Drainage Net

GEOLOGICAL SURVEY PROFESSIONAL f APER 282-A
Ephemeral Streams 

Hydraulic Factors and 

Their Relation to the 

Drainage Net

By LUNA B. LEOPOLD and JOHN P. MILLER

PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 282-A

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1956
UNITED STATES DEPARTMENT OF THE INTERIOR 

CECIL D. ANDRUS, Secretary

GEOLOGICAL SURVEY

H. William Menard, Director

Fin* printing 19M 

Second printing 1K9 

Third printing 1MB

For sale by the Branch of Distribution, U.S. Geological Survey, 

1200 South Eads Street, Arlington, VA 22202
CONTENTS

Page 

Symbols. __________________________________________ iv

Abstract. __________________________________________ 1

Introduction and acknowledgments ___________________ 1

Geographic setting and basic measurements ____________ 2

Measurement of hydraulic variables in ephemeral streams, 4 

General features of flow _________________________ 4

Problems of measurement._______________________ 6

Changes of width, depth, velocity, and load at individual channel cross sections ___________________ 7

Changes of hydraulic characteristics in the downstream direction._ ____________________________ 13

Interrelation of drainage net and hydraulic factors._____ 16 

Relation of stream order to stream number, stream

length, and drainage area___-__--___-__________ 16

Page 

Equations relating hydraulic and physiographic

factors....________-_______.__-____--__---_-_ 19

Some relations of hydraulic and physiographic factors to

the longitudinal profile.___________________________ 24

Channel roughness and particle size.______________ 24

Effect of change of particle size and velocity on

stream gradient.____.__._____-_______-____--__ 26

Equilibrium in ephemeral streams _____________________ 28

Mutual adjustment of hydraulic factors._____.._._ 28

The discontinuous gully.. __--____-___---____--_- 29

Summary, _________________________________________ 33

References. ___-_-_______-___--_----______---_----.- 34

Appendix._________________________________________ 35

Index,_________________-_____-_-___-__---__---_--_ 37

ILLUSTRATIONS

FIGURE 1. Location map_ ___________________________

2. Channel of Arroyo Caliente, a fifth order 

stream ________________________________

3. Channel of Arroyo San Cristobal, a 10th order 

stream ________________________________

4. Passage of a small bore in rising stage of a 

flash flow in the Rio Puerco, N. Mex., 

September 19, 1941. A, time 1:14 p. m.; 

B, time 1:14% p. m.; C, time 1:14% p. in.; 

D, time 1:15 p. m.; E, time l:15}lo p. m.___

5. Water pouring into open pipe or tunnel, Rio 

Pescado, N. Mex.______________________

6. Relations between hydraulic variables, Rio 

Galisteo at Domingo, N. Mex.___________

7. Relations between hydraulic variables, Rio 

Puerco near Cabezon, N. Mex____________

8. Relations between hydraulic variables in 

ephemeral streams._____________________

9. Suspended-sediment rating curves, Rio Galisteo at Domingo, N. Mex., and Rio Puerco 

near Cabezon, N. Mex__________________

10. Suspended-sediment rating curves for ephemeral streams, _ _ _ ______________________

11. Downstream change of hydraulic variables in 

ephemeral streams._____________________

2

3

3

5

6

8

9

10

11

12

14

Page 

FIGURE 12. Drainage basin of typical ephemeral arroyo,_ 17

13. Relation of stream length and number of

streams to stream order. ________________ 18

14. First-order tributaries in Arroyo Caliente-___ 19

15. Arroyo de los Frijoles, an eighth-order stream. 19

16. Relation of drainage area to stream order____ 20

17. Relation of channel slope to stream order, ___ 21

18. Relation of channel slope to channel width,-. 22

19. Relation of channel width to stream order. __ 23

20. Relation of discharge to stream order.______ 23

21. Relation of flood discharge to drainage area__ 24

22. Downstream interrelations of depth, velocity,

slope, and roughness ____________________ 25

23. Downstream change of bed-particle size in

ephemeral streams,_________-___-_--__,_ 27

24. Downstream change of stream gradient in

selected basins _________________________ 27

25. Typical continuous arroyo, Rio Puerco near

Manuelito, N. Mex_____________________ 29

26. Stages in development of an arroyo from a dis-.

continuous gully------------------------ 31

27. Profile of a discontinuous gully, Arroyo Falta. 33

in
IV

SYMBOLS

a coefficient in relation of width to discharge

AA drainage area

b exponent in relation of width to discharge

c coefficient in relation of depth to discharge

d mean depth defined as ratio of cross-sectional area

to width

Ds size of sediment particle, expressed as median size 

/ exponent in relation of depth to discharge 

j exponent in relation of suspended-sediment load to

discharge

k numerical coefficient having a specific but undetermined value 

I stream length 

L suspended-sediment load in units of weight per unit

of time 

TO exponent in relation of velocity to discharge

nr a roughness parameter

N number of streams in a basin

0 stream order

p coefficient in relation of suspended load to discharge

Q discharge in volume per unit of time

#2.3 mean annual flood (equaled or exceeded on the

average once every 2.3 years) 

r a numerical coefficient 

s stream slope 

t a numerical coefficient 

v mean velocity defined as quotient of discharge

divided by cross-sectional area 

w mean width of water surface 

y exponent in relation of roughness parameter to

discharge 

z exponent in relation of slope to discharge
PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

EPHEMERAL STREAMS—HYDRAULIC FACTORS AND THEIR RELATION

TO THE DRAINAGE NET

By LUNA B. LEOPOLD and JOHN P. MILLER

ABSTRACT

The hydraulic factors of width, depth, velocity, and suspendedsediment load of ephemeral streams near Santa Fe, N. Mex., 

were measured during flood flow. Later, channel slope was 

measured. These flood-flow data, in conjunction with an analysis of drainage-basin configuration by the methods proposed by 

Horton, are used to determine the generalized interrelation of 

stream order and hydraulic variables. The method developed 

for determining this interrelation allows an integration of the 

geographic and physiographic characteristics of a drainage basin 

with the channel characteristics; specifically, the interrelation 

of the length, number, and drainage area of streams of various 

sizes with their respective discharge, width, depth, velocity, 

slope, channel roughness, and suspended-sediment load.

These interrelations show that stream order is related to stream 

length, number of streams, drainage area, and discharge by 

simple exponential functions. The relation of discharge to 

width, depth, velocity, slope, and other hydraulic factors can be 

approximated by simple power functions. Thus, any pair of 

these factors is related by exponential or power functions.

The data indicate that suspended-load measurements made 

during various stages of a few individual floods provide a close 

approximation to the suspended-load rating curve obtained from 

periodic measurements taken at a sediment station over a period 

of years.

The analysis of the hydraulic data shows that in the ephemeral 

streams studied, velocity increases downstream at a faster rate 

than in perennial rivers. This appears to be associated with an 

increase in suspended-load concentration downstream in these 

ephemeral channels.

The tendency for stream channels to maintain a quasiequilibrium with imposed discharge and load is shown to be 

characteristic of ephemeral channels in the headwaters of the 

drainage basin, even to the most headward rill.

The formation of a discontinuous gully is analyzed in terms 

of this tendency. This type of gully is characterized by a 

gradient of the channel bed flatter than the gradient of the 

valley floor in which the gully is cut. The low gradient is 

explained as a hydraulic adaptation of channel slope to quasiequilibrium with the narrow gully width. Examples indicate 

that as a discontinuous gully widens, its gradient steepens and 

its length consequently increases. Thus a series of discontinuous gullies tends to coalesce into a continuous trench having 

a bed gradient nearly equal to the slope of the original valley 

floor.

INTRODUCTION AND ACKNOWLEDGMENTS

The flow of water in natural channels may be 

described as perennial, intermittent, or ephemeral. A 

perennial stream carries some flow at all times. An 

intermittent stream is one in which, at low flow, dry 

reaches alternate with flowing ones along the stream 

length. Those which carry water only during storms, 

and are therefore called ephemeral, are generally smaller 

but are much more numerous than perennial ones. 

From the divide to the mouth of a drainage basin, the 

increase in channel size is accompanied by a decrease 

in the number of channels.

Drainage channels are more apparent and more 

abundant in arid than in humid regions where vegetation hides small rills. Vegetation also tends to increase 

the length of overland flow, thereby decreasing channel 

density. For both kinds of areas, however, the land 

presents to the eye a contrast depending on the perspective one assumes. In arid areas there is the panorama 

of broad, apparently smooth surfaces sweeping up to 

the abrupt fronts of mountain ranges or mesa escarpments. A closer examination may show that the whole 

countryside is actually cut into a myriad of rills, gullies, 

and arroyos. Humid regions, on the other hand, are 

typically characterized by rounded, rolling topography 

which is covered by natural or cultivated vegetation. 

Although many details of drainage nets are commonly 

concealed beneath vegetation, careful search discloses 

the presence of many more channels than can be inferred from a distant view.

Despite the fact that the channels of ephemeral 

streams are generally recognized to have an important 

part in the erosion of the land and resultant production 

of fluvial landforms, they have not received careful or 

concentrated investigation.. The present report is 

devoted to a preliminary quantitative description of 

flow characteristics, channel properties, and configuration of the drainage net in arid regions only. However,
PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

it is believed that the principles established may be 

applicable to humid regions as well.

Practical considerations, along with personal bent, 

dictated that channels of arid rather than humid regions 

should receive our attention first. "Accelerated erosion" in the West is a problem of great social and economic importance. The arroyos that trenched alluvial 

valleys of the Southwest beginning in the latter part of 

the 19th century, some of which even now are eroding 

apace, are spectacular examples of ephemeral streams. 

Conservationists, range managers, geologists, and a host 

of others have debated the cause of gullying, have 

experimented with methods of gully control, and have 

tried to prognosticate the eventual status of the channel 

system. But in all such efforts practically no attempt 

has been made, to study the process of gully erosion 

itself, to describe the hydraulic conditions in the eroding 

channels, or to understand the nature of the equilibrium 

which was upset by grazing and climatic change. It 

seems bootless to spend large sums on the control of 

gullies and arroyos without concomitant attempts to 

increase knowledge of their hydraulic characteristics.

The present study provides new data that are pertinent to a fuller understanding of ephemeral streams. 

Specifically, geometric and hydraulic properties of 

channels are related to the drainage-net .configuration.

The plan of presentation is as follows: First, the channels in the area studied are described and the methods 

of investigation outlined. Next, the measurements of 

hydraulic variables, including channel shape, discharge, 

and sediment load are presented. A method of integrating hydraulic characteristics with properties of the 

drainage net is shown to increase markedly the utility 

of available data. Finally, the role of equilibrium in 

ephemeral channels, as related to the accelerated erosion 

problem, is discussed.

Many colleagues and friends gave us continuing 

advice and assistance in the study, and particular 

thanks are due M. Gordon Wolman, Thomas Maddock, 

Jr., John T. Hack, and Walter B. Langbein. The 

field work was made considerably easier in many ways 

by the cooperation of Paul C. Benedict and Berkeley 

Johnson. Charles E. Stearns visited us in the field 

and later discussed the manuscript with us.

We are indebted to John T. Hack and Stanley 

Schumm for permission to include in this report some 

of their unpublished data.

GEOGRAPHIC SETTING AND BASIC MEASUREMENTS

Nearly all the data included in this report were collected in the basins of the Rio Galisteo and Rio Santa 

Fe, tributaries to the Rio Grande in semiarid central 

New Mexico (see fig. 1).

NEW MEXICO

INDEX MAP

FIGURE 1.—Location map showing area in New Mexico where detailed studies were made.
EPHEMERAL STREAMS

As one approaches Santa Fe from the west, it appears 

that the Sangre de Cristo Mountains are abutted by a 

single relatively broad remnant of an erosion surface 

cut on poorly consolidated sand and gravel of the Santa 

Fe formation of Miocene and Pliocene age. From the 

top of any of the lava-capped mesas near the Rio Grande, 

one obtains a splendid view of this surface which slopes 

gently upward to the east toward the mountains. 

Actually, the plain, which from afar appears smooth, 

consists of several erosion surfaces differing but little 

in elevation. Furthermore, close inspection shows that 

the relief is in fact greater than it appeared from a more 

distant view. Rolling hills, dissected by gullies and 

rills and interlaced with sandy, flat-floored washes, are 

nearly everywhere at hand. The general appearance 

of streams in the area studied can be visualized by inspection of the photographs in figures 2 and 3. (For 

definition of stream order see p. 16.) The network of 

drainage channels ramifies upstream into increasing 

numbers of successively smaller gullies and rills extending almost to the divides.

As elsewhere in arid regions, these channels present 

a variety of forms. They range from tiny rills biting 

back into mesa escarpments to deep trenches or flumelike arroyos incised in otherwise flat alluvial valleys. 

Despite these striking differences in appearance, they 

all have certain common characteristics besides the 

ephemeral nature of their flow. First, vegetation in or

FIGURE 2.—Channel of Arroyo Caliente, a small tributary to Arroyo de los Frijoles 

near Santa Fe, X. Mex. The channel shown is typical of a fifth-order stream in the 

area studied.

FIGURE 3.—Channel of Arroyo San Cristobal just below Highway 41,1 mile south of 

Galisteo, NT. Mex. This stream is of 10th order. Mudballs in the channel bed 

attained a diameter of 2}-i feet. High-water marks indicated recent flows flooded 

thp willows in upper right and nearly reached the top of the vertical bank seen just 

to the left of the plinetable.

along the channel is so sparse that it exerts almost no 

influence on the form of the channel. Second, because 

large amounts of fine debris are readily available for 

transportation both in iriterstream areas and in the 

channel itself, the sediment concentration during flows 

greatly exceeds that of streams in more humid areas. 

Third, downcutting appears to proceed slowly after the 

initial development of gullying in a given reach. In a 

developing arroyo system, channels quickly achieve 

maximum depth and thenceforth little change in depth 

occurs despite a considerable thickness of easily removed 

material.

An arroyo in the Southwest discharges water only 

when a moderately heavy rain falls on the drainage 

basin. This is typically a summer phenomenon since 

flow-producing rain falls only from thunderstorms. 

Winter rains are of too low intensity to provide surface 

runoff and for this reason arroyos practically never flow 

in winter.

