Numerical Model of the Salt-Wedge 

Reach of the Duwamish River Estuary, 

King County, Washington

GEOLOGICAL SURVEY PROFESSIONAL PAPER 990

Prepared in cooperation with the 

Municipality of Metropolitan Seattle
Numerical Model of the Salt-Wedge 

Reach of the Duwamish River Estuary, 

King County, Washington

By Edmund A. Prych, W. L. Haushild, and J. D. Stoner

GEOLOGICAL SURVEY PROFESSIONAL PAPER 990

Prepared in cooperation with the 

Municipality of Metropolitan Seattle

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1976
UNITED STATES DEPARTMENT OF THE INTERIOR 

THOMAS S. KLEPPE, SVm/rm

GEOLOGICAL SURVEY 

V. E. McKelvey, Director

Library of Congress Cataloging in Publication Data

Prych, Edmund. A.

Numerical model of the salt-wedge reach of the Duwamish River Estuary, King County, Washington.

(Geological Survey Professional Paper 990)

Bibliography: p. 25-26.

Supt.ofDocs.no.: 119.16:990

1. Oceanography-Washington (State)-Duwamish River estuary-Mathematical models. 2. Estuarine pollution-Washington

(State)-Duwamish River estuary-Mathematical models. 3. Phytoplankton-Washington (State)-Duwamish River estuary.

I. Haushild, W. L., joint author. II. Stoner, J. D., joint author. III. Seattle. IV. Title. V. Series: United States

Geological Survey Professional Paper 990. 

GC856.P77 551.4'66'32 76-608136

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

Washington, D.C. 20402

Stock Number 024-001-02886-7
CONTENTS

Page

Abstract ___________________________ 1

Introduction __________________________________ 1

Acknowledgments _________________________ 3

Flow model _____________________________ 3

General description.._____________________ 3

Flow in the wedge ___________________________ 4

Flow in the upper layer____________________ 5

Transport model _______________________ g

General description _____________________ 6

Transport in the wedge __________________ g

Transport in the upper layer ______________ 6

Advection __________________________ 6

Diffusion _________________________ 8

Constituent models ___________________________ 8

General description______________________ 8

Salinity ______________________________ 8

Temperature _____________________________ 8

Phytoplankton _________________________________ 10

Biochemical oxygen demand _______________ 11

Dissolved oxygen _______________________ 12

Application of the model to the Duwamish River

estuary _____________________________ 12

General ________________________ 12

Input data ______________________________ 12

Flow model _____________________________ 12

Constituent models______________________ 14

Boundary conditions _________________ 14

Page

Application of the model to the Duwamish

River estuary Continued 

Input data Continued

Constituent models Continued

Salinity _________________________ 14 

Temperature _________________ ___ 14

Phytoplankton ___ _____ ____ 15

Biochemical oxygen demand __________________ 16

Dissolved oxygen ______ __ _ 17

Model verification ________ _ _ ___ _ 17

General remarks _____________ __ ___ 17

Temperature __________ __ _______ 20

Chlorophyll a ____________________________________ 21

Dissolved oxygen _________ ___ _____ 23

Prediction of future dissolved-oxygen concentrations ___ 23 

Summary and conclusions--______--- _ 24

References _______-____ __ __ ____ 25

Supplemental information

Phytoplankton _______________ __ __ 28

General ____________ __ -- 28

Sample collection, preparation, and analysis __ ____ 28 

Abundant phytoplankton taxa___ __ _ _ 28 

Production and consumption of oxygen by phytoplankton_________________ 29 

Estimated influence of nutrients on phytoplankton 

growth __________ __ __ 31

Herbivores--___________- - 33

ILLUSTRATIONS

Page

FIGURE 1. Map of Green-Duwamish River study area_________________________ _____ __ _ __ 2

2-5. Diagrams showing:

2. Longitudinal profiles of salinity, temperature, and dissolved-oxygen concentration in Duwamish River estuary ______________________ ________________________ __ 3

3. Probable circulation pattern in Duwamish River estuary __________________________ 4

4. Definition of elements in Duwamish River estuary model __________ __ _ 4

5. Definition of elements in the volume-transfer matrix, U ___ - 7

6. Graph showing solar-radiation and temperature functions for computing phytoplankton growth rates __ 11 

7-23. Graphs showing computed and observed data for Duwamish River estuary for:

7. Water temperatures and tide stages during July 1967 __ _ _____ __ 17

8. Water temperatures and tide stages during August 1967 ______ _ -- 17

9. Chlorophyll a concentrations during July 1967, 16th Avenue South Bridge 18

10. Chlorophyll a concentrations during July 1967, First Avenue South and Spokane Street Bridges 18

11. Chlorophyll a concentrations during August 1967 _______________ _ - - -- 18

12. Dissolved-oxygen concentrations during July 1967, 16th Avenue South Bridge - -- 19

13. Dissolved-oxygen concentrations during July 1967, Spokane Street Bridge _____ 19

14. Dissolved-oxygen concentrations during August 1967, 16th Avenue South Bridge _ ______ __ 19

15. Dissolved-oxygen concentrations during August 1967, Spokane Street Bridge -___ -- 19

16. Water temperatures and tide stages during July 1968 ____________ __ _ 20

17. Water temperatures and tide stages during August 1968 __ _ -- -- 20

18. Chlorophyll a concentrations during July 1968 _____________ __ _ -- 20

19. Chlorophyll a concentrations during August 1968 ____________________ _ __ 20

20. Dissolved-oxygen concentrations during July 1968 ______ - - 21
IV CONTENTS

Page

FIGURES 7-23. Graphs showing computed and observed data for Duwamish River estuary for Continued

21. Dissolved-oxygen concentrations during August 1968 _________________________________ 21

22. Water temperatures and tide stages during September 1971 ____________________________ 21

23. Dissolved-oxygen concentrations during September 1971________________________________ 22 

24-29. Graphs showing:

24. Computed dissolved-oxygen concentrations in top sublayer of Duwamish River estuary during September

1971 for 1971 and future RTF effluent discharges ________________________________ 24

25. Longitudinal distributions ofCyclotella sp ________________________________________ 31

26. Longitudinal distribution of oval flagellates.._______________________________________ 32

27. Longitudinal distribution of "coccoids" plus "coccoid" clusters ___________________________ 32

28. Relation between concentrations of excess dissolved oxygen and chlorophyll a __________________ 33

29. Relations between Michaelis-Menton factors and concentrations of nutrients __________________ 33

TABLES

TABLE

BOD/ 

BODM

BODS 

BODT

7.

Page

1. Data required for computing flow in the Duwamish River estuary ______________________________ 13

2. Relation of width to elevation at cross sections for computing geometry of the Duwamish River estuary ______ 13

3. Equations for computing location of wedge toe in Duwamish River estuary_______ _________________ 13

4. Data required for the phytoplankton model ____________________________________________ 15

5. Data for the BOD Model _______________________________________________________ 16

6. Miscellaneous BOD inflows to the Duwamish River estuary from upstream BODT, and downstream, BODM, from First

Avenue South Bridge _________________________________________________ 16

Data required for the DO model___________________________________________________ 17

8. Estimated decreases in the monthly averages of computed DO concentrations in the Duwamish River estuary during

June-September 1971 for an increase in the RTF effluent discharge to the probable future maximum _____ 24

9. Abundant taxa in samples collected at four stations in Elliott Bay near Seattle during 1967-69 ___________ 29

10. Abundant taxa in freshwater samples collected in 1967 at two stations on the Green-Duwamish River _______ 29

11. Five most abundant taxa in freshwater samples obtained at two stations on the Green-Duwamish River during

March-August 1967 _______________________________________________________ 29

12. Abundant taxa during two blooms in the Green-Duwamish River ______________________________ 30

13. Concentration of herbivores in water samples from the Green-Duwamish River for 3 days during phytoplankton

blooms in 1967 and 1968 _____________________________________________________ 33

DEFINITION OF SYMBOLS

average cross-sectional area of wedge element i, in 

square feet.

average width of interface between a wedge element i 

and the upper layer, in feet.

5-day biochemical oxygen demand of sublayer element 

i, j, in milligrams per litre.

5-day biochemical oxygen demand of sublayer element 

i, j before a computation, in milligrams per litre. 

5-day biochemical oxygen demand of freshwater inflow 

to the estuary, in milligrams per litre, 

miscellaneous 5-day biochemical oxygen demand added 

to the estuary downstream from First Avenue South 

Bridge, in pounds per day.

5-day biochemical oxygen demand of Green River water 

at Tukwila gage, in milligrams per litre. 

5-day biochemical oxygen demand of Renton Treatment 

Plant effluent, in milligrams per litre. 

5-day biochemical oxygen demand of seawater, in milligrams per litre.

miscellaneous 5-day biochemical oxygen demand added 

to the estuary upstream from First Avenue South 

Bridge, in pounds per day. 

constituent concentration.

constituent concentration when the Michaelis-Menton 

factor for that constituent equals 0.5.

Cf

cr

Cs

ct

CPf 

CPS 

CPWi 

CP'

constituent concentration in wedge element i. 

constituent concentration in wedge element i before 

computation.

constituent concentration in sublayer element i, j. 

constituent concentration in sublayer element i, j before a computation.

constituent concentration in freshwater inflow at wedge 

toe.

a coefficient used to compute incident long-wave radiation, dimensionless. 

constituent concentration in seawater. 

constituent concentration in water flowing into upstream end of sublayer./.

chlorophyll a concentration in sublayer element i, j, in 

micrograms per litre.

chlorophyll a concentration in sublayer element i, j before a computation, in micrograms per litre, 

chlorophyll a concentration of freshwater inflow to the 

estuary, in micrograms per litre.

chlorophyll a concentration of seawater, in micrograms 

per litre.

chlorophyll a concentration in wedge element i, in micrograms per litre.

chlorophyll a concentration in wedge element i before a 

computation, in micrograms per litre.
CONTENTS V

D

DOc

DOj_ j 

DO' j j 

DOSj; y 

DOg

E

E(T) 

Ea

FT

G

Go

H

Hs 

H*i

i

iT j

K20

k 

kc

upper layer thickness at the mouth of the estuary, in

feet.

average thickness of a sublayer element in segment i,

in feet.

oxygen consumed per unit decay of biological oxygen

demand, dimensionless.

oxygen produced or consumed during a unit change in

chlorophyll a concentration by growth or respiration, in

milligrams oxygen per microgram chlorophyll a.

dissolved-oxygen concentration of sublayer element i, j,

in milligrams per litre.

dissolved-oxygen concentration of sublayer element i, j

before a computation, in milligrams per litre.

dissolved-oxygen-saturation concentration of sublayer

element i, j, in milligrams per litre.

dissolved-oxygen concentration of seawater, in milligrams per litre.

dissolved-oxygen concentration in wedge element i, in

milligrams per litre.

dissolved-oxygen concentration in wedge element i before a computation, in milligrams per litre.

a constant of proportionality used in computing longitudinal mixing in wedge, dimensionless.

saturation vapor pressure in air at temperature T, in

millibars.

vapor pressure in air, in millibars.

solar-radiation function in phytoplankton growth equation, dimensionless.

