AUG 5 19 83

PQ

(M

<£>

p* 

c3 OH

2

OH

oo

"eS

Hydraulic Geometry of a 

Small Tidal Estuary

GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-B

CO

O2

-u

w 

2o 

o

Hydraulic Geometry of a

By ROBERT M. MYRICK and LUNA B. LEOPOLD

PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-B

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1963

PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

HYDRAULIC GEOMETRY OF A SMALL TIDAL ESTUARY

By ROBERT M. MYRICK and LUNA B. LEOPOLD

ABSTRACT

A tidal channel in a marsh bordering the Potomac River near 

Alexandria, Va., was mapped, and current-meter measurements 

of discharge were made at various locations and at various 

stages in the tidal cycle. These measurements allowed analysis 

of the change of width, depth, and velocity with discharge at 

various cross sections and along the length of the channel.

There is also presented a theoretical development of some, 

of these same relations based on hydraulic principles and on the 

assumption of a uniform distribution of energy and a minimum 

rate of work in the system as a whole.

The change of width, depth, and velocity with discharge downstream developed from the field data checked closely with the 

theoretically derived values.

The estuarine channel differs from a terrestrial one in that 

discharge at any section in an estuary varies depending on how 

the flow shaped the entire length of the channel between the 

point in question and the main body of tidal water. The result 

is that a tidal channel changes more rapidly in width and less 

rapidly in depth as discharge changes downstream than does a 

terrestrial channel.

INTRODUCTION

The shape and characteristics of a river channel are 

believed to be determined and maintained by what has 

been called by irrigation engineers the dominant discharge. Whereas in an irrigation canal the usual or design discharge is ordinarily the modal flow, the designation of a dominant discharge in a river is not as easy. 

The effective discharge most influential in river morphology and thus equivalent to the dominant discharge 

of canals is believed to be the flow at bankfull stage. 

This view was strengthened by the findings of Wolman 

and Leopold (1957), and substantiated later by Nixon 

(1959), that the bankfull discharge has a recurrence 

interval of 1 to 2 years and is similar among rivers 

in quite different physiographic settings.

Wolman and Miller (1960) extended and generalized 

this into a concept of effective force in geomorphology. 

The greatest amount of geomorphic work, they postulate, depends not only on the effectiveness of each single 

event but also on the frequency of the events of different magnitudes. The combination of effectiveness and 

frequency that was most influential in maintaining river 

channel form and pattern was the bankfull discharge.

A tidal estuary is, in a sense, merely one of the extremes in the continuum of river channels. The channel 

system of an estuary bears some close similarities to that 

of a river system. It differs, however, in at least one 

extremely important respect and that is the frequency 

of the bankfull stage. If, as some investigators believe, 

the bankfull stage is the one that is most effective in the 

maintenance of a river channel and is equaled or exceeded once every year or two, one may ask Avhy the 

channel system of a tidal estuary so much resembles 

a river network, despite the fact that bankfull stage is 

attained about twice a day. Observations and measurements of flow characteristics at different sections along 

a tidal estuary, particularly at bankfull stage, may help 

to illuminate the position of estuarine channel systems 

in the continuum of all natural channels.

The bankfull stage of a tidal estuary may be defined 

as the stage at which the water incipiently flows over 

the adjacent marshlands. Unlike other streams, an 

estuarine channel having no runoff from the uplands 

experiences zero velocity at the nadir of ebb tide. 

Velocity increases progressively as the water deepens 

but again becomes zero at the crest of flood tide. From 

this fact arises the question concerning how the hydraulic factors vary along the length of the channel 

system. There is a collateral question concerning the 

relation of the bankfull stage to the effective force in 

channel formation and maintenance.

To understand the hydraulic geometry of estuaries 

better and to compare it with that of an upland channel, 

this preliminary investigation was undertaken.

There are other collateral questions concerning the 

differences and similarities between upland streams and 

tidal ones. For example, does the tidal channel exhibit 

alternations of shallows and deeps which in upland 

channels we have referred to as pools and riffles? Does 

the drainage network of the tidal system divide into 

tributaries of various sizes in such a manner that their 

lengths and numbers bear a logarithmic relation to 

stream order as in upland channels?

Some evidence on these points was obtained in this 

investigation, enough to suggest that further investigation would be lucrative. But on all the points studied,

Bl
B2

the present data are limited, and any generalizations we 

suggest on the basis of these data should be viewed as 

working hypotheses needing further proof.

We are grateful to Estella B. Leopold and Anne 

Davis for their work on the samples obtained for pollen 

analysis, to H. L. Svenson for notes on the taxonomy 

of plants in the study area, and to Alfred C. Redfield, 

for a review of the manuscript and permission to include some of his unpublished data.

AREA OF INVESTIGATION

Against the west bank of the Potomac River, nearly 

adjacent to the city of Alexandria, Va., there was until 

the winter of 1960 a small area of tidal marsh relatively 

undisturbed by construction or dredging. It can be 

seen in the location map of figure 1 that the southern 

of three estuarine channels carries away nearly all the 

surface runoff originating in the high ground adjacent 

to the marsh. The central of the three channels, called 

by us Wrecked Recorder Creek, drains no upland area 

and is the one studied. The marshland tributary to this 

channel has an area of about 120 acres. The marsh area 

and channels draining it receive water only as rain falling directly on the tributary marsh and as flow induced 

by tidal fluctuation. The water exchanged with the 

Potomac estuary is here quite fresh (not salty).

The size of the channel was such that it could be 

measured with relative ease. Where it enters the 

Potomac River, the channel width is about 170 feet. 

The total length of channel from mouth to the most 

distant headwater tributary is approximately 5,000 feet. 

A tide gage operated by the U.S. Coast and Geodetic 

Survey is slightly less than 2 miles upstream from the 

mouth of the channel system studied.

Cross sections were measured at six locations along 

the channel system, as will be described. The estuarine 

channel was 134 feet wide at the most downstream section and 18 feet wide at the uppermost secti6n.

Tidal fluctuations affect the whole length of Chesapeake Bay, and tidewater ends at the foot of the steep 

reach where the Potomac River descends over the Fall 

Line in the vicinity of Washington, B.C. The tidal 

estuary studied is thus about 11 river miles downstream 

from the head of tidewater.

