the characteristics of the watershed control the paths and rates of movement of water and the types and amounts of its particulate and dissolved constituents as they move through the stream network. Hence, watershed geology, topography, and land cover regulate the magnitude, timing, and sediment load of streamflow. As William Morris Davis stated, “One may fairly extend the ‘river’
all over its [watershed], and up to its very divides. Ordinarily treated, the river is like the veins of a leaf; broadly viewed, it is like the entire leaf” (Davis 1899, p. 495).
2.1.2 Stream Networks
The drainage of the earth’s land surfaces is accomplished by stream networks—
the veins of the leaf in Davis’s metaphor—and it is important to keep in mind that stream reaches are embedded in those networks. Stream networks evolve in response
(a)
(b)
1st order 2nd order 3rdorder 4th order
0
480 465 450 435
N 420
390 405 375 360 345 330 300 315
285 270
Weir 255
________________ Stream _ _ _ _ _ _ _ _ _ _ Divide
500 meters Elevation in meters above mean sea level
Contour interval: 15 meters
Figure 2.1 A watershed is topographically defined as the area that contributes all the water that passes through a given cross section of a stream. (a) The divide defining the watershed of Glenn Creek, Fox, Alaska, above a streamflow measurement site (weir) is shown as the long-dashed outline, and the divides of two tributaries as shorter-long-dashed lines. (b) The watershed of a fourth-order stream showing the Strahler system of stream-order designation.
NATURAL STREAMS 23 to climate change, earth-surface processes, and tectonic processes, and network characteristics affect various dynamic aspects of stream response and geochemical processes. Knighton (1998) provided an excellent review of the evolution of stream networks, Dingman (2002) summarized their relation to hydrological processes, and Rodriguez-Iturbe and Rinaldo (1997) presented an exhaustive exploration of the subject.
2.1.2.1 Network Patterns
Network patterns, the types of spatial arrangement of river channels in the landscape, are determined by land slope and geological structure (Twidale 2004). Most drainage networks form a dendritic pattern like those of figures 2.1b and 2.2a: there is no preferred orientation of stream segments, and interstream angles at stream junctions are less than 90◦ and point downstream. The dendritic pattern occurs where there are no strong geological controls that create zones or directions of strongly varying susceptibility to chemical or physical erosion. Zones or directions more susceptible to erosion may display parallel, trellis, rectangular, or annular patterns (figure 2.2b–e).
The distributary pattern (figure 2.2f ) usually occurs where streams flow out of mountains onto flatter areas to form alluvial fans, or on deltas that form where streams enter lakes or the ocean. Regional geological structures may also cause patterns of any of these shapes to be arranged in radial or centripetal “metapatterns”
(figure 2.2g,h). The presence of these patterns and metapatterns on maps, aerial photographs, or satellite images can provide useful clues for inferring the underlying geology (table 2.1).
2.1.2.2 Quantitative Description
Figure 2.1b shows the most common approach to quantitatively describing stream networks (Strahler 1952). Streams with no tributaries are designated first-order streams; the confluence of two first-order streams is the beginning of a second-order stream; the confluence of two second-second-order streams produces a third-second-order stream, and so forth. When a stream of a given order receives a tributary of lower order, its order does not change. The order of a drainage basin is the order of the stream at the basin outlet. The actual size of the streams desig-nated a particular order depends on the scale of the map or image used,1 the climate and geology of the region, and the conventions used in designating stream channels.
Within a given drainage basin, the numbers, average lengths, and average drainage areas of streams of successive orders usually show consistent relations of the form shown in figure 2.3. These relations are called the laws of drainage-network composition and are summarized in table 2.2. Networks that follow these laws—that is, that have bifurcation ratios, length ratios, and drainage-area ratios in the ranges shown—can be generated by random numbers, so it seems that the evolution of natural stream networks is essentially governed by the operation of chance (Leopold et al. 1964; Leopold 1994). Table 2.3 summarizes the numbers, average lengths, and average drainage areas of streams of various orders.
(a) (b)
(c) (d)
Dendritic
Parallel
Trellis
Rectangular
(f)
Distributary
(g)
Radial
(h)
Centripetal (e)
Annular
Figure 2.2 Drainage-network patterns (see table 2.1). Panels a–e are from Morisawa (1985).
NATURAL STREAMS 25 Table 2.1 Stream-network patterns and metapatterns and their relation to geological controls.