Michaelis-Menton factor for reducing the phytoplankton growth rate because of the deficiency of some constituent, dimensionless.

temperature function in phytoplankton growth equation, dimensionless.

growth rate of phytoplankton, in reciprocal hours.

maximum growth rate of phytoplankton, in reciprocal

hours.

rate of heat transfer to water from atmosphere, in

gram-calories per square centimetre per hour.

rate of heat transfer to water by evaporation, conduction, and outgoing long-wave radiation, when the water

surface temperature is T&, in gram-calories per square

centimetre per hour.

rate of incoming long-wave radiation, in gram-calories

per square centimetre per hour.

rate of solar energy penetrating water surface, in

gram-calories per square centimetre per hour.

solar energy passing through a unit area of the bottom

of sublayer element i, j during a time step, in gramcalories per square centimetre.

a subscript denoting segment number.

number of segments in the estuary.

number of the time step.

a subscript denoting sublayer number.

decay rate for biochemical oxygen demand, in reciprocal days.

decay rate for biochemical oxygen demand at 20°C, in

reciprocal days.

column subscript in volume transfer matrix.

change in solar-radiation attenuation coefficient per

unit change in chlorophyll a concentration, in reciprocal feet per micrograms per litre.

solar-radiation attenuation coefficient for sublayer

element i, j, in reciprocal feet.

solar-radiation attenuation coefficient for upper layer

in absence of phytoplankton, in reciprocal feet.

k<> w = solar-radiation attenuation coefficient for wedge in absence of phytoplankton, in reciprocal feet.

kwi = solar-radiation attenuation coefficient for wedge element i, in reciprocal feet.

L = depth-averaged solar-radiation intensity in an element, 

in gram-calories per square centimetre per hour.

L0 = solar-radiation intensity for maximum phytoplankton 

growth, in gram-calories per square centimetre per 

hour.

Nt = number of time steps in a day.

P = atmospheric pressure, in millibars.

Pt = tidal-prism thickness, in feet.

Qe . . = upward flow through base of sublayer element i,j, in 

cubic feet per second.

Qf = mean daily freshwater discharge into head of estuary, 

in cubic feet per second.

Q'f. = fraction of Qf that flows into sublayer j at toe of wedge, 

dimensionless.

Qij = flow rate into the upstream face of sublayer element i,j, 

in cubic feet per second.

QH = mean daily discharge of Green River at Tukwilla gage, 

in cubic feet per second.

QRTP = niean daily discharge of effluent from Renton Treatment Plant, in cubic feet per second.

Qs = rate at which saltwater leaves the wedge and is transported upstream from the toe, in cubic feet per second.

Q's . = fraction of Qs that returns to sublayer j at the toe of the 

wedge, dimensionless.

R = phytoplankton respiration rate, in reciprocal hours.

R20 = phytoplankton respiration rate at 20°C, in reciprocal 

hours.

Ra = the mean daily relative humidity, in percent.

Rv . = the lesser of the volume ratio V^ + i/Vj and unity, dimensionless.

R'v . = the lesser of the volume ratio VJVi+ i and unity, dimensionless.

r = atmospheric-reaeration coefficient, in reciprocal days.

Si, j = salinity of sublayer element i, j, in parts per thousand.

S0 = slope of the interface between wedge and upper layer in 

feet per foot.

SR = total daily incident solar radiation, in gram-calories per 

square centimetre.

S'ft = fractional amount of total solar radiation penetrating 

water surface, dimensionless.

Ta = mean daily air temperature, in degrees Celsius.

TI, = a reference temperature, in degrees Celsius.

T i, j = temperature of sublayer element i, j, in degrees Celsius.

T'ij = temperature of sublayer element i,j before a computation, in degrees Celsius.

Ts = water-surface temperature, in degrees Celsius.

Twi = temperature of wedge element i, in degrees Celsius.

T'wi = temperature of wedge element i before a computation, 

in degrees Celsius.

U = volume-transfer matrix for a sublayer, in cubic feet.

Ui, k = term in matrix U which equals volume in sublayer element i, j at the end of a time step that was in element 

(i-lO+k),j at beginning of the time step, in cubic feet.

Ue = entrainment velocity across the interface between

wedge and upper layer, in feet per second.

U'ej = vertical velocity at base of sublayer j divided by entrainment velocity, Ue , dimensionless. 

Vi = volume of a wedge element, in cubic feet. 

V'i = volume of a wedge element for a preceding time step, in

cubic feet. 

Vs . = volume of a sublayer element, in cubic feet.
VI CONTENTS

V'

w

X

xb

Xt 

X,,

y

ABOD,

volume of a sublayer element in preceding time step, in 

cubic feet.

velocity of water in top sublayer relative to water in 

wedge, in feet per second, 

mean daily wind speed, in feet per second, 

settling velocity of phytoplankton, in feet per second, 

longitudi lal coordinate measured upstream from estuary mouth, in feet.

longitudinal coordinate of a cross section bounding a 

volume that shall move into a sublayer element (see fig. 

5), in feet.

longitudinal coordinate of a cross section bounding a 

volume that shall move into a sublayer element (see fig. 

5), in feet.

longitudinal coordinate of center of wedge element i, in 

feet.

longitudinal coordinate of wedge toe, in feet, 

longitudinal coordinate of downstream face of wedge 

element i, in feet.

longitudinal coordinate of the downstream face of 

wedge element i, for preceding time step, in feet, 

latent heat of vaporization at 0°C, in gram-calories per 

gram.

change in latent heat of vaporization per unit change in 

temperature in gram-calories per gram per degree Celsius.

constant in the Bowen ratio relating heat transfer by 

conduction to that by evaporation, dimensionless. 

change in biochemical oxygen demand of sublayer element i, j due to decay during a time step, in milligrams 

per litre.

A* 

8

e 

«y

£ 

i?

change during a time step in concentration of a constituent in wedge element i due to mixing with water 

from element i + 1.

change in chlorophyll a concentration in sublayer element i, j due to growth and respiration during a time 

step, in micrograms per litre.

change in chlorophyll a concentration in wedge element 

i due to growth and respiration during a time step, in 

micrograms per litre.

oxygen-consumption rate in wedge, in milligrams per 

litre per hour.

length of a wedge element i, in feet, 

length of time step in finite difference model, in seconds.

difference between kelvin and Celsius temperature 

scales, in degrees, 

emissivity of water, dimensionless. 

vertical turbulent diffusion coefficient between sublayers, in square feet per second.

constant in equation for saturation vapor pressure, dimensionless.

constant in equation for saturation vapor pressure, in 

degrees kelvin. 

time angle, in radians.

mass transfer coefficient for evaporation in grams per 

square centimetre per hour per millibar per foot per 

second.

constant in equation for saturation vapor pressure, in 

millibars.

Stefan-Boltsman constant, in gram-calories per square 

centimetre per hour per degree kelvin.

ENGLISH-METRIC EQUIVALENTS

Multiply

feet (ft)

square feet (ft2)

cubic feet (ft3)

feet per second (ft/s)

square feet per second

(ft2/s) 

cubic feet per second

(ft3/s) 

miles (mi) 

pounds per day (Ib/day)

By To obtain

0.3048 metres (m) 

.0929 square metres (m2 ) 

.02832 cubic metres (m3) 

.3048 metres per second (m/s) 

.0929 square metres per second

(m2/s)

.02832 cubic metres per second 

(m3/s)

1.609 kilometres (km) 

.4536 kilograms per day (kg/day)
NUMERICAL MODEL OF THE SALT-WEDGE REACH OF THE 

DUWAMISH RIVER ESTUARY, KING COUNTY,

WASHINGTON

By EDMUND A. PRYCH, W. L. HAUSHILD, and J. D. STONER

ABSTRACT

A numerical model of a salt-wedge estuary developed by Fischer 

(1974) has been expanded and used to calculate the distributions of 

salinity, temperature, chlorophyll a concentration, biochemical oxygen demand, and dissolved-oxygen concentration in the Duwamish 

River estuary, King County, Wash. With this model, which was 

calibrated and verified with observed data, computed temperatures 

usually agreed within 2° Celsius of observed temperatures. During a 

phytoplankton bloom in the summer of 1968, the computed chlorophyll a concentrations increased and decreased with the observed 

concentrations; however, during two blooms in 1967 the computed 

high concentrations persisted farther downstream and lasted a few 

days longer than the observed concentrations. The computed and 

observed dissolved-oxygen concentrations usually agreed within 2 

milligrams per litre, except during phytoplankton blooms. During 

the blooms, the differences were often larger, especially when the 

computed chlorophyll a concentrations were larger than the observed concentrations.

The model was used to predict the dissolved-oxygen concentrations 

in the Duwamish River estuary when the Renton Treatment Plant 

sewage-effluent discharge is increased to its proposed maximum of 

223 cubic feet per second (6.31 cubic metres per second). The computed monthly average dissolved-oxygen concentrations in the estuary decreased by a maximum of 2 milligrams per litre when compared with computations for the summer of 1971, when the effluent 

discharge averaged 37 cubic feet per second (1.05 cubic metres per 

second). The increase in effluent discharge is not expected to cause 

large changes in phytoplankton concentrations in the estuary.

INTRODUCTION

This report describes a numerical model designed for 

predicting concentrations of DO (dissolved oxygen) and 

other constituents in the Duwamish River estuary, a 

salt-wedge estuary (fig. 1). Modeling the upper layer 

(the layer overlying the salt wedge) is emphasized, but 

because concentrations of constituents in the wedge 

affect those in the upper layer, modeling of the salt 

wedge also is described.

Earlier, Fischer (1974) presented a generalized 

numerical model for predicting constituent concentrations in salt-wedge estuaries. Stoner, Haushild, and 

McConnell (1974) then used Fischer's model to predict 

DO concentrations in the salt wedge of the Duwamish 

River estuary. They verified the model with observed 

DO concentrations in the wedge and with observed salinity distributions in the upper layer. In the present

study, the model for the Duwamish River estuary was 

extended to predict DO concentrations in the upper 

layer of the Duwamish River estuary. To accomplish 

this goal, it is necessary for the model to also predict 

the temperature, the 5-day BOD (biochemical oxygen 

demand), and the phytoplankton concentrations in the 

estuary.

This report is one of several resulting from a study of 

the effects on estuary water quality of changes in the 

treatment and the quantity of sewage discharged to 

the Duwamish River estuary. The effects of effluent 

from RTF (Renton Treatment Plant), which provides 

secondary treatment, are of particular interest. The 

study is being conducted by the U.S. Geological Survey 

in cooperation with Metro (Municipality of Metropolitan Seattle). As part of its general plan for the treatment and disposal of sewage from the Metropolitan 

Seattle area, Metro operates the RTF and relies on the 

Duwamish River estuary to transport an increasing 

quantity of RTF effluent to Elliott Bay and Puget 

Sound. If the RTF effluent is detrimental to the estuary 

water quality especially for its use by anadromous 

fish regulatory Federal and State agencies may prohibit or limit the discharge of the effluent and (or) require additional treatment of the sewage at the plant. 