Summer thunderstorms in New Mexico typically produce rain over 5 to 50 square miles, but the larger coverage ordinarily results from movement of the storm. 

The storm itself covers, on the average, about 10 square 

miles. Depth-area curves of thunderstorm rainfall in 

Xew Mexico published by Leopold (1942) show that 

usually not more than 3 to 4 inches of rain falls over 3 

square miles and the amount decreases to 1 inch over 

50 to 100 square miles. It was our experience that 

flash flow in arroyos seldom occurs if the rainfall at the
4 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

storm center is less than 1.5 inches. Another rule of 

thumb is that for arroyos to flow, the curtain of rain 

as viewed from a distance must be so dense that one 

cannot see through it.

During three field seasons we were constantly on the 

watch for thunderstorms and when one appeared close 

enough to be reached in less than half an hour, we hurried to get to it. By seriously chasing individual 

storms, and even though helped by a rather intimate 

knowledge of local roads and geography, we were able 

in this period to obtain hydraulic measurements less 

than a dozen times. Most of them were for locations 

near Santa Fe, with the others at scattered points in 

Colorado, Wyoming, and Nebraska. As will be described later, the hydraulic characteristics measured 

during storms include channel width, depth, velocity, 

and suspended-sediment load.

It was our plan to combine the measurements of 

hydraulic factors with data obtained in the field when 

the channels were dry, and with information taken from 

maps and aerial photographs. Slope was measured 

with a telescopic alidade, and channel width was determined by taping or pacing. At the locations where 

flow was measured, material comprising the channel 

bottoms was sampled. Drainage areas and stream 

orders were determined from the excellent planimetric 

maps compiled by the Soil Conservation Service from 

aerial photographs and available for i parts of central 

New Mexico.

MEASUREMENTS OF HYDRAULIC VARIABLES IN 

EPHEMERAL STREAMS

GENERAL FEATURES OF FLOW

The ephemeral nature of flow in arroyos is their most 

impressive characteristic. Flash floods are the rule. 

A typical dry arroyo reaches peak discharge in less than 

10 minutes, the high flood flow seldom lasts more than 

10 minutes, and flow decreases to an insignificant 

amount in less than 2 hours.

A typical flood rise of small magnitude is shown by 

the series of photographs in figure 4. This series was 

taken by Herbert W. Yeo, whose diligence in obtaining 

sediment records added much to early knowledge of the 

suspended load carried by New Mexico streams. His 

work was the prelude to the establishment of regular 

sediment-sampling stations in that State. These photographs present a graphic account of what we observed 

several times during attempts to measure arroyo flow.

If a stream rises from nearly a dry condition to a 

depth of several feet within 10 minutes, one might well 

speak of a wall of water coming down the channel. 

Bores as large as 2 feet in height have been seen by 

the authors. However, observations made during the

course of this study cause us to conclude that a "wall 

of water" is less common than a rapid increase in 

stage attained through a succession of small surges or 

bores each a few inches in height. The rise shown in 

the photographs of figure 4 is believed to be typical in 

that the initial front does not amount to more than 

half of the peak depth. Jahns (1949) stated that 

debris flows may have nearly vertical but slow-moving 

fronts. As he pointed out, desert floods may include 

all of the gradations from mudflows to nearly clear 

water. We believe, however, that the faster moving 

fronts are seldom vertical "walls of water," but a 

succession of small bores as described above; Jahns' 

photographs also support this contention.

A flood in Canada Ancha Arroyo, July 26, 1952, 

provided an exceptional opportunity to observe the 

surges or bores. At maximum flow the width was 

about 100 feet, mean depth was estimated to be 1 

foot, and mean velocity slightly exceeded 5 feet per 

second. During the 5 minutes immediately preceding 

peak stage, a series of bores each % to 1 foot high 

moved down the channel at a velocity estimated to be 

greater than that of the water itself.

The approach of the third bore made it apparent 

that they were spaced rather regularly in time. Thereafter, we measured with a stopwatch the intervals 

between successive bores which were 31, 35, 34, 48, 

and 60 seconds respectively. Between surges the 

water stage decreased somewhat, as judged by submergence and reemergence of a gravel bar in the 

channel. Furthermore, the peak stage was much less 

than the sum of the heights of the eight individual 

wave fronts.

The nearly constant period between five of the eight 

surges seems to rule out the possibility that they 

resulted from successive arrivals of flood peaks from 

different upstream tributaries. Rather the bores are 

a type of momentum wave associated with the hydraulics of the channel itself.

One of our first opportunities to measure the flow in 

an arroyo was in a narrow and steep-walled channel 

cut in fine alluvium. After the stage had fallen and 

the channel was nearly dry, we heard a succession of 

"plops" which attracted attention. The source of the 

noises turned out to be falling chunks of bank hitting 

the nearly dry stream bed. It became obvious that 

bank caving followed the flood recession.

On every subsequent opportunity, examples of 

undercutting and bank caving due to high velocity 

water were specifically sought, but never experienced. 

Yet, without exception, caving of wetted banks intp 

the channel after the flood was observed. Moreover, 

on walking any dry arroyo one will find debris fallen
EPHEMERAL STREAMS

A. Time 1:14 p. m. Upstream view. Bore is covering the small sand island. B. Time about 1:14H p.m. Bore advances faster in deep than in shallow part of channel.

C. Time about 1:14?4 p. m. Floating debris typical of rising flood stage can be seen

on surface.

D. Time about 1:15 p. m. View across channel parallel to bore face. 

slope of water surface as shown by shadow of vertical bank.

Note the

JSW

•<SI*

E. Time about l:15Mo p. m. View diagonally downstream at agitated and debrisstrewn area just behind the bore.

355397—56———2

FIGURE 4. — Passage of a small bore in the rising stage of an 

ephemeral flow in an arroyo channel, Rio Puerco, a tributary 

to the Rio Grande. Location is 8 miles north of Puerco 

Station, N. Mex., September 19, 1941. Soil Conservation 

Service photographs.

from the vertical walls and lying crumbled but uneroded in the channel. This debris is picked up and 

washed away by subsequent flows and is undoubtedly 

an important source of debris load.

Flash floods in arroyos, therefore, appear to do but 

insignificant amounts of bank cutting as a direct result 

of impingement of flow on the banks. Wetting of the 

banks, however, results in subsequent collapse of 

arcuate slabs of alluvium which tumble into the channel 

to become important additions to the load of later 

floods.
6 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

Collapse of gully walls is greatly facilitated by piping 

or tunnels which develop in the gully walls and lead 

waters from the adjoining surface to the channel by an 

underground route (see fig. 5). It was our observation

FIGURE 5.—Water pouring into open pipe or tunnel leading downward to arroyo 

channel. The upper opening of this piping hole is at least 15 feet from wall of gully. 

Eio Pescado near Ramah, N. Mex., August 17, 1946.

that only a small proportion of total flow in a gully 

reaches the gully channel by direct overpour of the 

vertical banks. Piping tunnels and tributary gullies 

and rills deliver the bulk of the discharge.

The manner in which relatively large pebbles or 

cobbles move during flash flows is particularly worthy 

of comment. The bed material in the ephemeral 

streams studied in the Santa Fe area characteristically 

is composed of a matrix of moderately well sorted 

coarse sand, but it includes a certain number of cobbles, 

rocks, and even some small boulders. The cross section of flowing water during flash floods is wide and 

shallow, but the velocity of the water is high. Despite 

the small depth of flow, the large cobbles are effectively 

moved by rolling. Mudballs move in a similar fashion, 

rotating about the longest axis. Even cobbles which 

are irregular in shape and subangular roll along the 

stream bed for long distances without stopping. Cobbles were observed to roll spasmodically but rapidly 

even when the water was no deeper than half the diameter 

of the rolling object. At this depth the water seems to 

splash up on the upstream side of the cobble and plunge 

over its top, so providing a torque. It is indeed common to see particles, small and large, sticking well out

of the general water surface and rolling rapidly downstream with only temporary interruptions.

PROBLEMS OF MEASUREMENT

Attempts to obtain precise measurements of arroyo 

floods are fraught with many difficulties and inherent 

dangers. It was necessary to adopt unorthodox 

methods that yield data which are admittedly crude. 

Nevertheless, the data themselves are unique, and 

they appear to be adequate for the kinds of analyses 

undertaken.

Three factors militate against good measurements of 

the rising stage of arroyo floods. First, the stage rises 

to peak so quickly that one can seldom be present 

during the few minutes of rise even when he is trying 

to. Second, peak stage is dangerous for a person 

wading in the flood because of high velocity and the 

occurrence of surges or bores. Finally, peak flow of 

consequence occurs generally near the storm center 

where lightning is a deterrent to wading operations. 

Hence, most of the hydraulic data presented here were 

obtained during the falling stage of the individual floods. 

All measurements were made by wading; velocities 

were measured with a Price current meter.

When rocks as much as half a foot in diameter batter 

one's feet and meter in a current flowing 6 feet per 

second, and when the sand is constantly undermined 

from under one's heels, short-cut methods inevitably 

are adopted. For reasons which follow, our discharge 

measurements of arroyo flow must be considered rough 

approximations. Instead of using 20 to 30 measuring 

points or "verticals" across the channel, only 10 to 15 

were used. The duration of current-meter observation at each point was reduced from 40 to 20 seconds, 

and the meter was set at 0.6 depth in most cases. It 

is standard procedure to make adjustments for variation in water level during the measurement, but stage 

could be measured only crudely. We traversed back 

and forth along the tag-line without interruption during 

the falling flood. Each traverse of the 10 or 15 sections 

across the channel was considered a measurement of 

discharge, and the mean stage during the traverse was 

assumed to apply to that measurement.

Suspended-sediment samples were collected with the 

DH 48 hand-sampler or, in a few cases, by dipping a 

bottle without the aid of a hand-sampler. The samples 

were collected near the midpoint in time of discharge 

measurements. Usually two samples, each depthintegrated, were collected at two points in the channel 

cross section. The concentration of sediment was determined separately in the laboratory. The average 

of the two concentrations was considered to be representative of the flow during the discharge measurement.
EPHEMERAL STREAMS

CHANGES OP WIDTH, DEPTH, VELOCITY, AND LOAD 

AT INDIVIDUAL CHANNEL CROSS SECTIONS

Data for establishing the relation of width, depth, 

velocity, and load to discharge were obtained from two 

sources: our own wading measurements and gaging 

station records. Although there are several gaging 

stations in central New Mexico on streams that are 

essentially ephemeral, the stations command basins of 

considerable size. Thus, our wading measurements 

provide data not otherwise available and also extend 

considerably the ranges of the several hydraulic variables studied. The gaging station records used for 

comparison with the data from smaller ephemeral 

streams are for the Rio Galisteo at Domingo (drainage 

area about 670 square miles) and Rio Puerco near 

Cabezon (drainage area 360 square miles.)

Figure 6 shows the relation of width, depth, and velocity to discharge at the gaging station on the Rio Galisteo 

at Domingo. Figure 7 shows the similar graphs for the 

gaging station on the Rio Puerco near Cabezon. When 

graphs of this kind were first presented by Leopold and 

Maddock (1953) the scatter of points was poorly understood. Wolman (1954a) showed that in a stream where 

bed scour and fill are minor, the scatter of points is 

markedly reduced if the successive measurements are 

made at identically the same cross section. Clarification of this matter is important in that the scatter need 

not be attributed to lack of adjustment between the 

channel and the imposed load and discharge.

More scatter of points occurs on the at-a-station 

curves for streams having sandy or silty beds which are 

easily eroded than for streams flowing in gravelly channels. Except for the effect of varying position of measurement, the remaining scatter apparently is caused by 

relatively rapid adjustment of channel shape to changes 

in suspended-load concentration. These changes in 

concentration result from the varying location, intensity, or other characteristics of storms and the consequent varying amount of erosion on the watershed 

lands.

Data obtained from measurements of flash flow in 

small ephemeral streams are plotted in the graphs of 

figure 8, which show variation of width, depth, and 

velocity with changing discharge at each cross section. 

The same figure includes the lines representing the data 

from the two gaging stations discussed earlier as typical 

of the large ephemeral streams in central New Mexico.

When one sees the wide, flat-bottomed, sandy channels characteristic of the ephemeral streams in central 

New Mexico, he might obtain the impression that even 

at very low discharges the water flows over the full 

width of the channel. Actually, as shown in figure 8, 

width of the flowing water increases progressively with 

discharge in a quite uniform manner. The rate of

increase is about the same as for midwestern perennial 

rivers. Values of b in the equation relating width, w, 

to discharge, Q,

at a given cross section range from .09 to .44, and have a 

median of .26. This is the same as the average value of 

b for river data studied by Leopold and Maddock (1953, 

p. 9). For a discharge of 100 cfs, the width of flowing 

water in the various channels measured ranged from 

about 30 to 90 feet.

The rate of increase of depth with discharge at a given 

cross section, shown by the slope of the lines, tends to be 

generally similar for the several channel cross sections 

measured. The slopes ranged from .61 to .24, with a 

median of .33; the slopes represent the values of the 

exponent, /, in the equation relating depth, d, to discharge,

For rivers, this exponent has an average value of .40. 

The values here are for mean depth, defined as the 

quotient of the cross-sectional area of flowing water 

divided by the width of flowing water. The available 

data are insufficient to conclude that depth increases 

with discharge less rapidly in the ephemeral streams 

studied than in rivers. It should be noted that, as in 

rivers, the depth in various arroyo channels for a 

particular discharge varies considerably. For instance, 

at a discharge 100 cfs, for which six examples are available, the depth ranges from 0.25 to 0.8 feet.

Mean velocity (defined as discharge divided by 

cross-sectional area) increases with discharge at about 

the same rate in both the ephemeral streams and perennial rivers. The median value of m in the equation 

relating velocity, v, to discharge,

v=kQm

is .32 for the arroyos as compared with .34 for rivers 

previously studied. At a discharge of 100 cfs, mean 

velocity in the various channels measured ranged from 

2.9 to 5.0 feet per second.