Fieldwork was carried on during late spring and summer of 1960. Shortly thereafter, dredging operations 

for commercial gravel production destroyed the channel 

system.

The marsh area appears uniformly flat to the eye. In 

early spring the dried vegetation has been bent over by 

snow, so that.at low tide one can walk with ease over 

nearly the full unchanneled area. There are some small 

areas within the marsh on which timber is the primary 

cover. Though some of these timbered areas appear 

to have a slightly higher elevation than average for the 

marsh, other timber patches appear to be flooded 

regularly.

By midsummer the marsh vegetation is thick and 

green, standing at least knee high. The photographs in 

figure 2 show typical channels and vegetation in early 

summer.

MOUNT VERNON S.3 Ml 77°02'30"

38°45'

1 MILE

FIGURE 1. Location map of Wrecked Recorder Creek, a tributary to 

the Potomac River, near Alexandria, Va.

EXPLANATION FOR FIGURE 2

Photographs of typical reaches of Wrecked Recorder Creek, June 10, 1960. A., Small tributary 400 feet below section C, right bank, stage 

4.8 feet, falling. B, Section F, viewed from 20 feet downstream, stage 4.6 feet, falling. C, View looking downstream 250 feet upstream 

from section D, stage 4.6 feet, falling. D, Section D, looking upstream, stage 4.6 feet, falling. E, Left bank and entrance of small tributary at Iron Hulk Gage near section A, stage 4.92 feet, falling. F, View downstream to mouth of Wrecked Recorder Creek from position 

200 feet downstream from section B; Potomac River barely visible in distance ; stage 4.45 feet, falling.
HYDRAULIC GEOMETRY OF A SMALL TIDAL ESTUARY B3
B4 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

Along the margins of the channel, vegetation is absent 

below an elevation of 1 foot lower than the bankfull 

stage. At low tide there is exposed along the walls of 

nearly all channels a brown silt or clayey silt, containing some organic material. On the bottom along the 

c'enterline of the channel, the bed material is highly 

organic and black and has a texture not immediately 

obvious in the field.

Table 1 includes some typical data on particle-size distribution of samples taken at five of the established cross 

sections. The median size of these materials tended to 

be in the range from silt to fine sand. The table also 

shows the percentage of organic matter in the sample 

as determined from ignition-loss measurements.

TABLE 1. Particle-size analysis of channel-bank materials taken 

1-2 feet below level of marsh surface, and ignition loss in 

percent

[Data show percentage finer]

Particle size (mm'.)

1.0---...' ..... ..

.50-... .......................

.25...........................

.125... ......................

.0625.... __ ....... ____ ..

.0312.........................

.0156..... 

.0078....... ................

.0039..... 

.0019---.. ... _

Percent ignition loss at 700° F..

Section designation (sample from right bank)

A

100 

61 

54 

40 

35 

30 

24 

12 

5 

2

11.3

C

100 

85 

73 

60 

59 

49 

41 

18 

5 

2

10.6

D

100 

77 

59 

43 

40 

29 

22 

9 

3 

1

12.7

E

100 

92 

78 

64 

62 

48 

36 

20 

12 

3

10.9

F

100 

91 

75 

57 

48 

31 

28 

21 

11 

2

13.0

The dominant vegetation consists of cattails, Typha 

latifolia, and probably also T. angustifolia. Both 

species are to be expected in localities such as this, but 

T. angustifolia is in general dominant in areas of salty 

water. The basal part of some stands of Typha is whitened in spring, probably showing the limit of tide.

Another plant common to the area is the arrow arum, 

Peltandra virginica. Some of the plants are unusually 

luxuriant and might be taken for the yellow water lily, 

Nuphar advena, especially where partly submerged.

The trees growing in some areas slightly higher in 

elevation than average marsh surface include green ash, 

Fraxinus lanceolata, which is a common tree in such 

places in the tidewater region. The willows are probably 8alix nigra, common along the Potomac in the 

Washington area.

Underlying most of the marsh area is a thick layer 

of gravel, presumably Pleistocene in age, which is being 

dredged for use in highway fills and other construction. 

The operators of the dredge stated that gravel layers 

occur 10 to 40 feet below the surface of the marsh.

A few samples were taken for pollen analysis at section D at depths ranging from the marsh surface to 

14 inches below it. These samples were brown silty

clay which in the field appeared to be undisturbed by 

any recent lateral movement of the channel. A summary of the pollen analysis is presented in table 2. The 

following interpretation of these samples is quoted 

from an unpublished report by Dr. Estella B. Leopold 

(written communication to authors concerning Paleontology and Stratigraphy Branch shipment WR-60- 

11D).

The samples show a decrease of pine and a corresponding 

increase of hickory from the base to the top of the sampled 

section. These features, plus the presence of warm-temperate 

tree types and a small amount of spruce pollen represent an assemblage which is similar to that of the late pine zone or the 

postglacial pollen Zone B of Deevey (1939). Assuming the samples are postglacial rather than interglacial in age, an interpolation from a marsh section taken near Blackbird, Delaware, 

would suggest an age of at least 5000 to 7000 years B. P.

The small amount of oak, the large amount of pine and the 

presence of spruce make the samples quite unlike deciduous 

tree Zones C-l through C-3 as represented at Blackbird, Delaware, as well as in several postglacial sections from Delaware, 

Virginia, and New Jersey.

TABLE 2. Pollen analysis of samples taken along channel bank 

at section D

Tree pollen (in percent) :

Tilia.......... ............................

Shrub and herb pollen (in percent) :

Undetermined dicots _________ do ....

Ferns and fern allies (actual counts):

Samples assigned USOS 

Paleobotanical location Nos.

