Discharge and 

Flow Distribution, 

Columbia River Estuary

GEOLOGICAL SURVEY PROFESSIONAL PAPER 4 3 3-P

Prepared in cooperation with the 

U.S. Atomic Energy Commission
Discharge and 

Flow Distribution, 

Columbia River Estuary

By G. A. LUTZ, D. W. HUBBELL, and H. H. STEVENS, JR.

TRANSPORT OF RADIONUC L I D E S BY STREAMS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 433-P

Prepared in cooperation with the 

U.S. Atomic Energy Commission

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1975
UNITED STATES DEPARTMENT OF THE INTERIOR

GEOLOGICAL SURVEY 

V. E. McKelvey, Director

Library of Congress Cataloging in Publication Data

Lutz, Garson Alvin, 1913-

Discharge and flow distribution, Columbia River Estuary.

(Transport of radionuclides by streams)

(Geological Survey Professional Paper 433-P)

Bibliography: p.

Supt. of Docs, no.: I 19.16:433-P

1. Stream measurements Columbia River estuary, Or. and Wash. 2. Columbia River estuary, Or. and Wash. 3. Stream

measurements Mathematical models. 

I. Hubbel, David Wellington, 1925- joint author. II. Stevens, Herbert Humphreys, 1927- joint author. III. United States

Atomic Energy Commission. IV. Title. V. Series. VI. Series: United States Geological Survey Professional Paper 433-P 

QE75.P9 no. 433-P [GB1227.C6] 557.3'08s [551.4'83'09797] 75-619061

For sale by the Superintendent of Documents, U.S. Government Printing Office

Washington, D.C. 20402 

Stock Number 024-001-02214-1
CONTENTS

Abstract __________________________

Introduction and acknowledgments __________

Generalized description of the estuary ________

Discharge-measurement equipment and techniques 

Mathematical-model theory ______________

Development of mathematical models ________

Beaver Army Terminal reach ___________

Calibration procedure _____________

Mathematical-model discharge results __.

Astoria reach ____________________

Calibration procedure _____________

Page

PI

1

3

3

6

7

10

11

13

Development of mathematical models Continued 

Astoria reach Continued

Mathematical-model discharge results ________

Flow characteristics in the lower estuary ____________

Lateral circulation at the discharge-measuring cross 

section, Astoria reach ___________________

Flow-predominance patterns at the discharge-measuring 

cross section, Astoria reach ________________

Flow-predominance patterns in the south channel,

CRM 5.6-22.5 _____ _________________ 

Summary ______________________________

References cited __________________________

Page

19

19

19

22

25

27

ILLUSTRATIONS

Page

FIGURE 1. Map showing location of the lower Columbia River and estuary _____________________________ P2

2. Graphs showing observed vertical distributions of salinity, normal velocity, and flow direction at CRM 14 at various

times in tidal cycles during periods of high and moderate upland flows ______________________ 4

3. Photograph showing velocity-azimuth-depth assembly ___________________________________ 5

4. Photograph showing data-acquisition system ________________________________________ 5

5. Definition sketch of an element of channel _______________________ _________________ 7

6. Map of Beaver Army Terminal reach, showing cross sections where channel geometry was defined ________ 9

7. Graph showing variation of the flow-resistance coefficient with daily mean discharge, Columbia River at Beaver Army

Terminal near Quincy, Oreg. _______________________________________________ 11

8. Map of Astoria reach, showing cross sections where channel geometry was defined __________________ 12

9-11. Graphs showing:

9. Change in rj/ij min , with relative time at Astoria, Oreg. _______________________________ 14

10. Comparison between measured and computed discharges for low instantaneous discharges at Astoria, Oreg.,

on April 13, 1970 __________________________________ __ 15

11. Comparison between measured and computed discharges for high instantaneous discharges at Astoria, Oreg.,

on March 7, 1970 _________________________________________________ 16

12. Hydrographs showing daily mean discharges at Beaver Army Terminal and Astoria, Oreg., May-September

1968 _________________________________________________ 18

13-15. Graphs showing:

13. Vertical distribution of flow predominance in the "north," middle, and south channels at the measuring cross

section at Astoria, CRM 14, for various mean discharges _________________________ 21

14. Variation of flow predominance with mean discharge at the measuring cross section at Astoria, CRM 14 22

15. Graph showing flow predominance along the longitudinal axis of the south channel during a half tidal

cycle _________________________________--_----------_--_______ 23

16-17. Graphs showing salinity distribution along the longitudinal axis of the south channel at various times in the tidal 

cycle on:

16. September 14, 1969 __________________________________________________ 24

17. May 23, 1970____________________________________________ 26

18. Graph showing percentage error in computed instantaneous discharge at Astoria, Oreg., plotted against comparable

measured discharge ____________________________________________________ 30

ill
IV CONTENTS

TABLES

Page

TABLE 1. Summary of discharge measurements at Beaver Army Terminal near Quincy, Oreg. _________________ P9

2. Areas and top widths of average cross sections representative of the two subreaches in the Beaver reach _____ 10

3. Areas and top widths of average cross sections representative of the two subreaches in the Astoria reach _ _ _ _ 13

4. Summary of discharge measurements at Astoria, Oreg. __________________________________ 13

5. Lateral distribution of ebb and flood discharge in the measuring cross section at Astoria, CRM 14 ________ 19

6. Variation of flow predominance with depth at selected locations in the measuring cross section at Astoria, CRM 14 20

7. Error analysis of computed discharges at Astoria, Oreg. _________________________________ 31

APPENDIXES

APPENDIX 1 . Daily mean discharge of the Columbia River at Astoria, Oreg. ______________-

2. Determination of percentage error between measured and computed discharges at Astoria

Page

30

ENGLISH-METRIC CONVERSIONS

Physical quantity Multiply English units

Length

Area _________

Volume _______

Velocity _______

Water discharge _.

( Feet (ft)

| Yards (yd) 

L Miles (mi)

Square miles (mi2 ) ____.

Cubic yards (yd3 ) ______

Feet per second (ft/s) _ _. 

Cubic feet per second (ft3/s]

By

f 0.3048 

\30.48 

.9144 

1.609

2.589

.7646

.3048

.02832

To obtain metric units

Metres (m) 

Centimetres (cm) 

Metres (m) 

Kilometres (km)

Square kilometres (km2) 

Cubic metres (m3 ) 

Metres per second (m/s) 

Cubic metres per second (m;'/s)
TRANSPORT OF RADIONUCLIDES BY STREAMS

DISCHARGE AND FLOW DISTRIBUTION, 

COLUMBIA RIVER ESTUARY

By G. A. LUTZ, D. W. HUBBELL, and H. H. STEVENS, JR.

ABSTRACT

Low-level radioactive wastes were discharged into the Columbia, 

River at the Hanford Reservation, U.S. Atomic Energy Commission, 

near Richland, Wash., from 1944 until early 1971. The various 

radionuclides that made up the waste in the river associated with sediment and biota or remained in solution and were subsequently distributed throughout the estuary and into the Pacific Ocean. To provide 

information on the amount of radionuclides being transported through 

the estuary, continuous records of water discharge were obtained near 

both the upper and the lower ends of the estuary during the period 

1968-70.

Complex velocity distributions, mainly due to salinity gradients, 

made it impossible to use conventional methods of measuring discharge in the lower part of the estuary; however, a new technique, 

MOVD (measurement of velocity distribution by moving boat), was 

developed for determining the magnitude and direction of the water 

velocity throughout the entire depth at a vertical. Repetitive measurements at a series of verticals in cross sections at the Beaver Army Terminal, Oreg., Columbia River mile 53.3, and at Astoria, Oreg., Columbia River mile 14, defined flow hydrographs at these locations during 

half tidal cycles on a number of occasions.

The defined flow hydrographs, in turn, were used to calibrate 

mathematical models for computing continuous records of discharge. 

The models, which were necessary mainly because of the influence of 

the tide, consisted of partial differential equations that were written to 

express the conservation of mass and momentum in one-dimensional 

unsteady homogeneous-density open-channel flow and were solved by 

the method of characteristics.

At Beaver Army Terminal, which is in the freshwater part of the estuary, application of the mathematical model was fairly straight 

forward, and discharges computed every 15 minutes for May 1968 

through June 1970 are considered to be very accurate.

At Astoria, where salinity gradients are present and channel 

geometry is complex, it was necessary with the model to vary the flow 

resistance coefficient throughout each ebb and flood period in order to 

compute discharge hydrographs having the same shapes as the 

hydrographs defined by measurements. In addition, in order to compute hydrographs that compared closely in magnitude with defined 

hydrographs, a factor had to be applied daily to adjust measured 

water-surface slopes that are used in the model. The factor was determined by trial by comparing model discharges with daily mean discharges from a simple volumetric relation based on the daily mean discharge at Beaver Army Terminal and the net daily storage of water 

between Astoria and Beaver Army Terminal. Monthly mean discharges determined from model discharges computed every 15 minutes 

from March 1968 through June 1970 by this technique compare closely 

with monthly mean discharges determined by the routing technique 

used by the Northwest Water Resources Data Center. Computations

show that daily mean discharges at Astoria follow a cyclic pattern; 

hence, they cannot be used as indicators of the upland freshwater discharge.

When the daily mean discharge at Beaver Army Terminal is less 

than about 165,000 ftVs (cubic feet per second) or 4,670 mVs (cubic 

metres per second), the south (navigational) channel at Astoria conveys proportionately more flow during the ebb than during the flood, 

and the remainder of the cross section conveys proportionately more 

flow during the flood than the ebb. This produces a net clockwise circulation of water between the two channels. For daily mean discharges 

from about 165,000 to 190,000 ft3/s (4,670 to 5,380 mVs), there is a net 

counterclockwise circulation, and for those over 190,000 ftVs (5,380 

m3/s) the circulation also is clockwise.

In the north channel at Astoria, flow near the bottom is 

predominantly landward for mean discharges at the measurement 

cross section less than about 340,000 ftVs (9,630 m3/s), whereas flow in 

the upper layers is predominantly seaward for all discharges. The pattern in the south (navigational) channel is roughly similar to that in 

the north channel, but in the middle channel the flow-predominance 

pattern is more complex.

In the navigational channel on September 14, 1969, when the daily 

mean discharge at Astoria was 208,000 ftVs (5,890 m3/s), saltwater 

reached Columbia River mile 22 and the flow was predominantly 

seaward for all depths upstream from Columbia River mile 17.5. 

Downstream from that point, flow in the lower half of the depth was 

predominantly landward, whereas flow in the upper half was 

predominantly seaward. However, on May 23, 1970, when the daily 

mean discharge at Astoria was 437,000 ft3/s (12,400 mVs), seawater 

reached only Columbia River mile 14, and the flow was predominantly 

seaward throughout the entire study reach, which extended from 

Columbia River miles 5.6 to 16.2.

INTRODUCTION 

AND ACKNOWLEDGMENTS

Low-level radioactive wastes were discharged into the 

Columbia River at the Hanford Reservation, U.S. 

Atomic Energy Commission, near Richland, Wash. (fig. 

1), from the time of the initiation of nuclear-reactor 

operations in 1944 until early in 1971. The wastes 

resulted primarily from the neutron activation of 

chemical constituents in treated Columbia River water 

that was used to cool the nuclear reactors. Once the 

radionuclides were released to the river environment, 

they remained in solution or became associated with

Pi
PACIFIC 

OCEAN

BEAVER 

ARMY TERMINAL

DISCHARGE 

CROSS SECTION 

(UPPER GAGE)

INDEX MAP OF PART OF THE COLUMBIA RIVER BASIN

10 20 K

FIGURE 1. Lower Columbia River and estuary.
DISCHARGE AND FLOW DISTRIBUTION, COLUMBIA RIVER ESTUARY P3

sediment or stream biota. Although much of the radioactivity decayed, some radionuclides were transported by 

the associated media downstream and ultimately 

reached the Columbia River estuary and, thence, the 

Pacific Ocean.

In late 1963 the U.S. Geological Survey undertook, in 

cooperation with the U.S. Atomic Energy Commission, 

an investigation of the movement of radionuclides in the 

Columbia River estuary downstream from CRM 65.8 

(Columbia River mile) at Longview, Wash. (fig. 1). One 

of the purposes of the investigation was to define rates of 

transport of radionuclides at cross sections along the 

longitudinal axis of the estuary. To accomplish this end, 

it was necessary to obtain continuous records of total 

water discharge at the cross sections of interest. This 

report briefly describes the equipment and techniques 

used to obtain measurements of the flow, discusses the 

mathematical models utilized to determine discharges 

at two different cross sections, presents discharge data, 

and suggests the nature of circulation patterns in the 

vicinity of Astoria, Oreg., CRM 14.

In the early phases of the investigation, Chintu Lai 

aided in the selection of the farthest upstream reach, 

which is in the vicinity of the former Beaver Army Terminal, Oreg., CRM 53.3 (fig. 1), and prepared a computer program for processing data obtained during discharge measurements in the field. E. A. Prych contributed substantially to the physical installation of 

water-stage recording equipment utilized throughout the 

investigation, to the development of the dischargemeasuring system, and to some of the early techniques 

for processing measurement data.

GENERALIZED DESCRIPTION 

OF THE ESTUARY

The Columbia River estuary receives freshwater flow 

from throughout a drainage area of 259,000 mi2 (671,000 

km2 ). Based on long-term records at The Dalles, Oreg., 

CRM 189 (U.S. Geological Survey, 1971), the flow at 

Vancouver, Wash., CRM 107 (fig. 1), averages about 

200,000 ftVs (5,660 mVs). During 1963-70 daily mean 

discharges at Vancouver ranged from 78,900 to 675,000 

ftVs (2,230 to 19,100 mVs; U.S. Geological Survey, 1971).

Tides of the mixed-type characteristic of the Pacific 

coast, two high waters and two low waters during a 24.8- 

hour period, produce landward (upstream) flows in the 

estuary that have been measured as far upstream as 

Prescott, Oreg., CRM 72 (U.S. Geological Survey, 1970). 