As shown previously (Leopold and Maddock, 1953, 

p. 8),

because Q=iydv

then Q=aQ»XcQfXkQm 

and b-\-j-\-m=l

Even without weighing or adjustment, the sum of the 

respective median values of these three exponents for 

arroyo data equals .91. With so few data available 

there is no rational way to adjust the median reported 

for each exponent. Refinement of these values must 

await collection of additional data.
8 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

10 100 1000

> DISCHARGE, IN CUBIC FEET PER SECOND

FIGURE 6.—Relation of width, depth, and velocity to discharge at the stream gaging station on Rio Qalisteo at Domingo, N. Mex. Crosses indicate slope-area measurements.
EPHEMERAL STREAMS 9

O

o

LU 

CO

o: LU

Q_

o 

o

LU

> .5

.- ^-\ •v.»»^»

1 10 100

DISCHARGE, IN CUBIC FEET PER SECOND

FIGURE 7.—Relation of width, depth, and velocity to discharge at the stream gaging station on the Rio Puerco near Cabezon, N. Mex.
10 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

500

D Tributary to Hermanas Arroyo near Us Dos 

+ Ancha Chiquita near El Rancho Montoso 

• Canada Ancha at El Rancho Montoso 

X Rio Santa Fe at Old Albuquerque Road 

O Rio Galisteo at Cerrillos 

V Arroyo de los Chamisos at Mt. Camel Chapel 

	Meet John Wash near Casper, Wyoming

Arroyo measurements by 

Leopold and Miller

U. S. Geological Survey 

stream-gaging station

10 100 1000 10,000

DISCHARGE, IN CUBIC FEET PER SECOND

FIGURE 8.—Relation of width, depth, and velocity to discharge at several cross sections in western ephemeral stream channels. These represent at-a-station relations.
EPHEMERAL STREAMS 11

Figure 9 presents the data for suspended load at 

various discharges (the suspended-sediment rating 

curve) measured at two Geological Survey sediment 

stations. The same section is used for both discharge 

and suspended-load measurements at the Domingo 

station on the Rio Galisteo. For the other example, the 

sediment station is located 11 miles downstream from 

the gage on the Rio Puerco at Cabezon. Because no 

major tributaries enter in this reach of the stream, the 

sediment and discharge records may be considered 

essentially comparable.

The data from sediment measurements in small 

ephemeral streams are shown on figure 10, along with 

curves for the two sediment stations. In each arroyo 

cross section we measured only a single flood, whereas 

in regular sediment operations each measurement

represents generally a different hydrograph rise or 

different day. Nevertheless, lines drawn through our 

meager data agree very well with the sediment-rating 

curves of the regular sediment stations. Furthermore, 

the scatter of our data obtained by wading measurements (fig. 10) is no greater than that for the stations 

(fig. 9). This suggests that the suspended-load rating 

curve at a given cross section in a western ephemeral 

stream can be approximated by a series of measurements during one or two floods.

The relation of suspended load to discharge has been 

expressed as

L=pQ?

where L is suspended load in tons per day (Leopold and 

Maddock, 1953). Values of j (slope of the suspendedsediment rating curve) for arroyos range from 1.09 to

100,000

10,000

1000

Rio Galisteo at Domingo, New Mexico Rio Puerco at Cabezon, New Mexico

100

10 100 500 100 1000

DISCHARGE, IN CUBIC FEET PER SECOND

FIGURE 9.—Suspended-sediment rating curves for Rio Galisteo at Domingo, N. Mex., and for Rio Puerco near Cabezon, N. Mex., measured one-fourth mile above the

mouth of Chico Arroyo.
12 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

EXPLANATION 

Hermanas Arroyo near Las Dos 

Canada Ancha at El'-Rancho Montoso 

Rio Santa Fe at Old Albuquerque Road 

Rio Galisteo at Cerrillos

Arroyo de los Chamisos at Mt. Carmel Chapel 

Sowbelly Arroyo near Chadron, South Dakota 

Sedalia Gully near Sedalia, Colorado

Arroyo measurements by 

Leopold and Miller

U. S. Geological Survey 

sediment station

10 100 

DISCHARGE, IN CUBIC FEET PER SECOND

1000

FIGURE 10.—Change of suspended-sediment load with discharge at various cross'sections'during'individual storms. Comparison is made with similar curves

for Geological Survey sediment stations.
EPHEMERAL STREAMS 13

1.58, with a median of 1.29. For perennial rivers, the 

average value of j lies between 1.5 and 2.0. As this 

exponent is greater than unity, load must increase 

faster than discharge, or in other words, suspended-load 

concentration increases with increasing discharge.

CHANGES OF HYDRAULIC CHABACTEBISTICS IN THE 

DOWNSTBEAM DIBECTION

The graphs just presented show how width, depth, 

velocity, and suspended load vary with changing discharge at individual channel cross sections. Each 

discharge on these at-a-station curves represents an 

occurrence of different frequency. The hydraulic variables (width, depth, and others) at several cross sections 

along the length of a stream may appropriately be 

compared only for some constant frequency of discharge. 

Thus, the change of a variable in the downstream 

direction can be determined by drawing a line through 

those points on the at-a-station graphs which represent 

a discharge of equal frequency at all cross sections. 

This construction is part of the hydraulic geometry.

As the arroyos were measured whenever and wherever 

an opportunity was afforded, it is not possible to estimate the frequency or recurrence interval of the discharge recorded in the individual floods. We feel 

certain that none of the arroyo flows gaged were from 

unusually severe storms.

Lacking other alternatives, we assume that the lines 

defined by plotting all measurements of each variable 

against discharge represent the downstream changes 

at some constant frequency of discharge. The plot 

of width against discharge (fig. 11) lends credence to 

this assumption. Previous work has shown that the 

most nearly constant relation among the hydraulic 

variables in natural stream channels is the rate of 

increase of width with discharge in the downstream 

direction. This is the value of the exponent b, in

w=aQt>

The average value of b for rivers and regime canals 

is .5. In figure 11 the slope of the width-discharge 

relations for all our arroyo measurements is nearly .5 

regardless of exactly how one chooses to draw the line 

through the points. Because of the general agreement 

in the value of this exponent, it is believed that the 

plots in figure 11 approximate the downstream relations at a constant, though unknown, frequency of 

discharge.

The values of/ and ra, respectively the downstream 

rate of increase of depth and of velocity, are .3 and .2 

for arroyos as compared with .4 and .1 for perennial 

streams. No great reliance should be placed on the 

exact value of these exponents for arroyos, considering

355397—56———3

the nature of the data and the limited number of 

observations. Indeed, the average values of the same 

exponents in river data need verification with additional 

data. Tentatively, however, we conclude that for 

arroyos the downstream increase of depth is less and 

velocity greater than had been found for river data. 

An explanation of these differences appears to be found 

in a consideration of the sediment load.

Parallelism of the individual at-a-station graphs 

simplifies development of downstream relations for 

the sediment data. Regardless of the frequency chosen, 

any line representing the increase of suspended load 

with increasing discharge in the downstream direction 

would nearly coincide with the at-a-station lines already 

plotted on figure 10. Such a line has a slope of about 

1.3, which is the median of the slopes of the individual 

at-a-station graphs. Although the scatter allows some 

latitude of choice, it is apparent that any line through 

the data plotted on figure 10 must have a slope greater 

than unity. This leads to a conclusion of considerable 

interest.

From analyzing data for rivers, Leopold and Maddock 

concluded that in the downstream direction suspendedsediment load increased less fast than discharge. By 

a rather roundabout analysis, they found the mean 

value of j to be 0.8. Sediment stations are not arranged along the length of any single river in sufficient 

number to analyze directly the change of sediment 

load in the downstream direction. Whereas the j 

value of 0.8 also could be justified by deduction, the 

fact remains that those authors did not have so good 

a check on the downstream change of sediment load 

as is available in the present data.

The arroyo data indicate that suspended-sediment 

load increases downstream more rapidly than discharge 

and, therefore, suspended-sediment concentration increases downstream. From analysis of data for perennial rivers the opposite conclusion was reached; namely, 

that suspended-sediment concentration decreases in 

the downstream direction.

A downstream increase in sediment concentration 

requires (1) that runoff entering the river at downstream 

points carry larger sediment concentrations than in 

the headwaters, or (2) that the water pick up progressively more and more sediment off the stream bed as 

it flows down the channel, or (3) that water be progressively lost by percolation into the channel.

For an area of uniform rock and climate, slopes 

tend to be steeper near the headwaters than downstream. Drainage density tends to decrease downstream, and this means that the length of overland 

flow contributing to headwater rills is less than to 

minor channels which enter the main stream in down-
14 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

stream areas. The flatter slopes and greater lengths Our measurements in ephemeral streams were conof overland flow in the lower parts of drainage basins fined to relatively small drainage basins, all of which

seem to militate against a downstream increase in were characterized by fairly uniform lithology. It

sediment concentration contributed to the channel appears necessary then, to assume that the observed

system from the watershed. increase in concentration results either from a gradual

1000

1 10

DISCHARGE, IN CUBIC FEET PER SECOND

100 1000 10,000

FIGURE 11.—Change in width, depth, and velocity with discharge as discharge increases downstream with drainage area. Points plotted include all individual measurements made on ephemeral channels in New Mexico.
EPHEMERAL STREAMS 15

and progressive pickup of sediment off the stream bed 

or from loss of water. The relative aniounts of these 

two is not known. We presume that the downstream 

increase in sediment concentration measured by us 

can be attributed primarily to percolation into the 

channel bed, but this needs further verification.

Many ephemeral streams in the study area have degraded somewhat since 1860, when a combination of 

climate factors and overgrazing set off an epicycle of 

erosion. Most of the gullying occurred relatively soon 

after this erosion epicycle began. The beginning was by 

no means simultaneous in all valleys, for some had begun 

to erode in 1860 and others as late as 1915 (Leopold, 

1921). Many are still actively cutting, at least in the 

headwaters. So, there is some reason to suppose that 

the channels measured might still be undergoing some 

deepening which would imply a progressive pickup of 

debris along the length of the channel. To the extent 

that this is true, it would account for a downstream increase of suspended-load concentration.

It is well known that arroyo flows gradually lose 

water by percolation into the stream bed, but the 

amounts of such loss have been studied only in a few 

instances (Babcock and Gushing, 1941). These investigators found that the average rate of water seepage into 

the channel of a dry wash in Arizona during flash floods 

was about 1 foot of water depth per day. R. F. Hadley 

(personal communication) observed a flow in Twentymile Creek near Lusk, Wyo., which he estimated as 40 

cfs in an upper reach but which was entirely absorbed 

by the channel in a distance of 6 miles. This channel 

loss is particularly important in the ephemeral channels 

in semiarid areas where stream beds are often underlain 

by considerable thickness of sandy or gravelly alluvium.

Though it is not possible from our data to allocate to 

these two processes proportions representing their relative importance, the fact remains that it is logical that 

suspended-sediment concentration should increase downstream in these channels. It should be expected, therefore, that other channel characteristics would show some 

tendency to adjust to this condition. The manner of 

adjustment should be s.hown by the average values of 

the exponents representing the change of channel factors 

with discharge, and those values will be summarized and 

discussed in the following paragraphs.

Before summarizing the salient features of the hydraulic geometry of ephemeral streams described in this 

report, it should be stressed again that the median 

values of the exponents relating width, depth, velocity, 

and load to discharge are derived from a small number 

of measurements. These median values should, therefore, be considered as merely indicative of magnitude. 

With this qualification, the data presented here show 

that the general principles laid out in Professional Paper 

252 find verification in western ephemeral streams as 

well as in u single eastern stream, Brandywine Creek, 

Pa. (Wolman, 1954a). At cross sections in ephemeral 

stream channels the relations of width, depth, and 

velocity each to discharge are generally similar to those 

found in the perennial rivers previously studied, as can 

be seen by the tabulation below. The basic equations 

are listed in the left column, and the main body of the 

table presents values of exponents in the equations for 

ephemeral channels and for the average river.

Average downstream Average at-a-station 

relations relations

Average Average

Ephemeral river Ephemeral river

channels channels 1 channels channels 1

= kQ

6-0.5 

/= .3 

m = .2 

j=l.3

6 = 0.5 6 = 0.26 2 6= 0.26

/= .4 /= .33 2 6= .40

m= .1 m= .32 2 6= .34

j= .8 ; = 1.3» ; = 1.5-2.0

TM= ' 25 

1 Leopold and Maddock (1953). 2 Unadjusted median values.

To summarize, at discharge of constant frequency in 

both perennial rivers and ephemeral streams, width increases downstream as the square root of discharge. 

Velocity, however, increases more rapidly downstream 

in ephemeral streams, and this tendency is accompanied 

by a less rapid increase in depth. This more rapid downstream increase in velocity can be interpreted as a response to the downstream increase in suspendedsediment concentration. Indeed, the observed relations are in agreement with the principle enunciated by 

Leopold and Maddock (1953, p. 26) that "the rate of 

increase of suspended-sediment load with discharge is 

a function of the ratio :

rate of increase of velocity with discharge 

rate of increase of depth with discharge

Those authors state (p. 26) the generalization in this 

way: "for any given value of 6 * * * the value of j increases with an increase in the m to / ratio." The 

agreement with this principle can be visualized by comparing the downstream values of these exponents in the 

ephemeral channels with those in the average river data. 

So, it can be seen that the channel factors in the 

ephemeral channels appear to differ from those of the 

average river in a manner which is in accord with the 

observed change of suspended-load concentration.
16 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

INTERRELATION OF DRAINAGE NET AND HYDRAULIC 

FACTORS

RELATION OF STREAM ORDER TO STREAM NUMBER, 

STREAM LENGTH, AND DRAINAGE AREA

The quantitative description of drainage nets developed by Horton (1945) related stream order to the 

number, average length, and average slope of streams 

in a drainage basin. Our purpose in this section is to 

show how this useful tool may be extended to include 

the hydraulic as well as drainage-net characteristics.

All the data required for the Horton type of drainagenet description can be obtained from maps. As maps 

of several different scales were required for our own 

analysis, some explanation of the procedure is in order.