D 1599-0 D1599-B D1599-A

Depth (inches) below surface

0-2

0.7 

35.5 

.7

12.8 

.7

1.4 

9.2 

.7

2.1 

.7 

17.7 

1.4 

.7 

.7 

7.1 

7.8

(141)

(13)

7-9

2.6 

41.8 

.7

.7 

5.2

.7 

1.9 

1.3

1.9 

9.8

2.6

17.6

7.8 

5.2

(153)

(D 

(13) 

(D

12-14

2.5 

50.0 

1.5 

.5

8.4 

2.5 

1.0 

1.9 

6.9

1.9 

5.4

1.9 

3.9

5.9 

1.0 

.5 

.5 

1.0 

2.5

(202)

(10) 

(D

This preliminary study of pollen from a few samples 

indicates that the marsh has existed for a considerable 

time, at least 5,000 years. This knowledge is useful in 

that it implies sufficient time for establishment of a 

quasi-equilibrium between forces of erosion and deposition in the construction and maintenance of the channel 

system.
HYDRAULIC GEOMETRY OF A SMALL TIDAL ESTUARY B5

METHOD OF STUDY

The principal observational data consisted of the 

gage height and discharge at selected locations along 

the length of the estuarine channel. The location of the 

six measuring sections is shown in figure 3. This figure 

shows in detail only one tributary, the one on which 

section F is located. Kecording gages were installed 

near the upper and lower sections, and at each there 

was established a semipermanent staff gage. A level 

survey established the elevations of the staff gages 

above an arbitrarily chosen datum. At each of the six 

cross sections, several complete discharge measurements 

were made by means of a current meter operated from 

a canoe. Later a continuously recording current meter 

and, concurrently, a recording water-stage apparatus 

were installed and operated, first at one section and then 

at another, over a period of several weeks at each.

A longitudinal profile of the bed extending from 

section A to the headwater tributary was obtained by

sounding and is shown in figure 4. Some samples of 

bed and bank material for particle-size analysis were 

collected.

CHANNEL SHAPE AND DRAINAGE NETWORK

Before embarking on a discussion of the hydraulics 

of the tidal channels, a description of the drainage network in conjunction with the planimetric maps presented in figures 3 and 5 will provide the reader with 

some picture of the channels studied. Cross sections 

up to the level of the marsh that is, to bankf ull stage 

are presented in figure 6 for each of the six principal 

measurement locations. In table 3 the principal characteristics of these channel cross sections are compared 

with river cross sections considered typical for river 

channels of similar size in the vicinity of Washington, 

D.C. The table compares width-depth ratios, bankfull 

discharge, and some other channel parameters.

100

Stage and velocity recorder

D 

Staff gage

PIODRE 3. Map of Wrecked Recorder Creek at Potomac River, south of Alexandria, Va., showing location of cross sections and recording

Instruments.

G75126 O 63 2
B6 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

7.2 feet gage datum

cc ~ 

< 2

1000 2000 

DISTANCE, IN FEET

3000

FIGURE 4. Longitudinal profile of bed of Wrecked Recorder Creek; vertical scale is depth in feet below average level of marsh,

equal to 7.2 feet on the arbitrary gage datum of the present study.

4000.

which is also

Table 3 shows that the width of estuarine channels 

increases with respect to discharge faster than in upland channels. The table does not include enough 

examples to demonstrate conclusively any generalization about width-depth ratio. Tidal channels are often 

shallow near the mouth owing to deposition where an 

estuarine channel joins a large body of water.

TABLE 3. Characteristics of cross sections of tidal channel and 

some nearby rivers for comparison

Wrecked Recorder Creek near Alexandria. Va.:

B........... .....................

C.. ........................... ...

D...... .......................... E.. ............... -.:... ... ...

F. ...............................

Watts Branch near Rockville, Md .......

Seneca Creek near Dawsonville, Md.....

Width

(ft)

134

82

84

45

32

18

15

86

100

65

38

Depth

(ft)

4.13

4.77

3.83

3.38

3.00

2.38

3.4

4.2

5.35

3.4

2.0

Width/

depth

32.4

17.2

21.9

13.3

10.7

7.6

4.4

21.6

18.5

19.1

19.0

Bankfull

discharge

(cfs)

566

461

338

216

136 '47

170

1,320

2,700

1,000

310

1 For flood tide having a maximum stage of 8.6 ft, and a range in stage of 3.5 ft; this 

is one combination of range of stage and maximum stage at which maximum velocity 

during the tidal cycle occurs when the channel is bankfull.

The principal parts of the channel network were 

mapped by planetable, and the entire headwater network of one small tributary was sketched by paceTandcompass methods. The maps of figures 3 and 5 are 

sufficient to make a preliminary Horton analysis (Horton, 1945), relating stream order to the number and

average length of channels. The smallest channel without a tributary is considered to be of first order and is 

labeled 1 in figure 5. Channels having only first-order 

tributaries are labeled 2, meaning that they are of second 

order. The smallest channel where cross-section and 

discharge measurements were made is a channel of 

fourth order (fig. 5). The main channel in the network, 

on which are located sections A to C, is of fifth order. 

It is recognized that a map of only one example does 

not represent the variety of conditions that may occur 

in the field. Yet it is informative to note the relations 

of stream order to average length of stream shown in 

figure 7. As can be seen there, first-order channels O '

average about 20 feet in length, and the fifth-order main 

channel has a length of 5,000 feet. The length-order 

relations for the estuary are compared with the same 

relation for ephemeral streams (Leopold and Miller, 

1956, p. 18) and for perennial channels in Pennsylvania 

(Brush, 1961, p. 156). This estuary increases in length 

with stream order faster than do either ephemeral or 

perennial river channels.

The ratio of the number of streams of a given order 

to the number of the next higher order (called by Horton the bifurcation ratio) ranges for river channel networks from 3 to 4, often with an average of 3.5. Because only one tributary of the study estuary was 

mapped in detail, the estimate of the number of streams 

of various orders is crude. From the limited data for 

Wrecked Recorder estuary, this ratio averages 3.4. This
HYDRAULIC GEOMETRY OF A SMALL TIDAL ESTUARY B7

figure may not be typical of other estuaries, and other 

similar studies would be informative; yet, as can be 

seen in figure 8, this one sample is very similar to two 

other types of channels in very different environments.

Iris Bank 

staff gage

25

Section F

50 100 FEET

FIGURE 5. Detailed map of one tributary of Wrecked Recorder 

'Creek near Alexandria, Va., including all minor channels. 

Order number of each channel is shown, some of which are 

estimated (est).