At the mouth, CRM 0, the mean tidal range (U.S. Coast 

and Geodetic Survey, 1969, p. 172-173) is 5.6 feet (1.7 

m), and the diurnal range is 7.5 feet (2.3 m). Tide ranges 

are amplified somewhat within the estuary and are a 

maximum in the vicinity of Youngs Bay, CRM 12 (fig. 

1); the mean and diurnal tide ranges at this location are 

6.7 and 8.6 feet (2.0 and 2.6 m), respectively. In the

vicinity of the Beaver Army Terminal, Oreg., CRM 

53.3, the mean the diurnal tide ranges are approximately 

4.3 and 5.2 feet (1.3 and 1.6 m), respectively. During 

periods of low and moderate runoff, tides influence flow 

in the Columbia River as far upstream as Bonneville 

Dam, CRM 145 (fig. 1); at Vancouver, CRM 107, the 

tidal influence persists for all flows to as much as 500.000 

ftVs (14,200 mVs).

The combination of moderate tide ranges and a large 

freshwater discharge produces a dynamic environment 

within the estuary. At low upland flows, cold saline 

water from the ocean intrudes as far upstream as Harrington Point, Wash., CRM 23 (fig. 1). At times of high 

upland flow, salinity intrusion is limited to the lower 

part of the estuary. For instance, on May 23, 1970, when 

the daily mean discharge through the cross section at 

CRM 14 was about 437,000 ft3/s (12,400 mVs), 

measurable salinities extended upstream only to about 

CRM 14. As a result of the wide variation in the patterns 

of salinity intrusion, salinity gradients in the three coordinate directions vary significantly throughout the year. 

According to the classification of Pritchard (1955), the 

Columbia River estuary is in the general category of a 

coastal-plain estuary that exhibits characteristics most 

of the time of a type-B (partly mixed) estuary.

Velocity distributions in both the vertical and lateral 

directions vary significantly with time as a result of tidal 

motion, large freshwater discharges, and salinity (density) gradients. Several observed velocity distributions 

in the vertical direction are shown in figure 2. The lack of 

uniformity, with time and position, in the distribution of 

velocity complicates the definition of flow and necessitates detailed velocity measurements in both the vertical and lateral directions.

DISCHARGE-MEASUREMENT 

EQUIPMENT AND TECHNIQUES

In order to define the temporal variation of discharge 

through a cross section in the estuary it is necessary to 

define both the vertical and lateral distributions of 

velocity by repetitive measurements throughout the 

tidal cycle. Because of the cycle changes in velocity with 

time, it is essential that individual velocity measurements be obtained rapidly. A unique measuring system 

suitable for operation on a continuously moving boat was 

developed (Prych and others, 1967) to meet this requirement. With the system, the magnitude and direction of 

water velocities relative to the boat are measured 

throughout the depth with a modified version of a VADA 

(velocity-azimuth-depth assembly) unit (Lockett and 

Kidby, 1961; Barron, 1963) that is fitted with a modified 

Ott cosine-propeller current meter (fig. 3). Concurrently, 

the heading, the speed over the bottom, and the drift 

angle of the boat are determined from a compass and an
P4 TRANSPORT OF RADIONUCLIDES BY STREAMS

10 12 14 16 18 20

I I TTT

June 5,1967

I I I I I

I/I I I

-6 -5 -4 -3 -2-1 01

SALINITY, IN PARTS PER THOUSAND 

2 4 6 8 10 12 14 16 18 20

June 6, 1967

I I I I

TTT

0005 hours 

I I I

10 12 14 16 18 20

June 6,1967

I I I

0405 hours 

I I I I I

-5-4-3-2-1012345 

NORMAL VELOCITY, IN FEET PER SECOND

6 -6 -5 -4 -3 -2 -1

 10 ft:

NORMAL VELOCITY, IN METRES PER SECOND

I I I I I I I I I I I I I I I I I 

-120 -60 0 60 120 180 -180 -120 -60

FLOW DIRECTION, IN DEGREES

I I I I I I I I I I I I I I I 

0 60 120 180 -180 -120 -60 0

0 2 4 6 8 10 12 14 16 18 20

SALINITY, IN PARTS PER THOUSAND 

0 2 4 6 8 10 12 14 16 18 20 0 24 6 8 10 12 14 16 18 20

-6-5-4-3-2-1 0 1 2 34 5 -6-5-4-3-2-1 0 1 2 34 5 6-6 -5 -4-3-2-1 0 1 2 3 

NORMAL VELOCITY, IN FEET PER SECOND

NORMAL VELOCITY IN METRES PER SECOND

-120 -60 -120 -60 0 60 120 

FLOW DIRECTION, IN DEGREES

180 -180 -120

Salinity

EXPLANATION

Normal velocity Flow direction

FIGURE 2. Observed vertical distributions of salinity, normal velocity, and flow 

direction at CRM 14 at various times in tidal cycles during periods of high 

(June 1967) and moderate (April 1970) upland flows. Indicated normal velocities are 

components normal to the measurement cross section of absolute velocity vectors; negative values designate upstream flow. Flow directions give the angle of approach of the

flow to the cross section; positive and negative values pertain to angles measured clockwise and counterclockwise, respectively, from the southward heading of the cross section; hence, an angle of 90° indicates downstream flow normal to the cross section. 

Tide height during each set of distributions can be deduced by considering the bottom 

(solid horizontal line) to be at a fixed elevation.

on-board sonic navigation unit. Outputs from all sensors, including the velocity meter, are analog signals that 

either are or can be displayed visually on strip-chart 

recorders (fig. 4), and all signals are sampled on command, virtually instantaneously, by a moderately high

speed scanning digital voltmeter. Voltages, in turn, are 

recorded (fig. 4) on magnetic tape (except the signal 

from the depth transducer that indicates the height of 

the VADA unit above the bed; this signal is recorded 

only as an analog signal on strip chart).
DISCHARGE AND FLOW DISTRIBUTION, COLUMBIA RIVER ESTUARY P5

FIGURE 3. Velocity-azimuth-depth assembly (from Prych and others, 1967).

For the definition of the velocity distribution at a vertical, the boat is positioned and kept as stationary as 

possible by heading into the current and using power. 

Ordinarily, an attempt is made to maintain slight 

headway during the measurement. The VADA unit is 

lowered to within 2 to 3 feet (0.6 to 0.9 m) of the 

streambed, allowed to stabilize, and then raised slowly 

to the surface. The vertical velocity of the unit is 

regulated so that the angle of attack of the water relative 

to the meter is less than 20°. During the upward 

traverse, the analog signals are sampled and recorded 

every 3 seconds. Later, the recorded data (relative speed 

and direction of the current, absolute speed and drift 

angle of the boat, and boat heading) are combined by 

vector addition, using a digital computer, to yield the 

absolute velocity (speed and direction) of the current at 

each data point throughout the depth.

Discharge is measured by observing the entire velocity 

profile at a series of verticals along a cross section. Ordinarily, measurements are made at about 20 verticals 

during each traverse of the cross section. As soon as a 

traverse is completed, the boat is moved to the opposite 

side of the estuary, and the measurement sequence is 

repeated. The exact location of each measurement vertical is determined by triangulation with hydrographic 

sextant; hence, measurement sequences can only be 

made during daylight hours.

For the computation of discharge, the velocity component normal to the cross section at each data point in a 

vertical is used to calculate a discharge per unit width 

for the increment of depth represented by the point. The 

incremental discharges are summed over the depth to 

give a discharge per unit width for the entire vertical. 

The discharge for the section represented by the vertical

FIGURE 4. Data-acquisition system. The strip-chart recorders provide continuous 

real-time displays of sensor signals, and the magnetic-tape unit digitizes data for computer 

processing.

is determined by multiplying the section width, which is 

computed according to the standard midsection method, 

by the discharge per unit width for the vertical. Because 

discharge changes rapidly with time, section discharges 

are not summed to give a total discharge; rather, the 

cross section is divided into several parts and the discharge through each part is determined by summing the 

section discharges within the part. Hydrographs are then 

obtained by plotting the discharge for each part against 

the corresponding mean times when the verticals were 

measured. The total discharge hydrograph is defined by 

summing the hydrographs for the individual parts.

In addition to this "standard" method, which 

hereinafter will be called the MOVD (measurement of 

velocity distribution by moving boat) method, discharge 

measurements sometimes were made at the Beaver 

Army Terminal cross section by a modification of the 

moving-boat technique developed by Smoot and Novak 

(1969). With the Smoot-Novak method, a boat is 

traversed across the width of the measuring section by 

"crabbing" along a range line that is normal to the flow 

direction. During the traverse, a current-meter-vane 

unit, which is mounted to the boat 3 to 4 feet (0.9 to 1.2 

m) below the water surface by a rod and angle-indicator 

assembly, freely and continuously alines in a direction 

parallel to the vector sum of the lateral velocity of the 

boat and the velocity of flow. Periodically noting the 

current-meter velocity and the vane direction permits 

the velocity of flow normal to the cross section at the 

current-meter depth, V, to be determined at a series of 

data points across the width from

V=Vy sin a,
P6 TRANSPORT OF RADIONUCLIDES BY STREAMS

where

and

the instantaneous flow velocity relative to 

the boat as measured by the current meter;

a is the complementary angle between the crosssection line and the current-meter direction. 

The lateral positions of the data points, where depth as 

well as velocity and current-meter direction are 

observed, are established each time the filament of water 

passing the current meter reaches a preselected length, 

Ly. That is, observations are made whenever the current 

meter has turned a preselected number of revolutions. 

(Because the relationship between velocity and rate of 

rotation of the current meter is linear, a given number of 

revolutions is equivalent to a particular filament length.) 

If the flow is normal to the cross section, 

the width between data points, LB, equals Lv cos a. 

Section discharges are computed by the standard 

midsection method from values of V, LB, and the 

depth, d. Total discharge is obtained by correcting 

(multiplying) the sum of the section discharges by (1) a 

width-correction factor, which is the ratio of the 

measured width to the sum of the computed section 

widths (this factor adjusts for variations in boat course 

and for variations in the flow direction relative to cross 

section), and by (2) a velocity-correction factor, which is 

the ratio of the mean velocity in a vertical to velocity at 

the current-meter depth (this ratio is determined from 

detailed velocity-distribution data at several verticals). 

The modified technique used at the Beaver Army Terminal is essentially the same as the Smoot-Novak 

method; however, it was designed to permit use of the 

VADA and the navigation units. During the lateral 

traverse, the VADA unit was suspended 15 feet (4.6 m) 

below the water surface and lateral data points were established at 10-yard (9.1-m) intervals as indicated by 

the navigation unit. All variables were measured in the 

same way as in the MOVD method, and the velocity of 

flow normal to the cross section at the 15 foot (4.6-m) 

level was determined by vector addition in the same way 

as in the MOVD method. The widths between data 

points, except the sections adjacent to the shores, were 

assumed to be constant and were determined by dividing 

the measured width between the first and last stations 

by one less than the total number of data points. Section 

discharges were computed by the midsection method 

and summed to give a total discharge. This total was 

altered by a velocity-correction factor that, in effect, adjusted the velocity at the 15-foot (4.6-m) level to represent the mean velocity in the vertical. The factor, c, was 

determined as

c=0.82(d/;y)0 - 2 , 

where

y is the height of the velocity meter above the bed, 

by assuming (1) the parabolic velocity-distribution law

where

Vy is the velocity at a height, y;

C is a constant for given conditions (C cancels in

the derivation of c); and

n is a constant assumed to be equal to 5.0 for a 

rough boundary (Savini and Bodhaine, 1971, 

P. 12), _ 

and (2) the condition that the mean velocity, V, in the 

vertical occurs at y=0.368 d. The correction factor, c, 

was applied as a constant for all verticals by taking d 

equal to the mean depth in the cross section and y equal 

to the mean depth minus 15 feet (4.6 m).

MATHEMATICAL-MODEL THEORY

Whenever unsteady flow conditions exist, such as in 

an estuary, a unique relation between water-surface 

elevation (stage) and discharge does not exist, and continuous records of discharge cannot be obtained by conventional methods of discharge computation. In 1961, 

Baltzer and Shen developed a mathematical model for 

computing discharge for unsteady flow conditions. Two 

other methods for computing discharges for unsteady 

flow conditions have been developed by Lai (1965a; 

1965b). All three mathematical models are based on the 

equations of continuity and motion expressed in one 

spatial dimension and simplified by approximations to 

represent open-channel flow.

The equation of continuity, or conservation-of-mass 

equation, states that the net change in discharge in a 

reach is equal to the change in storage in the reach 

that is (fig. 5, definition sketch),

mass in mass out + side inflow = increase of

mass within the reach, 

or

pAu- d(pAu). ~\ , . ' r,dZ. -%r**\+l*i*x=fBai**'

where

p is the density of water;

A is the channel cross-section area;

u is the flow velocity in the longitudinal direction;

Z is the elevation of water surface;

t is time;

q is the lateral inflow per unit length;

x is a distance measured along the x axis; and

B is the top width of the cross section. 

The equation of motion, or conservation-ofmomentum equation, states that the resultant force 

acting on an element of water in a given direction is 

equal to the time rate of change of momentum of the 

water contained in the element. Applying Newton's Law
DISCHARGE AND FLOW DISTRIBUTION, COLUMBIA RIVER ESTUARY P7

 pAu + ^-. (pAu) Ax

A. Profile

B. Cross section

FIGURE 5. Definition sketch of an element of channel (from Lai, 1965a). See text for 

description of symbols.

of momentum, the equation of motion can be obtained 

as

du. du. dZ. k

dt dx dx #4/3 uu

where

g is acceleration due to gravity; 

R is the hydraulic radius; and 

k is a parameter depending on the flow-resistance coefficient, 77 which is similar to Manning's n.