A primary reason for developing the mathematical 

model of this report was to predict the effects of proposed changes in the quantity and quality of the RTF 

effluent on the estuary water quality.

The estuary reach included in the model study extends from the estuary mouth to the wedge toe. The 

wedge toe is defined as the upstream limit of a specified 

salinity. The location of the toe depends on tide stage 

and river discharge; therefore, the location of the model's upstream boundary is a function of time. During 

the summer low-flow periods, the toe ranges from a 

little downstream of 16th Avenue South Bridge to a 

little downstream of East Marginal Way Bridge. (See 

fig. 1.) The distance from the mouth to the wedge toe 

during the low-flow periods averages about 5 mi (8 

km). All flows are computed by using equations for the 

conservation of fluid volume and empirical formulas
NUMERICAL MODEL OF THE SALT-WEDGE REACH OF THE DUWAMISH RIVER ESTUARY

Elliott Bay 

(PUGET SOUND) 122 20'

East Waterway 

West Waterway

S E ATTLE

Spokane Street 

Bridge, surface 

and bottom

EXPLANATION

I S

First Avenue 

South Bridge

Stream gaging station

Qual ity-of-water site, 

chemical, temperature 

and biological measurements

With monitor 

Without monitor

16th Avenue South 

Bridge, surface 

and bottom

EATTLE CITY LIMITS

Boeing Bridge

East Marginal 

Way Bridge

PUGET 

SOUND

2 MILES

J

1

3 KILOMETRES

I____

TuKwila 

Gaging Station

Renton 

Junction monitor

FIGURE 1. Green-Duwamish River study area.
FLOW MODEL

developed from data for the Duwamish River estuary; 

the dynamic equations of motion are not used. The 

wedge is assumed to be vertically and laterally 

homogeneous. The upper layer is divided into three 

sublayers that are also assumed to be vertically and 

laterally homogeneous. Resolution of constituent concentrations in the longitudinal direction is limited by 

the length of the segments into which the estuary is 

divided; segment lengths average about 600 ft (200 m).

The model of the Duwamish River estuary was verified by data from the June-September periods of 

1967-69 and 1971. The summer months are periods of 

low river discharges when the probability of low DO 

concentrations are highest. Finally, the model was 

used to predict constituent concentrations in the estuary in the future when the discharge from RTF is 

expected to be much larger than at present.

Information on river discharges, salinities, and other 

constituent concentrations, and physical characteristics of the Duwamish River estuary are given in several previous reports, including Dawson and Tilley 

(1972), Santos and Stoner (1972), Stoner (1972), 

Stoner, Haushild, and McConnell (1974), and Welch 

(1969). The interested reader is referred to these reports for background information and data other than 

that included in this report.

ACKNOWLEDGMENTS

The authors express their appreceiation to Glen D. 

Farris, superintendent, Water Quality and Industrial 

Waste Division, Municipality of Metropolitan Seattle, 

and his staff for contributing information and many 

data needed for the model study and for their continuing support and encouragement in the study of the estuary's water quality. The authors acknowledge the 

contribution made by Hugo B. Fischer, Professor of 

Civil Engineering, University of California at Berkeley and part-time employee of the U.S. Geological Survey, in conceiving and developing the salt-wedge estuary model subsequently useful in modeling the 

Duwamish River estuary and for advice in development of the present model. An interim version of a 

computer program for constituent transport in the 

upper layer of the Duwamish River estuary, by 

Richard H. French, graduate student at the University 

of California at Berkeley, expedited the development of 

the present Duwamish River estuary model. The authors depended considerably on K. V. Slack of the U.S. 

Geological Survey for guidance and advice in modeling 

the phytoplankton and their effects on dissolved oxygen, for suggestions about presenting the results of the 

study of phytoplankton and herbivores, and for furnishing literature pertinent to the biological modeling.

The authors also thank H. E. Jobson, and M. J. 

Sebetich of the Geological Survey for reviewing this 

manuscript and for their constructive criticisms.

FLOW MODEL

GENERAL DESCRIPTION

Fischer (1974) developed the general salt-wedge estuary model which was used in the present study. The 

typical salt-wedge estuary is characterized by a lower 

layer, or wedge, of nearly undiluted seawater of uniform salinity. For the Duwamish River estuary, the 

term "sea" is interpreted as Elliott Bay (fig. 1), the 

body of water outside the mouth of the estuary.

Figure 2 shows observed salinity, temperature, and 

DO-concentration distributions in the Duwamish 

River estuary. The wedge in the Duwamish River estuary is defined as that volume of water with a salinity 

greater than 25 ppt (parts per thousand). By comparison, the salinity of the water flowing into the estuary 

from Elliott Bay is about 28 ppt. The 25 ppt salinity 

was chosen because the locus of points with this salinity was fairly easy to define and is a good approximation of the interface between the nearly homogeneous 

wedge and the stratified upper layer. The interface is

RIVER KILOMETRES 

6

RIVER MILES

FIGURE 2. Observed longitudinal profiles of salinity, temperature, 

and dissolved-oxygen concentration in Duwamish River estuary 

during the low high tide of September 13, 1968.
NUMERICAL MODEL OF THE SALT-WEDGE REACH OF THE DUWAMISH RIVER ESTUARY

relatively stable except near the toe. The location of 

the toe, defined as the upstream extent of the 25 ppt 

salinity, was fairly easy to define; however, the upstream extent of salinities slightly less than 25 ppt 

moved erratically, suggesting more intense mixing upstream from the toe.

Stoner (1972) found that during a tidal cycle, saltwater in the wedge near the toe flows upstream at a net 

rate of about 200 ft3/s (5.7 m3/s). This saltwater is 

transported upstream from the wedge toe, is mixed 

vertically, and is returned through the upper layer to 

the sea. The increase in salinity of the upper layer 

toward the sea (fig. 2) suggests entrainment from the 

wedge to the upper layer over the entire interface. A 

nearly uniform salinity in the wedge, approximately 

equal to that in Elliott Bay, indicates that freshwater 

is not transferred into the wedge from the upper layer. 

The net circulation pattern in the estuary is probably 

as shown in figure 3.

The waters in both the upper layer and the wedge 

oscillate upstream and downstream in the estuary during a rising and falling tide. In addition, water in the 

upper layer has a net downstream motion because the 

upper layer carries to the sea both the water discharged to the estuary by the river and the water entrained from the wedge. Water in the wedge has a net 

upstream motion because the wedge loses water to the 

upper layer by entrainment. Thus, water entering the 

upper layer is transported to the sea, and on its way it 

is mixed with water entrained from the wedge. Water 

entering the wedge on a rising tide either returns to 

the sea on a falling tide or remains in the wedge and 

moves upstream until it is entrained into the upper 

layer, where it then is returned to the sea. In addition 

to these advective transport processes there also are 

dispersive transport processes in the estuary; they are 

discussed in the section "Transport model."

FIGURE 3. Probable circulation pattern in Duwamish River estuary. Arrows indicate direction of mean flow.

For purposes of computation, the estuary is divided 

into control volumes called elements (fig. 4). The elements are constructed by first dividing the estuary vertically into the wedge and the upper layer, and the 

upper layer is in turn divided into three equal-depth 

sublayers. Next, the estuary is divided longitudinally 

by vertical planes that are perpendicular to the longitudinal axis of the estuary. Each volume between 

adjacent vertical planes is called a segment and contains one wedge element and three sublayer elements. 

Typically, there are about 35 segments in the estuary. 

The segments are numbered longitudinally from the 

toe to the mouth (z = l to imax), and the sublayers are 

numbered vertically from bottom to top (j =1,2,3).

All segments move with the velocity of the wedge 

water so that there is no net flow between wedge elements. However, because of the relative motion between the water in the wedge and the water in the 

sublayers, there is a net flow across the vertical boundaries between elements in the sublayers.

All horizontal flows are computed from conservation 

of volume equations for the water. Input information 

required for computing water flow in the model includes the geometry of the estuary, location of the 

wedge toe, flow rates across the vertical boundary at 

the wedge toe, vertical entrainment velocities, elevation of the water surface, thickness of the upper layer 

at the estuary mouth, and the slope of the interface.

In this, as in most numerical models, computations 

are made for finite steps in time, At. This model uses 

time steps of 15 minutes. The model computes average 

flow, mixing and other process rates, and the resulting 

changes in constituent concentrations during each 

time step.

FLOW IN THE WEDGE

In the model, the water surface is assumed to remain

Wedge toe

LONGITUDINAL COORDINATE,

FIGURE 4. Schematic diagram defining elements in Duwamish 

River estuary model. See text for definition of symbols.
FLOW MODEL

horizontal and the interface is assumed to have a constant slope, S0 , at all times. During a given day, the 

upper-layer thickness at the mouth, D, is assumed to be 

constant even though the wedge becomes deeper and 

shallower with a rising and falling tide. The wedge toe 

moves upstream and downstream with the tide. Data 

from the estuary show that these are reasonable assumptions for the periods modeled in this study.

At the start of computations for a time step, the new 

space occupied by the wedge is filled with the wedge 

elements from the end of the preceding time step. 

These elements, or water volumes, are deformed to fit 

the wedge geometry for the time step. Starting with 

segment number one at the toe, the model calculates 

for each wedge element a new length A%j, new longitudinal coordinates of the center xmi , and 

downstream face xui, a new average cross-sectional 

area Ait and a new average interface width BI. The 

model computes the widths, areas, and other geometrical properties by interpolation between data at reference cross sections where the channel geometry is defined by input data. If the total volume of all the previously designated wedge elements is insufficient to fill 

the wedge, a new element of seawater is added at the 

estuary mouth. On the other hand, if the wedge is filled 

before all the designated elements are used, the fluid in 

the extra elements is assumed to have flowed into the 

sea, and the model deletes these downstream elements 

from the computations.

Whenever the number of wedge elements exceeds 50, 

or whenever the shortest wedge element is less than 

200 ft (60 m) long, the number of wedge elements is 

decreased by combining the shortest element with its 

shortest neighbor. Also, the element adjacent to the 

wedge toe is combined with its neighbor whenever its 

volume is less than the volume that would flow out of it 

during the time step. The corresponding sublayer elements of two segments are combined at the same time 

the wedge elements are combined.

Near the end of all computations associated with a 

time step, the volume of each wedge element is decreased to account for entrainment. The model makes 

the calculation,

(1)

where V\ and Vi are the wedge-element volumes before and after this subtraction is made, and Ue is the 

entrainment velocity across the interface. In addition, 

the volume of the wedge element adjacent to the toe is 

decreased by the amount Qskt, where Qs is the net 

upstream flow rate of saltwater in the wedge in the 

vicinity of the toe. 

Because the elements in the wedge move with the

water, there is no net flow across the boundaries between wedge elements. However, longitudinal constituent transport between wedge elements may occur 

by longitudinal dispersion; the modeling of this process 

is discussed in the section "Transport in the Wedge."

