Morphology, Materials, and Flows
2. Nominal diameter: The diameter of a sphere that has the same volume as the particle, equal to (d max · d int · d min ) 1/3
2.6 Variables and Their Spatial and Temporal Variability .1 Principal Variables and Time and Space Scales
The principal variables discussed in this chapter, and in subsequent chapters, are summarized in table 2.10. Table 2.11 categorizes these variables as either measurable or derived. All these quantities vary on a range of spatial and temporal scales, and there is a general correlation between the size of a fluvial feature and the time scale at which it varies (table 2.12, figure 2.39).
Table 2.10 Measurable and derived variables characterizing stream morphology, materials, and flows.
Symbol Variable Dimensions
A Cross-sectional area of flow [L2]
ABF Bankfull cross-sectional area of flow [L2]
AD Drainage area [L2]
AD(ω) Average drainage area of streams of order ω [L2] dp Particle diameter greater than p% of particles [L]
DD Drainage density [1]
K Reach hydraulic conductance (equation 2.22) [1]
L Discharge of particulate sediment [F T−1]
N(braids) Average number of braids in a cross section [1]
N(ω) Number of streams of order ω [1]
Q Discharge [L3T−1]
QBF Bankfull discharge [L3T−1]
r Cross-section shape exponent (equation 2.20) [1]
rm Radius of curvature of meanders [L]
RA Area ratio (table 2.2) [1]
RB Bifurcation ratio (table 2.2) [1]
RL Length ratio (table 2.2) [1]
S0 Channel slope [1]
Ss Water-surface slope [1]
Sv Valley slope [1]
u Point velocity [L T−1]
U Cross-section or reach average velocity [L T−1]
UBF Bankfull cross-section or reach average velocity [L T−1] W Cross-section or reach average water-surface width [L]
WBF Cross-section or reach average bankfull water-surface width [L]
X Streamwise distance [L]
X(ω) Average length of streams of order ω [L]
Y Cross-section or reach average depth [L]
YBF Bankfull cross-section or reach average depth [L]
X Increment of streamwise distance [L]
Xv Increment of valley distance [L]
Z0 Difference in channel-bed elevation [L]
Zs Difference in water-surface elevation [L]
ZsBF Difference in water-surface elevation at bankfull [L]
Sinuosity [1]
m Meander wavelength [L]
X Total stream length [L]
Angle of bank slope [1]
Angle of repose of bank material [1]
Maximum depth in cross section [L]
BF Maximum depth in cross section at bankfull [L]
ω Stream order [1]
82
Table 2.11 Classification of measurable and derived variables characterizing stream morphology, materials, and flows.a
Derived
Domain Extent Measurable variables variables
Stream network Area or watershed N(ω), X(ω), AD(ω), X, AD RB, RL, RA, DD
aSee table 2.10 for symbol definitions.
Table 2.12 Space and time scales of fluvial features.
Dimensions Major controlling
Spatial scale (km, km2) Feature factors Time scale
Mega >103, >106 Major watersheds,
Macro 10–103, 102–106 Large watersheds, major
Meso 0.5–10, 0.25–102 Meanders, changes in
Micro 0.1–0.5, 0.01–0.25 Local erosion and deposition,
Reach 0.01–0.1, <0.01 Local erosion and deposition
84 FLUVIAL HYDRAULICS
Figure 2.39 Relation of length scale of various aspects of channel form to time scale of adjustment. From Fluvial Forms and Processes (Knighton 1998); reproduced with permission of Edward Arnold Ltd.
A sense of the complex network of interrelationships among these variables is conveyed in figure 2.40. This text is mostly concerned with phenomena at the reach scale, which would typically be in the range of a few meters to a few kilometers, and thus with typical time scales of up to 1 year. At this scale, we can characterize the variables of interest as follows:
Fixed quantities: Average discharge, bankfull discharge, timing of flows, discharge associated with various flood frequencies, bed-material size (d), and channel slope (S0) are determined by watershed size and regional geology, topography, and climate. The planform (), reach and cross-section dimensions (WBF, YBF), and shape ( , r) are determined by complex interactions among those factors. In general, channel dimensions increase downstream in a given watershed, and channel slope and bed-material size decrease.