Type Description Geological control Figure
Dendritic Treelike, no preferred channel orientation, acute interstream angles
None 2.2a
Parallel Main channels regularly spaced and subparallel to parallel, very acute interstream angles
Closely spaced faults, monoclines, or isoclinal folds
2.2b
Trellis Channels oriented in two mutually perpendicular
Rectangular Channels oriented in two mutually perpendicular
Annular Main streams in approximately circular pattern, nearly
Distributary Single channel splits into two or more channels that do not rejoin
Volcanic cone or dome of intrusive igneous rock
2.2g Centripetal
(metapattern)
Stream networks flow inward to a central basin
Calderas, craters, tectonic basins
2.2h
After Summerfield (1991) and Twidale (2004).
A stream network can also be quantitatively described by designating the junctions of streams as nodes and the channel segments between nodes as links. Links connecting to only one node (i.e., first-order streams) are called exterior links; the others are interior links. The magnitude of a drainage-basin network is the total number of exterior links it contains; thus, the network of figure 2.1b is of magnitude 43.
Typically, the number of links of a given order is about half the number for the next lowest order (Kirkby 1993).
The spatial intensity of the drainage network, or degree of dissection of the terrain by streams, is quantitatively characterized by the drainage density, DD, which is the total length of streams draining that area, X, divided by the area, AD:
DD≡X AD
. (2.1)
Drainage density thus has dimensions [L−1].
26 FLUVIAL HYDRAULICS
100
N(ω) = 615·exp(−1.33·ω)
50
NUMBER OF STREAMS
10 5
1
STREAM ORDER
1 2 3 4 5
100
AD(ω) = 0.18·exp(1.48·ω)
50
MEAN DRAINAGE AREA, km2
10 5 (a)
(c)
(b)
1
STREAM ORDER
1 2 3 4 5
ω L(ω) = 0.21·exp(0.97·ω)
MEAN STREAM LENGTH, km
10 5
1 0.5
STREAM ORDER
1 2 3 4 5
Figure 2.3 Plots of (a) numbers, N(ω), (b) average lengths, L(ω), and (c) average drainage areas, AD(ω), versus order, ω, for a fifth-order drainage basin in England, illustrating the laws of drainage-network composition (table 2.2). After Knighton (1998).
Drainage density values range from less than 2 km−1 to more than 100 km−1. Drainage density has been found to be related to average precipitation, with low values in arid and humid areas and the largest values in semiarid regions (Knighton 1998). In a given climate, an area of similar geology tends to have a characteristic value; higher values of DD are generally found on less permeable soils, where channel incision by overland flow is more common, and lower values on more permeable materials. However, it is important to understand that the value of DD
NATURAL STREAMS 27 Table 2.2 The laws of drainage-network composition.a
Average value and usual
Law of Definition Mathematical form rangeb
Stream numbers
aRB, bifurcation ratio; RL, length ratio; RA, drainage-area ratio; N(ω), number of streams of order ω; X(ω), average length of streams of order ω; AD, average drainage area of streams of order ω.
bGlobal average for orders 3–6 computed by Vörösmarty et al. (2000a, p. 23), considered to best “represent the geomorphic characteristics of natural basins.”
Table 2.3 Orders, numbers, average lengths, and average areas of the world’s streams.
Ordera Number Average length (km) Average area (km2)
1 14,500,000 0.78 1.6
aValues for orders 6–11 taken from Vörösmarty et al. (2000c) assuming that first-order streams at the scale of their study correspond to “true” sixth-order streams (Wollheim 2005). Values for orders 1–5 are computed using the global average bifurcation, length, and area ratios computed by Vörösmarty et al. (2000c): RB= 3.70; RL= 2.55; RA= 4.55.
for a given region will increase as the scale of the map on which measurements are made increases.
2.1.3 Watershed-Scale Longitudinal Profile
The longitudinal profile of a stream is a plot of the elevation of its channel bed versus streamwise distance. The profile can be represented as a relation between elevation (Z) and distance (X), or between slope, S0(≡ −dZ/dX) and distance. Downstream
28 FLUVIAL HYDRAULICS
distance can be used directly as the independent variable or may be replaced by drainage area, which increases with downstream distance, or by average or bankfull discharge, which usually increases with distance.
At the watershed scale, longitudinal profiles of streams from highest point to mouth are usually concave-upward, although some approach straight lines, and commonly there are some segments of the profile that are convex (figure 2.4).