FLOW IN THE UPPER LAYER

The location and length of the elements in the sublayers are determined by the geometry of the elements 

in the wedge. The thickness, di, of each sublayer element in a segment is the same:

di = (D+S0 xmi)/3. (2)

For simplicity, the widths of all sublayer elements in a 

segment are assumed to equal the interface width Bj. 

The volume, Vsi, of a sublayer element is therefore 

given as

Because of the relative motion between the water in 

the wedge and the water in the upper layer, there is 

usually a net flow between adjacent elements in a sublayer. An expression for the flow between elements in a 

sublayer is obtained by writing an equation for the 

conservation of water volume for a sublayer element. 

The resulting expression is

) + (4)

where Q^ 7 is the flow rate into the upstream face of 

sublayer element i, j; Qe> j is the upward flow through 

the base of element i, j; and V's,- and Vsi are the volumes of the sublayer element for the preceding and 

present times steps, respectively. The upward flows 

Qe it j are computed with the relation

Qe - 7.= U'ejUclxiBi, (5)

where U'S j is the vertical velocity at the base of sublayer j divided by the entrainment velocity across the 

interface Ue . Vertical flow through the water surface is 

prevented by defining C7'e4=0.

The use of equation 4 requires inflow to each sublayer at the wedge toe Qi,j. The model uses

where Q/-is the mean daily discharge of freshwater into 

the head of the estuary, and Q '/ and Q 's are the fractions of Qf and Qs that flow into sublayer./.

The parameters Ue , U'ej, Q'fj, and Q'S j are input information to the model. The selection of numerical 

values for these and other parameters for the 

Duwamish River estuary is discussed in the section 

"Input Data."
NUMERICAL MODEL OF THE SALT-WEDGE REACH OF THE DUWAMISH RIVER ESTUARY

TRANSPORT MODEL

GENERAL DESCRIPTION

This section describes the procedures used in the 

model to calculate changes in constituent concentrations due to advection and diffusion. The effects of 

other processes on constituent concentrations, such as 

the effects of photosynthesis on DO concentration and 

of solar radiation on temperature, are discussed in the 

section "Constituent Models."

The water-transport processes modeled are longitudinal advection and dispersion in the wedge, entrainment from the wedge to the upper layer and from each 

sublayer to the sublayer above (vertical advection), 

vertical turbulent diffusion in the upper layer and longitudinal advection in the upper layer. The assumption 

of lateral and vertical homogeneity in the wedge prohibits modeling vertical or lateral transport processes 

within the wedge, and the assumption of lateral 

homogeneity in the upper layer prohibits modeling 

lateral transport processes there.

That part of the longitudinal dispersion in the upper 

layer that is caused by the combined action of vertical 

differences in velocity and vertical mixing is represented in the model because both these phenomena are 

modeled; whereas, that part of the longitudinal dispersion that is caused by the action of lateral differences 

in velocity and lateral mixing is not modeled explicitly. 

However, the finite-difference representations of the 

advective processes in the model introduce additional 

longitudinal dispersion in the upper layer; this additional dispersion is often called "numerical dispersion." (For example, see Bella and Grenney, 1970.)

During each time step the model computes the 

changes in concentrations due to transport and other 

processes one at a time. The calculations are performed 

in the following order:

1. Position elements in the wedge.

2. Change concentrations in the wedge to account for 

longitudinal dispersion in the wedge.

3. Change concentrations in the upper layer to account 

for entrainment from the wedge and advection in 

the upper layer.

4. Change concentrations in the upper layer to account 

for vertical diffusion.

5. Change concentrations in the wedge and the upper 

layer to account for processes other than water 

transport.

TRANSPORT IN THE WEDGE

After the positions of the elements in the wedge are 

determined by the method described previously in the 

section "Flow in the Wedge," the model need only calculate changes in constituent concentrations due to

longitudinal dispersion. No advection calculations are 

required for the wedge because there is no net flow 

between wedge elements.

The model computes the changes due to dispersion 

with a procedure developed by Fischer (1972, 1974). 

The change in concentration caused by the mixing between adjacent wedge elements during a time step is 

assumed proportional to the width of the elements and 

to the distance traveled by the elements during the 

time step and inversely proportional to the square of 

the element lengths. The change in constituent concentration, ACji+1 , of wedge element i due to mixing 

with water from element i + 1 is calculated as

 A",. j

i + 1

, (7)

where E is a dimensionless constant of proportionality 

with a value of about 0.2; xui and x' ui are longitudingal 

coordinates of the boundary between wedge elements i 

and i + l for the present and previous time steps, respectively; C'i is the constituent concentration in 

wedge element i before the dispersion computation; 

andJRvj is the lesser of Vi+i/Vi and unity. To avoid occasional instabilities in the computations, an upper limit 

of 0.5 is imposed on the term in square brackets in 

equation 7. The corresponding change in concentration 

of a constituent in wedge element i + 1 due to mixing 

with water from wedge element i is

i i

where R \i equals the lesser of the values VyFj+i and 

unity.

After calculating the quantities AC/+1 and AC/+1 for 

all i, the model computes a new constituent concentration, Cj, for each wedge element, using the expression

i = C\+ ACV+1 (9)

where C'i and C; are constituent concentrations before 

and after this computation.

TRANSPORT IN THE UPPER LAYER

ADVECTION

The present model treats longitudinal advection a 

little differently than does the original model of 

Fischer (1974). The present model uses an extension of 

the upstream difference scheme or the method of 

characteristics that is computationally stable when a 

water particle moves more than one element length 

during a time step.

The model computes a volume transfer matrix, U, of
TRANSPORT MODEL

dimension /max by 21, for each sublayer. Each term Ut, k 

in the matrix equals the volume in element i, j at the 

end of a time step that was advected horizontally from 

element (i-W+k), j during the time step (fig. 5). 

Presently, the movement of water is limited to nine 

elements in the downstream (fc = l-9) or upstream 

(£ = 11-19) directions. The term with the subscript 

k = W is the volume that remains in element i during 

the time step. Terms with fc=20 and 21 represent volumes that originate seaward of the estuary mouth and 

upstream of the wedge toe, respectively.

The model evaluates terms in a row of the matrix U 

as follows. If the flow is into the upstream face of sublayer element i,j(Qi,j>Q), one sums the sublayer volumes from the preceding time step, V'8 , either wholly 

or in parts, upstream from the element until the accumulated volume equals the inflow volume Qijht. If 

the upstream boundary of the model is reached before 

the desired volume is obtained, a temporary element 

with sufficient volume to make up the deficit is attached to the upstream end of the sublayer. Starting at 

xa the location of the cross section at the end of the 

accumulated volume in figure 5 the volumes V'8 are 

summed again, this time proceeding in the downstream direction until the accumulated sum equals 

Vsi -(Qeij -Qetj+i )A£, or until the upstream face of 

element i is reached. The location of the cross section at 

the end of this summation is designated as Xb- The 

parts of the volumes V'8 contained in the interval Xb to 

xa are assigned to the appropriate terms in Uit k- If flow 

is into the downstream face of element i,j (Qi+ij<Q\ 

similar computations are performed on the elements 

downstream from element i,j.

The volume remaining in an element during a time 

step, Ui,io, is computed by subtracting the outflow volumes from the element volume and assuring that the 

difference is not negative:

(lOa)

or

t, 10 = 0 if the above is negative (10b)

With the above information the constituent concentrations can be changed to account for advection during a time step. The program uses the equation,

19

(11)

These volumes move 

to element/'.

Estuory mouth *b "a Wedge toe

LONGITUDINAL COORDINATE,*

FIGURE 5. Elements in the volume-transfer matrix, U. See text for 

definition of symbols.

where c'ij and Cij are the constituent concentrations in 

sublayer elements before and after this computation 

(whenj = l, the wedge concentration C; replaces the 

term c'ij-i) and C§ and ctj are the boundary conditions 

for the constituent concentration at the estuary mouth 

and at the wedge toe, respectively. (They are discussed 

in the section "Boundary Conditions.") The matrix U 

appearing in equation 11 is different for each of the 

three sublayers.

In those cases where C7/,io and U^Q or U^n are the 

only two nonzero terms in a row of the matrix U, this 

numerical scheme for modeling longitudinal advection 

is identical to the common explicit upstream finitedifference scheme. When there are other nonzero 

terms, the schemes are different, and the present 

scheme is stable while the common upstream difference scheme is not.

When water volumes from two or more elements 

combine in one element, as would occur in the example 

shown in figure 5, the inherent mixing of the volumes 

is numerical mixing in the longitudinal direction. The 

numerical scheme used for computing vertical advection similarly introduces numerical mixing in the vertical direction.

The boundary conditions for the constituent concentrations (the concentrations in the inflowing water) at 

the estuary mouth, C§, are the same for the entire 

upper layer and the wedge. These boundary conditions 

are input data for the model.

Boundary conditions for constituent concentrations 

at the wedge toe are required only for the upper layer. 

The model computes these boundary conditions with 

the equation
NUMERICAL MODEL OF THE SALT-WEDGE REACH OF THE DUWAMISH RIVER ESTUARY

Qs +c Q

(12)

which determines the concentration of a mixture of (1) 

water from the wedge at the toe with a concentration of 

Ci and (2) freshwater entering the upper layer at the 

toe with a concentration of c /-. The value of c / for each 

constituent is input data for the model.

DIFFUSION

The final water-transport computation in a time step 

is for vertical turbulent diffusion between sublayers; 

there is no diffusion between the wedge and the upper 

layer in the model. To compute vertical transport by 

turbulent diffusion, the model uses a constant vertical 

diffusion coefficient, e y , in the equation,

n^ f*

where c'ij and GIJ now represent constituent concentrations before and after this computation.

CONSTITUENT MODELS 

GENERAL DESCRIPTION

This section describes processes, other than water 

transport, that affect the constituent concentrations. 

The constituents discussed are salinity, temperature 

(heat), chlorophyll a (phytoplankton), 5-day BOD, and 

DO. Although information on the DO concentrations is 

the goal of the computations, the concentrations of the 

other constituents must also be computed because they 

affect the DO concentrations.

SALINITY

Salt enters the estuary from the sea and is transported through the estuary by the processes already 

described. No other sources of salt are considered. The 

model computes salinities of the sublayer elements in 

the upper layer, but the salinity in the entire wedge is 

assumed to equal the salinity of the inflowing seawater. The model treats sea salt as a conservative substance; computations for chemical reactions or other 

processes are not necessary.

TEMPERATURE

The model computes the temperature of each of the 

elements in the upper layer, TIJ, and also in the wedge,

TWi. In addition to heat transfer by advection and diffusion within the estuary, heat transfer occurs between 

the top sublayer and the atmosphere, and all sublayers 

and the wedge are heated (though not equally) by solar 

radiation.