The assumptions made in the derivation of the above 

two basic partial differential equations for describing unsteady flow are (1) the flow is moderately unsteady, (2) 

the fluid is of homogeneous density, (3) the channel is 

prismatic and has a very mild bottom slope, (4) the 

velocity of flow is uniform over the cross section, (5) the

variables in the equations and their derivatives are continuous functions with respect to distance and time, and 

(6) the flow-resistance coefficient is the same as that for 

steady flow and, hence, can be approximated from the 

Chezy or Manning formulas.

Because an explicit analytical solution for the set of 

partial differential equations is not possible, Baltzer and 

Shen (1961) developed a power-series method of solution. The method is based on a Maclaurin-series expansion of the two equations about a reference point along 

the waterway at a given instant in time. The expansion 

is made in terms of the stage at the reference point and 

the associated stage at a nearby point, and then it is 

rewritten in finite-difference form with respect to time in 

order to obtain the change in discharge during a finite increment of time; discharge is computed from appropriate boundary conditions in a stepwise manner 

throughout time. Lai (1965a) developed another means 

of solution by applying the method of characteristics 

which transforms the basic partial differential equations 

into four ordinary differential equations. These four 

equations are treated as finite-difference equations for 

which unique solutions of the flow at successive times 

are readily possible with the known boundary conditions. He also (Lai, 1965b) utilized the implicit method 

in which the two basic partial differential equations are 

directly transformed into corresponding finite-difference 

equations; by setting up as many equations as there are 

unknown dependent variables, the equations are solved 

simultaneously for all unknowns at the advanced (or 

forward) time level, using the appropriate boundary conditions, to obtain flow at subsequent times.

The three methods of computing discharges for unsteady flow conditions are designed for use on high-speed 

digital computers. All three methods have produced accurate results in the freshwater parts of tide-affected 

streams or in well-mixed tidal estuaries (Baltzer and 

Lai, 1968); however, depending on the physical properties of the site, one method may be more appropriate 

than the others.

Recently, two-dimensional flow models have been 

developed to account for variations in either the lateral 

or vertical directions, as well as in the longitudinal direction. These models usually must be solved with extensive quantities of input data and by using computers 

having extremely large core memory. Also, like the onedimensional models, they represent flow in well-mixed 

estuaries and are not adequate to describe complicated 

saltwater circulation.

DEVELOPMENT OF 

MATHEMATICAL MODELS

To meet the need for radionuclide and suspendedsediment transport rates through the upper and lower
P8 TRANSPORT OF RADIONUCLIDES BY STREAMS

parts of the estuary, measurement cross sections where 

discharges would be defined were established at Beaver 

Army Terminal and at Astoria (fig. 1). At both these 

locations tides produce unsteady flow; hence, the most 

feasible means for obtaining records of water discharge 

was to apply transient-flow models. When data collection for discharge determinations began in 1966, only 

one-dimensional models, as previously described, were 

available, and computer memory systems were relatively 

limited. As a result, the field program was designed to 

provide essential data for only the one-dimensional approach.

Physical conditions at the Beaver Army Terminal can 

be represented moderately well by a one-dimensional 

model; however, at Astoria, significant salinity gradients 

and complex flow patterns introduce serious deviations 

from one dimensionality. Because of the differences, the 

development and application of the model for each location was substantially different.

In order to adapt and use on a continuous basis any of 

the three mathematical models previously described, 

certain boundary values and other data are required. 

Boundary values include (1) frequent periodic 

simultaneous stages at the extremities of a reach that is 

sufficiently long to permit the accurage determination of 

the fall (water-surface slope), yet is not so long as to 

violate limitations implicit in the mathematical solutions; and (2) for all stages, values of the channel width, 

depth, and area that are representative of the reach. In 

addition, reach length and inflow or outflow within the 

reach must be known, and actual hydrographs of discharge at a cross section in the reach (preferably at oner 

end) must be abailable to define parameters of the 

model.

BEAVER ARMY TERMINAL REACH

The Beaver Army Terminal (hereinafter referred to as 

"Beaver") is a deactivated military reservation located 

on the Oregon side of the Columbia River at CRM 53.3 

(fig. 1). The cross section at Beaver is at one of the narrowest parts of the estuary (about 2,300 ft, or 700 m, 

wide), is devoid of any islands or extensive flats, is 

reasonably uniform in depth, and has a fairly uniform 

lateral velocity distribution. As such, it is an excellent 

measurement and sampling section. To develop a 

mathematical model of discharge at Beaver, two sites 

had to be selected for the collection of water-stage data. 

The upstream gage was installed on the main wharf at 

Beaver and the downstream gage was installed at Cape 

Horn, Wash., CRM 47.6 (fig. 1). Both stations were 

equipped with digital and graphic water-stage recorders. 

A-C synchronous motor-driven timers actuated the 

digital recorders essentially simultaneously (within 30 

seconds of each other) every 15 minutes. A float and 

stilling-well system was used to operate the gage at

Beaver and a "bubble gage" (Barren, 1963) was used at 

Cape Horn. Because at times the water level within the 

reach is lower than mean sea level and the digital 

recorders used require the stage data to be always 

positive, "MSL 1929" -10 feet (3.048 m) was selected 

as the gage datum for the reach. The reach between the 

two gages (fig. 6) includes Wallace Island and surficial 

areas that are regularly inundated by the tide. However, 

the reach is relatively straight, and most of the flow is 

conveyed in the navigational channel.

The geometry of the reach was defined from lateral 

profile data obtained at 11 cross sections (fig. 6) that 

were spaced roughly equidistant apart. The configuration of the wetted perimeter of each cross section was 

determined by sounding depths with a recording 

fathometer and measuring horizontal distances with the 

navigation unit in the deep parts of the channel and with 

a Tellurometer in the shallow parts where only a small 

skiff could go. The profile above the waterline was 

defined by using a hand level, level rod, and survey 

chain. Data from all defined cross sections were combined according to the procedures outlined by Davidian 

(1964) to provide relations between the water-surface 

elevation, cross-sectional area, and width for average 

cross sections representative of the entire reach or of 

several subreaches. Lengths between sections of the 

reach were measured along the centerline of the main 

channel on a U.S. Coast and Geodetic Survey navigation 

chart of the Columbia River.

Discharge hydrographs were defined just downstream 

from the upper gage at Beaver by applying the MOVD 

method and the modified Smoot-Novak method to 

measure the flow. In order to obtain optimum information, measurements were scheduled so that both a complete ebb and a complete flood period would be defined 

during the measurement sequence.

Discharge measurements at Beaver began on March 

24, 1966, and were concluded on April 1, 1970. During 

this period, 17 hydrographs of discharge, or a total of 119 

individual lateral traverses, were made during varying 

flow conditions. A summary is given in table 1. Examination of table 1 shows that most of the hydrographs 

were well defined over a time span of about 10 hours. 

Those of less than about 8 hours resulted because of 

equipment malfunction or bad weather conditions. Both 

occurrences could cause a late start or an early termination of the measurement sequence.

CALIBRATION PROCEDURE

The term "calibration" in this report refers to the 

process of determining values for the parameters in a 

mathematical model that provide the best possible fit 

between measured discharge hydrographs (hydrographs 

defined from discharge measurements) and hydrographs 

computed with the model. To calibrate the
DISCHARGE AND FLOW DISTRIBUTION, COLUMBIA RIVER ESTUARY P9

123° 15'

46° 10' 

FIGURE 6. Beaver Army Terminal reach, showing cross sections where channel geometry wa« defined. Numbers within river boundaries indicate depths, in feet, at Columbia River datura.

Base from U.S. Coast and Goedetic Survey; Columbia River Chart 6152, Oregon-Washington, 1968.

TABLE 1. Summary of discharge measurements at Beaver Army 

Terminal near Quincy, Oreg.

Number of lateral traverses

Date

1966 Mar. 24

May 19

Oct 21

1967

July 25 ----- -

Aug. 20 -- 

1968 Feb. 15

Mav 7

Oct 22

1969

1970

iwnvn Modified 

^l?hS Smoot-Novak method method

 7

 0

 5

 2

 5

 5

 5

0 

0 

0 

0 

0

4 

1 

1 

5

5 

3

7 

7

7 

0 

3

0

Period of 

defined 

hydrograph 

(hours)

9.8 

10 

10 

10 

5.3

2 

11 

9.5 

8.3

4.5 

10.5 

11.5 

7.3

5 

10.3 

10.5

8

Daily mean 

discharge 

at Beaver

(ftVs)

210,000 

304,000 

306,000 

198,000 

106,000

637,000 

250,000 

130,000 

81,000

163,000 

148,000 

297,000 

152,000

404,000 

450,000 

181,000

172,000

mathematical models adopted for the Beaver reach 

values for two parameters had to be determined. The 

first was the datum correction, which is the factor that is

applied to the readings from the two water-stage 

recorders to adjust the slope to achieve the best fit 

between measured and computed discharges. The 

datum correction is the algebraic sum of the absolute 

difference in elevation between the gage datums (elevation at which the gage reading is zero) and whatever 

other correction is necessary to obtain accurate discharges with the model. Ordinarily, levels are run 

between the two gages to evaluate or eliminate any difference between the gage datums; however, because the 

gages for the Beaver reach were located on opposite 

banks of the estuary, this was not practicable. Instead, 

each gage datum was referenced only to the U.S. Coast 

and Geodetic Survey bench mark located nearest to the 

gage site, and the required datum correction was determined by trial and error from the model without 

reference to the magnitudes of the separate elements in 

the correction. The other parameter that was evaluated 

is the flow-resistance coefficient, T;. Although the value 

of i) is dependent mainly on the flow resistance, it is affected like the datum correction by discrepancies 

between the mathmatical formulations and actual
P10 TRANSPORT OF RADIONUCLIDES BY STREAMS

physical conditions. Manning's n can be used as an initial estimate of rj for a given hydrograph; however, a 

trial-and-error method of calibration must be used to obtain the final value of 17.

The first attempt at developing a suitable model was 

made by using the power-series method. However, 

because the magnitude of the flood and ebb discharges 

and the general shape of the computed hydrographs did 

not correspond sufficiently closely to the measured 

hydrographs, it was concluded that the complex channel 

geometry precluded treating the Beaver reach as a single 

entity.

The second method tried was the characteristic 

method. The advantage of the characteristic method is 

that discrepancies caused by complex channel geometry 

can be minimized by dividing the total reach into several 

subreaches (Lai, 1967). For the Beaver model, the reach 

was divided into two subreaches. The upper subreach extends from the gage at Beaver downstream to the upstream tip of Wallace Island (fig. 6). This reach is virtually a single channel, although a midchannel bar is 

periodically exposed at the downstream end. The lower 

subreach extends from the upstream tip of Wallace 

Island downstream to the Cape Horn gage. This reach 

consists of the main river channel and a smaller channel 

formed by Wallace Island (fig. 6). Extensive flats on the 

north side of Wallace Island are exposed during periods 

of low tide. Table 2 lists for various elevations the areas 

and top widths of cross sections that are representative 

of the two subreaches and which were computed from 

data obtained at the 11 cross sections in figure 6.

Calibration of the model was done by the trial-anderror method. First, by repeated trials with various 17 

values, it was found that measured and computed 

hydrographs compared most consistently when the 

datum correction was taken as 0.10 foot (3.0 cm). To correct this condition, 0.10 foot (3.0 cm) was added to Cape 

Horn stages. Then, values of 17 that provided the closest 

comparisons between measured and computed 

hydrographs were extablished. Figure 7 shows the 

relationship for the upper subreach between final 17 

values and daily mean discharge. For daily mean discharges less than 225,000 ft3/s (6,370 m3/s), 77 equals 

0.0297. As the daily mean discharge increases from 

225,000 to 430,000 ft3/s (6,370 to 12,200 m3/s), 17 decreases 

from 0.0297 to 0.0248. When the daily mean discharge is 

greater than 430,000 ft3/s (12,200 m3/s), 17 equals 0.0248. 

For lower subreach, 17 is equal to the 77 value of the upper 

subreach plus 0.0004. The causes for the variation in 17 

with daily mean discharge were not investigated; 

however, it seems reasonable that important contributing factors were the variation in the Reynolds 

number of the flow and the apparent change in the 

heights and lengths of the bed forms in the channel with 

discharge.

TABLE 2. Areas and top widths of average cross sections representative of the two subreaches in the Beaver reach

Upper Subreach Lower Subreach

Depth1 

(ft)

0

2.0

4.0

6.0

8.0

10.0

13.0

14.0

16.8

22.0

24.0

Top 

width

(ft)

2,560

2,650

2,730

2,780

2,860

2,950

3,090

3,110

3,200

3,320

3,360

Crosssectional 

area

(sqft)

78,200

83,600

88,800

94,300

99,800

105,700

114,800

117,800

126,800

143,600

150,300

Depth1

(ft)

0

2.0

4.0

6.0

10.0

12.0

13.0

14.0

16.0

24.0

---

Top 

width

(ft)

2,350

2,400

2,470

2,600

2,760

2,840

2,890

2,920

2,980

3,180

 

Crosssectional 

area

(sqft)

73,800

78,500

83,400

88,500

99,200

104,800

107,700

110,600

116,500

141,100

 

 Depth of water above gage datum ("MSL 1929" 10.0 ft).

MATHEMATICAL-MODEL DISCHARGE RESULTS

Values of the parameters defined by the calibration, 

together with appropriate stage data, were used in the 

model to generate a record of instantaneous discharge 

every 15 minutes for the period May 1, 1968, through 

June 30, 1970. Daily mean discharges obtained from this 

record by algebraically averaging the 96 instantaneous 

values for each 24-hour day have been published by the 

U.S. Geological Survey (1971, p. 275-277). To check the 

accuracy of the computed discharges, monthly mean discharges determined from the model discharges were 

compared with monthly mean discharges computed by 

the Northwest Water Resources Data Center. The 

method used by the Data Center to obtain a monthly 

mean discharge was developed by Orem (1968) and is 

believed to be accurate within 2 or 3 percent. 