Figure 12 presents planimetric maps of a sample area 

near the city of Santa Fe. The map on the right, which 

shows the drainage net in a typical watershed about 

9 miles long by 2 miles wide, was compiled from planimetric maps made by the Soil Conservation Service 

from aerial photographs at an original scale of 2 inches 

to the mile. The left map shows in more detail the 

drainage net in one small tributary which for purposes 

of this report we will refer to as Arroyo Caliente. This 

map was made by pace and compass after a planetable 

traverse had been run for control. Each tributary rill 

was paced out to its farthest upstream extension in 

order that the map would include all recognizable 

channels.

The orders of various channels in the basin of Arroyo 

Caliente are indicated by numbers appearing in the 

upper part of the left map of figure 12. A small 

unbranched tributary is labeled "order 1," and the 

stream receiving that tributary is labeled ''order 2." 

All streams of orders 3, 4, and 5 are labeled with appropriate numbers near their respective mouths.

On the right map of figure 12, the little basin called 

Arroyo Caliente is one of the minor tributaries which 

even on this small-scale map appears to be unbranched 

like the tributary just west of it. If only this map 

were available, one would conclude that Arroyo Caliente 

is a first-order stream. This points up an important 

qualification to the Horton scheme of stream-order 

classification; namely, that the definition of a first-order 

stream depends on the scale of the map used. The 

first-order stream, by definition, should be the smallest 

unbranched channel on the ground. The designation 

of which stream is master and which is tributary is 

somewhat arbitrary, but we have followed the guide 

suggested by Horton (1945, p. 281).

The largest drainage basin which is included in the 

present analysis is that of the Rio Galisteo (fig. 1). At 

its mouth this basin drains about 670 square miles. 

Such an area contains a very large number of small 

tributaries. It was desired to estimate the number of

tributaries of various sizes, their lengths, and other 

characteristics. The task of counting and measuring 

each individually would be inordinately great but 

approximate answers could be obtained by a sampling 

process. Arroyo Caliente is one of the samples used.

The detailed map of Arroyo Caliente was used to 

determine the number, the average length, and average 

drainage area of each order of stream in its basin. At 

its mouth, Arroyo Caliente is of fifth order.

The Arroyo de los Frijoles basin shown in small scale 

at the right in figure 12 was used as another sample and 

similar measurements were made. The small, unbranched tributaries on this 1 mile to 2 inch map, of 

which Arroyo Caliente is one, would be designated 

order 1, in accordance with the definition of stream 

order. The detailed study of Arroyo Caliente showed, 

however, that on the ground this tributary which had 

appeared unbranched was, in reality, composed of a 

drainage network of still smaller tributaries. Arroyo 

Caliente and other channels which appeared as order 1 

on the right-hand map are, on the average, actually of 

5th order. Thus the true order of any stream determined from the right-hand map of figure 12 is increased 

by adding 4, so that an order 1 stream on that map 

becomes order 1+4, or 5.

This provides a way of combining maps of different 

scales to carry the numbering of stream order from one 

map to the other. It can be seen in the plot of stream 

length against stream order (left graph of figure 13) 

that this relation in stream orders 1 to 5 (average values 

of Arroyo Caliente) fits well with data from the smallscale maps after the values of stream order were 

adjusted as described above.

Similarly, this principle was used to obtain the estimate of the number of streams of each order in an llth 

order basin in the area studied, as shown in the right 

diagram in figure 13. An actual count of the number of 

streams of highest order was made. From the smallscale maps the order of the Rio Galisteo at its mouth 

(fig. 1) was determined to be 7, which when adjusted 

by 4, indicates the true order of 11. Because this is the 

only stream of order 11 in the area studied, the graph 

must go through the value of 1 on the ordinate scale at 

an abscissa value of 11. The mean relation was drawn 

for the numbers of streams of highest orders and extrapolated to determine the number of streams of order 5, 

the smallest tributaries shown on the small-scale map. 

The graph of number of streams of orders 1 to 5 in the 

Arroyo Caliente basin which included only one stream 

of order 5 was superimposed on the graph determined 

from the 2 inches to 1 mile map and placed so that the 

points representing order 5 coincided. By extending 

the graph to order 1, the number of 1st order tributaries 

in the llth order basin could be estimated.
ARROYO D

DRAIN

ARROYO CALIENTE . 

BASIN \

\

0 200 feet \

FIGURE 12.—Drainage basin of typical ephemeral arroyo near Santa Fe, N. Mex. Left, basin of Arroyo C aliente, a tributary to Arroyo de los Frijoles; right, basin o

Frijoles showing location of the tributary Arroyo C aliente.
18 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

1,000,000

EXPLANATION 

Average values, Arroyo Caliente 

Arroyo de los Frijoles 

Arroyo de las Trampas to Rio Santa Fe

Rio Santa Fe, Rio Galisteo, and 

Canada Ancha (del oriente)

100,000

100

10 •/J •/

f-.

f

o

010-———

/

. 

O

C/3

en\—C/3

10,000

1000

1234 56 7 8 9 10 11

100

10 11

STREAM ORDER

FIOTTRE 13.—Relation of stream length and number of streams to stream order in basins of llth order in central New Mexico. Loft, stream length plotted against stream

order; right, number of individual streams of a given order plotted against order.

This same general procedure was followed for several 

other sample drainage basins. Order 11 was the highest 

found, this being for the main stem of Rio Galisteo. 

The number and average length of streams in each order 

were determined and the results are included in 

figure 13.

It will be noted that the plots of stream order against 

stream length and stream order against number of

streams are straight lines on semilogarithmic paper, as 

Horton discovered. Lengths range from about 50 feet 

for the 1st order tributaries to 54 miles for the Rio 

Galisteo.

In order to visualize better the types of channels 

studied and their relation to stream order, the reader is 

referred to the photographs in figure 14, which show 

first-order tributaries in the basin of Arroyo Caliente.
EPHEMERAL STREAMS 19

iffi&Wig&S&J** ! ?**• */;^3f ;jpi;*P^

Because maximum stream length is a function of

fe* '•Tk-^, ^-^Ka

FIGUEE 14.—First-order tributaries in basin of Arroyo Caliente. These show the most headward extensions of the smallest tributary rills in the area.

and figure 15 which pictures Arroyo de los Frijoles at drainage-basin area, it is not unexpected that the relaa place where its size is typical of an 8th order stream. tion of drainage area to stream order is also a straight 

Channels of order 5 and order 10 can be seen in figures iine on semilogarithmic paper, as can be seen in figure 

2 and 3, respectively. 16 The smallest unbranched tributaries, which are

rills about 8 inches wide and 1 to 4 inches deep, drain 

on the average about .00006 square miles or .04 acre. 

In the 670-square-mile basin of Rio Galisteo there are 

roughly 190,000 such first-order tributaries, as estimated from figure 13.

EQUATIONS RELATING TO HYDRAULIC AND PHYSIOGRAPHIC FACTORS

From the previous work of Horton or from our data 

plotted in figures 13 and 16, it is apparent that stream 

order, 0, bears a relation to number of streams, N, in 

the form

0=k log AT or Ooclog N (1)

FIGURE 15.—Arroyo de los Frijoles at place where it typifies a stream of eighth order.

and a similar relation to stream length, /, slope, s, and 

drainage area, Ad ,

Ooclog I (2)

Ooclog s (3)

Ooclog AA (4)
20 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

1000

1234567

STREAM ORDER

FIGURE 16.—Relation of drainage area to stream order for ephemeral channels in 

central New Mexico. Number beside point is serial identification keyed to data 

in appendix.

It follows, then, that there must exist a power-function 

relation between each two of these variables other than 

order, in the form

, socAkd , sod"

where k is some exponent having a particular value in 

each equation. Moreover, because it is known that

discharge, Q, bears a relation to drainage area of the 

form

(5)

then discharge must be related to stream order according 

to

Ooclog Q (6)

Reiterating some of the hydraulic relations discussed 

earlier :

(7)

(8)

(9)

(10)

(11)

(12) 

Where

vocQm 

LocQ*

soc^ 

n'ocQ"

Q = discharge 

w— width 

(/—depth 

v= velocity 

L = suspended load 

s= slope

n'=a roughness parameter 

6, /, m, j, z, and y are numerical exponents

It follows from equation (6) and equations (7) to 

(12) that there definitely is a relation between stream 

order and width, depth, velocity, suspended load, 

slope, and roughness of the form

Ooc log (width, depth, etc.)

Also, there are several interrelations among variables 

in equations (1) to (4), and among those in equations 

(7) to (12). An example of such a relation is that 

which would exist between sediment load, L, and 

drainage area, Ad.

Because

and

then

LocQ*

QocAd k

LocA?

Thus, it can be seen that a whole series of hydraulic 

and drainage-network factors are interrelated in the 

form of power or exponential functions. These equations can be added to the several others derived by
EPHEMERAL STREAMS 21

Horton (1945, p. 291) which, in his words, "supplement Playfair's law and make it more definite and 

more quantitative. They also show that the nice 

adjustment goes far beyond the matter of declivities." 

The equations listed above merely state the condition 

of proportionality; for them to be definitive the constants involved must be determined. In addition to the 

values of certain exponents already discussed, data 

collected during this investigation established the relations among order, slope, and width, as will now be 

described.

Although channel depth, velocity, and discharge at 

a particular cross section can be measured only when 

the stream is flowing, the important hydraulic variables, channel width and channel slope, can be estimated even in a dry stream bed. During the many 

days when no thunderstorms were occurring in the 

area studied, our field work included measurement of 

width and slope in the dry channels. The procedure 

used will be described briefly.

The width was defined as width near bankfull stage. 

In Eastern United States where the development of 

the river flood plain is the rule rather than the exception, the bankfull width is relatively easy to define and 

measure. It is the width which the water surface 

would reach when at a stage equal to the level of the 

flood plain. In dry arroyos where flood plains are the 

exception and alluvial terraces exceedingly common, 

it is difficult to point specifically at an elevation which 

might be called bankfull. Nevertheless, field inspection allows fairly consistent estimates of width corresponding to an effective or dominant discharge, even 

by different observers. The positions on the stream 

banks representing the two ends of the cross section 

were chosen independently by each of us, and the 

recorded widths represent a compromise between our 

individual judgments.

Channel slope is somewhat easier to determine. 

Although there are local dry pools or deeps resulting 

from both definite patterns of flow and random channel 

irregularities, a smooth profile of channel bed drawn 

through several points in a reach provides an estimate 

of channel slope which is reproducible by successive 

measurements. Our procedure was to place the planetable in the center of the dry stream bed and take a 

series of sights both upstream and down as far as the 

stadia rod could be read with a leveled instrument. 

In most instances this gave a measurement of slope 

through a reach of about 600 feet.

Width and slope were measured at more than 100 

channel cross sections. Because of the widespread 

geographic distribution of measured cross sections and 

because of the lack of adequate maps, we have not

determined the order of the streams on which many of 

these are located. The graphs of figures 17 and 19

6 7 

STREAM ORDER

FIGURE 17.— Relation of channel slope to stream order, showing the individual 

measurements and the mean relation (solid line) derived from other parameters. 

Number beside point is serial identification keyed to data in appendix.

were plotted from data on cross sections of streams of 

known order. As expected, order is related to width 

and slope by equations of the type.

O=k log w and O—k log s

The relation between width and slope can be determined 

by equating the relations above or by direct plotting 

of the data as in figure 18. To represent the relation 

in larger streams, a straight line on logarithmic paper 

has been drawn through the numbered points (fig. 18). 

The equation of this line is

This means that for the area studied, channel slope 

decreases downstream approximately as the reciprocal 

of the square root of channel width. This expresses an 

interrelation and does not imply a direct dependence between these two parameters.

Establishing the relations among width, slope, and 

order merely requires collecting and plotting the appropriate data. It should be possible to combine those 

factors which are most easily and definitively measured, 

and thereby arrive at approximations of other factors 

which are more difficult to measure. Such a scheme 

would expand the usefulness of the techniques now 

available. An example to illustrate this possibility will 

now be cited.
22 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

1.0

O

.10

.01

—

? ? 7^

7 V ? V

*- v " "'

V

V

——

V

7

C

f

D

D

D —— D D DC

_

——

—

T 

w _

f ?T I ^

s

D

1^. J^

-4—^^^--

iB 0 X

f

£• ———————————

w X44

A-S*fc^

D _ ^^ D ^^^

L.I

D

° B

DD

1 1

41

X*S

38

bb

EXPLANATION _

O Various

X Caliente

arroyos

Arroyo

T 5th of July Wash

near Santa Fe

principal channels (coarse sand)

(steep rills in small gravel)

D Caliente Arroyo (rills in fine gravel and coarse sand) __

V Road-cut rills (sand and fine gravel)

• Natural rills (sand and fine gravel)

xvx 40

J^W 29_

31C

X 34

) ^^O

33X C

020

5 *

17

N

n

t>

>•s

S

1

w

'c)(

C

22°S

O 14

5 21 270 o 26

>28

^

025

^ 5 f

D 15

8Q07 o12 oT^N^4

o13 °'~ un

>

"X

X

O2

—

——

—

—

—

—

— o1 ~~

^

.001

.1 1 10 100 1000

CHANNEL WIDTH, IN FEET

FICTRE 18.—Relation of channel slope to channel width. Data from moderate and large arroyos and differentiated from rills by the different symbols. Number beside

point is serial identification keyed to data in appendix.

The manner in which discharge increases with 

stream order is of particular concern, as it provides the 

link between the Horton analysis and the hydraulic 

geometry. For perennial streams, gaging-station data 

are available in quantity for the larger stream orders. 

The determination of stream order at a gaging station 

is laborious but can be made, and order can be plotted 

directly against discharge of a given frequency derived 

from gaging-station data. In arid regions there are 

relatively few gaging stations, and for the most part 

they are located on the larger streams. To obtain a 

relation between order and discharge for ephemeral 

streams requires discharge measurements of the smaller 

streams. But even these measurements do not provide 

at a given cross section a specific value of discharge 

which can be plotted against stream order, for it 

must be remembered that discharge at any given location may fluctuate through a wide range and that 

sufficient record is required to allow some kind of frequency analysis.