STAGE-DISCHARGE RELATIONS

The relation of water stage to discharge has been 

determined for tidal estuaries in many localities, and 

the characteristic difference between the form of the 

curve on ebb and flood tide is well known. Nearly all 

previous observations, however, have been made on 

relatively large estuaries. With the exception of the 

studies made by Gilbert (1917, p. 108) in a tidal marsh 

near San Francisco, no published records are known 

to us for small tidal channels having no drainage area 

in the upland. The present investigation adds to Gilbert's work only the comparison of the relations between 

different cross sections located along the length of the 

estuarine channel.

Velocity, and thus discharge, at a given stage was 

dependent both on the maximum stage attained by the 

particular tidal cycle and on the range of stage in the 

tidal cycle. Thus, every tidal cycle requires a slightly 

different stage-discharge curve8.

The dependence of velocity on range of tide is known 

from other investigations. Bradley (1957, p. 664), also 

working in a relatively small estuary, found that the 

near-bed velocity varied directly with the range of tide. 

His excellent work does not help solve the particular 

questions here investigated. All his flow measurements 

were made close to the estuary bed. Further, he was 

concerned with a small bay rather than with riverlike 

estuarine channels.

It was impossible for a party of two to make a complete set of discharge measurements at all sections along 

the stream simultaneously through a single tidal cycle. 

We chose, therefore, to make as many discharge measurements as possible through a single cycle at each 

section, and on different days, making similar measurements on different cross sections. Now the cross-sectional area of each section for any given stage is known 

from the surveys of the section. The problem is to determine the mean velocity for the whole section at 

selected stages in order to construct a relation of discharge to stage. For two tidal cycles the relation of 

velocity to stage is shown in figure 9, using section D 

as an example. Such graphs alone, however, do not 

give sufficient information to determine the influence 

of either tidal range or maximum stage.

To incorporate these additional variables, two sections were chosen for a more intensive series of velocity 

measurements covering different tidal cycles. At sections D and F a recording current meter was installed, 

along with a device that recorded water stage simultaneously. Thus, at a particular distance above the bottom, velocity was measured near the center of the channel continuously for two weeks to sample tidal cycles 

from spring to neap tides.
B8 PHYSIOGRAPHIC AND HYDRAULIC STUDIES .OF RIVERS

VERTICAL SCALE 

rO

-2.0

L 4.0 FEET

10

U.

10 20 30 FEET

Section F HORIZONTAL SCALE

Cross sections viewed 

looking downstream

FIGURE 6. Cross sections viewed looking downstream of each of the principal measuring places along the channel of Wrecked Recorder Creek.

Vertical scale is feet below average level of marsh.

The location and depth of the recording current 

meter were selected on the basis of velocity readings 

from previous measurements to provide a position 

where point velocity best correlated with mean velocity. 

Owing to the variation in water depth throughout the 

tidal cycle, the depth below the water surface where 

the meter was installed changed. The position of the 

recording meter varied from three-tenths to seventenths of the full depth below water surface at section 

Z>, and from two-tenths to eight-tenths at section F. 

The velocities recorded at the single point during a 

given cycle correlated reasonably well with measured 

mean velocities for the whole section. From this correlation, the relation of mean velocity to stage and to 

range of stage could be constructed.

For each cross section, curves of the type shown in 

figure 10 were developed to show the stage-velocity relation for each of several values of range in stage. The 

curves shown by full lines in figure 10 are merely 

selected examples which comprise three values of maximum stage, each with a different range in stage. For 

one of these examples, an additional graph is shown 

(dashed line) representing the condition of another 

value of range in stage but identical maximum stage. 

It will be visualized that there may be a very large 

number of curves making up such a family as that 

shown in figure 10.

The graphs drawn in the interpolation procedure are 

considered reasonably reliable except for very low velocities, but even there the error in velocity does not 

exceed 10 percent. Minor variations in individual cycles were neglected in the construction of the graphs.

Most data in tidal studies have been presented as 

graphs of stage or velocity as a function of time in a 

tidal cycle (for example, Gilbert, 1917, p. 124; Ahnert, 

1960, p. 397). Our data plotted in that usual way 

exhibit the usual features (fig. 11). Minor variations 

occur in individual tidal cycles; velocity decreases 

toward zero with time in a somewhat irregular manner in the early part of the ebb tide.
HYDRAULIC GEOMETRY OF A SMALL TIDAL ESTUARY B9

10,000

1000

100

STREAM ORDER

FIGURE 7. Relation of stream length to stream order for Wrecked 

-Recorder Creek and for some other drainage areas. The data 

from Brush (1961) were extended to acc'ount for differences In 

map scale, and his order 1 was comparable to order 5 In the 

present study. Dots represent measurement data from this study.

BANKFULL DISCHARGE

The evaluation of the banks relative to gage datum 

was determined by visual observations at several sections 

in the reach during the flood tides. The stage at which 

the flow started to inundate the marsh ranged from 7.1 

to 7.3 feet, and averaged 7.2 feet (relative to our arbitrary datum) for the reach. The difference between 

the elevation of the marsh at the headwaters and at 

the mouth is very small.

In upland river systems, the higher the discharge the 

less frequently it is experienced. Except for very low 

flows, this inverse relation of discharge rate and frequency of occurrence is a general characteristic of 

rivers.

In tidal channels, on the other hand, zero discharge

occurs twice in every tidal cycle, and thus there are also 

two occurrences of high discharge in each cycle. There 

remain different frequencies of the different values of 

the peak discharge from cycle to cycle, but the range 

of values possible is much smaller than the range of 

possible values of flood flows in an upland fiver of the 

same size.

Because of this difference, it may be fruitful to examine the frequency of occurrence of the factors which 

do vary.

Figure 12 shows the percentage of time the high and 

low tides equal or exceed a given stage at the Alexandria, 

Va., tide gage. The ordinate shows the variation in 

terms' of the datum uesd in Wrecked Eecorder Creek.