The program computes the heat transfer at the 

water surface by using the procedure given by Yotsukura, Jackman, and Faust (1973). The rate of heat 

transfer between the atmosphere and the water per 

unit area of water surface, H, is approximated by the 

linear expression,

dH

(14)

where

HS = rate of solar energy penetrating a unit

area of water surface; 

HL = rate of incoming long-wave radiation

per unit area water surface;

Hb = heat transfer rate per unit area of water 

surface due to outgoing long-wave 

radiation, evaporation, and conduction when the water surface temperature is Tb ;

Tb a reference temperature; and 

Ts = water surface temperature. 

Positive values of these quantities denote heat transfer from the atmosphere to the water, and negative 

values denote heat transfer from the water to the atmosphere.

The program computes the quantities Hb and dHb/ 

dTb once at the beginning of every day using daily average meteorological data and Tb =Tij 3. Meteorological 

data required by the model are

Ra = mean daily relative humidity, in percent;

SR = total daily incident solar radiation, in 

gram-calories per square centimetre 

(g-cal/cm2 ); 

Ta = mean daily air temperature, in degrees

Celsius; and 

Wa = mean daily wind speed, in feet per second.

Also required is coefficient Cr, which is used to compute 

the incoming long-wave radiation.

The heat transfer rate Hb and its derivative are computed using the following equations, which were given 

by Yotsukura, Jackman, and Faust (1973):

H = - -[xWa (a-(3Tb HE(Tb )-Ea)] 

-[y\PWa (a-(3Tb }(Tb -Ta )] (15)
CONSTITUENT MODELS 9

and

(16)

The three bracketed terms on the right side of equation 

15 represent heat-transfer rates due to outgoing 

long-wave radiation, evaporation, and conduction, respectively. The variables appearing in equations 15 

and 16 are denned as follows:

Ea = vapor pressure in air, in millibars; 

E(Tb ) = saturation vapor pressure in air at

temperature Tb , in millibars; 

P = atmospheric pressure, assumed to be

1,013 millibars; 

a = 595.9 g-cal/g, latent heat of vaporization

at 0°C;

P = 0.545 g-cal-g-i^C- 1 , change in latent 

heat of vaporization per unit change in 

temperature;

y = 0.61xlO-3/°C, constant in the Bowen 

ratio relating heat transfer by conduction to evaporation; 

8 = 273 K (kelvin), constant to convert degrees Celsius to kelvin units; 

e = 0.97, emissivity of water; 

X = 3.6x10-4 g-cm-2 hr-i-mb-i (ft/s)-i, 

mass-transfer coefficient for evaporation; and 

a =4.88x10-9 g-cal-cm-2- hr-i-K-i,

Stefan-Boltsman constant.

With the exception of X the above coefficients were 

taken from Yotsukura, Jackman, and Faust (1973). 

The present value of X is based on more recent work by 

H. E. Jobson (oral commun., 1974) and is about twice 

the value given by Yotsukura, Jackman, and Faust 

(1973).

The saturation vapor pressure at temperature Tb 

and its derivative are approximated with the expressions

E(Tb ) = )

and

dE(Tb )

= E(Tb )

(17)

(18)

where £ = 23.38 mbar; 77= 5303.3 K; and £= 18.1.

The model approximates heat transfer to the water 

by incoming long-wave radiation, HL, by the following 

equation, proposed by Koberg (1964):

HL = (Cr +0.0263VEa)o-(Ta + 8)4. (19)

The coefficient Cr is a function of the air temperature 

and the ratio of the actual to the clear-sky incident 

solar radiation. Koberg (1964) presents Cr in graphical 

form; it has a numerical value of the order 0.7.

The model computes the quantities HL, Hb , and 

dHb IdTb for each day using mean-daily meteorological 

data and the values of Tlj3 at the start of the day for Tb . 

During each time step the temperature of each element 

in the top sublayer is changed according to the relation

(20)

where T'^ and T^ are the temperatures of an element 

before and after this heat-transfer computation. The 

numerical constant in equation 20 converts the dimensions of the element thickness, d j, from feet to centimetres.

The amount of solar radiation penetrating a unit 

area of water surface during a time step, Hs, is calculated with equation 2 la or 21b,

Hs = 0, 0<0 or 0>7r, 

where 0 is the angle

(21a)

(21b)

(22)

where lt is the number of the time step; Nt is the total 

number of time steps in a day; and S'R is the fraction of 

the incident solar radiation that penetrates the water 

surface.

The model computes the effect of solar heating on 

each sublayer element in the upper layer by taking the 

difference between the solar energy entering a unit 

area of the top, H*ij+i, and leaving the bottom, H*ij, 

of each element. These quantities are related by the 

equation

(23)

where kij is the solar-radiation attenuation coefficient 

for a sublayer element. Thus, during every time step 

the model changes the temperature of each sublayer 

element by the equation

Tij = T'ij + (H* u+i-#*w)/(30.48 di) , (24)

where T'ij and Ttj are now the temperatures before 

and after a solar-heating calculation. 

The model changes the temperatures of elements in
10 NUMERICAL MODEL OF THE SALT-WEDGE REACH OF THE DUWAMISH RIVER ESTUARY

the wedge, Twi , according to equation 25, which was 

derived with the assumption that all solar energy passing through the wedge-upper layer interface heats the 

wedge elements:

Tw = T' i Bi /(30.48 A,), (25)

where T'wi and Twi are the temperatures before and 

after the solar-heating calculation.

PHYTOPLANKTON

In addition to calculating the transport of phytoplankton by advection and diffusion, the model considers the growth, respiration, and settling of the 

phytoplankton. In this report growth is defined as the 

increase in phytoplankton biomass by both reproduction and by the increase in size of the individual 

plankters. Respiration includes all those processes that 

cause a decrease in phytoplankton biomass and all 

those processes by which living and dead (decomposing) phytoplankton use oxygen. All phytoplankton are 

represented in the model as chlorophyll a. Chlorophyll 

a concentrations are denoted in the upper layer by 

CPitj, and in the wedge by CPwi. Because of insufficient 

data, no attempt is made to distinguish between different taxa, even though it is known that species of both 

freshwater and saltwater origin bloom in the estuary 

(see section "Supplemental Information" under "Phytoplankton"), and that the different species probably 

respond differently to changes in salinity, temperature, and other parameters.

Growth and respiration are modeled by adding the 

amount

or

(26)

(27)

to the chlorophyll a concentrations in the sublayer or 

the wedge elements, respectively. The quantities CP'ij 

and CP'Wi represent chlorophyll a concentrations before this addition. The variables R and G are respiration and growth rates. The respiration rate used by the 

model doubles with every 10°C rise in temperature (see 

for example McKee and Wolf, 1963, p. 284) and is given

by

R = JR 2oexp[0.0693(T-20)], (28)

where R2o is R at 20°C and T is the local water temperature. The growth rate is a function of the local temperature and solar-radiation intensity,

G = FTFL G0 , (29) 

where G0 is the maximum growth rate and FT and FL

are functions of temperature and solar-radiation intensity, respectively. The temperature function used 

by the model is

FT = 0, (T<10°)

FT = 0.1T-1.0, (100<Ts£20°) (30)

FT = 1.0, (T>20°). 

The solar-radiation function is

FT = exp (1 - (31)

where L is the depth-averaged solar-radiation intensity in the wedge or sublayer element where the 

growth rate is desired andL0 isL for maximum growth. 

Both FT and FL are shown in figure 6.

Equation 30 is the authors' approximation of the effect of temperature on phytoplankton growth; it is 

based on data by Jitts, McAllister, Stephens, and 

Strickland (1964) for some marine phytoplankton and 

on knowledge that phytoplankton blooms are normally 

observed in the Duwamish estuary when the freshwater temperatures are about 20°C or higher.

Equation 31, which describes the effect of solar radiation on phytoplankton growth, was previously used 

by Steele (1962) for computing the effect of light on 

photosynthesis.

The model computes an average value of L for a sublayer element with the equation

L =

tit

(32)

and for a wedge element with

L =

Htl llit (33)

where kit j and kwi are solar-radiation attenuation coefficients for the sublayer and wedge elements.

The attenuation coefficients for the sublayer and 

wedge elements are approximated by

and

kwi = kow +k C CPwi,

(34)

(35)

where k°u and k°w are the attenuation coefficients of the 

sublayers and the wedge in the absence of phytoplankton (chlorophyll a) and kc is the increase in the attenu-
CONSTITUENT MODELS 11

TEMPERATURE, IN DEGREES CELSIUS

10 20

9=0.5

1234 

t/i0 ,DIMENSIONLESS SOLAR-RADIATION INTENSITY

FIGURE 6. Solar-radiation and temperature functions for computing phytoplankton growth rates.

ation coefficient due to a unit increase in chlorophyll a 

concentration.

The effects of nutrient concentrations on phytoplankton growth were not modeled in this study because present nutrient concentrations in the 

Duwamish River estuary are believed high enough, 

even during phytoplankton blooms, not to have an important effect on the growth rate. Data on phytoplankton in the estuary and a discussion of the probable 

effects of nutrients on phytoplankton growth rates in 

the estuary appear in the section "Estimated Influence 

of Nutrients on Phytoplankton Growth."

The net amount of oxygen produced in the upper layer by the phytoplankton through photosynthesis and 

respiration is assumed to be proportional to ACP^j. 

(See equation 45.)

Because of settling of phytoplankton, the wedge receives chlorophyll a from the upper layer and loses 

chlorophyll a by deposition of phytoplankton on the 

channel bottom; resuspension from the bottom is not 

modeled. For every time step, the model performs the 

following settling computation for each wedge element, equation 36, and each sublayer element, equation 37:

CPW = CP'W + )w A (36)

and

ij = CP'ij + (CP'Lj+1 -CP'ij ) wlt/di, (37)

where CP'Wi and CPWi , and CP'ij+i and CPij are the 

chlorophyll a concentrations before and after the settling computation and w is a constant settling velocity.

BIOCHEMICAL OXYGEN DEMAND

The present model computes the BOD (5-day 

biochemical oxygen demand) in the upper layer of the 

estuary. BOD was not modeled in the wedge because 

the processes that transport BOD into the wedge are

largely unknown. A constant BOD, equal to the BOD 

of the seawater, was assumed for the entire wedge. 

Data from the Duwamish River estuary (Welch, 1969) 

show that the BOD is typically 1-2 mg/1 (milligrams 

per litre) and exceeds 5 mg/1 only during phytoplankton blooms. The present low BOD has only a small 

effect on the dissolved oxygen in the upper layer of the 

estuary. However, because increased discharges from 

RTF probably will increase the BOD in the estuary, 

BOD is modeled for the upper layer. Only the carbonaceous oxygen demand was modeled because the 

time of travel from the RTF outfall to the mouth of the 

estuary is about 5 days. The nitrogenous oxygen demand, which was not modeled, becomes important only 

for longer periods.

The sources of BOD in the model are those coming 

into the estuary from the river, the sea, and miscellaneous sources such as small industrial or domestic 

sewage discharges within the estuary. The miscellaneous sources downstream from First Avenue South 

Bridge are combined and distributed uniformly along 

the lengths of the three most seaward elements of the 

top sublayer.