If the discharge chosen has some specific relation 

to the position of the cross section along the length 

of the stream, an approximation to constant frequency 

might result. It is proposed to use that discharge 

which corresponds to the average width representative 

of the stream order at the point in question. To obtain 

this value, the relations already demonstrated can be 

utilized. Specifically, the relation between width and 

order indicated on figure 19 will be combined with the 

relation of downstream increase of width with increasing discharge. The latter is, fortunately, one of the most 

consistent of the graphs representing the hydraulic 

variables. From figure 19 the stream width for each 

order can be obtained and, by use of those values of 

width, the corresponding discharges can be read from 

the width-discharge graph of figure 11. In such a 

manner the relation between stream order and discharge 

may be derived, and it is plotted as the unbroken line 

on figure 20.
1000 r—

EPHEMERAL STREAMS

10,000

23

<s —

5678 10 11 12

STREAM ORDER

FIGURE 19.—Relation of stream width to stream order in arroyos. Number beside 

point is serial number in appendix.

A second approach to the discharge-order relation 

makes use of flood-frequency analysis. On streams of 

small and moderate size in central New Mexico there 

are only a few gaging stations that have a relatively 

long period of record. Despite the paucity of data, 

flood-frequency curves for all gaging stations considered 

applicable to the area being studied were obtained from 

an unpublished study by H. H. Hudson. Also, the 

records of some additional stations were analyzed by 

the authors. In eastern streams the bankfull stage is, 

attained about once a year. A conservative quantity 

little affected by the length of record is the average of 

the highest flood each year of record. This is called 

the mean annual flood. It has a recurrence interval 

of 2.3 years, and thus we assume roughly approximates 

the discharge at bankfull stage. The relation between 

2.3-year flood discharge and drainage area for gaging 

stations in central New Mexico is presented on figure 

21.

It is known by hydrologists that the discharge of a 

flood of a given frequency increases somewhat less 

rapidly than drainage basin size. It is typical for the 

relation of flood discharge to drainage area to plot as a

Derived from Q vs 

drainage area 

relations and 

drainage area vs 

stream order 

relations

1 2 3 4 5 6 7 8 

STREAM ORDER

FIGURE 20.— Relation of discharge to stream order derived by two separate types of

analyses.

straight line on logarithmic paper, expressed by the 

equation

where k is a constant which for the mean annual flood 

has a value of between 0.7 and 0.8. The slope of line 

on figure 21 is consistent with values known from other 

areas. In drainage basins of the same size, those 

studied in the West produce floods of about one-fifth 

the magnitude of the typical one in the East.

By combining the discharge-area graph with the 

order-area graph, discharge can be related to order. 

The drainage area corresponding to each stream order 

is read from figure 16, and the value of discharge for 

an equal drainage area is determined from figure 21. 

The resulting plot is shown as the dashed line in figure 

20.

Thus, the relation between stream order and discharge, which is the link between the Horton analysis 

and hydraulic geometry, has been derived for a central 

New Alexico area in two ways which are at least somewhat independent. Comparison between the results is 

indicated by the two lines in figure 20. Only the slopes
24 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

75'

10,000

Hudson curve for stations in 

Taqs and Santa Fe Counties 

defined as "Area 2"

10

100 1000 10,000

DRAINAGE AREA, IN SQUARE MILES 

FIOTRE 21.—Relation of mean annual flood discharge (equaled or exceeded in 2.3 years) to drainage area. Number beside point is serial number in appendix.

of the two lines should be alike, and indeed, they are 

similar. The intercepts should not be expected to be 

exactly the same because the discharge determined 

from flood-frequency data was chosen to represent a 

frequency of 2.3 years. The frequency of that discharge which corresponds to full channel width is unknown and need not be identical to the discharge 

having a 2.3-year recurrence.

It should be recognized that each of the graphical 

relations presented has considerable scatter owing to 

the nature of the measurements and the inherent 

variability of these factors in the field. It should not 

be inferred that a relation such as that presented between discharge and stream order is considered precise. 

Rather we are concerned with explaining a methodology 

by which generalized relations can be obtained, and with 

demonstrating the nature of the interrelations between 

a variety of hydraulic and physiographic factors.

In summary, it has been shown that the ephemeral 

streams in New Mexico are characterized by a uniform

downstream increase of width, depth, and velocity 

with stream order, and also with drainage-basin size. 

Increasing size of drainage basins is accompanied by a 

downstream increase in discharge. Furthermore, the 

interrelations among all of these factors, both hydraulic 

and physiographic, may be expressed in simple terms, 

either as exponential or power functions. The method 

of combining the hydraulic variables with factors 

measured on a map or obtained in the field from a dry 

stream bed allows a simple means of obtaining interrelations which cannot be measured directly.

SOME RELATIONS OF HYDRAULIC AND PHYSIOGRAPHIC FACTORS TO THE LONGITUDINAL PROFILE

CHANNEL ROUGHNESS AND PARTICLE SIZE

The longitudinal profile of a river is ordinarily drawn 

as a graph showing elevation as a function of horizontal 

distance from the headwaters. Hack (1955) showed 

that if slope were substituted for elevation as the ordinate, the gradual downstream decrease in stream gradi-
EPHEMERAL STREAMS 25

ent makes the graph a straight line on logarithmic 

paper. Furthermore, the slope of this line is a measure 

of the concavity of the profile.

Hack demonstrated for several streams in the 

Shenandoah Valley that the rate of decrease of bedmaterial size in the downstream direction is related to 

the degree of prof le concavity. The greater rates of 

particle-size decrease downstream are associated with 

the more concave profiles.

The relation of channel characteristics to the longitudinal profiles of streams was discussed in an initial 

way by Leopold and Maddock (1953). They stated 

that under most conditions in a river there is a tendency 

for channel roughness to be conservative and to change 

downstream less rapidly than some of the other 

hydraulic variables. They say (1953, p. 51) that "the 

suspended load and its change downstream characteristic of natural rivers require a particular rate of increase 

of velocity and depth downstream. Under the conditions of a nearly constant roughness, to provide the 

required velocity-depth relations, slope must generally 

decrease downstream; it is for this reason that the 

longitudinal profile of nearly all natural streams 

carrying sediment is concave to the sky."

The data presented in the present paper provide some 

indication of how the channel factors are related to the 

changing size of bed material, or more specifically, how 

the idea expressed by Leopold and Maddock can be 

integrated with the later work of Hack to provide a 

more nearly complete picture of the interaction of geologic and hydraulic factors.

The hydraulic variables slope and roughness can be 

related to the other factors in an approximate way by 

considering a Manning-type equation,

0=1.5 n'

in which d is mean depth (approximately equal to 

hydraulic radius), 5 is slope, and n' is a roughness factor. 

It is necessary in our data to use the slope of the stream 

bed as an approximation to the slope of the energy grade 

line. Let n' be the corresponding roughness parameter. 

If these factors are expressed as functions of discharge 

in the form used by Wolman (1954a)

n'=rQ"

and substituting the function of discharge (equations 

7-12) for each variable, then

it follows that

If the downstream change of width is related to discharge as wacQ* (shown to be a characteristic of all 

streams studied), because 6+/+m=l, then m+/=0.5. 

Thus, when m increases, / decreases. 

The equation

is represented graphically by the lines in figure 22, 

assuming that m+/=: 0.5. The average relations of the 

hydraulic variables as functions of Q in the downstream 

direction give values of f=A and m=.l. For the

y, Rate of change of 

roughness, in n'<x.O

EXPLANATION

1 Ephemeral streams, N. Mex.

2 Kansas-Republican Rivers, Kans.

3 Yellowstone-Missouri Rivers

4 Tombigbee River, Miss.

5 Maumee River, Ind.

6 Brandywine Creek, Pa.

7 Watts Branch, Md.

8 Assumed average for 

perennial rivers

Note: b 

In relation w*.Q , 

b assumed equal to0.5

-1.1 —

•1.2

0 .1 .2 .3 .4 .5 .6 

f, RATE OF CHANGE OF DEPTH, IN d«Qf

.5 .4 .3 .2 .1 0 

m, RATE OF CHANGE OF VELOCITY, IN rocOTM

FIGURE 22.—Downstream relations of depth, velocity, slope, and roughness expressed 

as values of exponents in equations relating these factors to discharge.
26 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

average value of z, Leopold (1953, p. 619) suggested — .49, 

but analysis of additional data leads us to revise this to

— .95. 1 The average downstream relations, therefore, 

can be represented by point 8 marked with a triangle in 

figure 22. This indicates that the average value of y is

—0.3, which means that channel roughness decreases 

slightly downstream for discharge of constant frequency. 

This apparently is related to some progressive downstream decrease in bed-particle size resulting from 

sorting and abrasion.

From the graph it can be seen that if width, depth, 

and velocity changed downstream at rates respectively 

equal to those in the average river (6=.5,/=.4, m=.l) 

and if roughness (measured by Manning-type n') 

remained constant downstream (y—0), then slope would 

decrease downstream with increasing discharge in the 

form

In few natural rivers, however, does roughness remain 

constant downstream.

Values of / and z for several rivers are plotted in 

figure 22, and the corresponding values of the exponent 

y, expressing rate of change of roughness, can be read 

from the scale given by the sloping lines. Point 2, 

representing the Kansas-Republican River system, indicates a near-average value for rate of change of depth 

(/=0.43). The nearly constant value of roughness 

downstream (y=0) in that river system is presumably 

related to the fact that bed-particle size decreases a 

relatively small amount, ranging from sand near the 

headwaters to silt farther downstream. Consequently, 

the river gradient decreases downstream less rapidly 

than in the other rivers; that is, the value of z is about 

— .4, whereas most of the other rivers have values 

between —.8 and —1.1.

The headwaters of the Yellowstone and Bighorn 

Rivers rise in coarse gravel which decreases to fine sand 

in the main Missouri, and this river has a correspondingly rapid change of gradient as indicated by the large 

negative value of z in figure 22. Thus it appears that 

an important effect of decreasing particle size downstream is to decrease the roughness downstream.

EFFECT OF CHANGE OF PARTICLE SIZE AND 

VELOCITY ON STREAM GRADIENT

Channel roughness is not determined entirely by 

particle size. That our roughness factor is not necessarily proportional to the sixth root of bed particle 

size — as would be expected from the Strickler equation 

n=ckl/6 (see O'Brien and Hickox, 1937, p. 314) — 

follows from the fact that n' involves not only the grain 

roughness but also resistance attributable to channel 

configuration and other factors. However, for load

composed of coarse sand or gravel, channel roughness 

is materially influenced by tjie size of grains. The 

manner in which particles of finer materials are piled 

into dunes and riffles becomes an important determinant of channel resistance.

For a given rate of change of depth and velocity, the 

rate of decrease of stream slope (concavity of the profile) is adjusted to the* rate of decrease of particle size 

as expressed by roughness. In figure 22 this may be 

seen by comparing points 2 and 6. Each has a value 

of/ approximately equal to 0.45; that is, in these two 

streams the downstream rates of increase of depth with 

discharge are nearly identical. River 2 is characterized 

by nearly constant roughness downstream (y=0), 

whereas river 6 has a rapid downstream decrease of 

roughness (y~ — 0.3). Likewise, the slope of river 6 

decreases much more rapidly downstream (2« — 1.1) 

than river 2, (z~— 0.4).

During the initial field work in New Mexico particlesize measurements of bed material were made by sieving 

scoop samples. The results were quite inconsistent 

owing to the large range of size over the bed even 

within short reaches. In 1954 the senior author returned to the field area with M. Gordon Wolman to obtain new measurements. Using a grid-system sampling 

method (described by Wolman, 1954b), bed-material 

size was measured at those reaches where discharge 

measurements had been obtained earlier, and these 

data are summarized in appendix D.

These data were plotted against discharge corresponding to the stream order at the places of measurement, and the graph is presented as figure 23. The line 

drawn through the points shows the downstream relation of discharge to grain size, and can be expressed as

where D50 is the median grain size; that is, 50 percent 

of the grains are finer.

The streams in central New Mexico are characterized 

by the following set of values: /— 0.3, 2=— 0.24; and 

the coordinates of these values in figure 22 are indicated

1 Other ways of deriving an approximate value for this exponent can be obtained from the equation presented by Hack (1955) 

relating stream length, I, and slope, s,

socr 1

and his equation relating length to drainage area

Combining these with the generalization typified by figure 21 

that bankfull flood discharge

then
EPHEMERAL STREAMS 27

.05 1 .51 5 10 50 100 500 1000 5000

DISCHARGE, IN CUBIC FEET PER SECOND

FIGURE 23.—Downstream change of median size of bed particles with discharge, ephemeral streams near Santa Fe, N. Mex. Discharge determined from discharge-order relation,

solid line, figure 20. Number beside point is location serial number in appendixes A to D.

by point 1. Ail these streams had only a small decrease 

of particle size downstream, and if particle size alone 

controlled roughness, point 1 would be expected to fall 

near the line y=Q (constant roughness). That the data 

indicate some downstream decrease of roughness 

(y^— 0.12) presumably indicates that in addition to 

the slight downstream increase in particle size, there is 

also some effect of bed configuration. Possibly also 

width-depth ratio itself tends to influence channel 

resistance.

The manner in which the analysis given above verifies and integrates previous work of Leopold and Maddock (1953) and Hack (1955) can now be seen. In 

figure 24 the relation between stream gradient and

drainage area is presented for a selected set of streams 

in the United States, including both ephemeral and 

perennial streams representing both eastern and western 

conditions. The two examples from Hack are for two 

eastern streams having different rates of change of 

bed-material size. In Calf pasture River the size of 

bed material is constant downstream; in Gillis Falls 

Branch the size of bed material increases downstream. 

Comparison of the graph for ephemeral streams with 

those for Calf pasture River and Gillis Falls Branch 

would lead one to surmise that the New Mexico streams 

have an increase in particle size in the downstream direction. However, our measurements showed that 

particle size tended to decrease slightly downstream.

I" Ephemeral channels in New Mexico, 

I particle size about constant downstream

Gillis Falls Brench, Maryland, 

particle size increasing downstream 

(Data from J. T. Hack)

,001 .01 .1

DRAINAGE AREA, IN SQUARE MILES 

FIGUEE 24.—Downstream change of stream gradient in selected drainage basins.
28 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

Other very small ephemeral streams are similar to 

the New Mexico example in that the slope of the graph 

is relatively small compared with that of Calfpasture 

River. Note the line representing the rills developed 

on an artificial fill at Perth Amboy, N. J., which was 

studied by Schumm. 2 Although analyses of particle size 

on Perth Amboy streams are not available, Schumm's 

report implied that particle size tended to remain more 

or less constant or decrease slightly in the downstream 

direction.