Throughout the period of record, monthly mean discharges from the model consistently compared within 

±5 percent of the Data Center values. Presumably, the 

computed daily mean discharges are in this same general 

accuracy range. Inasmuch as the agreement between 

measured and computed hydrographs obtained during 

the calibration is good and the average relation for the 

variation of 17 (fig. 7) is moderately well defined, it also 

seems reasonable to conclude that the computed instantaneous discharges are fairly accurate.

Continuous-flow hydrographs for the period of discharge record at Beaver exhibit the following 

characteristics:

For daily mean discharges of less than about 125,000 

ft3/s (3,540 m3/s), flow reverses (flow upstream) during at least one of the flood periods during the day, 

and reversals are very common for both flood 

periods.

For daily mean discharges of about 125,000 to 185,000 

ft3/s (3,540 to 5,240 m3/s), flow almost always 

reverses during the strongest flood. 

For daily mean discharges of about 185,000 to 225,000 

ftVs (5,240 to 6,370 m3/s), the flow reverses occasionally during the strongest flood.
DISCHARGE AND FLOW DISTRIBUTION, COLUMBIA RIVER ESTUARY Pll

700

600

500

400

300

200

100

- 18

-16

DC

1-12 ui

- 10

14

- 6 -J

- 2

0.024 0.026 0.028 0.030

FIGURE 7. Variation of the flow-resistance, coefficient, ij, with daily mean discharge, 

Columbia Kiver at Beaver Army Terminal near Qumcy, Oreg.

4. For daily mean discharges greater than about 225,000 

ft3/s (6,370 mVs), the flow rarely reverses.

A review of the daily mean discharges shows that, on the 

average, flow reversals occur during about 10 months of 

the year and at all times except during the period of high 

upland flow during the spring.

During the calibration, it was found that a gage correction of 0.01 foot (0.3 cm) would change the daily mean 

discharge by about 5,000 ft3/s (142 m3/s) or about 2 percent of the average daily mean discharge of 225,000 ft3/s 

(6,370 m3/s) for the period of discharge record at Beaver. 

Although the stage records from both the float-stilling 

well system and the bubble gage are accurate at best to 

only ±0.01 foot, (0.3 cm), it is reasonable to assume that, 

on the average throughout a tidal cycle, the difference 

between gage readings would be in error no more than 

about 0.01 foot (0.3 cm).

ASTORIA REACH

The reach at Astoria extends from Tongue Point 

(CRM 17.5) downstream to the west side of Astoria 

(CRM 13.1), as shown in figure 1. The flow characteristics of the Astoria reach are complicated by both longitudinal and lateral salinity gradients that result from 

saltwater intrusion into the reach. Generally, salinity is 

present during at least one of the floodtides each day, 

and usually during both, except when runoff is high in 

the late spring. Also the geometry of the reach is complex 

(fig. 8); the navigational channel is along the south 

shore, the center part is laced diagonally with shallow 

minor channels and with flats that are exposed at low 

tide, and a deep channel along the north shore at the 

lower end of the reach divides and becomes shallow 

toward the upstream end of the reach. The width-tolength ratio of the reach is close to 1.0.

The upstream gage was located at the U.S. Coast 

Guard Tongue Point Station and consisted of a digital 

recorder coupled to a permanent tide-station gage 

recorder that is operated by the U.S. Coast and Geodetic 

Survey. The gage is float-stilling-well type and is maintained daily by Coast Guard personnel. The lower gage 

was installed on Pier 1 of the Port of Astoria Docks. This 

gage also was a float-stilling-well type with digital and 

graphic water-stage recorders. A-C synchronous motordriven timers were used to obtain virtually simultaneous 

stage readings at both gages every 15 minutes. The 

datum to which the gages were set was the same as for 

the Beaver reach; that is, "MSL 1929" -10 feet (3.048 

m). Channel cross-sectional geometry was defined from 

eight cross sections (fig. 8) equally spaced along the 

reach. The shape and dimensions of each section were 

determined from Coast and Geodetic Survey 

navigational charts rather than from field-survey 

measurements. Data from the eight sections were com-
P12 TRANSPORT OF RADIONUCLIDES BY STREAMS

123° 50'

46 15'

46° 10'

FIGURE 8 Astoria reach showing cross sections where channel geometry was defined. Numbers within estuary boundaries indicate depths, in feet, at mean lower low 

water. Base from U.S. Coast and Goedetic Survey; Columbia River Chart 6151, Oregon-Washington, 1968.
DISCHARGE AND FLOW DISTRIBUTION, COLUMBIA RIVER ESTUARY P13

bined to define the geometric characteristics of cross sections that are representative of upper and lower subreaches which roughly divide the total reach in half 

(table 3). The length of the reach was obtained by 

measuring the distance between the two gages along the 

centerline of the navigational channel.

Discharge hydrographs for calibration of the 

mathematical model were obtained at the measuring 

cross section at Astoria (fig. 8) exclusively by the MOVD 

method. With this method, the complex velocity profiles 

produced by salinity gradients are defined throughout 

the entire depth; thus, regardless of conditions, accurate 

section discharges can be measured. As at Beaver, 

measurements were scheduled so that both ebb and 

flood discharges would be defined during the measurement sequence.

Discharge measurements at Astoria began on April 13, 

1966, and were concluded on May 19, 1970. During this 

period, 20 discharge hydrographs, or a total of 82 lateral 

traverses, were made during varying flow conditions. A 

summary of the discharge measurements is given in 

table 4. At Astoria, it was considered a successful day if 

four or more complete traverses could be made. On the 

average, it would take about 2 hours for a complete 

round-trip lateral traverse. Of the 2 hours, about IVz

TABLE 3. Areas and top widths of average cross sections representative of the two subreaches in the Astoria reach

Upper Subreach

Depth'

(ft)

0 

2.0 

4.0 

6.0 

7.0 

8.0 

24.0

Top 

width 

(ft)

17,500 

19,700 

23,100 

24,300 

25,600 

27,200 

27,200

Crosssectional 

area 

(sqft)

226,000 

263,000 

306,000 

353,000 

378,000 

405,000 

840,000

Lower Subreach

Depth' 

(ft)

0 

2.0 

4.0 

6.0 

8.0 

24.0

Top 

width 

(ft)

16,900 

17,500 

19,700 

20,400 

21,900 

21,900

Crosssectional 

area

(sqft)

286,000 

321,000 

358,000 

398,000 

440,000 

791,000

'Depth of water above gage datum ("MSL 1929" -10 ft).

TABLE 4. Summary of discharge measurements at Astoria, Oreg.

D

1966

July

July

1967 

July

1968 

Feb

M«v

July

Oft

T)o/,

1969

1970

ate

10

25

no

OQ

17 

15 

11

10

16

26

12

17

1ft

24 

Number of 

lateral 

traverses

 4

- 5

 5

Period of 

defined 

hydrograph 

(hours)

5

9.5

ft Q

7

ft ft

9.5

£ c

11.5

11 ft

7 1

ft Q

in Q

9.5

Daily mean 

discharge 

at Astoria 

(ftVs)

qqi nnn

OQ7 AAfi

nrjc nnn

1 19 nnn

462 000

97ft nnn

OCA AAA

ifla nnn

hours were spent obtaining the measurements, and the 

other one-half hour was spent traveling back across the 

4-mile-wide (6-km-wide) cross section. Lateral traverses 

were always started on the north side of the estuary so 

time differences between corresponding verticals of the 

different traverses would be the same.

Weather conditions played a very important part in 

how successful a measurement sequence would be. 

Heavy morning fogs caused delays in starting, and winds 

caused cancellations or delays in the starting time and 

early terminations of the measurements. Equipment 

malfunctions also caused late starts, early terminations, 

or delays during the discharge measurements.

CALIBRATION PROCEDURE

From the beginning of the study, there was speculation as to whether any one-demensional model could be 

adapted to provide reasonably accurate discharges 

throughout the tidal cycle at Astoria because of the complex geometry of the reach, the existence of salinity 

gradients, and the presence at particular times in 

each tidal cycle of diametrically opposed flow both 

within a vertical and laterally within the cross section. In 

order to provide the most flexibility, the model based on 

solution of the flow equations by the method of 

characteristics (Lai, 1967) was adopted and computations were made within the framework of a two-subreach 

system. The initial calibration procedure for the Astoria 

reach was the same as that used for the Beaver reach; 

that is, an attempt was made to determine values for the 

datum correction and flow resistance, TJ, by the trialand-error procedure. After evaluating many trials with 

various combinations of datum corrections and T/'S, it 

was concluded that (1) a constant datum correction 

could not be determined, and (2) use of rj=0.025 

produced flow hydrographs with maximum and 

minimum discharges that corresponded closely to comparable measured data, but whose shapes deviated consistently from measured hydrographs.

Baltzer and Lai (1968) showed, with data from 

Threemile Slough near Rio Vista, Calif., that rj increases 

sharply as the Reynolds number approaches zero. 

Because the Reynolds number is a function of and varies 

mainly with water velocity, in effect, they showed that as 

the velocity (or discharge) approaches zero, q increases. 

Because the tidal pattern at Astoria is such that the discharge passes through zero four times each day, the 

variation of T; is a significant consideration. To ascertain 

how T; varies with time at Astoria, the model was solved 

for ri by using measured discharges and an arbitrary 

datum correction. Computed rj values, normalized by 

dividing by the minimum computed T; value, are plotted 

in figure 9 against the corresponding relative times, 

T/AT, in the period between zero discharges. The graph 

in figure 9 for ebb periods shows that for relative times
P14 TRANSPORT OF RADIONUCLIDES BY STREAMS

16

14

12

10

Ebb

*>s 

EXPLANATION

« May 10,1968

A June 16,1968

a July 26,1968

o Sept. 12,1968

A Oct. 17, 1968

« Dec. 18,1968

A Apr. 24,1969

 Mar. 7,1970

A Apr. 13,1970

 May 19,1970

-] i i r

Flood

0.5 1.0 0

RELATIVE TIME BETWEEN ZERO DISCHARGES

FIGURE 9. Change in 1/1minwith relative time at Astoria, Oreg.

0.5 1.0

between 0.35 and 0.70, the value of rj is constant. For the 

remaining 65 percent of the time, 77 increases as T/AT approaches zero and 1.0. For flood periods (fig. 9), 77 is constant and a minimum for T/AT values between 0.23 and 

0.47 or for 24 percent of the time. As with ebb periods, 

whenever rj is not at the minimum value it progressively 

increases as T/AT approaches zero and 1.0. It should be 

noted that the data in figure 9 cannot be taken as absolute because (1) the best datum correction for each 

hydrograph may not have been used in the computations; and (2) no deliberate provisions were taken to correct the model for salinity gradients, Coriolis effects, 

wind, and other factors that could have an effect on the 

discharge. However, the general trends in the figure can 

be assumed to be representative.

In order to incorporate the variation of 77, the 

mathematical model was programmed to accept the 

curves shown in figure 9. With this change, close comparisons between measured and computed hydrographs

(figs. 10, 11) could be obtained by using 77 mjn=0.025, 

provided a different datum correction was used for each 

hydrograph. Apparently, the necessity for a variable 

datum correction is mainly a reflection of the fact that a 

variable water-surface slope (fall) correction is necessary 

to adjust the model to compensate for the influence of 

salinity gradients, complex geometry, and other factors 

not expressed in the fundamental equations.

Because a direct independent means for determining 

the required datum correction was not available, an indirect approach was adopted. First, a simple volumetric 

model to compute daily mean discharges at Astoria was 

developed. This model equated the daily mean discharge 

at Astoria with the algebraic sum of the daily mean discharge at Beaver and the daily rate of net change in the 

volume of storage within the reach between Beaver and 

Astoria. That is,

e.-e.-H^. a)
DISCHARGE AND FLOW DISTRIBUTION, COLUMBIA RIVER ESTUARY P15

DAY HR

13 21*

2

<t

6

e

IE

12

|i»

16

18

20

22

MEAS. DIS.

*

179000

3050EO

i«i»80QO

570000

690000

A05000

913000

IGIOOOO

1103000

1 130000

121*9000

1295000

1337000

I35500&

1365000

1360000

131*7000

1315000

I2750CO

1230000

1 173000

1 1 10008

101*3000

955000

670000

775000

680000

570000

1*60000

31*5000

233000

90000

-1*5000

-175000

-301000

-i*<*OOGO

-555000

-650000

-737000

-79500C

-839000

-860000

-868000

-860000

-81*0000

-805000

-763000

-715000

-655000

-58COOO

-500008

-1*10300

-315000

-210003

-90000

1*0800

18200.3

CDMP. DIS.

0 -1500 

I I 

I I

556829 t

l*38<*62

2781*71

182861

99295

2299

-I3«*1*I2

-?I8I75

-335387 *

-1*57130

-506153

-577286

-663828

-703338

-733659

-7501*09

-735952 t

-739267

-621799

-523383

-1*17185

-261650

-t33i*2P

7509

151*200 

301355

<*Qi*663

1*88902

616866

720657

806696

90991*5

989180 t

101*1*859

1099886

1 162210

12021*35

123 1921

126991*3

1271*068

128711*1 *

1302615

1260580

1255371

1220291

1 153285

105221*6

998539

901165 +

8I3«*20

761613

637859

500071

1*29658

27587«»

1 8661*2

|i*66i*9 +

10031*

-681*16

-l«*«*307

-2«»52i*9

-3191*58

-I*! 21*35

-506770

-573938 »

-65561*1

-721867

-75171*2

-82161*5

-839523

-838890

-83 1 189

-816991 *

-770267

-701*374

-610851

-501836

-1*22176

-31*1*327

-19901*1*

-91182 « 

32325

20 H*3i*

311*332

381*961

1*5631 1

525820

61 0517

681*361 t

753165

821*01*4

886637

921284

91*961*6

DISCHARGE, IN THOUSANDS OF CUBIC FEET PER SECOND

-5BQ 500 1500 2500 

Till 

I I I I I I I I 

t »fl 

I 

* 

III-

*

0

0

0

0

0

0

to * * *

0

0

0

0

0

0

0

0 t * * +

0

0

0

0

0

0

« 8 + * *

0

* 0

* 0

» 0

* 0

* 0

» 0

* + * 0 « »

»0

*

0»

0»

0*

0 »

0 »

« + o» * »

0*

0»

0»

0 »

0 

0 *

0 »

« * 0 * * *

0 »

0 »

0 

0 »

0 *

0 »

0 »

* o » * * *

0 »

*

*

»0

* 0

* 0

* 0

» 0* t t *

* 0

* 0

» 0

*0

*

*

0»

o» * * » »

0*

0*

*

*

»

0*

»

t » t « *

*

*

0

0

0

0

0

t tfl » »

0

0

0

0

0

FIGURE 10. Comparison between measured (*) and computed (0) discharges for low instantaneous discharges at Astoria, Oreg., on April 13, 1970.
P16 TRANSPORT OF RADIONUCLIDES BY STREAMS

DAY HR MEAS. DIS.