For every time step, the model performs the computation

-6

BOD-3 = BOD f 3 +BODM/(Nt 62.4x10

'max

I Vs ), (38)

n i 9 n 71 "'max Z

for i = jmax -2 to jmax, where BOD M is the miscellaneous BOD, in pounds per day, added to the estuary 

downstream from the First Avenue South Bridge; the 

numerical constant in equation 38 converts the units of 

the term in which it appears to milligrams per litre; 

and VSn is the volume of a sublayer element. 

The BOD of the freshwater flow at the wedge toe, 

is computed within the model by the equation

BODf =

BOD+Q^BOD^ + 0.185 BOD, R ^RTP RTF

-. (39)

RTF

The quantity BOD f has units of milligrams per litre 

and is the BOD of a mixture of (1) the mean daily flow 

of the Green River at the Tukwila gage, QR, with a 

concentration of BOD#; (2) the mean daily flow rate 

from RTF, QRTP, with a concentration of BODRTp; and 

(3) miscellaneous BOD inputs to the estuary upstream 

from First Avenue South Bridge, BOD r, expressed in 

pounds of BOD per day.

The model treats the decay of BOD as a first-order 

reaction and computes the change in BOD due to decay
12 NUMERICAL MODEL OF THE SALT-WEDGE REACH OF THE DUWAMISH RIVER ESTUARY

with the equation

ABODzJ = - (40)

where ABOD/j is the change in BOD of an element due 

to decay during a time step and ^BOD is a 

temperature-dependent decay rate. The oxygen consumed by the BOD is assumed to be proportional to 

ABOD;j. (See equation 45.)

The decay rate at 20°C, -K"20BOD> is data input to the 

model, and the decay rate at other temperatures is 

computed by the equation

tfBOD = #20BOD exp [0.046CT-200)], (41)

which is derived from information given by Fair, 

Geyer, and Okun (1971, p. 645).

DISSOLVED OXYGEN

The processes in the model that affect the DO 

(dissolved-oxygen) concentrations are the exchange of 

oxygen between the water and the atmosphere, commonly called atmospheric reaeration; the production 

and consumption of oxygen by phytoplankton; and the 

consumption of oxygen by BOD. These processes are 

modeled explicitly only in the upper layer. DO in the 

wedge is modeled by using an oxygen-consumption 

rate as was done by Stoner, Haushild, and McConnell 

(1974).

The effect of reaeration is computed only for the top 

sublayer by the linear equation

D0;;3 = Al, (42)

where the quantities DO '^3 and DOj; a are DO concentrations of elements in the top sublayer before and 

after the reaeration computation; r is a temperaturedependent reaeration coefficient; and DOS; 3 is the saturation concentration of the top sublayer element in 

segment i. The DO saturation concentration is computed by the equation

DOsi)3 = (487-2.65Si3)/(33.5+rij3), (43)

where Sij is the salinity in parts per thousand and TIJ 

is the temperature, in degrees Celsius, of sublayer 

element i,j.The equation was taken from a report by 

the Thames Survey Committee and the Water Pollution Research Laboratory (1964, p. 349), but one coefficient was changed to give a better fit to the data of G. 

C. Whipple and M. C. Whipple (American Public 

Health Association and others, 1971, p. 480).

The model computes the reaeration coefficient, r, by 

the equation

r = |/rf. 5/3) exp [0.024(7^,3-20)], (44) 

where i>j is the velocity of the water in the top sublayer

relative to the velocity of that in the wedge. The dimension of r in the equation is days -1 when vi is in feet 

per second and G?J is in feet. The first term in parentheses is an adaptation of an equation for the reaeration coefficient in homogeneous streams by M. A. 

Churchill, H. L. Elmore, and R. A. Buckingham, and 

the exponential function is a temperature correction 

made by H. G. Becker; both are reported by Fair, 

Geyer, and Okun (1971, p. 651).

To account for the effects of BOD and phytoplankton, 

the model changes the DO concentrations of sublayer 

elements according to the equation

DO, , = DO'; H-DOc ACP; ,+DOBOD ABOD , (45) <"ij ij <">j i,j

where the differences ACPjj and ABODjj are given by 

equations 26 and 40, respectively, and the coefficients 

DOc and DOBOD are the changes in DO concentration 

associated with unit changes in chlorophyll a concentration and BOD.

The DO concentration of each wedge element, DO^., 

is changed for every time step by the equation

where ADO^ is a rate of oxygen consumption in the 

wedge, which includes the effects of all dissolved-, 

suspended-, and benthic-oxygen demands.

APPLICATION OF THE MODEL TO THE 

DUWAMISH RIVER ESTUARY

GENERAL

Computations for the model of the Duwamish River 

estuary were made using historical input data for parts 

of the periods June-September 1967-69 and 1971. A 

second set of computations was made using most of the 

input data for 1971 and increasing the effluent discharge from RTP to the maximum magnitude expected 

in the future.

Parameters in the model that could be varied were 

adjusted to give the best agreement between computed 

and observed data for the 1968 period. These parameters were not changed for the other years.

INPUT DATA

FLOW MODEL

The items required by the model for computing flow 

in the estuary are listed in table 1. Values or equations 

for some items are given either in this table or in tables 

2 and 3. The estuary geometry, wedge-toe location, tide 

stages, tidal-prism thickness, upper-layer thickness at 

mouth, slope of upper layer-wedge interface, and 

freshwater inflow are identical to those used by Stoner, 

Haushild, and McConnell (1974). Data for the cross
APPLICATION OF THE MODEL TO THE DUWAMISH RIVER ESTUARY 13

TABLE 1. Data required for computing flow in the Duwamish River

estuary

Item Value

Geometry of the estuary _ _ 

Location of wedge toe _____ 

Tide stage ______________

Tidal-prism thickness (ft) _

Thickness of upper layer at 

mouth (ft) _________

Slope of upper layer-wedge 

interface (ft/ft) __________

Freshwater inflow at head of 

estuary (ft3/s)__________

Distribution of freshwater inflow among sublayers ____

Average rate of transport of 

saltwater upstream from 

wedge toe (fts/s)________

Distribution of saltwater inflow among sublayers ____

Entrainment velocity across 

upper layer-wedge interface (ft/s) ________________

Dimensions vertical velocities at base of sublayers 

Vertical diffusion coefficient

(ft2/s) __________________

See table 2. 

See table 3. 

See section "Input Data,"

paragraph 2. 

Pt ; see section "Input Data,"

paragraph 2.

D=2.5+0.3P,+0.003Qf. 

S0 = 0.00006.

Qf, see section "Input Data," 

paragraph 3.

Q'f.=0.l, 0.4, 0.5 U =1,2, 3).

Qs =200.

Qs =0.65, 0.20, 0.15(j =1, 2, 3).

U = -6.5xlO- 7 +3.25xlO-6 P, 

+4.5xlO- 8 Qf.

Ue. =1.0,0.8,0.5 (j = l,2,3). 

e v =0.0001.

TABLE 2. Relation of width to elevation at cross sections for computing 

geometry of the Duwamish River estuary

[Elevations are in feet above mean lower low water, and widths are in feet. The mouth is 

denned as 3,500 ft downstream from the Spokane Street Bridge.]

Cross-section 1, at mouth: 1

Elevation_______ -31 -20 -9 2 15 

Width ________ 175 302 378 476 640

Cross-section 2, 2,000 feet upstream from mouth:

Elevation_______ -31 -20 -9 2 15 

Width __________ 175 302 378 476 640

Cross-section 3, 4,000 feet upstream from estuary mouth:

Elevation________ -34 -22 -10 2 15

Width ____________ 50 319 485 590 706

Cross-section 4, 5,300 feet upstream from estuary mouth:

Elevation_______ -49 -33 -17 -3 15 

Width ____________ 11 640 984 1,068 1,148

Cross-section 5, 9,900 feet upstream from estuary mouth:

Elevation_______ -31 -20 -9 2 15 

Width ________ 304 425 528 642 700

Cross-section 6, 14,700 feet upstream from estuary mouth:

Elevation_______ -30 -19 -8 3 15 

Width ___________ 153 229 274 358 500

Cross-section 7, 19,000 feet upstream from estuary mouth:

Elevation _________ -19 -8 -1 6 15 

Width ____________ 128 368 435 506 587

Cross-section 8, 24,700 feet upstream from estuary mouth:

Elevation_______ -19 -8 -1 6 15 

Width ____________ 223 281 346 427 500

Cross-section 9, 28,700 feet upstream from estuary mouth:

Elevation_______ -12 -2 4 10 15 

Width ____________ 478 572 622 668 690

Cross-section 10, 31,100 feet upstream from estuary mouth:

Elevation_______ -11 -2 3 8 15 

Width _________ 229 242 252 267 280

Cross-section 11, 38,200 feet upstream from estuary mouth:

Elevation_______ -10 -1 4 9 15 

Width ______________ 120 141 162 183 200

'For modeling, this cross section was assumed the same as cross-section 2.

TABLE 3. Equations for computing location of wedge toe in 

Duwamish River estuary

[Equations were fit to data collected mostly when Qf was between 200 and 600 ft3/s, with some 

data between 600 and 1,100 ft3/s]

xt = longitudinal coordinate of wedge toe, in feet upstream from

estuary mouth.

Y~ = tide stage, in feet above mean lower low water. 

Qf = mean daily freshwater discharge, in cubic feet per second. 

Xl = 25,151 +1,626 Y-79.95Y2 

X2 = 22,188+1,210 Y-36.10Y2 

Z3 - 21,276+768 Y+0.60 Y2 

X4 = 20,400+638 Y 

X5 = 20,200+526 Y

xt = 19400,

*_=*! yss-1.6, Q,=s200 

yss-1.6, 200<Qf<400

yss-1.6, 400=£Qf<600 

yss-1.6, 600=sQf<1100 

yss-1.6, 1100=sQf<2000

sections were obtained from 1971 maps by the U.S. 

Army Corps of Engineers of the dredged part of the 

estuary and from measurements by the U.S. Geological 

Survey of cross sections upstream from the dredged 

channel. The slope of the interface and the equations 

used to compute the location of the wedge toe and the 

thickness of the upper layer at the mouth were determined from observed data for the Duwamish River estuary. The errors in the upper layer thickness and 

wedge-toe location computed with the equations and 

the error in the interface slope probably average less 

than 25 percent.

Tide stages at hourly intervals are input data for the 

model; they are computed using the data for Seattle, 

Wash., and the procedures published by the [U.S.] Environmental Science Services Administration (1967- 

71). The model computes the tide stage at each time 

step by linear interpolation between the hourly stages. 

The tidal-prism thickness, Pt, is used by the model to 

compute the upper layer thickness at the mouth and 

the entrainment velocity across the upper layer-wedge 

interface. Pt is defined here as the difference between 

the sum of the two daily high and the sum of the two 

daily low tide stages.

The mean daily freshwater inflow to the estuary, Qf, 

is the sum of the mean daily flow of the Green River at 

the Tukwila gaging station and the mean daily outflow 

from RTF, which was obtained from Metro records. 