Independent variables: Discharge (Q) and sediment discharge (L) are delivered to a reach from upstream. These vary with time but, at the reach scale and for short time periods, can be considered essentially constant, specified independent variables.
Dependent variables: Width (W ), average depth (Y ), average velocity (U), and sediment transport (L) out of the reach are the principal dependent variables. Their values are determined by the imposed discharge and the geometric and material properties of the reach and change as discharge changes spatially and temporally.
As discussed in detail in chapter 6, local conductance (K) in general changes with discharge but may be considered an independent variable to the extent the Q− K relation is known. Local water-surface slope (Ss) may also change with discharge (discussed in chapter 11) but may often be considered a constant equal to the channel slope.
NATURAL STREAMS 85
QBF Sediment Input Bank Material Composition and Strength Φ
Figure 2.40 Interrelations among variables in the fluvial system. Arrows indicate direction of influence. Dashed lines indicate interrelations that are not fully diagrammed. Note that the figure contains some variables that have not yet been discussed (e.g., bedforms, stream power, and frictional resistance); these will be introduced in later chapters. Modified from Knighton (1998).
2.6.2 Channel Adjustment, Equilibrium, and the Graded Stream
It has been recognized at least since the writings of James Hutton in the late eighteenth century that the elements of the landscape are in a quasi-equilibrium state, implying relatively rapid mutual adjustment to changing conditions. John Playfair clearly articulated Hutton’s observations as applied to streams and their valleys in 1802:
Every river appears to consist of a main trunk, fed from a variety of branches, each running in a valley proportioned to its size, and all of them together forming a system of valleys, communicating with one another, and having such a nice adjustment of their declivities, that none of them joins the principal valley, either on too high or too low a level, a circumstance which would be infinitely improbable if each of these valleys were not the work of the stream which flows in it. (quoted in Summerfield 1991, p. 4)
In the fluvial geomorphological literature, this observation evolved into the concept of the graded stream, which was most notably articulated by J. Hoover Mackin:
A graded river is one in which, over a period of years, slope and channel characteristics are delicately adjusted to provide, with available discharge, just the velocity required
86 FLUVIAL HYDRAULICS
for the transportation of the load supplied from the drainage basin. The graded stream is a system in equilibrium; its diagnostic characteristic is that any change in any of the controlling factors will cause a displacement of the equilibrium in a direction that will tend to absorb the effect of the change. (Mackin 1948, p. 471)
Figure 2.40 gives a sense of the complicated interactions that are involved in responding to changes in the driving variables of climate, geological processes, and human activities. Until the middle of the twentieth century, geomorphologists tended to emphasize mutual adjustments among only three of these variables: sediment load, channel slope, and velocity, such that an increase in sediment delivery from upstream causes deposition, which causes local slope to increase, which causes velocity to increase, which increases sediment transport out of the reach, which reduces slope and velocity back toward the original conditions.
It has since become recognized that changes in slope usually occur only very slowly, and that the mutual adjustments that tend to maintain an equilibrium form involve other aspects of flow and channel geometry that respond more rapidly to change. Thus, Leopold and Bull (1979) suggested that the concept of the graded stream be restated to be more consistent with this recognition and with figure 2.40:
“A graded river is one in which, over a period of years, slope, velocity, depth, width, roughness, (planform) pattern and channel morphology . . . mutually adjust to provide the power and the efficiency necessary to provide the load supplied by the drainage basin without aggradation or degradation of the channel” (p. 195).
The following section describes hydraulic geometry, which is the general term for the quantitative description of the adjustment of hydraulic variables to temporal and spatial changes in discharge.
2.6.3 Hydraulic Geometry
Leopold and Maddock (1953) coined the term “hydraulic geometry” to refer collectively to the quantitative relations between various hydraulic variables and discharge:
At-a-station hydraulic geometry refers to the changes of hydraulic variables as discharge changes with time in a given reach.
Downstream hydraulic geometry refers to the changes of hydraulic variables as discharge changes with space in a given stream or stream network.