*

7 24

2

it

6

8 40000

-210003

-1*70000

-730000

-993000

 1 I8COOO

-131*9000

-l<*90000

10 -1611000

- I70COOO'

-1779000

-1820000

-1795000

-I7<*0000

-I6<*3000

-1500000

12 -1315000

-1 1 IOOOC

-875000

-620000

-328000

-15000

325000

660000

14 982000

1270000

151.1000

1775000

1965000

2075000

2171000

2230000

16 221*0000

2236000

2191*000

2160000

2093000

2025000

|q<*0000

1875000

18 171*6000

1625000

I<*95GOO

1355000

1201*000

I0300CO

86CCOO

660300

20 5I30CO

320000

124000

-50COO

22

COMP. DIS.

0

-1 185568

-953949

-768868

-464894

-176359

180063

500169

738587

891382

1120879

1249572

149265*

1586183

1868247

1964267

2168716

2205072

23227 1*0

2288833

2299599

2233739

21581*75

2052043

1974^91

1801991

1651460

11*89169

1270401

1048300

791882

505053

269412

70310

-178766

-4621 1*

-7(7537

-1007673

-1224767

-1439672

-1651563

-1755139

-1815276

-1818778

-1723970

-1646916

-15531 12

-1368756

-1 1R0525

-909I3C

-713555

-45651 1

-223061

83779

477243

6871 12

828326

1002154

1268686

1326055

1600907

1582740

1825455

1839822

2044441

1996823

2142791

2102134

2105568

2075647

2090655

2057344

2003301

1910726

1814926

1691705

1549529

1377837

1 163504

963597

731929

491 153

286546

102735

-178015

-405121

-702258

-945981

-1 1 22605

-1367663

-1577942

-1732265

-1881617

DISCHARGE, IN THOUSANDS OF CUBIC FEET PER SECOND

-1500 -500 500 1506 2500 

I I I I I I I I I I 

I I I I I I I I .... I I

+ 0 * * * *

0

0

0

0

0

0

0

« « + 0 * *

0

0

0

0

0

0

0

«. t * « 0 « 

0

0

e

0

0

0

0

» * « * 0 *

0

0

0

0

0

0

0

* * » « * *

»

*

*0

*

*

0 *

0 *

0 * « « * * *

0 *

»

* 0

» 0

* 0

* 0

* 0

« » a + * * *

* 0

* 0

* 0

* 0

» 0

* 0

* 0

» + + » * * *

0 *

0 »

0 »

0 »

0 *

0 *

«. * + « 0 » *

0 *

0 »

0*

*

»0

* 0

* 0

+ «. + + * 0 *

» 0

» 0

» 0

» 0

* 0

» 0

*0

+ * » + *

»

V

0 »

0

0

0

0

tO -f + * *

0

0

0

FIGURE 11. Comparison between measured (*) and computed (0) discharges for high instantaneous discharges at Astoria, Oreg., on March 7, 1970.
DISCHARGE AND FLOW DISTRIBUTION, COLUMBIA RIVER ESTUARY P17

where _ _

QA and QB are the daily mean discharges, in cubic feet per second, at Astoria and 

Beaver, respectively;

AV is the difference in the volume of

water within the reach, in cubic

feet, at the beginning and at the

end of each day; and

T is 86,400 seconds (equivalent to 24

hours).

AV is the product of the surface area of the reach, K', 

and the weighted average depth of the net change in 

storage, AS', which may be either positive or negative. 

K' was determined from a correlation with the stage at 

Tongue Point, STP , at the beginning of the day; the correlation was established with K' values obtained by solving equation 1 with values of QA for days when discharge 

hydrographs were defined at Astoria, comparable values 

of QB computed from the Beaver model, and values of 

AS' determined from

' = (2/3)ASrP+(l/2)ASB , (2)

where

ASrp and AS B are the differences in stage between the beginning and end of 

each day at Tongue Point and at 

Beaver, respectively.

Equation 2 was determined to give AS' values that, in 

turn, yielded K' values which gave the best correlation 

with the Tongue Point stages. The equation also roughly 

reflects the fact that approximately two-thirds of the 

surface area is located in the lower one-third of the 

reach.

Daily mean discharges at Astoria computed from 

equation 1 were used to calculate monthly mean discharges. These values were compared with monthly 

mean values computed by the Northwest Water 

Resources Data Center by a routing technique (Orem, 

1968). Because the agreement between the monthly 

mean discharges was consistently within ±5 percent, the 

individual daily mean discharges were assumed to be 

reasonably accurate. Then, for each day, trial discharges 

were computed every 15 minutes with the mathematical 

model, using different daily datum corrections, until the 

resultant daily mean discharge corresponded within ±5 

percent of the "known" daily mean discharge for each 

day determined from the volumetric model (eq 1).

MATHEMATICAL-MODEL DISCHARGE RESULTS

A continuous record of the discharge at Astoria every 

15 minutes was computed by the trial procedure 

throughout the period from March 1, 1968, to June 30, 

1970, except during periods of missing gage records. A 

summary of this record (app. 1) shows that daily mean 

discharges ranged from 630,000 to -7,400 ft3/s (17,800 to

-210 m3/s; minus sign indicates upstream flow). Instantaneous discharges frequently exceeded 2 million ft3/s 

(0.06 million m3/s) downstream and 1.5 million ft3/s (0.04 

million m3/s) upstream (fig. 11). Near-surface velocities 

observed during the discharge measurements ranged up 

to 8 ft/s (2.4 m/s) during the ebb and were as high as 6 

ft/s (1.8 m/s) during the flood. However, the timeaverage velocity in the cross section is about 3 ft/s (0.9 

m/s) during an ebb and about 2 ft/s (0.6 m/s) during a

flood.

Daily mean discharges determined from instantaneous discharges computed with the mathematical 

model (solution by the method of characteristics) are 

considered to be reasonably accurate because they were 

adjusted by trial to be within ±5 percent of the results 

obtained with the volumetric model. However, because 

the datum (slope) correction had to be applied as a daily 

factor and could not be varied throughout the tidal cycle, 

the computed instantaneous discharges are generally 

less accurate than the daily mean discharges. A 

statistical analysis of the percentage differences between 

discharges computed with the model and comparable 

discharges from the measured hydrographs is presented 

in appendix 2. The analysis indicates that the 

probability is 0.90 that 90 percent of all computed instantaneous discharges greater than 1,250,000 ft3/s 

(35,400 m3/s; both ebb or flood discharges) will be within 

about ±25 percent of the actual discharge; the error 

decreases as the discharge increases. (See fig. 18 and 

table 7.) For discharges less than 1,250,000 ft3/s (35,400 

m3/s), the percentage error increases rapidly as discharge 

decreases. Flows less than 1,250,000 ft3/s (35,400 m3/s) 

generally occur only during those periods in the tidal cycle when discharge either is increasing or decreasing 

rapidly after having been zero. Hence, even though the 

proportion of time such flows occur may be fairly large, 

their contribution to the total volume of flow passing the 

cross section during an entire ebb or flood is generally 

small. For this reason, the large errors, as such, are not 

significant when the volume of flow during a partial tidal 

cycle is a prime concern.

During calibration of the Astoria reach, it was determined that the model was very sensitive to datum corrections. On the average, a correction of 0.01 foot (0.3 

cm) would change the daily mean discharge by about 

25,000 ft3/s (710 m3/s). This extreme sensitivity is 

believed to have resulted primarily because the reach 

was not long enough and the cross-sectional area was 

very large. Although the reach length was 4.4 miles (7.1 

km), the maximum water-surface fall was only slightly 

over 1.0 foot (30 cm), and the average fall was only about 

0.65 foot (20 cm). Hence, a datum correction of 0.01 foot 

(0.3 cm) caused an appreciable change in the fall, and 

this caused a significant change in computed velocity 

that, in turn, was magnified into a large change in discharge because of the extensive size of the cross section.
P18 TRANSPORT OF RADIONUCLIDES BY STREAMS

600

Q

Z

o 

oLU 

W

DC 500

LU

400

Z

w 300

O

I 1-

g 200

< 

I 

o

LU 100

i i i i n i i i i r i i i i i n i i i i r

 i i i i r

EXPLANATION

O Full Moon

 New Moon

 Astoria

 Beaver Army 

Terminal

May June July August 

FIGURE 12. Daily mean discharges at Beaver Army Terminal and Astoria, Oreg., May-September 1968.

September

In tidal flow, the daily mean discharge is not necessarily indicative of the magnitude of upland flow as it is 

in nontidal flow. According to the volumetric model, and 

daily mean discharge at the station may be either less or 

greater than the upland discharge; the difference is 

caused by the net change in storage within the reach upstream from the station. The effect of changes in storage 

is shown in figure 12 by the hydrographs of the daily 

mean discharges at Beaver and Astoria. Although there 

is little inflow into the Columbia River between Beaver 

and Astoria, daily mean discharges fluctuate cyclically 

through a much greater range at Astoria than they do at 

Beaver because of the greater change in the amount of 

storage per unit change in stage at Astoria. The 

character of the cyclic pattern of the hydrograph for 

Astoria is affected significantly by the combined effects 

of (1) the daily 50-minute shift in the position of the tide 

curve relative to absolute clock time due to the difference in the length of lunar (24.8 hours) and solar (24 

hours) days, and (2) the cyclic variation in maximum 

and minimum tide heights that occur with a frequency of 

about 14.3 days (spring tides occur a day or two after a 

full moon or a new moon, and neap tides occur about 7 

days later). Essentially, the daily mean discharge increases from one day to the next, so long as AV (eq 1) increases or becomes less negative that is, it increases so 

long as AS' continues to increase algebraically. Daily

mean discharges progressively decrease when the opposite trend in AS' occurs. Because the moon is the major force-producing body that influences the timing and 

magnitude of the tides, moon phases correlate well with 

the cyclic variations in daily mean discharges. (See fig. 

12.)

As stated earlier, the net change in storage during the 

day due to tide effects can be severe enough to produce 

daily mean discharges that are in the upstream direction 

at Astoria. For instance, on September 21-22, 1969, 

when the daily mean discharge at Beaver was about 

100,000 ftVs (2,830 m3/s), the tides were such that the 

daily mean discharges at Astoria for these 2 days 

were 3,200 and 7,400 ft3/s (91 and 210 m3/s) upstream, 

respectively. At Beaver, similar cycling (but not net upstream daily flow) occurs; however, it appears that for 

daily mean discharges greater than about 450,000 ft3/s 

(12,700 m3/s), the cycling effect is essentially eliminated.

The cyclic variations in daily mean discharges 

significantly affect the degree to which the discharge 

during any short period of time (days) can be used to 

represent a longer term discharge. However, because the 

average period for the cyclic pattern is about 14 days, a 

monthly mean discharge will cover approximately 2 cycles and, therefore, should largely eliminate the effects of 

the ocean tides.
DISCHARGE AND FLOW DISTRIBUTION, COLUMBIA RIVER ESTUARY P19

FLOW CHARACTERISTICS 

IN THE LOWER ESTUARY

The adaptation and application of the onedimensional mathematical model for determining instantaneous discharges at Astoria was made possible by 

using, both implicitly and explicitly, information on flow 

and salinity distributions at the measuring cross section 

and elsewhere in the lower estuary. This same information also has been utilized to examine gross circulation 

and mixing patterns in the estuary.

LATERAL CIRCULATION AT THE

DISCHARGE-MEASURING CROSS SECTION,

ASTORIA REACH

To better understand the general pattern of flow in the 

estuary, the net circulation that results from differences 

in the lateral flow distribution of flood and ebb discharges was investigated at the measuring cross section 

at Astoria. Flow volumes from the dischargemeasurement sequences were used for this purpose.

For measured hydrographs in which both an ebb and 

flood period are defined, the quantities of water flowing 

upstream and downstream in the north and south channels1 were determined from the individual hydrographs 

for each part of the cross section (p. P5) that had been 

used to obtain the total flow hydrograph. The percentage 

of the total ebb volume that passed through each channel was then determined; a similar computation was 

made for the flood volumes. The various percentages are 

presented in table 5.

The results in table 5 reveal some interesting facts 

about the flow characteristics and lateral circulation 

patterns at the measuring cross section. On the average, 

the north channel conveys 76 percent of the floodflow 

and 69 percent of the ebbflow, or 73 percent of the total 

flow. More importantly, however, comparison of the differences in the percentages of flow through each channel 

during the ebbs and floods indicates the net lateral circulation during the half tidal cycle defined by the 

hydrograph. For instance, on September 15, 1967 (table 

5), the proportion of the ebbflow which passed through 

the north channel was 9 percent less than the proportion 

which passed through during the floodflow (9 percent 

loss). Conversely, the proportion in the south chanel was 

9 percent greater. This indicates a net clockwise circulation wherein some of the floodwater that flowed into the 

estuary via the north channel returned to the ocean via 

the south channel. On October 17, 1968, the data indicated a net counterclockwise circulation. Examination 

of table 5 shows that the lateral circulation of water cor-

'In this analysis, the part of the estuary that includes the shallow midestuary area and the 

northernmost channel (fig. 8) is referred to as the north channel, and the navigational channel 

is called the south channel.