Other freshwater inflows, such as that from the relic 

Black River, are negligibly small.
14 NUMERICAL MODEL OF THE SALT-WEDGE REACH OF THE DUWAMISH RIVER ESTUARY

The values of the other required parameters, Q'/y, Qs, 

Q'sj, Ue , U'ej, and ey given in table 1 were determined 

by trial and error. They were varied, within some logical limits, until computed salinities agreed with observed salinities for the three sublayers at 16th Avenue South, First Avenue South, and Spokane Street 

Bridges. In order to improve the agreement between 

computed and observed salinities at Spokane Street 

Bridge, the values of some of these parameters differ a 

little from those used by Stoner, Haushild, and 

McConnell (1974). These investigators did not verify 

model salinities with the data from Spokane Street 

Bridge because calculations of advective transport in 

the upper layer near the estuary mouth sometimes 

were unstable in the model they used. In general, the 

agreement between the computed and observed 

salinities at 16th Avenue South and First Avenue 

South Bridges is equal for the present and the earlier 

reported values of these parameters.

The values of U^ computed by the equation in table 1 

are 25 percent higher than comparable values used by 

Stoner, Haushild, and McConnell (1974); the dimensionless quantities, Q'/y, Q'8j, and U'ep also differ by 

about 25 percent. Vertical diffusion was not in the earlier model but was used in the present model to increase the computed salinities in the top sublayer near 

the mouth without further changing the flow ra.te 

there. A value for the vertical turbulent diffusion coefficient, y , was estimated by computing the average 

depth value for a homogeneous density flow (Jobson 

and Sayre, 1970) with a depth and velocity typical of 

the upper layer in the Duwamish River estuary and 

multiplying this value by about 0.01 to account for the 

stabilizing effect of the density stratification in the estuary. The reducing factor could only be estimated 

within one order of magnitude by extrapolating data 

summarized by Nelson (1972).

CONSTITUENT MODELS 

BOUNDARY CONDITIONS

The modeling of each constituent requires that the 

concentrations in the water flowing into the estuary at 

its mouth and at the toe of the wedge be known. These 

concentrations, called boundary conditions, are necessary input to the model and were determined from data 

for estuary water sampled by project personnel or automatic monitors at the locations shown in figure 1. 

The monitors, presently (1975) operated by Metro, record the conductivity, temperature, DO concentration, 

and other data at 6- to 12-minute intervals.

In the model, the boundary condition is the same for 

the wedge and the entire upper layer at the estuary 

mouth. Constituent concentrations there were usually

determined from the data for water sampled near the 

bottom at Spokane Street Bridge.

Boundary conditions at the wedge toe are required 

for only the upper layer. They are computed in the 

model by equation 12, which requires freshwater concentrations, cf, as input data to the model. Because 

transport and other processes act on the brackish 

water upstream from the wedge toe, it is seldom correct 

to equate Cf to the concentration observed in the freshwater part of the river. In this study, values of cf for 

temperature, DO, and chlorophyll a were calculated by 

using data collected in the vicinity of the wedge toe in 

the equation

(47)

cf =

Q'/Qf

This equation was derived by solving equation 12 for cf 

and deleting the j subscripts. The concentration ct in 

equation 47 is the constituent concentration at the 3-ft 

(1-m) depth at the computed location of the wedge toe 

and was determined by interpolation between observed 

concentrations for the same depth at 16th Avenue 

South Bridge and East Marginal Way Bridge, and occasionally at Boeing Bridge. The observed concentration near the bottom at 16th Avenue South Bridge was 

substituted for Ci. Values of Q's and Q'ffor the 3-ft 

(1-m) depth were estimated by interpolation in the vertical direction between the centers of the sublayers.

Thus, except for salinity or for some other conservative constituents with slowly varying or steady concentrations for which there are no sources in the reach 

between the freshwater part of the river and the wedge 

toe, the concentrations Cf are not necessarily the concentrations in the true freshwater inflow. The concentrations Cf are artificial concentrations computed from 

available observed concentrations so that when they 

are used in equation 12, the computed sublayer concentrations at the wedge toe, ctj, approximate the 

observed concentrations there. Changes in concentrations in the brackish waters between the end of 

freshwater flow and the toe are incorporated in the 

definition of Cf.

SALINITY

The only additional input data needed to model salinity are the salinities of the seawater and the river 

water. In the Duwamish River estuary they are 25 and 

0 ppt, respectively.

TEMPERATURE

Input data required by the temperature model are 

the temperatures of the seawater and of the freshwater 

inflow to the estuary, and the meteorological data that 

is used to compute heat transfer at the water surface.
APPLICATION OF THE MODEL TO THE DUWAMISH RIVER ESTUARY 15

The temperature of the sea is taken as the daily 

minimum temperature measured by the bottom 

monitor at Spokane Street Bridge. The temperature of 

the freshwater is hourly input data. These are 

computed by the method described earlier from 

data from the surface monitors at the 16th Avenue 

South and East Marginal Way Bridges.

The meteorological data required by the model are 

the total incident solar radiation for the day and the 

mean daily values of air temperature, wind speed, relative humidity, and a coefficient for computing incident long-wave radiation. These data were determined 

from information collected at the Seattle-Tacoma International Airport by the [U.S.] Environmental Science Services Administration (1967-69) and the [U.S.] 

National Oceanic and Atmospheric Administration 

(1971). The coefficient for computing incident longwave radiation was calculated using the information 

given by Koberg (1964) and the other meteorological 

data.

PHYTOPLANKTON

The data required by the phytoplankton model, 

which computes chlorophyll a concentrations, are 

listed in table 4. The chlorophyll a concentration in the 

seawater is typical of what was found in samples from 

Elliott Bay. The chlorophyll a concentrations of the 

freshwater are daily input data for the model. These 

concentrations were calculated using equation 47 and 

data for water sampled from the 3-ft (1-m) depth at 

16th Avenue South, Boeing, and East Marginal Way 

Bridges, and from near the bottom at 16th Avenue 

South Bridge. Data are available for samples collected 

twice daily about two times a week during the years 

1967-69. Although hourly data are preferred, the data 

were sufficient only to estimate daily values of CPf.

Usually, data were not available to evaluate directly 

many of the remaining parameters required in the 

phytoplankton model. Values in table 4 often were 

selected because they gave the best agreements between the computed and observed chlorophyll a and 

DO concentrations for the summer of 1968, No systematic attempt was made to determine the combination 

of parameters that give a statistically best agreement 

between computed and observed data; instead, decisions were made by visually comparing the plotted 

data. Attempts were always made to select values of 

the parameters within the range of those published in 

reports on research of the individual phenomena or in 

other reports on ecosystem models (for example, 

DiToro and others, 1971; or Chen and Orlob, 1972).

The phytoplankton settling velocity was chosen so 

that computed chlorophyll a concentrations agreed 

with observed concentrations in the wedge. The tabulated value is at the lower end of the range of the valTABLE 4. Data required for the phytoplankton model

Item Symbol Value

Chlorophyll a concentration

of seawater _____ CPS

Chlorophyll a concentration 

of freshwater inflow at

toe CPf Variable

k°

"'in

0.00002 ft/s

0.25 flr

0.125 ft- 1

0.005 ft'VOig chlorophyll a/1)

10 g-c 

0.004

Phytoplankton settling velocity __________________

Solar-radiation-attenuation 

coefficient in upper layer 

in the absence of phytoplankton (chlorophyll a)

Solar-radiation-attenuation 

coefficient in wedge in 

the absence of phytoplankton (chlorophyll a)

Change in solar radiationattenuation coefficient 

due to a unit change in 

chlorophyll a concentration - ____-_______

Maximum growth rate of

phytoplankton _____ G0 0.2 hr"

Solar-radiation intensity 

for maximum phytoplankton growth ______ L0

Respiration rate at 20°C __ R20

Oxygen produced during 

growth or consumed during respiration for a unit 

change in chlorophyll a 

concentration _____ DOc 0.125 mgO2//u,g chlorophyll a

ues reported by Hutchinson (1967, p. 277) for marine 

diatoms.

The solar-radiation attenuation coefficient in the absence of phytoplankton in the upper layer, k°u , was 

estimated, by using a few Secchi-disc depths in equation 48, which was suggested by Poole and Atkins (see 

Sverdrup and others, 1961, p. 92). The Secchi-disc 

depths averaged about 6 ft (2 m) and were obtained 

during periods of low chlorophyll a concentrations:

(48) K k° 

,, =

Secchi-disc depth .

The constant of 0.73 in place of 1.7 in this equation as 

given by Welch (1969, p. 15) was not used because the 

computed solar-radiation intensity at a depth of 10 ft 

(3 m) would be one-third the intensity at the surface. 

According to D. L. Todd of Metro, who scuba dives in 

the Duwamish River estuary, the light intensity seems 

much less at that depth (oral commun., Nov. 1973). 

Furthermore, if one used 0.73 in equation 48, the computed bottom of the photic zone (estimated by Ryther, 

1963, as the depth where the solar-radiation intensity 

is 1 percent of the intensity at the surface) would be 

about 40 ft (12 m). However, according to Welch (1969, 

p. 7), the photic zone is about 12 ft (4 m) deep.

Because Todd also noted that the water in the wedge 

was usually clearer than the water in the upper layer, 

the attenuation coefficient of the wedge in the absence
16 NUMERICAL MODEL OF THE SALT-WEDGE REACH OF THE DUWAMISH RIVER ESTUARY

of chlorophyll a was selected arbitrarily to be one-half 

the value used in the upper layer.

Data from the Duwamish River estuary were not 

available to estimate the increase in the attenuation 

coefficient due to the presence of phytoplankton. The 

chosen value of kc is an estimate based on data given 

by Megard (1973), Platt (1969), Ryther (1963), and 

Ryther and Yentsch (1957).

A maximum growth rate of 0.2 hr-i used in the 

model gave the best agreement between computed and 

observed chlorophyll a concentrations in the upper 

layer. This value is about twice the maximum value 

computed with data from Jitts, McAlister, Stephans, 

and Strickland (1964) after adjusting their data for the 

length of their light and dark cycles. A growth rate of 

0.2 hr-i is equivalent to about 3.5 doublings per 12- 

hour day, which is a little greater than the usual range 

of 1 to 3 doublings per day (Tailing, 1962, p. 744).

The solar-radiation (light) intensity for maximum 

growth, Lo = 10 g-cal-cm-2-hr-i, was selected so that 

maximum growth rates would occur during the clear 

summer days of the bloom periods. The computed daily 

average growth rate is a maximum at 3-ft (1-m) depth 

when the total daily solar radiation is 500 g-cal/cm2 

and the chlorophyll a concentration is about 45 /*,g/l 

(micrograms per litre). Jitts, McAlister, Stephans, and 

Strickland (1964) found that the optimum light intensity for cell division varied for different species and 

ranged from 4.5 to 18 g-cal-cm-2-hr- 1 . Strickland 

(1960, p. 11) reports that the optimum light intensity 

for photosynthesis is probably between 6 and 9 

g-cal-cm-2-hr- 1 . However, data by Ryther (1956) 

shows that optimum light intensities ranged from 

about 1 to 4 g-cal-cm-2-hr-i.