1000

oc

UJ

o 

oc. 

o

z

UJ

> 

o

V)

1

cc

100

EXPLANATION

Relation between indicated 

order streams in estuary:

O O

1st and 2d

STREAM ORDER

FIGURE 8. Relation of number of streams to stream order; data for 

Wrecked Recorder Creek are compared with average relations for 

an area of perennial streams and an area of ephemeral streams.
BIO PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

-1.0

VELOCITY, IN FEET PER SECOND 

0 0.5 1.0

4.0 -100 100

DISCHARGE, IN CUBIC FEET PER SECOND

FIGURE 9. Stage-discharge relation (light line) and str ge-velocity relation (heavy line) ; a nearly complete tidal cycle having a maximum 

stage of 6.8 feet is shown and the ebbing limb only is shown for a tidal cycle in which maximum stage was 7.6 feet. Data are for 

section D. iThe velocity shown is mean velocity in the cross section.

A comparison of our observations with the concurrent 

values of stage of the Potomac estuary at Alexandria 

showed that the stage at the two locations varies in a 

comparable manner, and this indicates that the frequency of stage at the two locations is comparable. To 

eliminate the effect of the spring and neap tides, a period 

of 2 lunar months, 58 days, was used in the preparation 

of the frequency data. The extremes shown by dashed 

lines were extended on the basis of a straight-line 

projection.

From the graph we find the median high tide for 

this area is 7.27 feet relative to our local datum. Bankfull stage, 7.2 feet, is, interestingly, approximately the 

median high tide and is therefore attained on the average every other tidal cycle or once a day. A terrestrial 

river reaches its bankfull stage but once every 1 or 2 

years.

This, though true, leaves an incomplete and perhaps 

false impression in the mind of the student of channel 

morphology. We are interested, after all, in the frequency of the effective discharge.

In an estuarine channel, at some value of stage near 

high tide, velocity must be zero, for the direction of flow 

must reverse. In Wrecked Recorder Creek, the velocity 

is,highest at a stage much lower than high tide.

The velocity and the stage corresponding to maximum 

discharge both increase with the maximum stage of 

of any given tidal cycle. The stage at which the maximum velocity occurs is not well defined by the available 

data; but as a rough approximation, a tide reaching a 

maximum stage of 8 to 8% feet is required for the maximum velocity to occur at a stage of 7.2' feet, which is the 

bankfull stage. Thus, because of the dependence of dis-
HYDRAULIC GEOMETRY OF A SMALL TIDAL ESTUARY Bll

-1.00 -0.5 0.5 1.00

VELOCITY, IN FEET PER SECOND

FIGUKB 10. Sample relations of stage to velocity at section D showing effects of maximum stage and of range In stage. The samples Include 

various maximum stages, 7.0, 7.3, 8.0 feet. Two samples are for the same maximum stage (8.0 feet) but Illustrate different ranges of 

stage, 3.5 feet and 2.0 feet, respectively.

charge on velocity at a given stage, the top of banks 

is also the level of maximum discharge when the high 

tide occurs at the stage between 8 and 8V2 feet.

From an analysis of records of the Alexandria tide 

gage, a high tide of 8.5 feet is equaled or exceeded about 

0.9 percent of the time, and 8.0 feet about 9 percent of 

the time. Nixon (1959) found that bankfull stage is 

equaled or exceeded 0.6 percent of the time in English 

rivers. These results are of the same order of magnitude. Whether this is coincidental or morphologically 

important cannot be stated in the present state of knowledge. It is a problem worth further work. It is hoped 

that further investigations on estuarine streams will 

consider the problem of the dominant discharge.

HYDRAULIC CHARACTERISTICS AT A CROSS SECTION

In terrestrial rivers, changes in width, depth, and 

velocity at any cross section accompany a change in 

discharge. These variables are related to discharge 

as simple power functions expressed by Leopold and 

Maddock (1953) in the following form:

w=aQi> 

d=cQf 

v=lcQM

where Q is discharge, w, d, and v represent -the watersurface width, mean depth, and mean velocity, respectively. The coefficients, a, c, and 7c, are constants, 

and 6, /, and m are exponents representative of the 

section.
B12 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

\

\

Stage versus time, Apr. 20,1960

Median height of tide 

Apr. 19 to June 17,1960

-, -1.00

velocity versus time 

./ Apr. 20,1960

- 1.00

12 P.M. 2

FIGUKE 11. Stage and velocity as functions of time at section D. High tide during the three cycles shown was below, at, and above bankfull

istage, 7.2 feet.

Hydraulic geometry is a phrase coined to express 

these and collateral equations at-a-station and in a 

downstream direction. In the present study, similar 

curves were constructed for a tidal estuary.

It has been previously stated in the discussion of 

the velocity-stage curve that an unlimited number of 

velocities are possible at any stage for a given cross section. The effect on at-a-station curves of range in stage 

and of high tide at one station or section is not large 

compared with differences between channels in different

areas. Figure 13 presents data for three flood tides 

in the study area. The figure also includes data from 

Gilbert's study (1917) and some data collected by the 

authors in Old Mill Creek, Del.

For Wrecked Recorder Creek, figure 13 shows the 

changes in width, depth, and mean velocity as functions 

of discharge in flood tide only at section Z>. Other sections would have similar curves, and the ebbing limb of 

a tide would differ principally in the direction of the 

hysteresis loop.
HYDRAULIC GEOMETRY OF A SMALL TIDAL ESTUARY B13

9.0 

8.0

ID 

Q 7.0

LJ

C3

LU£6.0

z

UJ

^5.0 

4.0

99.9 99 90 70 50 30 10 1 0.1

-

-

-

-

i i

*x

1 1

1 1

\

1 1

1

k

s>.X

^fe

^^Oi,

1

1

\

*

1

1

X

X

1 1

\

1 1

1 1

.s

1 1

-

-

-

-

0.1 10 30 50 70 90 

TIME, IN PERCENT

99 99.9

FIGURE 12. Duration curves (cumulative frequency) of values of 

stage at high and low tide, Alexandria, Va., tide gage. Datum 

is the arbitrary datum used at Wrecked Recorder Creek.

Figure 13, thus presents for Wrecked Recorder Creek 

two sets of curves; each set represents a different tidal 

cycle and is indicated separately by crosses and triangles. The -results of three current-meter measure-, 

ments, represented by dots, are shown for comparison. 