Leopold and Maddock (1953) and subsequent researchers have focused on the hydraulic geometry relations for the components of discharge and postulated that these could be quantitatively represented by simple power-law equations:
Width versus discharge:
W= a · Qb, (2.29)
Average depth versus discharge:
Y= c · Qf, (2.30)
Average velocity versus discharge:
U= k · Qm, (2.31)
NATURAL STREAMS 87 Because Q= W · Y · U, it must be true that
b+ f + m = 1 (2.32)
and
a· c · k = 1. (2.33)
The coefficients and exponents in equations 2.29–2.31 vary from reach to reach and differ for at-a-station and downstream relations in a given region.3 Leopold and Maddock (1953) and most subsequent writers have determined the values of these coefficients and exponents empirically (by regression analysis; see section 4.8.3.1) and have identified tendencies for the exponents to center around particular values (different for at-a-station and downstream relations). As discussed in the following subsections, many researchers have attempted to find physical reasons for these tendencies.
2.6.3.1 Temporal Changes: At-a-Station Hydraulic Geometry
Dependence on Cross-Section Geometry and Hydraulics In at-a-station hydraulic geometry, the symbols Q, W , Y , and U refer to instantaneous values of those quantities at a given cross section or reach. Figure 2.41 shows the ranges of values of the exponents b, f , and m reported in a number of field studies summarized by Rhodes (1977). Although there is wide variation, there is a tendency for at-a-station values to center on b≈ 0.11, f ≈ 0.44, m ≈ 0.45; as an example, figure 2.42 shows the at-a-station hydraulic geometry relations for the Boise River, for which b= 0.19, f = 0.45, and m = 0.35.
There have been several attempts to understand the factors that determine the exponent values, as reviewed by Ferguson (1986) and Knighton (1998). Ferguson (1986) showed conceptually that the exponents and coefficients for a given reach are determined by the channel cross-section geometry and hydraulic relations.
Following this reasoning, Dingman (2007a) used equation 2.20 along with generalized hydraulic relations to derive the relations shown in box 2.4. His analysis showed that the exponents depend only on the exponent r in the general equation for cross-section shape (equations 2.20 and 2B4.2) and the depth exponent p in the general hydraulic relation (equation 2B4.3). As shown in figure 2.41, the theoretical exponent values coincide with the central tendencies of the observed values. The effects of channel shape (r= 1, triangle; to r → ∞, rectangle) and different values of p on the exponents can be clearly seen in figure 2.41. Box 2.4 also shows the theoretical relations for the coefficients, which can take on a wide range of values depending on the channel dimensions, conductance, and slope as well as on r and p.
(Note that the coefficient values also depend on the units of measurement; the exponents do not.)
Application to Characterizing Stream Hydraulics It can be shown from equation 2.29 that dW /W= b · (dQ/Q), and analogously for equations 2.30 and 2.31; thus, the at-a-station hydraulic geometry relations give information on how small changes
88 FLUVIAL HYDRAULICS
Figure 2.41 Tri-axial diagram showing values of exponents b (width), f (depth), and m (velocity) in at-a-station hydraulic geometry relations (equations 2.29–2.31). The inner (solid) curve encloses most of the empirical values reported by Rhodes (1977); virtually all the values he reported are enclosed by the outer (dashed) curve. The lines radiating from the lower left vertex show the loci of points dictated by the value of the depth exponent p in the generalized hydraulic relation (equation 2B4.3). The lines radiating from the upper vertex show the loci of points dictated by the value of the exponent r in the generalized cross-section relation (equation 2.20).
in discharge are allocated among changes in width, depth, and velocity in a reach.
For example, if b= 0.23, f = 0.46, and m = 0.31, a 10% increase in discharge is accommodated by a 2.3% increase in width, a 4.6% increase in depth, and a 3.1%
increase in velocity.
The hydraulic geometry relations, in conjunction with the flow-duration curve, can also be used to construct curves that show the time variability of width, depth, velocity, or any other quantity that depends on discharge, using the method described in box 2.5 and figure 2.43. The information presented in such curves is invaluable for such water resource management concerns as characterizing the suitability of the reach as habitat for aquatic organisms, which typically depend on velocity and depth;
determining the frequency of overbank flooding, which is a function of depth; and evaluating the potential for stream-bed erosion at a bridge site, which is a function of velocity and depth (Dingman 2002).