TABLE 5. Lateral distribution of ebb and flood discharge in the 

measuring cross section at Astoria, CRM 14

[Flow period: E, ebb; F, flood]

Difference in percentage of flow

Date

9-15-67

5-10-68

9- -592

10-17-68

9-12-68

8-17-67

3- 7-70

4-13-70

12-18-68

5-19-70

4-24-69

6-16-68

6- -592

Daily 

mean

discharge 

at Beaver

(ftVs)

'129,000

157,000

'162,000

166,000

171,000

1 175,000

186,000

197,000

282,000

325,000

415,000

462,000

'551,000

Flow

period

[FIE

rFIE

rF IE

I-F 

LE

rEIFI-F 

LE

rF IE

rE IF

rF IE

rF IE

rE LF

[EL F

rF LE

Percent of flow

North

channel

79 

70

79 

72 273 

268

68 

80

79 

70

71 

71

73 

75

58 

73

80

74

79 

66

58 

88

65 

81 277 

261

South

channel

211 

30 J

211 

28-1

227l 

232-l

32i 

20-1

21i 

30J

29i 

29 J

271 

25\

421 

2?J

20l 

26J

21i 

34 J

42i

12.1

35i 

19.1

2 23i 2 39J

conveyed during 

ebb and flood

North

channel

-9

-7

-5

+ 12

+ 9

0

+2

-15

-6

-13

-30

-16

-16

South

channel

+9

+7

+5

-12

- 9

0

-2

+15

+6

+ 13

+30

+16

+ 16

'Daily mean discharge at Prescott, Oreg. 

-From data presented by Hansen (1965).

'Daily mean discharge at the mouth as computed by the U.S. Army Corps of 

Engineers (1960).

relates .with the discharge at Beaver. For discharges at 

Beaver of less than about 165,000 ft3/s (4,670 m3/s), the 

south channel conveys a proportionately higher flow during the ebb than the flood (gains water from the north 

channel during the ebbflow). For discharges from about 

165,000 to 190,000 ft3/s (4,670 to 5,380 m3/s), the north 

channel gains water from the south channel during the 

ebbflow, and for discharges at Beaver greater than about 

190,000 ft 3/s (5,380 m3/s), the south channel gains water 

from the north channel during the ebbflow.

Also shown in table 5 are percentages computed from 

Hansen's (1965, fig. 3, p. 954) interpretation of U.S. 

Army Corps of Engineers data collected at CRM 5.5 in 

June and September of 1959. The results from Hansen's 

work consistently agree with data from the measured 

hydrographs at the Astoria measuring cross section.

FLOW-PREDOMINANCE PATTERNS AT THE

DISCHARGE-MEASURING CROSS SECTION,

ASTORIA REACH

One especially useful way to characterize in simple 

terms the distribution of tidal flow in an estuary is by the 

method of flow predominance (Simmons, 1955). Application of this method to the analysis of velocity profile 

data from the measurement cross section at Astoria has 

provided an insight into the variation of the net vertical 

distribution of flow at various lateral locations across the 

section.
P20 TRANSPORT OF RADIONUCLIDES BY STREAMS

TABLE 6. Variation of flow predominance with depth at selected locations in the measuring cross section at Astoria, CRM 14

[Fraction of tidal cycle: A, half tidal cycle consisting of one flood and one ebb; B, full tidal cycle]

Flow predominance at height above the bed equal to the indicated percentage of the total depth

Date

1966 

Apr. 13 

July 6 

July 28

1967 

June 6 

June 20-21 

July 28 

Aug. 18-19 

Sept. 15

1968 

Feb. 13 

May 10 

June 16 

July 26 

Sept. 12 

Dec. 18

1969 

Apr. 25

1970 

Mar. 7 

Apr. 13 

Apr. 14 

Apr. 15 

May 19

Fraction 

of tidal 

cycle

A 

A 

A

B 

B 

A 

B 

A

A 

A 

A 

A 

A 

A

A

A 

A 

A 

A 

A

Mean , , 

discharge 

(ft3 /s)

294,000 

220,000 

93,000

453,000 

616,000 

150,000 

194,000 

-24,000

538,000 

84,000 

425,000 

57,000 

34,000 

468,000

368,000

280,000 

264,000 

226,000 

131,000 

133,000

"North" Channel

5

36 

41 

43

81

57

81 

48 

68 

24 

49 

69

51 

46

71 

2

25

48 

51 

48

82

65

86 

55 

72 

46 

62 

66

54 

32

58 

27

50

65 

65

54

65

53

79 

55 

76 

71 

68 

57

47 

53

80 

45

75

74 

80 

82

64

43

72 

61 

78 

79 

60 

56

51 

62

64 

57

95 5

78 22 

68 34 

89 71

65 35

56 19

73 -

65 6 

81 46 

79 60 

65 55 

60 

57 

/=Q

74 

59 ....

Middle Channel

25 50 75 95 5

44 49 58 61 42 

61 59 71 70 26 

19 34 82 53 3

 77

 93

62 60 71 56 

 60

23 38 70 74 44

37 52 56 62 45 

84 99 79 72 81 

36 61 61 78 52 

46 54 63 75 39

 60

£Q

 57

 51

 69

South Channel

25

60 

80 

10

82 

. 90

53

44

48 

81 

62 

33 

47

75

45 

64 

49

51

50

78 

76 

45

85 

92

61 

55

53 

82

64 

42 

61

75

41 

73 

81

63

75

82 

78 

88

83 

93

68 

45

61 

78 

74 

57 

62

84

50 

78 

88

56

95

76 

62 

80

80 

92

68

48

58 

77 

75 

62 

68

82

60 

84 

90

65

I/ Algebraic mean during the indicated fraction of tidal cycle.

Flow predominance at a point is determined by the 

following procedure: (1) A conventional plot of the 

velocity normal to the cross section versus time is made 

over a complete tidal cycle; (2) the areas, F, subtended 

by the flood curves, and the areas, E, subtended by the 

ebb curves of the plot are planimetered; and (3) the flow 

predominance is calculated from

flow predominance= 2E+2F X100.

Thus, flow predominance is the percentage of the total 

flow at a point that is downstream or toward the ocean. 

By having velocity data for several points in a vertical, a 

plot to define the distribution of flow predominance from 

surface to bottom at the vertical can be made. It is 

emphasized that this plot does not represent the distribution of discharge from surface to bottom, but only 

defines the direction and degree of predominance of flow 

throughout the vertical. For example, although a vertical 

may have a flow predominance of 50 percent at points 

near both the top and bottom, this does not mean that 

the same volume of water flowed past both points, 

because the top would probably have higher velocities 

than the bottom.

At the measurement cross section at Astoria, flow is 

conveyed through three broad parts termed "channels."2 

Flow in the "north" and south channels parallels the 

banks, but flow in the middle channel is in a diagonal 

direction. In each of these channels, a specific vertical 

was selected to characterize the flow of the entire channel. For each discharge-measurement sequence (table 4) 

that had sufficient velocity data, flow predominance was 

calculated at heights above the bed of 5, 25, 50, 75, and 

95 percent of the total depth at the selected vertical in 

each of the three channels. These data are given in table 

6. Inasmuch as the discharge measurement sequences 

were usually about 10 hours long, most of the flowpredominance values in table 6 are based on velocity 

data from only one-half of a tidal cycle instead of a complete tidal cycle. However, unless there were extreme 

differences between the high-high and the low-high, or 

the low-low and the high-low tides, flow-predominance 

values calculated using data for only one-half of a tidal 

cycle should be reasonably representative of the flowpredominance values for the complete tidal cycle. Also, 

the "mean discharge" given for each date in table 6 is the 

algebraic mean for exactly the same part of the tidal cy-

-In this analysis, the northernmost channel, the midestuary area, and the navigational 

channel are called the "north," middle, and south channels, respectively.
DISCHARGE AND FLOW DISTRIBUTION, COLUMBIA RIVER ESTUARY P21

Upstream Downstream

100

predominance predominance

50

O

< 100

May 10, 1968

Mean discharge ; 

80,000 ft 3/s 

(2,270 m3/s)

EXPLANATION

O "North" channel 

A Middle channel 

D South channel 

Uppermost points in each graph are nearest the water surface

Upstream Downstream Upstream Downstream

predominance predominance predominance predominance

50

April 13,1966

Mean discharge : 

294,000 ft3/s 

(8,330 m 3/s)

June 16,1968 

Mean discharge 

425,000 ft 3/s 

(12,000 mVs)

50 100 0 

PERCENTAGE OF FLOW DOWNSTREAM

100

FIGURE 13. Vertical distribution of flow predominance in the "north," middle, and south channels at the measuring cross section at Astoria, CRM 14, for

various mean discharges.

cle as was used to calculate the flow predominance. 

Therefore, the correlation between the two variables 

should be roughly the same as would be obtained from 

using-data for a full tidal cycle.

Flow-predominance curves for three different times 

are plotted in figure 13 to indicate how the flow distribution varies with the mean discharge at these verticals 

and within each channel. For zero upland discharge, normally the flow predominance at all depths would be 

close to 50 percent because there would be no possibility 

of stratification. The figure demonstrates that for some 

mean discharges the flow is predominantly upstream 

near the bottom and predominantly downstream near 

the surface.

The variation of flow predominance with mean discharge is shown more clearly in figure 14. In the "north" 

channel, at the bottom, flow is predominantly landward

for all mean discharges of less than about 340,000 ft3/s 

(9,630 m3/s) and is most predominantly landward (37 

percent seaward) at a mean discharge of 200,000 ft3/s 

(5,660 m3/s); as mean discharge increases, flow 

predominance gradually changes from landward to 

seaward and at 550,000 ft3/s (15,600 m3/s) it reaches a 

value of 75 percent seaward. At the 25 percent depth, 

flow predominance remains at 50 percent for mean discharges of less than about 370,000 ft3/s (10,500 m3/s) and 

increases gradually as mean discharge increases up to a 

value of 77 percent seaward at about 550,000 ft3/s (15,600 

m3/s). In the upper half of the vertical, seaward flow 

predominance increases rapidly to a value of 70 percent 

as the flow increases to a mean discharge of 100,000 ft3/s 

(2,830 m3/s). It then levels off and gradually increases to 

78 percent at 550,000 ft3/s (15,600 m3/s). For all depths 

within the vertical, flow predominance approached the
P22 TRANSPORT OF RADIONUCLIDES BY STREAMS

MEAN 

0

DISCHARGE, IN THOUSANDS OF CUBIC METRES PER SECOND 

2 4 6 8 10 12 14 16 18 20

100

50

I I I I

\ \ \ r

j___i

"North" channel

_____I_____I_____I_____I_____I

0 100 200 300 400 500 600 700 

MEAN DISCHARGE, IN THOUSANDS OF CUBIC FEET PER SECOND

FIGURE 14. Variation of flow predominance with mean discharge at the measuring 

cross section at Astoria, CRM 14. Numbers indicate height above bed expressed as a percentage of the total depth.

same level of about 77 percent at 550,000 ftVs (15,600 

m3/s), but at no time during the measurement did the 

flow ever reach 100 percent seaward.

The flow-predominance pattern for the south channel 

is very similar to that for the "north" channel. Near the 

bottom it is less than 50 percent for low mean discharges 

and decreases to a minimum of 40 percent at a mean discharge of 150,000 ftVs (4,250 m3/s). With mean discharges greater than 150,000 ftVs (4,250 m3/s), it progressively increases to 50 percent at a mean discharge of 

260,000 ftVs (7,360 m3/s) and is as high as 93 percent at a 

mean discharge of 700,000 ftVs (19,800 mVs). At the 25 

percent depth, flow predominance is close to 50 percent 

for mean discharges under 200,000 ftVs (5,660 mVs), and 

increases as mean discharge increases to 92 percent at 

700,000 ftVs (19,800 mVs). In the upper half of the vertical, flow predominance increases rapidly with increases 

in mean discharge up to about 200,000 ftVs (5,660 mVs) 

and then it gradually increases to 93 percent at 700,000 

ftVs (19,800 mVs). Although flow predominance in the 

south channel was not defined at mean discharges 

greater than about 700,000 ftVs (19,800 m3/s), by extrapolation it appears that for a mean discharge of about 

1 million ftVs (0.03 million mVs) the flow at CRM 14 

would be 100 percent seaward. This agrees with the 

mean discharge predicted for 100 percent seaward flow 

for the "north" channel.

Flow-predominance patterns for the middle channel 

are different from those in the "north" and south channels. At the bottom, flow predominance is seaward with 

mean discharges of less than 115,000 ftVs (3,260 m3/s); 

however, it is landward with mean discharges from 

115,000 to 425,000 ftVs (3,260 to 12,000 mVs). At 25 percent of depth, flow predominance is landward with mean 

discharges of less than 120,000 ftVs (3,400 m3/s), but it is 

seaward at higher mean discharges. The upper half of 

the vertical follows the same pattern as was observed in 

the "north" and south channels except the magnitude of 

the seaward predominance is not as large. The main 

reason for the difference in flow patterns between the 

middle channel and the "north" and south channels is 

that the flow in the "north" and south channels is 

generally perpendicular to the cross section, whereas the 

flow in the middle channel is not and at times parallels 

the cross section.

FLOW-PREDOMINANCE PATTERNS IN 

SOUTH CHANNEL, CRM 5.6-22.5

THE

For a more complete understanding of flow patterns in 

the estuary, data were collected at two different flow 

conditions to define sediment transport and flow 

characteristics along the longitudinal axis of the south 

channel. During a 3-day period when the tidal patterns 

and upland discharges were relatively constant, data 

were collected at seven stations that were located about 

2 miles (3.2 km) apart. On each day, measurements were 

made at three adjacent stations from a single vessel that 

repetitively occupied each station about every 2 hours 

during the daylight hours. On the second day the occupied stations were located in the middle part of the 

study reach and the upstream and downstream stations 

corresponded, respectively, to the downstream station 

on the first day and the upstream station on the last day. 