Widths for stages above 7.2 feet (bankfull) were determined by a straight-line extension of a width-stage 

curve, and discharges were based on these widths. The 

set of curves indicated by a dashed line represent a 

flow below bankfull stage during the complete tidal 

cycle.

The two tides give such similar results that the slopes 

for each set of curves are nearly identical. Each tidal 

cycle will give a separate set of curves, but the slopes are 

consistent.

It can be visualized that in a flood limb of a tidal 

cycle, width and depth increase progressively from zero 

discharge through maximum to zero discharge. Velocity, on the other hand, begins with zero, reaches a 

maximum value at or near the stage of maximum discharge, and decreases to zero at the highest value of 

depth when flow reverses. Because figure 13 presents 

a flood tide only, the arrows show only one direction 

of progression around the hysteresis loop.

The respective slopes of the lines representing relations of width, depth, and velocity to discharge were

nearly identical on flood and ebb limbs of a tide. Thus, 

it is possible to discuss the relation of these slopes as 

exponents in power function equations, and to compare 

the exponents with those applicable to terrestrial rivers. 

For Wrecked Recorder Creek, values at given cross 

sections during flood tide of the exponent constants, 

&, /, and m, are:

Increasing velocity Decreasing velocity 

Change in width_____________6=0.04 6= 0. 01 

Change in depth____________/= .175 /= .04 

Change in velocity__________m= .785 m= 1. 05

For an upland stream from Leopold and Maddock 

(1953, p. 26) the average values are:

6=0.26

/= .40 

ra= .34

In the values for the tidal channel, the left-hand column is undoubtedly'the more important from the standpoint of channel morphology. The increasing velocity 

is related to flow in the channel, whereas when the 

velocity decreases on a flood tide at least part of the 

discharge is usually governed by conditions of overflow 

rather than channel hydraulics.

The small value for the exponent b in the estuarine 

channel was expected because of the relatively vertical 

banks prominent throughout the reach except at the 

bends.

The median range in tide in the vicinity of the project area is 2.8 feet, although under extreme conditions 

a range of more than 4 feet may be expected. Increasing velocity occurs principally in the lower segment of 

the range in stage. As an example, on a high tide 

of 8.6 feet with a range in stage of 3.6 feet the maximum velocity occurs at a stage of 7.2 feet or 1.4 feet 

below the maximum stage.

It is not known whether the value for the exponent / 

for Wrecked Recorder Creek is also representative of 

the numerous areas Avhere a large range in stage occurs 

during each tidal cycle. Some data are available "for 

comparison as discussed below, but more examples are 

needed.

Old Mill Creek differed from Wrecked Recorder 

Creek by being longer and deeper. The former also 

drains some area of upland whereas the latter does not.

Gilbert (1917, pi. 33) presented a graph which included the velocity, discharge, and stage for Ravenswood Slough on the southwest shore of San Francisco 

Bay. The range in stage for one of the flood tides 

illustrated was over 7 feet. From his published data, 

width at any given stage could be measured, and depth
B14 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

0.01

10

UJ

u_

z

-

t

UJ

O 

1 0

~

:

-

f=-0.04 

i i i i i i i i

 s-

 -r""*" ^ _ 

^.175

i i i i i i i i

f=O.U

_^-B 

 ^'^"^

1 1 1 1 1 1 1 1

f=0.08 , _ _

i i i i i i i i i i i i i i i i

500

50

20

b=

10 100 1000 

DISCHARGE, IN CUBIC FEET PER SECOND

10,000 100,000

FIGURE 13. Changes at-a-station of width, depth, and velocity as functions of discharge. Data for Wrecked Recorder Creek were collected 

at section D. Also included are observations at Old Mill Creek estuary, near Lewes, Del., and Ravenswood Slough, San Francisco Bay, 

Calif.
HYDRAULIC GEOMETRY OF A SMALL TIDAL ESTUARY B15

versus discharge computed from the width and velocity. 

The error in reading values of velocity, width, and discharge from his data is less than 10 percent.

The three examples presented in this report are only 

a small sample of the numerous variations in tidal 

channels. Because of the vast difference in the three 

channels, it is surprising that the values of the exponents in the equations are as comparable as they are.

The following tabulation summarizes the observed 

values of exponents in the equations for width, depth, 

and velocity, respectively, relative to discharge at a 

cross section, and on the rising velocity of a flood tide.

Sec. D, Wrecked Recorder Creek ________

Old Mill Creek.............. __________

Ravenswood Slough ___________________

Average. _______________________

Average for terrestrial rivers. ___________

Change of 

width

6

0 04 

08 

14

09 

26

depth

/

0. 18 

. 14 

.08

. 13 

.40

velocity

m

0 78 

78 

78

78 

34

As can be seen in figure 13, the curves for width and 

depth versus discharge have slightly negative slopes 

as discharge decreases from its maximum value to zero 

at high tide. Thus the values of the exponents l> and / 

given above would be smaller rather than larger if 

averaged with those applicable to decreasing discharge 

on the floodtide. It, must be concluded then that the 

principal difference in the at-a-station hydraulic characteristics of estuarine and terrestrial streams is that 

the former have a much more rapidly changing velocity 

with discharge than do terrestrial rivers. This is compensated by less rapidly changing depth and width with 

discharge.

THEORETICAL ANALYSIS OF THE HYDRAULIC GEOMETRY IN THE DOWNSTREAM DIRECTION

By WALTER B. LANGBEIN

Pillsbury (1939, p. 228-230) defined an ideal estuary 

as one in which the tidal range, depth, and current are 

uniform through the length of the channel. By analysis 

of the motion of water in the channel as induced by the 

tides at the mouth and as retarded by friction, Pillsbury 

showed that in such an estuary the width decreases exponentially with the distance up the estuary; thus

w=w0e-nx cotv>

where 10 = width at distance x upstream from the mouth 

where width is wn , e is base for natural logarithms, <p is 

the lag (in degrees) between the time of maximum slope 

and the time of maximum velocity, and n is the quantity 

a/ V<7 D where a is the mean angular speed of the semidiurnal lunar tide= 0.00014 radians.per second, D is

mean depth at a given cross section, and g is the acceleration of gravity. This equation describes an estuary 

decreasing in width exponentially with distance.