10 100
1 10 100 1000
Discharge, Q (m3/s) (a)
(b)
(c)
Width,W (m)
W = 23.2·Q0.19
0.10 1.00 10.00
1 10 100 1000
Discharge, Q (m3/s)
Average Depth, Y (m)
Y = 0.133·Q0.45
0.10 1.00 10.00
1 10 100 1000
Discharge, Q (m3/s)
Average Velocity, U (m/s) U = 0.326·Q0.35
Figure 2.42 Log-log plots of (a) width, (b) average depth, and (c) velocity versus discharge for the Boise River at Twin Springs, ID, showing empirical at-a-station hydraulic-geometry relations established by regression analysis. Note that the fits are stronger at the higher discharges, and there is considerable scatter at lower flows, especially for the width relation.
BOX 2.4 Relations between the Exponents and Coefficients in At-a-Station Hydraulic Geometry Relations and Reach Properties
Starting with the basic continuity relation of equation 2.21,
Q= W · Y · U, (2B4.1)
Dingman (2007a) used the general cross-section geometry model of equation 2.20,
and a generalization of the hydraulic relation of equation 2.23,
Q= g1/2· K ·
to derive the following relations:
Width exponent b:
Velocity coefficient k:
p depth exponent in generalized hydraulic relation q slope exponent in generalized hydraulic relation r exponent in cross-section geometry relation S energy or surface slope
U average cross-sectional velocity w cross-channel distance from center W water-surface width
WBF bankfull water-surface width Y cross-sectional average water depth
≡ 1 + r + r · p.
BF bankfull maximum water depth in cross section
BOX 2.5 Construction of Duration Curves for Quantities That Are Functions of Discharge
In figure 2.43 the graph in the upper right quadrant is the flow-duration curve (FDC), established using methods described by Dingman (2002). The curve in the upper left-hand quadrant is the relation between width, depth, or velocity (or any other quantity that depends on discharge) and discharge.
The lower left quadrant is simply a 45◦, or 1:1, line.
The duration curve for width, depth, or velocity is constructed in the lower right quadrant by first selecting a number of points on the FDC covering the entire curve. From each point, a vertical line is then projected into the lower right quadrant, and a horizontal line is projected into the upper left quadrant to its intersection with the relation plotted there. A vertical line is projected from each intersection to intersect with the 1:1 line in the lower left quadrant.
Finally, horizontal lines are extended from those points to intersect with the vertical lines in the lower right quadrant. Those intersections define the relation between values of width, depth, or velocity and the corresponding exceedence probability, which defines the desired duration curve for width, depth, or velocity.
(Continued)
91
BOX 2.5 Continued
As noted in the text, the long-term average discharge, Q, is equal to the integral of the flow-duration curve:
Q= 1
0
Q(EP)· dEP. (2B5.1)
The curve constructed in the lower right quadrant of figure 2.43 is the duration curve for a quantity that is a function of Q. The long-term average value X of a quantity X that depends on discharge, X(Q), is likewise found by integrating its duration curve:
X= 1
0 X[Q(EP)] · dEP. (2B5.2)
Width, depth, or velocity Probability
DischargeWidth, depth, or velocity
Flow-duration curve
Hydraulic-geometry relation
1:1 line
Duration curve for width, depth, or velocity
Exceedence
Figure 2.43 Diagram demonstrating construction of duration curves for width, depth, or velocity from the flow-duration curve and at-a-station hydraulic-geometry relations, as described in box 2.5.
NATURAL STREAMS 93 2.6.3.2 Spatial Changes: Downstream Hydraulic
Geometry
In downstream hydraulic geometry, the hydraulic geometry relations of equations 2.29–2.31 characterize spatial changes in width, depth, and velocity through a river system at a given reference discharge, which is usually taken to be the bankfull discharge, QBF. The values of the exponents for the downstream relations determined empirically for many regions of the world have been found to vary less than those for the at-a-station relations and are typically near b= 0.5, f = 0.4, and m= 0.1. The coefficients depend on the reference discharge used (as well as the units of measurement) and vary widely depending largely on climate. Again, many attempts have been made to derive these values theoretically, mostly based on considerations similar to the stable-channel approach described in section 2.4.3.1, but there is no generally accepted explanation (for reviews, see Ferguson 1986;
Knighton 1998).
One practical application of the downstream relations is in estimating bankfull discharge, depth, and velocity using measurements of bankfull width remotely observed via satellite or air photographs (Bjerklie et al. 2003).