Measurement data were combined according to the
DISCHARGE AND FLOW DISTRIBUTION, COLUMBIA RIVER ESTUARY P23

100

Sept. 14, 1969

Daily mean discharge =

208,000 ft3/s (5,890 m3/s)

16 17 

COLUMBIA RIVER MILE

LJJ

CJ 100

50

-5 

May 23, 1970

Daily mean discharge =

437,000 ft3/s (12,400 m 3/s)

9 10 11 12 

COLUMBIA RIVER MILE

13 14 15 16 17

FIGURE 15. Flow predominance along the longitudinal axis of the south channel during a half tidal cycle. Numbers indicate height above the bed expressed as a percentage of the total depth.

relative time of collection in the tidal cycle to give distribution curves that show the time variations of 

variables at all stations on the second (middle) day.

For the first flow condition (September 14, 1969) the 

daily mean discharge at the Astoria cross section was 

208,000 ft3/s (5,890 m3/s) and the study reach extended 

from CRM 10.7 to CRM 22.5. From velocity profile data, 

flow-predominance values were calculated for the seven 

stations at heights above the bed of 5, 25, 50, 75, and 95 

percent of the total depth. A plot of these data (fig. 15) 

shows that upstream from CRM 17.5, the flow was 

predominantly downstream at all depths. Near the bottom, net flow was zero at CRM 17.5 and was 

predominantly upstream in the lower half of the study

reach. At middepth, the net flow was zero at CRM 16.6 

and downstream from that point it was predominantly 

landward. Flow in the upper layers remained 

predominantly downstream throughout the entire study 

reach.

For the second flow condition (May 23,1970) the daily 

mean discharge at the Astoria cross section was 437,000 

ft3/s (12,400 m3/s), and the study reach extended from 

CRM 5.6 to CRM 16.2. Velocity profiles collected at each 

of the 7 stations during this study also were used to compute flow predominance at heights above the bed of 5, 

25, 50, 75, and 95 percent of the total depth. Figure 15 

also shows these data. Near the bottom, the flow 

predominance ranged from 76 percent at the upstream
P24 TRANSPORT OF RADIONUCLIDES BY STREAMS

20

10

0400 0800 1200 

PACIFIC STANDARD TIME

1600 2000 2400

1000 hours

H

5 20

10 1200 hours

20

10

1400 hours

I

 All depths 

D Bottom 

A Middle 

O Top

10 12 14 16 18 

COLUMBIA RIVER MILE

20 22

FIGURE 16. Salinity distribution along the longitudinal axis of the south channel at various times in the tidal cycle on September 14, 1969.
DISCHARGE AND FLOW DISTRIBUTION, COLUMBIA RIVER ESTUARY P25

end of the reach to 70 percent at the downstream end. At 

the top, the flow predominance remained approximately 

a constant 80 percent over the entire reach.

In both studies, the flow predominance was determined with velocity data for only half of the tidal 

cycle the early morning ebb and the following flood. 

Tide ranges during these periods (figs. 16, 17) were such 

that the computed values of flow predominance on 

September 14 probably are lower than full-tidal-cycle 

values, and those computed for May 23 are higher. 

Despite this discrepancy, the data show that the point 

on the bed where the flow predominance is 50 percent 

(where the net flow is zero) varies longitudinally in 

response to the upland flow.

According to Simmons (1955), heaviest shoaling in 

partly mixed estuaries is between the high-tide and the 

low-tide positions of the upstream limit of saltwater intrusion, and flow predominance near the bottom in this 

zone is close to 50 percent. Shoaling occurs because the 

net downstream transport of sediment along the bottom 

is effectively stopped where there is a predominance of 

upstream currents and because suspended sediment 

tends to accumulate in the zone where net flow is near 

zero. This latter effect results from the tendency for sediment particles in the upper layers, where the flow is 

predominantly seaward, to settle during slack water into 

the lower layers where they are transported back upstream by the predominantly landward currents (fig. 

13); this action over many tidal cycles causes an accumulation of sediment at and near the bed in the zone 

where neither upstream or downstream flow 

predominates. In the Columbia River estuary, much of 

the accumulated sediment is resuspended during the 

high-velocity period of each flood and ebb, thereby 

creating locally a "turbidity maximum" wherein the 

concentration of suspended sediment is considerably 

higher than it is either upstream or downstream (Hubbell and others, 1971).

Salinity data from the two longitudinal studies (figs. 

16, 17) show that on September 14, during the time for 

which flow predominance was computed, saltwater intrusion ranged between about CRM 18 and CRM 21; on 

May 23 it varied between about CRM 5 and CRM 14. On 

the basis of these data, shoaling would be expected in the 

reach between CRM 5 and 21. In confirmation, Lockett 

(1967) pointed out that in the 11-mile (17.7-km) reach 

from Tongue Point (CRM 17.5) to Sand Island (CRM 

6.5) there has been 77 million cubic yards (59 million m3 ) 

of material dredged from the navigational channel during 1868 to 1958.

SUMMARY

Unsteady flow, caused mainly by tides, and complex 

velocity distributions, due mainly to salinity gradients,

precluded use of conventional methods of measuring discharge and of computed discharge records in the Columbia River estuary. However, discharge data were obtained by employing a moving-boat technique (MOVD) 

in which both the direction and magnitude of the water 

velocity are measured throughout the entire depth at a 

series of laterally spaced verticals in a cross section. Data 

from repetitive measurements at Astoria and the Beaver 

Army Terminal during half tidal cycles (about 10 hours) 

were used to define flow hydrographs at these locations, 

and the hydrographs, in turn, were used to adapt and 

calibrate one-dimensional mathematical models for 

calculating continuous records of discharge. The discharge models for both Beaver Army Terminal and 

Astoria were based on solution of partial differential 

equations that express the conservation of mass and 

momentum in one-dimensional unsteady homogeneousdensity open-channel flow according to the method of 

characteristics (Lai, 1965a) using measured watersurface slopes and channel geometry as boundary conditions.

The mathematical model for the Beaver Army Terminal, which is in the freshwater part of the estuary at 

CRM 53.3, was used to compute instantaneous discharges every 15 minutes; daily mean discharges, in 

turn, were determined from these values. The flowresistance coefficient, rj, in the model varied with discharge and was lowest at high flows. Discharges computed for the period May 1968 through June 1970 show 

that flow reversals at Beaver commonly occur during at 

least one of the twice-daily flood periods whenever the 

daily mean discharge is less than 185,000 ft3/s (5,240 

m3/s). For daily mean discharges of 185,000 to 225,000 

ft3/s (5,240 to 6,370 m3/s), the flow reverses occasionally 

during the strongest flood period, and for daily mean 

discharges greater than 225,000 ft3/s (6,370 m3/s) the flow 

rarely reverses.

At Astoria (CRM 14), application of the basic onedimensional model was complicated because of the complex geometry of the reach and the presence of both 

longitudinal and lateral salinity gradients. To obtain, 

with the model, hydrographs having an overall shape 

that corresponded to measured hydrographs, it was 

necessary to vary rj throughout each flood and ebb 

period. In addition, in order to compute hydrographs 

that compared closely with measured hydrographs, a 

factor had to be applied daily to adjust the measured 

water-surface slopes. The required factor for each day of 

record was established by adjusting the slopes by trial 

until the daily mean discharge calculated from the instantaneous discharges computed every 15 minutes by 

the mathematical model agreed within a few percent of 

the daily mean discharges determined from a simple 

volumetric model. Because monthly mean discharges 

determined with daily mean discharges from the
CONCENTRATION OF DISSOLVED SOLIDS (SALINITY), IN PARTS PER THOUSAND

ro * ro -* ho '

o o o o ooo oo o

1 I I T

O

STAGE AT 

ABOVE COLU

O NJ 

"1 I 

_i____i 

I 

o 

STAGE AT C

ABOVE COLU
DISCHARGE AND FLOW DISTRIBUTION, COLUMBIA RIVER ESTUARY P27

volumetric model compared within 5 percent of monthly 

mean discharges computed by the routing technique 

used by the Northwest Resorces Data Center (Orem, 

1968), instantaneous discharge computed every 15 

minutes with the mathematical model for the period 

March 1968 through June 1970 are considered to be fairly 

accurate.

Observations at Astoria indicate that near the surface, 

ebb velocities are as much as 8 ft/s (2.4 m/s) and flood 

velocities are as much as 6 ft/s (1.8 m/s). Time-averaged 

velocities in the cross section during each ebb and flood 

period, however, are about 3 and 2 ft/s (0.9 and 0.6 m/s), 

respectively. Because of the relatively large upland flow 

and moderate tide range, instantaneous ebb discharges 

frequently are greater than 2 million ft3/s (0.06 million 

m3/s) and instantaneous flood discharges often are over 

1.5 million ft3/s (0.04 million m3/s).

Daily mean discharges at Astoria fluctuate cyclically 

because of (1) the daily 50-minute shift in the position of 

the tide curve relative to absolute clock time, due to the 

difference in the length of lunar (24.8 hours) and solar 

(24.0 hours) days; and (2) the periodic variations in the 

maximum and minimum tide heights that occur with a 

frequency of about 14.3 days. Because of this cyclic pattern, daily mean discharges at Astoria are not a measure 

of the upland freshwater discharge. For instance, on 

September 22, 1969, the daily mean discharge at Beaver 

was about 100,000 ft3/s (2,830 m3/s) downstream, but at 

Astoria it was 7,400 ft3/s (210 m3/s) upstream.

Based on the change in the proportion of flow through 

the north (northernmost channel and shallow 

midestuary area combined) and south channels at CRM 

14 during ebbs and floods, for daily mean discharges at 

Beaver of less than about 165,000 ft3/s (4,670 m3/s) there 

is a net clockwise circulation in the lower estuary. That 

is, the proportion of the total quantity of water flowing 

upstream in the north channel during the flood is greater 

than the proportion of the total quantity of water flowing 

downstream during the ebb, whereas the reverse occurs 

in the south channel. For daily mean discharges from 

165,000 to about 190,000 ft3/s (4,670 to about 5,380 m3/s) 

the net circulation is counterclockwise, and for discharges greater than 190,000 ft3/s (5,380 m3/s) it is also 

clockwise. On the average, the north channel conveys 76 

percent of the flood flow and 69 percent of the ebb flow.

At the measuring cross section at Astoria, flow near 

the bottom in the "north" (northernmost) channel is 

predominantly landward for mean discharges (algebraic 

mean throughout a tidal cycle) less than about 340,000 

ft3/s (9,630 m3/s), whereas from middepth to the surface 

it is predominantly seaward with the highest 

predominance being near the surface. At a mean discharge of 550,000 ft 3 /s (15,600 m3 /s), the flow 

predominance at all depths is about 77 percent. In the 

south channel, the flow-predominance pattern is very

similar to that of the "north" channel. Near the bottom 

the flow is predominantly landward for mean discharges 

less than 260,000 ft3/s (7,360 m3/s), and in the upper half 

of the depth the flow is predominantly seaward, with the 

highest predominance near the surface. At a mean discharge of 700,000 ftVs (19,800 m3 /s), the flow 

predominance at all depths is about 92 percent. It is estimated that the flow predominance in the "north" and 

south channels would reach 100 percent seaward at a discharge of about 1 million ft3/s (0.03 million m3/s). The 

direction of flow in the "north" and south channels is 

parallel to the riverbank, but the direction of flow in the 

middle channel trends in a diagonal direction, and the 

channel is not well defined. As a result, in the middle 

channel, at the bottom, flow predominance is seaward 

for mean discharges less than 115,000 ft3/s (3,260 m3/s) 

and landward for discharges from 115,000 to 425,000 ft3/s 

(3,260 to 12,000 m3/s). At 25 percent of the depth, flow 

predominance is landward for discharges less than 

120,000 ft3/s (3,400 m3/s) but is seaward for higher flows. 

In the upper half of the depth, the flow predominance is 

seaward and increases as the discharge increases. At a 

mean discharge of about 450,000 ft3/s (12,700 m3/s), the 

flow predominance at all depths is about 70 percent.

Longitudinal flow characteristics in the south channel 

determined when the daily mean discharge at Astoria 

was 208,000 ft3/s (5,890 m3/s; Sept. 14, 1969), indicated 

that flow at all depths was predominantly seaward upstream of CRM 17.5. At CRM 17.5, flow predominance 

near the bottom became landward while the flow in the 

upper layers remained seaward. Salinity data showed 

that during the day the upstream extent of saltwater 

instrusion ranged between about CRM 18 and CRM 21. 

For a daily mean discharge at Astoria of 437,000 ft3/s 

(12,400 m3/s; May 23, 1970), the flow was predominantly seaward at all depths throughout the entire study reach, which extended from CRM 5.6 to 16.2. 

During the day saltwater intrusion extended upstream to 

CRM 14 and receded to about CRM 5. On the basis of 

these data, heavy shoaling should occur in the reach 

between CRM 5 and 21; dredging records of the U.S. 

Army Corps of Engineers confirm that this is the case.

REFERENCES CITED

Baltzer, R. A., and Lai, Chintu, 1968, Computer simulation of unsteady flows in waterways: Am. Soc. Civil Engineers, Hydraulics 

Div. Jour., v. 94, no. HY4, Proc. Paper 6048, p. 1083-1117.

Baltzer, R. A., and Shen, John, 1961, Flows of homogeneous density in 

tidal reaches: U.S. Geol. Survey open-file report, 107 p.

Barron, E. G., 1963, New instruments for surface-water investigations, 

in Selected techniques in water resources investigations, compiled 

by G. N. Mesnier, and K. T. Iseri: U.S. Geol. Survey WaterSupply Paper 1669-Z, p. Z1-Z12.

Davidian, Jacob, 1964, Computation of discharge in tidal reaches: U.S. 