The elevation at the mouth of a stream, usually where it enters a larger stream, a lake, or the ocean, is the stream’s base level.
This level is an important control of the longitudinal profile because streams adjust over time by erosion or deposition to provide a smooth transition to base level.
The relation between channel slope, S0(X), and downstream distance, X, for a given stream can usually be represented by empirical relations of one of the following forms:
S0(X)= S0(0)· exp(−k1· X), (2.2a) or
S0(X)= k2· X−m2, (2.2b)
or by a relation between slope and drainage area, AD,
S0(X)= k3· AD−m3, (2.2c)
where the coefficients and exponents vary from stream to stream depending on the underlying geology and the sediment size, sediment load, and water discharge provided by the drainage basin. Increasing values of k1, |m2|, or |m3| represent increasing concavity.
It is generally assumed that the smooth concave profiles modeled by equation 2.2a–c represent the “ideal” form that evolves over time in the absence of geological heterogeneities or disturbances. Deviations from this form that produce convexities in the profile are common and are due to 1) local areas of resistant rock formations, 2) introduction of coarser sediment or a large sediment deposit by a tributary or landslide, 3) tectonic uplift, or 4) a drop in base level. Pronounced steepenings due to these causes are called knickpoints.
Knighton (1998) reviewed many studies of longitudinal profiles and concluded, Channel slope is largely determined by 1) the quantity of flow contributed by the drainage basin and 2) the size of the channel material.
In almost all river systems, bankfull (or average) discharge increases downstream as a result of increasing drainage area contributing flow; thus, channel slope can be estimated as
S0(X)= k4· Q(X)−m4· d(X)m5, (2.3a) or
S0(X)= k5· AD(X)−m6· d(X)m7, (2.3b)
0 500 1000 1500 2000 2500 3000 3500
0 500 1000 1500 2000 2500
Distance (km)
(c)
Elevation (m)
Rio Grande 0
500 1000 1500 2000 2500
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Distance (km)
(a)
Elevation (m)
(b)
0 1000 2000 3000 4000 5000 6000
0 500 1000 1500 2000 2500 3000
Distance (km)
Elevation (m)
Indus Mississippi
Figure 2.4 Examples of longitudinal profiles of large rivers. All examples are basically concave-upward, even those in which discharge does not increase downstream (lower Indus, Murray, Rio Grande), but some have convex reaches, especially pronounced for the Rio Grande and Indus. Data provided by B. Fekete, Water Systems Analysis Group, University of New Hampshire. (continued)
(d)
0 1000 2000 3000 4000 5000 6000
Distance (km)
0 1000 2000 3000 4000 5000 6000
Distance (km)
Elevation (m)
Zaire (Congo)
(f) Figure 2.4 Continued
NATURAL STREAMS 31 where Q is some measure of discharge (e.g., bankfull or average discharge), d is some measure of sediment size (e.g., median sediment diameter), X is downstream distance, and ADis drainage area. The values of the empirical exponents m4through m7
vary from region to region. As discussed in the following section, d tends to decrease downstream in most stream systems; thus, relations of the form of equation 2.3 predict that the more rapid the downstream increase in Q or ADor the downstream decrease in d, the more concave the profile.
2.1.4 Downstream Decrease of Sediment Size
There is a general trend of downstream-decreasing bed-material sediment size in virtually all river systems (figure 2.5a), which is typically modeled as an exponential decay:
d(X)= d(0) · exp(−k6· X), (2.4)
where d(0) is the grain size at X= 0 and k6is an empirical coefficient that varies from stream to stream (values for various streams are tabulated by Knighton 1998). In many river systems, the exponential decay is “reset” where tributaries contributing coarse material enter a main stream (figure 2.5b). Interestingly, the rate of size decrease is especially pronounced in gravel-bed streams, and an abrupt transition from gravel to sand is often observed.
Two physical processes produce the size decrease: grain breakdown by abrasion and selective transport of finer sizes. Experimental studies have shown that abrasion does not produce the downstream-fining rates observed in most rivers (see, e.g., Ferguson et al. 1996), so selective transport is almost always the dominant process producing downstream sediment-size decrease.
Hoey and Ferguson (1994) were able to simulate the rates of sediment-size decrease observed in a Scottish river using a physically based model. Their results supported the strong correlation between downstream rates of slope decrease and of particle size, as reflected in equation 2.3.
2.2 Channel Planform: Major Stream Types