The difference in slope of the lines representing Calfpasture River and the New Mexico streams in figure 

24 is associated with difference in rates of downstream 

change of channel characteristics. It is presumed that 

in the ephemeral streams, the small value of z (slope 

of the line in the gradient-discharge relation) is related 

to the downstream increase in sediment concentration 

reported in a previous section of this paper. It is 

reasoned that to maintain a quasi-equilibrium with this 

increase in sediment concentration downstream, there 

must be an accompanying downstream increase of 

velocity somewhat larger than that which characterizes 

normal rivers. As can be seen in figure 22, the more 

rapid downstream increase in velocity (large value of 

ra) is provided by a relatively small value of z for any 

given value of the rate of change of roughness (value 

of y).

In summary, the longitudinal profile of a river is 

related to a combination of geologic and hydraulic 

factors. These hydraulic factors are, first, the rate of 

change of particle size of bed sediment downstream, 

which tends to govern the change of channel roughness; 

and, second, the relative rate of increase of velocity 

downstream, which appears to be controlled by the 

rate of change of suspended-sediment concentration 

downstream. Other things being equal, the less concave profiles are characterized by greater rates of increase of suspended-sediment concentration downstream 

or by less rapid decrease of bed-particle size in the 

downstream direction.

EQUILIBRIUM IN EPHEMERAL STREAMS 

MUTUAL ADJUSTMENT OF HYDRAULIC FACTORS

As has been discussed previously (Mackin, 1948; 

Leopold and Maddock, 1953), the factors which may 

be considered independent of the channel itself are discharge and quantity of sediment load, and to some 

degree, caliber of load. Slope, width, velocity, and 

depth are determined by or altered by forces operating 

within the channel. In the words of Rubey (1952, p. 

131), the dependent variables "are determined not 

separately but jointly"; that is to say, when circumstances require or impose a change in one or more of 

the factors, the others will become adjusted to maintain a quasi-equilibrium between forces of erosion and 

forces of deposition.

That some of the forces operating within the channel 

may be determined by previous history is generally 

recognized. There may be a difference in the "degrees of freedom" characteristic of the different hydraulic situations. As an example, one might consider 

a flume with a movable bed. If the sides of the flume 

are made of wood or metal so that the width is not 

adjustable, there must be an adjustment among all the 

remaining variables to a condition of mutual accommodation. The fixed width may be thought of as the 

restriction of one degree of freedom in the process of 

mutual accommodation.

If a particular valley had been carved during the 

Pleistocene by a relatively large river, and through 

change in climate or other physiographic factors the 

valley at present carries only a minor streamlet, it is 

likely that the small stream would not be able to recarve 

the valley bed to such an extent that the valley slope is 

materially altered. In such a circumstance the minor 

stream, initially at least, would have to accommodate 

itself to the slope of the valley in which it flows.

The organization of a drainage system and the remarkable adjustment of its various hydraulic factors 

have been amply demonstrated. These have been 

discussed by several authors, Rubey (1952), Leopold 

and Maddock (1953), and in detail by Wolman (1954a). 

The present paper presents further evidence that 

nearly all rivers approach a condition of quasi-equilibrium which was formerly attributed only to a graded 

stream. This postulate implies further that if the 

degrees of freedom of the hydraulic parameters are 

reduced, the remaining factors tend toward a mutual 

accommodation or quasi-equilibrium. It is our purpose 

here to demonstrate that this reasoning can be extended 

even to a headwater stream in youthful topography as 

far as the uppermost tributary rill.

In figure 18 a line has been drawn to represent the 

relation between channel slope and channel width for 

arroyos of large and moderate size. In the upper left 

part of this figure the open squares represent values of 

slope and width of the smallest unbranched rills in the 

headwater tributary basin of Arroyo Caliente. These 

rills averaged somewhat less than 1 foot in width. 

Gradients were measured by laying a board in the bed of 

each rill and determining its slope with a Brunton 

compass. The measurements may be considered crude 

and the scatter of the points on the graph may be partly 

attributed to the method of obtaining the data; nevertheless, the position of those points indicates that the

2 S. Schumm, 1954, Evolution of drainage systems on an eroding fill at Perth 

Amboy, N. J., Ph. D. thesis, Columbia Univ.
EPHEMERAL STREAMS 29

relation of channel slope to channel width is generally 

in line with that of the larger ephemeral channels.

The solid triangles apply to similar unbranched rills 

occurring in steeper tributaries in the same area which 

drained the unconsolidated gravels of a pediment 

remnant somewhat nearer the mountains than the 

Arroyo Caliente. The difference in stream slope 

between Arroyo Caliente and Fifth of July Wash can 

probably be attributed to the difference in size of the 

gravel characterizing the two small basins.

To determine whether this relation between slope and 

width would be maintained even by the very smallest 

rills, measurements were made of the smallest natural 

rills which could be found in the area. The values of 

slope and width of these miniature features are represented by the black circles in the far upper left part of 

figure 18. They fall in a position so nearly representing 

an extension of the line in the diagram that it may be 

inferred that the slope-width relation in the area studied 

applies as far upstream as the smallest observable rill.

In the same area are many places where modern 

highways have required deep road cuts through the 

same material as that constituting the drainage basins 

under study. On the steep road cuts little or no vegetation has become established, and numerous steep 

parallel rills of an average depth of 0.2 feet have developed. The width and slope of these road-cut rills were 

measured. The slope had changed but little from that 

determined by the blade of the highway grader. The 

rills on road cuts are shown by open triangles on figure 

18. These points are no more scattered from the mean 

line than any other data. It appears then, that if the 

degrees of freedom are reduced, in this instance by a 

prescribed slope, the width of the rill will be formed in 

accordance with the slope-width relations for natural 

streams having a larger number of degrees of freedom. 

This is interpreted as further evidence that channels 

cut by water carrying sediment tend to maintain a 

quasi-equilibrium even as far upstream as the most 

remote ephemeral rills.

THE DISCONTINUOUS GULLY

Beginning late in the 19th century the alluvial 

valleys of the West experienced an epicycle of erosion 

characterized by the development of valley trenches 

or arroyos (see fig. 25). These arroyos range in size 

from insignificant rills to canyons 600 feet wide, 50 

feet deep, and 150 miles long. The notorious Rio 

Puerco (del Oriente) in New Mexico has the latter 

dimensions. Some gullies are narrow enough to step 

across but are deep enough to lose a giraffe in. As 

Gregg (1844, p. 184) said more than a century ago,

FIGURE 25.—A typical large continuous arroyo trenching an alluvial valley in New 

Mexico: Rio Puerco (del oeste) near Manuelito, looking downstream.

The sides are usually perpendicular—indeed, often shelving 

at the base, and therefore utterly impassable . . . Though, 

to a stranger, the appearance would indicate the very head of a 

ravine, I would sometimes be compelled to follow its meandering 

course for miles without being able to double its "breaks."

It is characteristic of the large arroyos that depth 

remains quite uniform through very long reaches. 

This uniformity of depth means that the gradient of 

the channel bed had attained a slope almost parallel 

to the original valley floor. There is also another 

distinctive type of gully that is characterized by a 

vertical head-cut, a rapidly decreasing depth downstream, and a fan at the lower end. Such channels 

generally occur in groups arranged irregularly along 

the length of the drainageway, and because of this 

characteristic are called discontinuous gullies.

Bryan (1928) has presented evidence that when the 

Rio Puerco was first reached by the reconnaissance 

teams of the Army of the West in 1846, the valley floor 

was already being dissected by discontinuous gullies. 

Some of them were evidently large, for in August 1846, 

Lt. Simpson had to cut down the gully wall of the Rio 

Puerco in order to get his brass cannon across. Yet, 

at the same time, there were long reaches in the Puerco 

valley so smooth that the native grass was cut for hay, 

and water was diverted from the channel by felling a 

cottonwood tree to form a dam.

Discontinuous gullies have long presented a problem 

to the erosion-control engineer. In the first place, the 

mechanics of gully formation is very poorly understood. 

Although it is generally presumed that discontinuous 

gullies, at least in places, can coalesce and form a 

continuous channel, it is not known whether the nature
30 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

of formation of discontinuous and continuous arroyos 

is different. If the mechanics of the two are appreciably 

different, the choice of control measures presumably 

would be affected.

The salient characteristic of a discontinuous gully is 

the relatively small gradient, or slope, of its bed. It is 

this flat slope—less steep than the floor of the original 

ungullied alluvial valley—that makes the gully discontinuous, for the bed profile must at some downstream 

point intersect the profile of the original valley floor. 

At that point the gully depth has diminished to zero, 

and the flow of the gully spreads out over the valley 

floor and at least part of the load is deposited in a low 

fan.

Therefore, it is important to find an explanation for 

the low gradient. Any discussion of such gullies which 

does not provide an explanation of the mechanics by 

which this flat slope is developed would miss the dominant feature. Our approach to this aspect of the problem is essentially inductive, for there is a paucity of 

adequate data on essential elements in the hydraulic 

relations. However, as will be shown later, field characteristics appear to support the conclusion reached inductively.

Consider the wide, grassy floor of an alluvial valley 

in the semiarid West before the recent epicycle of erosion. Though the summer rains are of short duration 

and intensity, the flash flows spread widely over the 

valley, and channel storage consequently keeps the peak 

discharge at any point to a small value per unit of 

drainage area. The grassy floor offers resistance to the 

flow of water and sediment contributed to the valley 

floor from adjoining slopes. This resistance keeps the 

velocity low and thus the sediment concentration is 

small. The profile and cross section of such an ungullied 

valley floor are shown as diagram 1 of figure 26.

Assume that local weakening of vegetation allows an 

initial furrow, scarplet, or small basin to form by erosion. 

The cause of this lowered resistance to erosion could be 

grazing or trampling by stock, fire, or an exceptional 

local storm. The regional erosion in the West, which 

began about 1880, is considered to be a result of a combination of overgrazing and meteorologic shift.

Subsequent storms cause the head of this initial 

erosion feature to progress up-valley, and the debris 

excavated splays out at the downstream toe in the form 

of a low fan. As soon as a short channel is formed, 

terminating in a vertical head-cut, the concentration 

of water in the flumelike trench reduces channel 

storage. Consequently, from a storm of given size, the 

peak discharge passing through the channel is greater 

than would have been experienced on the ungullied 

valley floor. This increased peak discharge is accompanied by greater velocity and cutting power, and the

initial gully advances so rapidly during storms that 

growth of vegetation in the intervals between storms 

cannot heal it.

Water pours over the lip and develops a plunge pool 

at the toe. The original sod, even when weakened and 

incomplete as a protective cover, keeps the lip of the 

head-cut relatively stronger than the underlying 

alluvium and the latter is cut away by the turbulence 

in the plunge pool, leaving the root-bound lip overhanging the plunge pool. Undercutting by plunge-pool 

action during storm flow is greatly aided by—and 

perhaps is even less important than—slumping of the 

moistened headwall after the storm flow ends. This 

slumping is promoted by piping holes (see fig. 5) which 

develop on a diagonal line between the lower part of 

the vertical headwall and the valley floor many feet 

upstream from the head-cut. We have often observed 

the upper entrance to these piping holes on the swale 

floor 50 feet from the head-cut. Piping permits water 

to penetrate deeply into the material into which the 

head-cut is eating, and this moisture aids the process 

of sapping and slumping at the base of the headwall.

The plunge pool is always seen to be dug deeper 

than the level of the floor of the discontinuous gully 

just downstream from the plunge pool. This clearly 

means that the floor of the discontinuous gully is 

composed of a layer of newly deposited material which 

overlies the undisturbed alluvium. This deposit is 

laid in immediately below the plunge pool, and deposition proceeds upstream as fast as the plunge pool does.

These details of gully growth are important to an 

understanding of the mechanics, for they show that 

when a plunge pool is present, and it usually is, the 

flat slope of the discontinuous gully is a grade qf deposition. Furthermore, these details point up a significant 

difference between the action of flood flow in the 

channel of a normal river and that of a developing 

gully. The plunge-pool action tends to deepen the 

channel faster than to widen it. In the early stages 

of gully development, then, the channel is relatively 

narrow and deep.

At any time, the gully floor is built at such a slope 

that the material coming into the reach will be transported through the reach under the particular condition 

of roughness. At the early stage of gully development 

now under consideration, the channel has considerable 

depth but a restricted width. At the same time it is 

forming a deposit just downstream from the plunge 

pool and thus is currently forming its bed slope under 

conditions in which slope can be adjusted with relative 

rapidity, as compared to width. Under conditions 

which prevail in ordinary rivers the reverse is true; 

width adjusts rapidly during floods, but because of the 

large amounts* of material involved in appreciably
EPHEMERAL STREAMS 31

changing slope through long reaches of river, slope 

adjusts only slowly.

Analysis of river data showed that at a given discharge, the same suspended-sediment load will be 

carried by a narrow, deep channel at a particular 

velocity as in a wider, shallower channel at somewhat 

higher velocity (see fig. 15, p. 23, in Leopold and 

Maddock, 1953). The early stage of a discontinuous 

gully is considered to represent the narrow, deep condition, and at a later stage, lateral cutting and slumping 

of the arroyo walls lead to widening, with a consequent 

shallowing of the cross section of flowing water at a 

particular discharge. Now, under conditions of a given

and constant roughness, a relatively large depth and 

small velocity requires a small value of slope; that is, 

a flat gradient, as can be seen by considering the 

Manning-type equation

0=1.5 n'

As widening progresses during a later stage in gully 

development, to carry a particular sediment load at a 

given discharge requires a slightly increased velocity 

as the width-depth ratio increases. To achieve the 

larger velocity with decreasing depth at a given roughness means an increase in the value of slope.

Complete integration i

LONGITUDINAL SECTIONS CROSS SECTIONS THROUGH A-A'

FIGURE 26.—Stages of development of an arroyo from discontinuous gullies.
32 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

This line of reasoning which follows the principles 

derived from interrelation of width, depth, velocity, 

and suspended load at constant discharge in natural 

rivers does provide an explanation for the small gradient 

of the channel bed in the early and narrow stage of 

the discontinuous gully. It indicates why an increase 

in bed slope could be expected to be associated with 

the channel widening.

