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<br />
<br />26
<br />
<br />PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS
<br />
<br />average value of z,Leopold (1953, p, 619) suggested --.49,
<br />but analysis of additional data leads us to revise this to
<br />_,95.' The average downstream relations, therefore,
<br />can be represented by point 8 marked with a triangle in
<br />figure 22. This indicates that the average value of y is
<br />--0.3, which means that channel roughness decreases
<br />slightly downstream for discharge of constant frequency.
<br />This apparently is related to some progressive down-
<br />stream decrease in bed-particle size resulting from
<br />sorting and abrasion.
<br />From the graph it can be seen that if width, depth,
<br />and velocity changed downstream at rates respectively
<br />equal to those in the average river (b=.5,]=.4, m=.l)
<br />and if roughness (measured by Manning-type n')
<br />remained constant downstream (y=O), then slope would
<br />decrease dm,vnstream with increasing discharge in the
<br />form
<br />
<br />SCXQ-.33
<br />
<br />"
<br />In few natural rivers~ however, does roughness relnain
<br />eonstant downstream.
<br />Values of ] and z for several rivers are plotted in
<br />figure 22, and the corresponding values of the exponent
<br />Y J expressing rate of change of roughness) ean be read
<br />from the scale given by the sloping lines, Point 2,
<br />representing the Kansas-Republican River systeml indi-
<br />cates a near-average value for rate of change of depth
<br />(j=0.43). The nearly constant value of roughness
<br />downstream (y=O) in that river system is presumably
<br />related to the fact that bed-particle size decreases a
<br />relatively small amount, ranging from sand near the
<br />headwaters to silt farther downstream. Consequently,
<br />the river gradient decreases downstream less rapidly
<br />tban in the other rivers; that is, the value of z is about
<br />- .4, whereas most of the other rivers have values
<br />between -.8 and -1.1.
<br />The headwaters of the Yellowstone and Bighorn
<br />Rivers rise in coarse gravel \vhich decreases to fine sand
<br />in the main 11issouri) and this river has a correspond-
<br />ingly rapid change of gradient as indicated by the large
<br />negative value of z in figure 22. Thus it appears that
<br />an important effect of decreasing particle size down-
<br />stream is to decrease the roughness dO\l,rnstream.
<br />
<br />EFFECT OF CHANGE OF PARTICLE SIZE AND
<br />VELOCITY ON STREAM GRADIENT
<br />
<br />Channel roughness is not determined entirely by
<br />particle size. That our roughness factor is not neces-
<br />sarily proportional to the sixth root of bed particle
<br />size-as would be expected from the Strickler equation
<br />n=ck'/6 (see O'Brien and Hickox, 1937, p, 314)-
<br />follows from the fact that n' involves not only the grain
<br />roughness but also resistance attributable to channel'
<br />configuration and other factors. However, for load
<br />
<br />composed of coarse sand or gravel, channel roughness
<br />is materially influenced by the size of grains. The
<br />manner in which particles of finer materials are piled
<br />into dunes and riflles becomes an important determin-
<br />ant of channel resistance.
<br />For a given rate of change of depth and velocity, thc
<br />rate of dccrease of stream slope (concavity of the pro-
<br />file) is adjusted to the rate of decrease of particle size
<br />as expressed by rouglmess. In figure 22 this may be
<br />seen by comparing points 2 and 6. Each has a value
<br />of] approximately equal to 0.45; that is, in these two
<br />streams the downstream rates of increase of depth with
<br />discharge are nearly identical. River 2 is characterized
<br />by nearly constant roughness downstream (y=O),
<br />whereas river 6 has a rapid downstream decrease of
<br />roughness (y ~ -- 0.3). Likewise, the slope of river 6
<br />decreases much more rapidly downstream (z~ --1.1)
<br />than river 2, (z~ --0.4).
<br />During the initial field work in New Mexico particle-
<br />size measurements of bed material 'were made by sieving
<br />scoop samples. The results were quite inconsistent
<br />owing to the largc range of size over the bed even
<br />within short reaches. In 1954 the senior author re-
<br />turned to the field area with M. Gordon Wolman t{) ob-
<br />tain new measurements, Using a grid-system sampling
<br />method (described by Wolman, 1954b) , bed-material
<br />size was measured at those reaches where discharge
<br />measurements had becn obtained earlier, and these
<br />data are summarized in appendix D.
<br />These data were plotted against discharge corrc-
<br />sponding to the stream order at the places of measure-
<br />ment, and the graph is presented as figure 23. The line
<br />drawn through the points shows the downstream rela-
<br />tion of discharge to grain size, and can be expressed R3
<br />
<br />D50cx Q-'"
<br />
<br />wheTe Dso is the median grain size; that is, 50 peTcent
<br />of the grains are finer.
<br />The streams in central New Mexico are charactcrized
<br />by the following set of values: ]=0,3, z= --0.24; and
<br />the coordinates of these values in figure 22 are indicated
<br />
<br />1 Other ways of deriving an approximate value for this expo-
<br />nent can be obtained from the equation presented by Hack(1955)
<br />relating I:3tream length, l, and slope, s,
<br />
<br />s<x:l-1
<br />
<br />and his equation relating length to drainage area
<br />
<br />locA~.6
<br />
<br />Combining these with the generalization typified by figure 21
<br />that bankfull flood discharge
<br />
<br />Q2.3rx..I.\.:,/o
<br />
<br />then
<br />
<br />,CXQ-O.8
<br />2,3
<br />
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