Pillsbury showed that tidal estuaries tend to approximate this exponential form, generally with a sinuous 

alinement. The analysis is based on an assumption of 

a single channel estuary. Presumably, if applied to a 

branched estuary the width, w, must be the total width 

at distance x. There is, however, a well known tendency for natural channels to become shallower as they 

become narrower. It will 'be instructive to deduce the 

hydraulic geometry of an estuary in the terms defined 

by Leopold and Maddock (1953), and to introduce concurrently some of the principles of hydraulic probability described by Leopold and Langbein (1962). That 

report treats the distribution of energy in natural systems, in which known physical relations or constraints 

are insufficient to yield unique solutions in other words, 

a system with remaining degrees of freedom. From 

analogy to entropy production in thermodynamics, it 

was shown that a river system, in adjusting to the remaining degrees of freedom, would tend toward uniform distribution of energy, and a minimum rate of 

work in the system as a whole. Pillsbury, for example 

in postulating equal depth and velocity in an estuary introduced only the first of these conditions. This is incomplete. We shall explore the effect of minimizing 

total work as well.

The amount of work done by the tide in filling and 

emptying the estuary is a function of the volume of the 

tidal prism and of the hydraulic friction. Both these 

quantities are related to the width and depth of the 

estuary.

If change in depth is described by the function 

D=D0 e-kx where D0 is mean depth at the mouth of the 

estuary, an approximate equation describing the width 

of an estuary is as follows:

w=w0 e~ (ncot</>+k) x

According to this equation, the estuary will narrow 

and thus the tidal prism lessen more rapidly, the 

greater the rate of decrease in depth. Other hydraulic 

relations that govern flow in an estuary are as follows:

Continuity. Since Q=vdw 'at each cross section, the 

relation m + /+Z> = 1.0 must be satisfied.

Slope-depth relation. Slope in this case is the surface slope of the water surface created by the progress 

of the tide in the estuary. The slope varies sinusoidally during the tidal cycle. As Pillsbury shows, the 

value of s (amplitude oj: the slope) is a function only 

of the depth, thus:

s=Aa|^[gD 

where A is the tidal amplitude (half the range be-
B16 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

tween mean high water and mean low water), and a and 

D are as described previously.

The average value during one phase of the tidal cycle 

will be 0.63 S and

Since

Hence *=-*/

Velocity -depth relation. The amplitude of the tidal 

current is as follows :

sn <f>

where <f> is the angular lag of the primary current behind the hydraulic slope. The value of <£ is itself a 

function of depth and channel friction, and is evaluated 

by tables given by Pillsbury (1939, p. 123). Using a 

Chezy coefficient of 80 and mean depth of cross sections 

ranging from 2 to 10 feet, the value of the amplitude 

of the current is calculated to be

V =0.35 AD1 ' 5

Velocities during the tidal cycle will vary sinusoidally 

in proportion to the amplitude V. Mean velocities at 

different cross sections vary as the fifth root of the mean 

depths, therefore, m= //5.

Thus we have the three following hydraulic conditions :

m+/+6=1.0 

m=//5

There are four unknowns; the remaining statement 

will be supplied by the necessity that the estuary approach a state in which energy is as uniformly distributed as is consistent with the necessity that total work 

in the estuary as a whole be a minimum.

The first condition is satisfied if velocity and depth 

are uniform; hence, m+/ must approach zero. Since 

m and / are of the same sign, each of the parameters, m 

and /, also would approach zero. Since m-f /+6 = 1.0, 

(1 6) approaches zero.

The second condition, that of minimum total work, is 

met as the integral 22 Qsdx approaches zero. In the 

upland river, Q, the discharge, is the land drainage and 

is independent of the channel geometry, and slope is the 

only variable. The discharge in an estuary, however, 

is uniquely a function of its size and thus of its width, 

depth, and length, and so the problem is more complex.

The relative work performed during a tidal cycle in an 

estuary is as follows:

(1)

where P is the ratio of the integral XJD Qsdx for an estuary in which the depth varies as D0 e kx - to the same 

integral for an estuary in which the depth does not vary, 

that is, & = 0.

This equation shows that the relative work decreases 

as the depths decrease upstream from the mouth. As 

was shown previously, the decrease in depth affects the 

estuary width, and so affects the value of the exponent 

6 in the relation between width and discharge, w <x$ 6 , 

where w is the channel width at a given cross section 

and Q is the bankfull discharge. For an estuary in 

which depth does not decrease, the value of 6 is unity. 

For an estuary in which depths decrease upstream from 

the mouth, the value of l> decreases. The relation is 

as follows:

,_ (2)

Formulas 1 and 2 can be solved to determine the 

relation between the relative work P and the width 

exponent 6, thus:

n 1.36-0.59

A 01. i A AI 0.00 + 0.41

. , 

or as a more convenient

approximation :

P= 1.76 -0.7

As stated previously, the condition of uniform energy distribution requires (1 6) to approach zero. 

Thus 6 should be high. On the other hand, minimum 

work requires 1.76-0.7 to approach zero in other 

words, 6 should be low. Following the logic developed 

in the paper by Leopold and Langbein (1962), the most 

probable value of 6 is that for which the probability 

of the deviation of (1 6) from 6 equals the probability 

of a deviation of 1.76 0.7 from 6. These two probabilities are equal when

F,(b)=F2 (b)

crFi crF2

where F\ ( 6 ) = 1.76 - 0.7 and ^2 ( 6 ) = ( 1 - 6 ) . *F1 and 

aF2 are the standard deviations of those quantities. 

The absolute values of the standard deviations need 

not be known. Each can be expressed in terms of the 

standard deviation of 6, thus

0-^=1.7(76 and
HYDRAULIC GEOMETRY OF A SMALL TIDAL ESTUARY B17

Hence

1.7&-0.7 1-

or 5 = 0.71, which leads to the following set of values

of the hydraulic geometry exponents:

m= 0.05

/= .24

5= .71

s=-.12

These results derived without reference to field measurements will be compared in the next section with data 

obtained in the tidal creek studied.