The principles outlined apply to the changing relations at constant discharge, suspended load, and roughness. Before the individual discontinuous gullies coalesce to form a continuous, flumelike channel, there is 

still sufficient channel storage in the ungullied reaches 

of the valley to attenuate flood peaks originating far 

upstream. Thus,. the most effective flows in a given 

reach probably are derived from storm rainfall in the 

immediate vicinity of the reach in question. The 

suspended load in the reach is derived from the bank 

caving and headcutting of the individual discontinuous 

gully. In the early stages of gully development these 

conditions justify the use of the assumption of constant 

discharge and suspended load. Constant load and 

discharge imply constant sediment concentration which 

is one of the important determinants of roughness. 

So, the assumption of constant roughness is logical.

In developing an argument to explain the low gradient 

associated with the narrow and deep channel of the 

early stages of a discontinuous gully, it is inappropriate to reason from the downstream relation of slope 

to width shown in figure 18, for there is implicit in that 

diagram an increase in discharge associated with 

increasing width downstream. The stages in development of the discontinuous gully must be explained in 

terms of changes in channel characteristics which can 

take place at constant discharge.

In summary, it is reasoned that mutual accommodation causes the small width of the initial discontinuous 

gully to be associated with the development of a small 

slope, in fact so much flatter than that of the original 

valley floor that the two gradients intersect (diagram 2, 

fig. 26). The depth rapidly decreases downstream to 

the point of extinction, and below that point a small 

fan develops where the sediment is spread on the original valley floor.

These same forces operate to expand the gully headward and to deepen it, the combination of which makes 

the gully increase in total length. Because there are 

usually several gullies along the valley floor, the toe of 

one gully tends to extend itself downstream while the 

head-cut of the succeeding one is extending itself 

up-valley, decreasing the total length of unchanneled 

valley floor between them. This change is indicated 

by diagram 3 of figure 26.

As the process continues through time, a stage will 

be reached when the head-cut of the downstream gully 

meets the toe of the one upstream and the two discontinuous gullies coalesce. This stage is indicated by 

diagram 4. The lengthening and deepening of the 

gully is accompanied, however, by a gradual increase 

in width. As both width and depth continue to become 

greater, the channel reaches a size sufficient to contain 

without overflow on the valley floor the largest discharge 

which the basin upstream can produce. At this stage 

the effect of the large amount of channel storage which 

characterized the flat-floored alluvial valley in its 

ungullied condition has been lost and the discharge 

in a given reach increases greatly.

Finally a condition is reached in which the total width 

between the arroyo walls is sufficient to allow the stream 

to wander back and forth between the walls and the 

development of local patches of flood plain can begin. 

At that stage the gradual increase in width has required 

such an increase in slope that the gully bed becomes 

almost parallel to the original valley floor (diagram 5). 

This is characteristic of most of the large arroyos 

throughout the West. There may still remain a 

difference, however, between the total width of even a 

large gully and the width of flowing water characteristic of the condition when the ungullied valley floor 

carried the same discharge. Probably it is for this 

reason that through long reaches in the down-valley 

direction the depths of some arroyos gradually decrease. 

The hypothesis implies that gullies in the process of 

development could generally show the features indicated diagrammatically on figure 26.

To test this hypothesis several gullies in the Santa 

Fe area were mapped. The profile of figure 27 is a 

typical example. This gully is discontinuous, as can 

be seen by the fact that its depth becomes progressively shallower downstream to point of extinction at 

a distance of 2,750 feet. Immediately upstream from 

this point the gradient of the gully floor was .014, a 

much smaller figure than the original gradient of the 

valley floor, .028. Some deposition in the form of a 

fan can be seen between 2,750 and 3,200 feet on the 

distance scale, but only a small proportion of the total 

evacuated materials is ever deposited in the fan below 

a discontinuous gully; most of it proceeds downstream. 

The situation downstream from point 2,300 feet indeed 

bears a marked similarity to that shown in diagram 2 

of figure 26.

At 2,240 feet it can be seen that the gully bed drops 

more than 4 feet, and this is certainly the point of 

coalescense of two formerly separated discontinuous 

gullies, exactly as shown in diagram 4 of figure 26.
EPHEMERAL STREAMS 33

500 1000 1500 2000 

DISTANCE. IN FEET

2500 3000

FIGURE 27.—Profile of a discontinuous gully, Arroyo Falta, near Las Dos, 6 miles northwest of Santa Fe, N. Mex.

To extend the analogy between field conditions and 

the postulated stages of development, it was noted 

that another gully nearby and paralJel to the one shown 

in figure 27 could be characterized by diagram 5 (fig. 26) 

in that the depth remained nearly uniform and width 

had developed to a point where small patches of flood 

plain were beginning to appear within the gully walls.

So it appears that the discontinuous gully changes 

progressively through a series of stages to be transformed into a continuous gully of nearly uniform depth. 

The early stages are characterized by relatively flat 

slope of the discontinuous gully bed, but slope gradually 

increases with increasing channel width until the slope 

becomes nearly equal to that of the original ungullied 

valley floor.

The relation between the hydraulic variables and 

the tendency for establishment and maintenance of a 

quasi-equilibrium, even in headwater ephemeral channels, provide a logical explanation for the development 

of continuous gullies from discontinuous ones. Without the hydraulic argument, the transition from discontinuous to continuous channels could not be explained.

SUMMARY

The broad, sandy, and usually dry washes, and the 

trenchlike arroyos of the semiarid Southwest appear 

to be the antithesis of the perennial rivers of the humid

East. Probably the most striking fact which emerges 

with the collection and analysis of hydraulic data pertaining to these apparently different channel systems 

is the similarity between them. Differing in detail, to 

be sure, the interrelations among hydraulic and physiographic factors in ephemeral and perennial channel 

systems are generally similar. Differences observed 

are in the direction which might be expected from the 

hydrologic and physiographic character of the areas, 

and for this reason the observed differences in values 

of hydraulic factors can be assumed valid.

The same forces or processes operating in a river 

channel which lead to the condition of grade also exist 

in channels which are not graded. This leads to a 

tendency for adjustment among hydraulic parameters 

toward a condition of quasi-equilibrium between transporting capability and load delivered to the channel 

system. Even when complete adjustment is not 

achieved, the disequilibrium is spread through relatively 

long reaches of channel. Furthermore, when one of 

the ordinarily adjustable hydraulic factors is fixed or 

invariant, the other factors mutually adapt toward 

quasi-equilibrium under the prevailing conditions.

The ephemeral streams studied provide examples of 

each of these general situations. These channels are 

probably downcutting slowly, for they are located in 

the headwater parts of their drainage areas where over
34 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

short periods geologically, land reduction is in progress. 

Thus the streams do not fulfill the requirement of 

stability which the term "grade" usually implies 

(Wolman, 1954). Yet, the channels are adjustable. 

These ephemeral streams in the Southwest are characterized by increasing concentration of suspended 

sediment in the downstream direction, which results 

from loss of water by infiltration to the channel bed and 

to some lesser extent, from progressive pick-up of load 

from the stream bed. In response to this downstream 

change in sediment concentration, channel factors 

mutually adjust themselves toward quasi-equilibrium. 

One observable aspect of this adjustment is a greater 

downstream increase in velocity than is characteristic 

of normal perennial rivers.

These same processes of adjustment lead to a progressively increasing width and decreasing gradient 

downstream in a nearly uniform manner, not only in 

the larger streams but headward as far as the uppermost rills. As an example of adaptation to a temporarily fixed or predetermined value of one hydraulic 

factor, the width-slope relations in road-cut rills were 

investigated. It was shown that on a road-cut where 

slope was determined by a road grader rather than by 

natural processes, rills developed a width which fits 

the width-slope relations in natural ephemeral rills 

and channels in the same locality.

The same general principles were used to analyze 

the probable interrelation between hydraulic factors in 

gullies or discontinuous arroyos. This leads to the 

conclusion that the mechanics of gully development 

involve the same tendency for mutual adjustment 

of hydraulic factors, even in a channel undergoing 

progressive erosion. The analysis suggests that 

measurements of flash flows in active gullies and 

suspended-sediment load would be pertinent, even

essential, to an improved understanding of the manner 

and rate of gully erosion and effective measures for 

gully control.

REFERENCES

Babcock, H. M., and Gushing, E. M., 1942, Recharge to groundwater from floods in a typical desert wash, Final County, 

Ariz.: Am. Geophys. Union Trans., pt. 1, p. 49-55.

Bryan, Kirk, 1928, Historic evidence on changes in the channel of 

Rio Puerco, a tributary of the Rio Grande in N. Mex.: 

Jour. Geology, v. 36, p. 265-282.

Gregg, J., 1844, Commerce of the prairies, New York, H. G. 

Langley Co., v. 2.

Hack, J. T., 1956, Studies of longitudinal stream profiles in Virginia 

and Maryland: U. S. Geol. Survey Prof. Paper 294-B. (In 

preparation.)

Horton, R. E., 1945, Erosional development of streams and their 

drainage basins—hydrophysical approach to quantitative 

morphology: Geol. Soc. America, Bull. v. 56, p. 275-370.

Jahns, R. H., 1949, Desert floods: Engr. and Sci. Monthly, 

~ Pasadena, Calif., May issue, p. 10-15.

Leopold, Aldo, 1921, A plea for recognition of artificial works 

in forest erosion control policy: Jour. Forestry, v. 19, no. 3.

Leopold, L. B., 1942, Areal extent of intense rainfalls, New 

Mexico and Arizona: Am. Geophys. Union Trans., pt. 2, 

p. 558-566.

———— 1953, Downstream change of velocity in rivers: Am. Jour.

Sci., v. 251, p. 606-624. 

Leopold, L. B. and Maddock, Thomas, Jr., 1953, The hydraulic

geometry of stream channels and some physiographic

implications: U. S. Geol. Survey Prof. Paper 252. 

Mackin. J. H., 1948, Concept of the graded river: Geol. Soc.

Amer. Bull. 59, p. 463-512. 

O'Brien, M. P., and Hickox, G. H., 1937, Applied fluid mechanics,

New York, McGraw-Hill Book Co., 360 p. 

Rubey, W. W., 1952, The geology and mineral resources of

Hardin and Brussels quadrangles, Illinois: U. S. Geol.

Survey Prof. Paper 218. 

Wolman, M. G., 1954a, The natural channel of Brandywine

Creek, Pa., U. S. Geol. Survey, Prof. Paper 271.

———— 1954b, A method of sampling coarse river-bed material: 

Am. Geophys. Union Trans., v. 35, no. 6, p. 951-956.
EPHEMERAL STREAMS 35

APPENDIX

APPENDIX A.—Basic data on ephemeral channels near Santa Fe, N.

Point 

No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

19

20

Stream and location

Rio Qalisteo, at mouth _ . _ ._

at Hwy. 85..----.-....---...

at Cerrillos-... ...............

Arroyo San Cristobal, near 

Qalisteo _.. .— .. .

at ruin (above Hwy, 285) ....

Arroyo La Jara, 1% miles south 

ofOalisteo. — — — — — — . .

Apache Canyon, near mouth at

2.2-square-mile wash, 1 mile west

/$ l mile below mouth of Arat mouth of Arroyo do los

Arroyo de los Chamisos, at Old

Pecos Road (near Mt. Carmel 

Chapel)-. -__..___.-..._.-__-

Canada Ancha at Hwy. 84, near 

El Rancho Montoso, Seton

Ancha Chiquita, near Hwy. 84, 

0.4 mile north of El Rancho

Drainage 

area 

(square 

miles)

672

540

17

2.2

240

210

2.5

1.52

.093

Width

(feet)

615

275

240

180

100

90

120

50

90

22

Slope 

(feet per 

foot)

.0041

.0074

.0085

.007

.0071

m ^9

.0054

.0072

.0175

.0286

Order (adjusted)

11

g

8

8

Length 

from 

divide 

(miles)

46

9 9

2.2

.7

Point 

No.

21

22

23

24

25

26

27

28

29

31

32

Stream and location

SW}4 sec. 17. T. 17 N., R.9 flat "Short of Sand Reach" at

Pojuaque Creek, near San IldeAncha Arroyo, near CuyaCanada Ancha, % mile below

Alamo Creek, at NEV4SEV* sec. 

33, T. 18 N., R. 8E_._ — — —

Canada Ancha, in NW}4 sec. 33,

R.9E. — - — — ——— _ — —

Tributary to Hermanas Arroyo,

Lagunitas Arroyo, at Las Dos, 

SE}4 soc. 30, T. 18X.,R.9E._.

Drainage 

area 

(square 

miles)

8.85

07 A

9.72

.27

1.16

.063

.79

Width 

(feet)

on

90

OQQ

ooft

219

98

1Q

32

20

Slope 

(feet per 

foot)

0.0177

.0125

.0155

019

.014

.018

.032

.019

Order (adjusted)

8

10

9

8

7

7

Length 

from 

divide 

(miles)

7.7

6.1

.75

2.85

.3

1.3

APPENDIX B.—Arroyo Caliente l : channel data

Point

No.

33

34

35

36

37

38

39

40

42

43

44

45

46

47

Location

Do... -.-...--.. .17-.

Do----_---_---_15._

Do— - — —— — 13,.

Do— —— —— — 12-

Do- — — — —— 11-

Do----_--------- 9-.

Do--~---- — - 8-

Do--.---.-----.. 6^_

Do--_-_ --------- 4._

Do----------.---.3--

Do--.- — .-- — .2..

Do— — — — — 1-

basin.

Drainage area 

(square 

miles)

0.027

.025

.0018

.019

.013

.011

.0076

.0022

.00058

.00021

Width 

(feet)

16

6.5

8.4

7.2

2.8

10

5.6

3.8

2.8

2.0

1.0

Slope 

(feet 

per 

foot)

.020

.0255

.0405

.0395

.048

.08

.071

Order (adjusted)

2

2

Length 

from 

divide 

(miles)

.238

91ft

.189

.121

Length 

from 

divide 

(feet)

130

Point 

No.