Geol. Survey Surface Water Techniques, book 1, chap. 2, 49 p.
P28 TRANSPORT OF RADIONUCLIDES BY STREAMS

Dixon, W. J., and Massey, F. J., Jr., 1957, Introduction to statistical 

analysis [2d ed.]: New York, McGraw-Hill Book Co., p. 130.

Hansen, D. V., 1965, Currents and mixing in the Columbia River estuary, in Ocean Science and Ocean Engineering: Marine 

Technology and Am. Soc. Limnology and Oceanography Joint 

Conf., Washington, D.C., June 1965, Trans., p. 943-955.

Hubbell, D. W., Glenn, J. L., and Stevens, H. H., Jr., 1971, Studies of 

sediment transport in the Columbia River estuary: Tech. Conf. on 

Estuaries of the Pacific Northwest, Oregon State Univ., Corvallis, 

1971, Proc. Circ. 42, p. 190-226.

Lai, Chintu, 1965a, Flows of homogeneous density in tidal reaches, 

solution by the method of characteristics: U.S. Geol. Survey openfile report, 58 p.

____1965b, Flows of homogeneous density in tidal reaches, solution 

by the implicit method: U.S. Geol. Survey open-file report, 43 p.

 1967, Computation of transient flows in rivers and estuaries by 

the multiple-reach method of characteristics, in Geological Survey 

Research 1967: U.S. Geol. Survey Prof. Paper 575-D, p. 

D273-D280.

Lockett, J. B., 1967, Sediment transport and diffusion: Columbia 

River estuary and entrance: Am. Soc. Civil Engineers, Waterways 

and Harbors Div. Jour., v. 93, no. WW4, Proc. Paper 5601, p. 

167-175.

Lockett, J. B., and Kidby, H. A., 1961, Prototype measurements of the 

Columbia River estuary: Am. Soc. Civil Engineers, Hydraulics 

Div. Jour., v. 87, no. HY1, Proc. Paper 2710, p. 57-83.

Orem, H. M., 1968, Discharge in the lower Columbia River basin, 

1928-65: U.S. Geol. Survey Circ. 550, 24 p.

Pritchard, D. W., 1955, Estuarine circulation patterns: Am. Soc. Civil

Engineers Proc., v. 81, Paper 717, 11 p. 

Prych, E. A., Hubbell, D. W., and Glenn, J. L., 1967, New estuarine

measurement equipment and techniques: Am. Soc. Civil

Engineers, Waterways and Harbors Div. Jour., v. 93, no. WW2,

Proc. Paper 5219, p. 41-58. 

Savini, John, and Bodhaine, G. L., 1971, Analysis of current-meter

data at Columbia River gaging stations, Washington and Oregon:

U.S. Geol. Survey Water-Supply Paper 1869-F, 59 p. 

Simmons, H. B., 1955, Some effects of upland discharge on estuarine

hydraulics: Am. Soc. Civil Engineers Proc., v. 81, Paper 792, 20 p. 

Smoot, G. F., and Novak, C. E., 1969, Measurement of discharge by

the moving-boat method: U.S. Geol. Survey Techniques Water

Resources Inv., book 3, chap. All, 24 p. 

U.S. Army Corps of Engineers, 1960, Interim report on 1959 current

measuring program Columbia River at mouth, Oregon and

Washington: v. I-IV. 

U.S. Coast and Geodetic Survey, 1969, Tide tables high and low water

predictions, 1970, west coast of North and South American including the Hawaiian Islands: Washington, U.S. Govt. Printing

Office, p. 172-173. 

U.S. Geological Survey, 1970, Surface water records, part 1 of Water

resources data for Oregon, 1969: Portland, Oreg., Water Resources

Div., U.S. Geol. Survey, 367 p. 

___1971, Surface water records, part 1 of Water resources data for

Oregon, 1970: Portland, Oreg., Water Resources Div., U.S. Geol.

Survey, 378 p.
APPENDIX 1. Daily mean discharge of the Columbia River at Astoria, Oreg.

[Discharge in thousands of cubic feet per second]

Day Mar.

1 

2 

3 

4 

5 

6 

7 

8 

9 224

10 207

11 195

12 214

13 242

14 274

15 269

16 273

17 273

18 264

19 252

20 231

21 212

22 146

23 132

24 146

25 165

26 238

27 257

28 292

29 326 

30 276

31 215

Mean:

...

Max:

326

Min:

132

Apr.

222

260

236

206

249

195

182

159

151

150

190

224

204

198

235

224

177

85.7

96.7

96.2

92.2

47.1

127

150

166

 

...

...

260

47.1

May

__

...

 

...

 

 

...

---

 

112

149

187

246

272

234

173

179

153

119

139

134

195

257

276

338

312

330

379

397 

377

___

397

112

June ,

324

340

357

346

316

305

297

315

353

419

480

467

498

504

490

462

449

422

417

432

390

395

425

431

442

446

443

411

394

374

405

504

297

July

320

312

292

273

261

235

242

265

337

380

402

428

384

378

359

330

299

270

246

202

188

198

208

216

266

278

254

243

204

151

278

428

151

1968

Aug.

135

90.7

85.0

53.8

56.6

113

210

262

285

257

216

157

162

160

133

114

75.9

37.7

40.6

84.3

91.3

135

185

251

254

225

215

243

214

168

149

157

285

37.7

Sept.

118

58.0

58.7

105

150

222

250

242

223

226

217

168

131

149

142

90.2

67.2

65.6

82.1

160

211

238

221

226

.1/186

1/186

1/181

~ 147

74.9 

41.5

155

250

41.5

Oct.

64.9

66.3

108

205

193

250

239

205

165

164

143

189

182

138

199

151

98.0

104

122

204

229

279

269

248

270

220

154

69.2

60.4

144

130

170

279

60.4

Nov.

161

172

227

246

250

244

232

195

262

275

272

332

316

262

255

227

239

267

302

332

278

332

333

313

302

221

248

202

164

151

254

333

151

Dec.

234

236

244

276

340

379

318

362

288

276

313

343

278

260

225

286

272

350

366

322

312

262

214

199

215

203

224

259

274

324

1/314

283

379

199

Jan.

1/292

1/283

T/258

T/266

T/296

1/336

T/414 ~422

398

387

451

378

326

287

330

356

413

369

325

322

311

322

278

268

229

226

172

205

1 00

LOJ

1/236

i/220

308

451

172

Feb.

1/245

1/255

1/253

T/260

.1/285

304

258

228

275

220

228

261

323

357

342

346

232

211

222

268

278

203

183

189

170

183

149

149

246

357

149

Mar.

193

177

190

220

229

278

262

284

293

268

211

197

168

175

173

146

247

280

260

277

332

292

362

309

261

230

242

221

212

273

242

362

146

Apr.

293

359

369

422

504

424

309

337

374

383

352

345

437

441

460

432

450

473

426

437

456

380

408

442

417

400

388

395

399

401

404

504

293

May

416

439

444

449

421

386

351

318

349

396

410

445

471

528

534

579

549

558

555

558

540

522

516

516

466

463

479

446

450

491

483

469

579

318

June

499

510

493

485

479

444

408

402

400

390

415

440

448

474

501

470

472

437

391

391

339

260

233

220

220

220

254

310

386

440

394

510

220

1969

July .

457

460

421

390

352

323

316

311

281

263

308

325

300

308

322

1/309

1/286

1/263

1/240

T/260

1/237

1/194

1/167

1/160

1/149

82.0

124

184

9 CO

9 on

255

277

460

82.0

Aug.

216

187

136

117

137

99.7

101

70.3

99.1

105

137

152

234

232

230

223

186

181

167

144

109

50.4

23.3

47.5

93.3

165

228

263

224

190

164

152

263

23.3

Sept.

150

116

112

95.8

91.8

70.2

56.3

57.1

121

156

182

180

223

208

154

125

82.5

67.5

68.3

59.7

-3.2

-7.4

95.9

158

220

228

212

171

150

142

125

228

-7.4

Oct.

114

126

164

124

68.5

29.8

1/91.0

1/154

~ 201

273

238

236

186

164

135

157

139

118

126

66.2

1/39.0

1/115

1/163

1/237

1/214

1/186

~ 163

211

167

1 fQ

1D7

165

153

273

29.8

Nov.

137

116

70.9

23.0

82.9

183

186

286

284

259

219

234

210

161

164

170

168

114

78.5

99.8

200

230

220

231

217

202

194

172

151

69

175

286

23.0

Dec.

143

106

71.5

164

130

136

156

177

197

178

159

244

243

262

226

194

174

170

160

194

256

309

374

375

381

397

319

294

O 0.1 Zol

251

207

224

397

71.5

Jan.

190

160

147

206

217

246

228

205

223

234

205

169

140

276

279

286

259

298

331

361

430

431

506

601

630

569

608

621

Crtfl

_>Uo 

491

337

630

140

1/

1/

1/

I/ Estimated.
P30 TRANSPORT OF RADIONUCLIDES BY STREAMS

INSTANTANEOUS MEASURED DISCHARGE, IN THOUSANDS OF CUBIC METRES PER SECOND

0 10 20 30 40 50 60

200

160

120

80

40

»

-40

-80

120

160

7nn

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

A

a

A

-

0

-

- A

0*

-

'

A3

a

9

*" A

 0

O 0

A

" 

A

O

 A

1 I

1 1

 *

m 

3

A

C

°»

a

 B 

o

0 ° 'A

o

A

° A

^

*

O

A

&

e

1 1

0 

A 

a

A

f 0 A 

1 

A

°°« A

0 A &

J* 

"

 A °

4 *

C

0

a

i i

1 1 | 1 1

o 

A* 0

*

, -x.= 

 8 4

^3 *

* A A "

A

A 2

O AM

c

*o 

 . A

» a i 8 . tf

Jo_D ^C»

 ^ 0 A

0 A A (i

c

c

I I

o ^^

d» i '

QO QV ^

^» A^ °^0

A 0 *°A & O

0 0

c

1 1

1 1

EXPLANATIO

o May 10, 1968

A July 20, 1968

a Oct. 17, 1968

« Dec. 18, 1968

* Mar. 7, 1970

 Apr. 13, 1970

 May 19, 1970

a a Q

! ° ^

-^V~ ^

c

§ (

1 1

N

A

A A

A

A 

i

C

( (

1 1

-

~

-

-

""*

\ A^

ec

-

-

-

1 I

250 500 750 1,000 1,250 1,500 1,750 

INSTANTANEOUS MEASURED DISCHARGE, IN THOUSANDS OF CUBIC FEET PER SECOND

2,000 2,250

FIGURE 18. Percentage error in computed instantaneous discharge at Astoria, Oreg., plotted against comparable measured discharge.

APPENDIX 2. DETERMINATION OF PERCENTAGE

ERROR BETWEEN MEASURED 

AND COMPUTED DISCHARGES AT ASTORIA

In order to evaluate the accuracy of the mathematical 

model for computing instantaneous discharges at 

Astoria, differences between discharges determined 

every 15 minutes and comparable discharges from the 

measured hydrographs have been analyzed statistically 

to define the "tolerance interval" (Dixon and Massey, 

1957, p. 130), which indicates with a specified confidence 

the proportion of a population of computed discharges 

that have an error (difference) equal to or less than a

specified value. The tolerance interval is expressed as

y±Ks,

where

y is the mean error of a sample from the population;

K is the tolerance factor for a normal distribution 

and depends on the specified proportion of the 

population, sample size, and confidence coefficient; and

s is the standard deviation of the sample errors. 

Figure 18 shows the percent error in computed dis-
DISCHARGE AND FLOW DISTRIBUTION, COLUMBIA RIVER ESTUARY P31

charges plotted against the corresponding measured discharges for all hydrographs used in the calibration of the 

mathematical model. This graph demonstrates that the 

distribution of plus and minus errors is roughly symmetrical about the zero-error line and that the scatter 

decreases as the discharge increases. In order to 

eliminate the influence of discharge from the statistical 

analysis, the errors were divided into 10 ranges according 

to discharge, and values within each range were treated 

as an independent set of data. In addition, several sets of 

data were examined to confirm that the distribution of 

errors within any set approximately follows the normal 

error curve a requirement for the use of K.

Results from the analysis are presented in table 7. The 

tolerance limits define the end points of the tolerance interval and indicate, with 90 percent confidence, the 

percentage error that will not be equaled or exceeded by 

more than 10 percent of all computed discharges in a discharge range. Although errors are large when discharges 

are low, table 7 shows that for discharges greater than 

1,250,000 ft3/s (35,400 m3/s), which occur about 30 percent of the time, 90 percent of all instantaneous discharges might be expected to be in error less than about 

25 percent.

TABLE 7. Error analysis of computed discharges at Astoria, Oreg.

[Values are based on percentage errors].

Discharge range 1

(ftVs)

0- 225,000

226,000- 500,000

501,000- 750,000

751,000-1,000,000

1,001,000-1,250,000

1,251,000-1,500,000

1,501,000-1,750,000

1,751,000-2,000,000

2,001,000-2,250,000

2,251,000-2,500,000

Number

of

observations

38

43

51

57

45

51

24

11

12

3

Mean 

percentage 

error

(.V)

102.1 -0.6

-7.fi

-10.9 -5.4

-2.2

1.6 -3.3

1.7

7.7

Tolerance limits

K2

1.971

1.945

1.913

1.895

1.935

1.913

2.089

2.463

2.404

5.847

Standard

deviation

(s)

408.5

62.5

39.3

29.8

21.5

13.4

12.3

9.7

6.8

1.5

Lower 

limit

-703.1

-122.2

-82.8

-67.4

-47.0

-27.8

-24.1

-27.2

-14.6 -1.1

Upper 

limit

907.2

121.0

67.6

45.6

36.2

23.4

27.3

20.6

18.0

16.5

'Pertains to both ebb and flood discharges.

^Tolerance factor for normal distribution which defines with 90-percent confidence an interval about the sample mean that includes 

90 percent of the population. (From Dixon and Massey, 1957, table A-16.)

*U.S. GOVERNMENT PRINTING OFFICE: 1975-677-340/11