MEASUREMENTS OF THE HYDRAULIC GEOMETRY IN 

THE DOWNSTREAM DIRECTION

In the discussion of channel characteristics, the 

changes in width, depth, and velocity in the downstream direction are also important. A representative 

discharge for all cross sections is required for this analysis. The dominant or bankfull discharge is probably 

the most meaningful for terrestrial rivers. The estuarine channel presents the problem of what value of discharge is pertinent. It is reasonable to assume that the 

discharge occurring at the time of maximum velocity 

at a section is a pertinent discharge. This occurs nearly 

simultaneously at all sections in a channel as short as 

Wrecked Recorder Creek because the time lag along 

the length of the channel is small. It is this discharge 

that is used in the computation of the downstream 

curves in figure 14.

The data used in construction of the curves represent 

flood and ebb flow during a tidal cycle having a high 

tide of 8.6 feet, for which some measurements are summarized for floodtide in table 3.

From the slopes of the lines in figure 14 the values 

of the exponents in the hydraulic geometry may be 

determined. The curves represent the average of both 

flood and ebb flows.

Values of exponents m the hydraulic geometry, with respect to 

increasing discharge downstream

Exponent of 

Exponent of 

Exponent of 

Exponent of

width, ft_.__ 

depth, /. . . . 

velocity, m. 

slope.-__-_-

Theoretical 

(Langbein)

0. 71 

. 24 

.05 

-. 12

Tidal estuaries

Wrecked 

Recorder 

Creek 

(this 

report)

0. 77 

. 23 

.00

Barnstable 

Marsh, 

Mass. 1

0. 74 

. 17 

.09

Kivers

Average 

for rivers in midwestern 

United 

States >

0. 50 

.40 

. 10 -.49

' Preliminary estimates by Alfred C. Redfleld and Lincoln I-Iollister communicated 

by letter to Leopold, Mar. 30, 1962. 

» Leopold and Maddock (1953) and Leopold (1953).

Langbein's theoretically derived values of these exponents are compared below with the values determined 

from the present field investigation of Wrecked Recorder tidal stream. Also included are values from 

field data collected and analyzed by Redfield and Hoilister at Barnstable Marsh, Mass.

 o

£<"

_,*

>£

UJ

u. 

n R

-

m = 0.00

-

-

I i I i i i i i

x " x

1 1 1 1 1 1 1 1

5.0

1.0

100

10

Ebb tide 

Flood tide

10 100 

DISCHARGE, IN CUBIC FEET PER SECOND

1000

FIODBB 14. Downstream or along-t.he-channel changes In width, 

depth, and velocity as functions of discharge, at bankfull stage In 

Wrecked Recorder Creek.

The data from Redfield and Hollister are particularly 

valuable because, the channel studied ranged in width 

from 10 feet near the headwaters to 3,800 feet near the 

mouth. This is a much greater range in channel size 

than in our study and yet there is remarkable agreement 

of the results.

The agreement between the field results of this investigation and the theory is very-.satisfactory. The 

field .data, however, are few and the scatter is large. 

More field data are needed.

Both theory and field data show that in tidal estuaries, 

depth tends to be more conservative (a low value of /)
B18

than in upland rivers, so that the width-depth ratio 

varies rapidly downstream. At the mouth, a tidal 

estuary is wide and relatively shallow; at its head, it is 

narrow and relatively deep. The reason for this is that, 

in contrast to rivers, the discharge at any section is 

itself a dependent variable depending on how the flow 

shaped the channel in all the channel length between the 

point in question and the main bay or body of tidal 

water. In a terrestrial river, discharge is independent 

in that it is produced by the watershed and the channel 

is accommodated to it, rather than by such accommodation modifying the discharge itself.

It is this influence of the channel on discharge that 

makes the rates of change of width, depth, and velocity 

along the channel different than in terrestrial rivers.

o

REFERENCES

Ahnert, Frank, 1960, Estuarine meanders in the Chesapeake 

Bay area: Geog. Rev., v. 50, no. 3, p. 390-401.

Bradley, W. H., 1957, Physical and ecologic features of the 

Sagadahoc Bay Tidal Flat, Georgetown, Maine: Geol. Soc. 

America, Mem. 67, p. 641-682.

Brush, L. M., Jr., 1961, Drainage basins, channels, and flow 

characteristics of selected streams in central Pennsylvania: 

U.S. Geol. Survey Prof. Paper 282-F, p. 145-180.

Deevey, E. S., Jr., 1939, Studies on Connecticut lake sediments:

Am. Jour. Sea., v. 237, no. 10, p. 691-724. 

Gilbert, G. K., 1917, Hydraulic mining debris in the Sierra

Nevada: U.S. Geol. Survey Prof. Paper 105, 154 p. 

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. 

Leopold, L. B., 1953, Downstream changes of velocity in rivers:

Am. Jour. Sci., v. 251, no. 8, p. 606-624. 

Leopold, L. B., and Langbein, W. B., 1962, The concept of

entropy in landscape evolution: .U.S. Geol. Survey Prof.

Paper 500-A, p. 1-20. 

Leopold, L. B., and Maddock, Thomas, Jr., 1953. Hydraulic

geometry of stream1 channels and some physiographic implications: U.S. Geol. Survey Prof. Paper 252, 57 p. 

Leopold, L. B., and Miller, J. P., 1956, Ephemeral streams 

hydraulic factors and their relation to drainage net: U.S.

Geol. Survey Prof. Paper 282-A, p. 1-37. 

Nixon, Marshall, 1959, A study of bankfull discharge of rivers

in England and Wales: Inst. Civil Engineers Proc., v. 12,

p. 157-174. 

Pillsbury, G. B., 1939, Tidal hydraulics: Washington, U.S. Army

Corps of Engineers, Prof. Paper 34, 281 p. 

Wolman, M. G., and Leopold, L. B., 1957, River flood plains;

some observations on their formation: U.S. Geol. Survey

Prof. Paper 282-C, p. 87-109. 

Wolman, M. G., and Miller, J. P., 1960, Magnitude and frequency

of forces in geomorphic processes: Jour. Geology, v. 68, no.

1, p. 54-74.

U.S. GOVERNMENT PRINTING OFFICE; 1963 O 675126