49

cn

53

56

57

58

CQ

60

Location

Do-. --------------

Do. — .... . — -----

Do— —— —— — —

Do..-- — ---------

Do- — -------------

Do-__ — -----------

Do— —— — — - ——

Do- — . — -------

Do-.--. — — - —— .

Do-. — ----------

Do... --------------

Do-__ — — -_. -----

Do... -------------

Drainage area 

(square 

miles)

0 00021

.00021

.00047

.0016

. 00093

Width 

(feet)

0.5

1.6

.8

1.5

.8

2.0

2.2

2.3

1.5

.9

.8

3.1

Slope 

(feet 

per 

foot)

Order (adjusted)

1

1

1

3

3

Length 

from 

divide 

(miles)

Length 

from 

divide 

(feet)

380

450

535

650

810

1,010

1,190

1 Arroyo Caliente Is a tributary (not named on Geological-Survey maps) to Arroyo de los Frijoles, in SW}4 NE}4 sec. 17, T. 17 N., R. 9 E., within the limits of Aqua Fria 

quadrangle, New Mexico.

APPENDIX C.—Flood-frequency data for streams in central New Mexico

Point 

No.

61

62

63

65

66

67

Stream and location

Willow Creek at Park View

Drainage 

area 

(square 

miles)

CO

70

38

86

1 Q<1

Mean 

annual 

flood 

(cfs)

U,800

1350

1 oon

1235

1300

i 1, 550

Years of 

record

28

20

16

36

31

20

Point 

No.

68

RQ

71

73

74

75

Stream and location

Drainage 

area 

(square 

miles)

3,200

10

37

11

56

3.8

5,160

Mean 

annual 

flood

(cfs)

5,000

!62

] 270

1120

'200

i 11

12,600

Years of 

record

17

20

20

20

42

20

18

1 Data from unpublished compilations by H. H. Hudson.
36 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS 

APPENDIX D.—Distribution of bed-particle, size in selected channels

Point

No.

20 

17

19 

5 

31 

16 

76 

33 

42 

36 

52 

47 

2

Stream and location

Ancba Chiquita northeast of Hwy. 84, 0.4 mile

Arroyo de los Chamisos, at Old Pecos Bd. (near

Canada Ancha, at Hwy. 84 nr El Rancho Montoso.

Arroyo de los Frijoles, at mouth of Arroyo Caliente. 

Rio Oalisteo, at Hwy. 85-— ----------------------

Percent finer than (millimeters)—

0.25

2

9 

27 

1 

1 

11 

1

4 

6

0.50

14

6 

17 

42 

11 

4 

36 

7 

5

7 

11 

22

1

48

26 

39 

46 

37 

34 

71 

23 

23 

33 

21 

28 

70

2

78

52 

59 

60 

54 

61 

91 

56 

46 

52 

34 

62 

89

4

87

72 

83 

67 

70 

68 

96 

82 

54 

61 

45 

80 

91

8

93

86 

93 

79 

81 

76 

96 

90 

59 

63 

61 

92 

92

16

97

94 

96 

86 

89 

90 

100 

94 

72 

71 

72 

95 

94

32

98

97 

98 

94 

97 

96

96 

79 

78 

88 

98 

97

64

98

99 

100 

101 

99 

100

100 

90 

88 

99 

100 

99

128

100 

100

100

99 

96 

100

100

256

100 

100

50 percent 

finer 

than 

(feet)—

0.0034

.0066 

.0048 

.0039 

.0052 

.0046 

.0022 

.0057 

.0083 

.0066 

.0174 

.0050 

.0022

84 percent 

finer 

than 

(feet)—

0.010

.023 

.013 

.043 

.033 

.039 

.005 

.015 

.140 

.150 

.085 

.016 

.005

APPENDIX E.—Arroyo discharge measurements during flash-flood flow at particular cross sections in New Mexico and Wyoming

Stream and location

Meet John Wash, near Casper, Wyo. 2

Cafiada Ancha. near El Rancho 

Montoso, N. Mex.

Ancha Chiquita, near Rancbo 

Montoso, N. Mex.

Rio Santa Fe at Old Albuquerque Road, N. Mex.3

Rio Oalisteo at Cerrillos, 

N. Mex. 

Tributary to Hermanas Arroyo, near Las Dos, N. Mex. 

Arroyo de los Chamisos, near 

Mt. Carmel Chapel, N. 

Mex. 

Rio Oalisteo at Domingo *. . .

Rio Puerco at Cabezon *....-

Date

}Aug. 2,1953

}july 26,1952 

}Aug. 9, 1952 

}july 28,1952

Jjuly 7, 1952 

jjune 29,1952

ljuly 3, 1952

Time '

5:26 

5:33 

5:35

5:38 

5:41 

5:46 

1:30

1:40 

3:05 

3:08 

3:15 

2:50 

3:07 

3:49 

!8:30 

3:15 

3:35

f §:55 

i 6:05 

I 6:20

Discharge 

0 (cfs)

3.4 

2.3 

1.14 

.8 

.31 

.14 

40 

173 

2.8 

21.5 

11.2 

3.1 

126 

79 

40 

66 

800 

2.1 

.14 

120 

57 

9.5

Width 

w 

(feet)

4.5 

4.0 

3.6 

3.5 

2.7 

2.1 

58 

100 

18 

14 

13 

11 

80 

68 

57 

79 

110 

7 

2.8 

40 

36 

32

Crosssectional 

area 

(square 

(feet

1.35 

.9 

.61 

.51 

.38 

.22 

9.0 

35.5 

1.6 

4.7 

2.9 

1.3 

34 

26 

16 

26

.9 

.18 

22.4 

13.6

Mean 

depth

(feet)

0.3 

.22 

.17 

.15 

.14 

.10 

.16 

.35 

.09 

.34 

.22 

.11 

.42 

.39 

.28 

.32 

1.3 

.12 

.06 

.56 

.34 

.12

Mean 

velocity

(fps)

2.5 

2.6 

L9 

1.6 

.82 

.64 

4.4 

4.9 

1.8 

4.6 

3.9 

2.4 

3.7 

3.0 

2.5 

2.6 

6.5 

2.4 

.7 

5.4 

4.2 

2.8

Suspended 

load 

L (tons 

per day)

86 

38

10 

5.3 

.9 

5,800 

62,000 

260

17,500 

8,500 

3,600 

5,200 

110,000 

155 

3.9

10,500 

1,600

Suspended 

load 

concentration 

C. 

(ppm)

8,000 

4,100

3.270 

2,460 

1,700

At-a-station values

6

0.22

.39 

.13

.26

}.13 

} .32

1 .09

.32 

.34

.27

/

0.33

.23 

.61

.45

.56 

.24

.61

.39 

.35

.35

m

0.45

.28 

.32

.36

.29 

.43

.26

.24 

.34

.36

j

1.4 

1.27

1.34

1.23 

1.36

1.08

1.28 

1.58

Slope 

(feet 

per 

foot)

0. 0175 

.0286

.0151

.0058 

.032

.025 

.0055

Remarks

L computed for Q interpolated between 

individual measurements.

Estimated peak.

/Estimated from partial 

I measurement.

Slope measured near 

mouth.

1 All times are postmeridian except that for Rio Oalisteo., 8:30 a. m. 

3 Measurements made by M. G. wolman and Leopold.

3 Measurements made with assistance of C. E. Stearns.

4 U. S. Geological Survey gaging stations.

APPENDIX F.—Arroyo discharge measurements along length of stream during individual flood (considered to be downstream relations')

Stream and location

Sedalia Gully, near Sedalia, Colo., 30 miles south of 

Denver.

Sowbelly Creek, near Hat Creek near Harrison, Nebr ...

Date

July 30,1952

July 11,1951

Discharge 

0

(cfs)

0.14

.25

.3 

1.2

1.6

4.1

.03

.07

.44 

.74

1.15

1.18

Width 

w

(feet)

2.2

2.2

2.5 

3

5

5

1.1

1.4

2.4 

3.0

2.5

4.3

Mean 

depth

(feet)

0.10

.12

.11 

.14

.12

.17

.08

.09

.13 

.14

.18

.21

Mean 

velocity

0 

(fps)

0.57

.92

1.1

2.8

2.5

4.8

.33

.54

1.4 

1.7

2.4

1.4

Suspended 

load 

L

(tons per

day)

6.7

2.9

6.8 

33

63

440

2.4

5.0

38 

82

140

170

Downstream values

ft

0.29

.31

/

0.15

.20

m

0.58

.49

j

1.85

1.2

Slope

ft 

(feet per

foot)

0. 017-0. Oil
INDEX

Page 

Acknowledgments...___..._____.._.............___._....___ 2

Arroyo. See Gully.

Arroyo Caliente, N. Mex., channel data___-________._____ — — 35

Bank cutting. — __-_--____ — — — _ — -_........._._................ 4-5

Bed-particle size, distribution in selected channels............................ 36

equation relating to discharge...-.---.-.___.._..._._-_____.__._.. 26,27

relation to roughness in downstream direction......_................... 26

Current-meter and section measurements__- — -.-._.__.__............._ 6

Depth, equation relating to drainage area...__.-_--.________--._...._..___ 20

relation to discharge at cross section-.. ———.._____ — _-..--..... 7,8,9,10

relation to discharge in downstream direction..______..................... 13

table of values of exponent..—__ — — — - — --—_ — .____.__ — — __— 15

Discharge, equation relating to bed-particle size. _________________ — — — — 26,27

equation relating to drainage area-—____-_----- --__ — -_____-_.-___---. 20

increase with stream order.. ——— — — — —_ — _._.________._ —_..___ 22-24

measurements of arroyos in downstream direction during flood..-. — — — 36 

measurements of arroyos at cross sections during flash floods____-._....... 36

relation to stream order_---..____. — _ — - — - — — — _.____.-___ — — 23

Drainage area, relation of mean annual flood discharge_______-_____ —........ 24

Ephemeral channels near Santa Fe, N. Mex., basic data..-------------------- 35

Ephemeral streams, nature and characteristics.._.__ — ———— _.— __.______ 3

Equilibrium.----............................................................. 28-33

Exponents, average values of y and z......................................... 26

table of values...................................__....................... 15

Flood flows. See Streamflow.

Geographic setting of study_...__________....__________.__ 2-3

Gully, discontinuous, characteristics of-...................................... 29-30

discontinuous, formation of...... — — . — — .___ — ____.._ — — _.______ 30-33

low gradient of_ __........____-..-..--_........_....._....._ 30-31

Hack, J. T., cited.-- — — —_-_-..__._ — — — — — — ...._ — — .._.__— 24-25

equation quoted.-------------.._...........---...............--.......... 26

Horton, R. E., quoted__..__..___--__-_ —— — ___ — _....._...... 21

Leopold, L. B., and Haddock, Thomas, Jr., quoted..-- — —._....._.... 15,25

Pebbles and cobbles, movement during floods.-- — — _.................... 6

Perennial streams in central N. Mex., flood-frequency data.-— — ._ — — — 35

Purpose and nature of study_..._______——_-—— — _-—.-- — — - 1-2

Page 

Rainfall..-.--- —------- .......................—— .....—— — .. — -— 3-4

Rills..... — . — . — ..—-—————————. — ——— — — — — 29

Roughness, equation relating to drainage area. ——————. ————.. — ——— 20

Serial identification. See -point numbers in Appendix.

Slope, equation relating to drainage area——_ — — — _- — — —— -.... ——... 20

rate of change related to depth__--------- — _.... —..__----___.________ 25

rate of change related to velocity_--.-.-----.-„-.--- — ___ — _ — — — 25

relation to stream order— — ___________ _---.. — ---.-_ —-------------- 21

relationto width.--.--..----.-.---.-.-.----.----------- —----- 22,28-29

way of determining------.---___.---..__-------___------------------------ 21

Streamflow, bank cutting_...............-.--.-....-----..-..---- — ----- — . 4-5

bores. —-_— — _— — -- — — —------— — — — — — —-- — — — ----- —----------- 4, 5

floods..-.___--.-----_-----_____-------____-----.__---------------------- 4

movement of pebbles and cobbles......_____---.__._---------------------. 6

problems of measurement——_ — -____- — ____—-. — __. —. — _ — —... 6

Stream length, relation to stream order-.-----.-----.--.------—-------------- 18-19

Stream number (number of streams), method of counting—...... — ________ 16-18

Stream order, equations relating to discharge, width, depth, velocity, suspended load, slope, and roughness-. —— _-__ ——------ — -—- 20

increase with discharge-.-..-.-------------------------------------------- 22-24

method of classifying.____ — ___-- — _____-_ — _____ — — ___ — __ — -—- 16-18

relationto discharge—— ---__ — _____ —------------- — — _____ — ___ 23

relation to drainage area. — — _._——— — - — — — — — — — — — — — — - 19-20

relationto slope......_- — ______ — _ — — — — — — — — — —— — — — — 21

relation to width--------------------------------------------------------- 23

Suspended sediment, collection of samples--. — —-____ — — ---- — — -—-- 6

relation to discharge at cross section-- — — -- — - — — — — — — - — — — 11-13

relation to discharge in downstream direction.__________.. ——_________ 13-15

relation to drainage area—— - — - — -——————- ——- — — --- — — — — — 20

table of values of exponent-—.------------- — .-----.-- — .----------------- 15

Velocity, equation relating to drainage area-___ —... ——— — ——— - — _-___- 20

relation to discharge at cross section. -.______ — _____-— — ----------- 7,8,9,10

relation to discharge in downstream direction................- — ...„. — .. 13

table of values of exponent..._._ — — _____ — — _--- — _____ — — — 15

Width, equation relating to drainage area_-._ —— .__- — — — -------------- 20

relation to discharge at cross section. -_____—_____—— — __ —... 7,8,9,10,13

relation to discharge in downstream direction— __.__ — — .. — — -—-_—--— 13 

relationtoslope.——..—————————— — —— —— — — — —— 22,28-29

relation to stream order_ — ___- — _-_ — — — —...-----...--.-..----.- 23

table of values of exponent_...------.----.—---.-. — . — — —. — -___— 15

way of determining---.--.-------------------------------- — ------------- 21

37

U. S. GOVERNMENT PRINTING OFFICE: 1956