The respiration rate, R, was calculated by using an 

oxygen-use rate during respiration of 3.5 ml (millilitres) oxygen per hour per gram dry weight of 

phytoplankton (estimated from data in Gibbs, 1962, 

p. 64), a mass density of oxygen at 1.43 g/1 (grams per 

litre), a phytoplankton dry weight-to-chlorophyll a 

ratio of 100 (estimated from data in Strickland, 1960), 

and the tabulated value of DOC =0.125 mg (milligrams) oxygen per microgram chlorophyll a (obtained 

by the procedure described in the section "Production 

and Consumption of Oxygen by Phytoplankton." The 

data yield

#=3.5

mlOs

g dry weight/hour

1.43 g

1

100 g dry weight\ /^g chlorophyll a

A

" *-* J. / 

0.125 mg O2 

At maximum growth, the computed oxygen-

=0.004/hr.

production rate by photosynthesis is G0 DOc = 0.025 

mg oxygen per microgram chlorophyll a per hour. Data 

by Verduin (1956) yields 0.0064 mg oxygen per microgram chlorophyll a per hour. For a photosynthetic quotient (moles of CO2/moles of O2) equal to 1.2, the computed maximum production rate of carbon is 8 grams 

carbon per gram chlorophyll a per hour. Barlow, 

Lorenzen, and Myren (1963) report values between 8.7 

and 16.3 grams carbon per gram chlorophyll a per hour 

for a eutrophic estuary. However, both Welch (1969, 

p. 17) and Shimada (1958) report values of the order 4 

grams carbon per gram chlorophyll a per hour.

BIOCHEMICAL OXYGEN DEMAND

Data required by the BOD model are listed in tables 

5 and 6. The values for BOD§ and BOD/j were based on 

observed data. Mean daily values for the BOD of the 

RTF effluent were determined from plant records. The 

BOD added to the estuary by miscellaneous discharges 

(table 6) was estimated from data furnished by Metro 

(Cecil Whitmore, written and oral communs., 1973).

The decay rate, K20BOD =Q.25 day- 1 , was calculated 

with data obtained from laboratory analyses of RTF 

effluent by Metro (R.I. Matsuda, written commun., 

1973). The amount of oxygen consumed during the 

decay of a unit amount of BOD, DOBOD=l-4, is related 

to the decay rate through the definitions of a first-order 

reaction and the 5-day BOD. The relationship is

TABLE 5. Data for the BOD model

Symbol Value

BOD of seawater___________

BOD of Green River at Tukwila 

gage _ ___________ 

Mean daily BOD of RTF effluent ____________ __ __

Miscellaneous BOD added to estuary upstream of First Avenue South Bridge ____-___

Miscellaneous BOD added to estuary downstream of First 

Avenue South Bridge ______

BOD decay rate _________

Oxygen consumed per unit decay of BOD ________________

BODS 

BODR

BODn

v-20 A BOD

DOBOD

1 mg/1 

1 mg/1 

Variable

See table 6

See table 6 

0.25 day" 1

1.4

TABLE 6. Miscellaneous BOD inflows to the Duwamish River estuary from upstream, BOD^, and downstream, BOD M , from First 

Avenue South Bridge

[BOD is in pounds per day]

Year BODM

1967 _ __ _______

1968 ___ ___

1Q6Q

1971 _______ _____

__________ 2,757

____ _ _____ 824

____ _ _____ 762

____________ 762

6,685

6,685

6,400

0
APPLICATION OF THE MODEL TO THE DUWAMISH RIVER ESTUARY 17

=[l-exp (- 1-1 (49)

DISSOLVED OXYGEN

A list of the data required by the DO model appears 

in table 7. Both the DO concentrations of the saltwater, 

DOs, and of the freshwater, DOf, are hourly inputs-to 

the model. DOs is set equal to the concentration measured by the bottom monitor at Spokane Street Bridge, 

and DO/- is computed using equation 47 and data from 

the surface monitors at 16th Avenue South and East 

Marginal Way Bridges and the bottom monitor at 16th 

Avenue South Bridge.

The rate of oxygen consumption in the wedge, ADO^, 

is 25 percent greater than that reported by Stoner, 

Haushild, and McConnell (1974). The larger value was 

used because the entrainment velocity is higher and 

the residence time of water in the wedge is shorter in 

the present model than in the original model.

The method of selecting DOecD was described in the 

preceding section, and the procedure for estimating 

DOc is described in the section "Production and Consumption of Oxygen by Phytoplankton."

MODEL VERIFICATION

GENERAL REMARKS

Constituent concentrations for the Duwamish River 

estuary were computed by the model for most of the 

June-September periods of 1967, 1969, and 1971 and 

for most of the June-August period of 1968. In figures 

7-23, the computed temperatures, DO concentrations, 

and chlorophyll a concentrations in the upper layer for 

parts of these periods are compared with the appropriate observed data. The figures are grouped chronologically. Tide stages are plotted on the same figures 

as the temperature data.

No chlorophyll a data were available for 1971; therefore, the chlorophyll a concentration in the inflowing 

freshwater was assumed to be 1/xg/l. As a consequence,

TABLE 7. Data required for the DO model

Item Symbol Value Year

DO concentration of seawater__ DOS 

DO concentration of freshwater 

inflow at toe DO.

Oxygen consumed per unit decay of BOD DOBOD

Oxygen produced or consumed 

during the growth or respiration of phytoplankton _ DOC

Oxygen consumption in wedge ADOW

Variable

1.4

0.125 mg O2 /ng

chlorophyll a 

0.035 (mg/l)/hr 

0.029 (mg/l)/hr 

0.033 (mg/l)/hr 

0.020 (mg/l)/hr

 

1967 

1968 

1969 

1971

Observed, 3-foot (1-metre) depth 

Computed, top sublayer

FIGURE 7. Water temperatures and tide stages in Duwamish River 

estuary at times of high and low tides during July 1967.

Observed, 3-foot(l-metre)depth 

Compuled, top sublayer

10 15

AUGUST 1967

FIGURE 8. Water temperatures and tide stages in Duwamish River 

estuary at times of high and low tides during August 1967.
18 NUMERICAL MODEL OF THE SALT-WEDGE REACH OF THE DUWAMISH RIVER ESTUARY

UJ 

(£

H

_l

o:

UJ

o.

CO

Z

<

(£ 

§

o: 

o

Z

z 120

g no i-

;IOO

80

60

> 40

20

i

0 0 

i i i i 1 i i i i 1 i i i j-M-LJ ' 1 ' ' ' ' I ' ' ' ' I

1 L

Ii * i

r Li _ 1 i

Freshwater , i ' i

-

Computed top sublayer 

- Computed wedge --- 1 Observed: 1

3-foot(l-metre) depth A j 1

3 -feet (1-metre) above bed 1 i (1 .

lri!»

'I 1 ! I 1 ! Ill I

l/'/rJi / *\

. I6th Avenue '|| \/\

South Bridge fi ! ^ ' \ A

" A ^J^f^ \ '^^^ '

 trecyKf*-r'-}"H 4 4 4- l"+-~-h"f f 1 ~l "1 *"i '"i M'*'Y*T>'<- r''1^>l f"f"'Y~TI~

80

60

40

20

0

15 10 15 20 25 31

JULY 1967

FIGURE 9. Chlorophyll a concentrations in Duwamish River estuary at 16th Avenue South Bridge computed at times of high and

low tides and observed about twice a day two times per week

during July 1967.

UJ 

(£

1 

_J

t£ 

UJ

O.

CO

<

(£

0 

O

(£ 

O

Z

 z.

 £

0

5

H 100

UJ

z 80

o o

b 60

|j 40

o_ 

cc 20

O

-J

5 0

< i i < | i I i > | 1 1 1 1 | .1 i 1 i | I I i . | i i I i , 

Computed top sublayer ' t

~ Computed wedge - ll j. \

Observed: |1 f I , i\

3-foot(l-metre) depth A \\ n, M\ 4

3 -feet (1-metre) above bed P III A , I 1

1 l' I 1

f ' ' U A ^ i\l \'\ MY v\ First Avenue A ' \^

South Bridge /^ » \A /v. , /y v \ / \

*-"''VW'f , , . A /V^X^^ >J _^^-^-x..«:-^_.......... ... V---- -- -^-- ~4- ^^ ^v^-s_.%.--.4--.--~.--^>----

} ! < l| }

rjsii'fe r '' jih i\

Spokane Street A/ J | || jl [ll | r

Bridge Aj A lit ij |, ' l^'^^u 7

~z?sr~!~/~i ijf\ »i ^|,^(| \| ^'| | | -^ J- i- wj. L )l L ) ^i Wj j i j, J. L.

5 10 15 20 25 31

I£U

100

80

60

40

20

0

JULY 1967

FIGURE 10. Chlorophyll a concentrations in Duwamish River estuary at First Avenue South and Spokane Street Bridges computed at times of high and low tides and observed about twice a

day two times per week during July 1967.

the computed phytoplankton (chlorophyll a) concentrations were sufficiently low so that neither photosynthesis nor respiration affected the computed DO concentrations. Although comparisons are not shown, the

generally good agreements between computed and observed salinities were about the same as those of

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Computed wedge - H f 1 1 f 1 -

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3 -foot (1-metre) depth A A n A \\ 1 1 |\. j /| /i/' j l'| 1 1 : ft [ 

3 -feet (1-metre) above bed A / y v ' vj ^ ^\ ji V V [/ I/ I/ r I/

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>4\V?^1A^f >TfT 1 l.4|l jJln'lJl.^jtl hlln.KM-+

15 10 15 20 ^ 25 31

AUGUST 1967

FIGURE 11. Chlorophyll a concentrations in Duwamish River

40

20

0

80

60

40

20

0

estuary computed at times of high and low tides and observed about

twice a day two times per week during August 1967.

Stoner, Haushild, and McConnell (1974). Data for BOD

also are not shown; the computed and observed BOD

values were always low (about 2 mg/1 or less). The low

BOD had only a small effect on the computed DO concentrations.

For each period, graphs show the constituent concentrations in the freshwater inflow, which are input

data, and the observed and computed concentrations at

16th Avenue South and Spokane Street Bridges.

Chlorophyll a concentrations at First Avenue South

Bridge also are given. Although the model can supply

output concentrations for every 15 minutes, all graphs

were drawn through data plotted only for each high

and

the

low tide. At some low tides, the upstream end of

model (wedge toe) was downstream from 16th Avenue South Bridge. For these times, the computed concentrations at the wedge toe were plotted in place of

the computed data at 16th Avenue South Bridge. The

observed temperature and DO-concentration data are

from the automatic monitors that sample water from

about the 3 -ft (1-m) depth; the chlorophyll a data are

from samples taken from a similar depth and from

near the streambed. All computed data are for topsublayer elements which almost always contain the

observed data-collection points near the water surface.

Because 16th Avenue South Bridge is in the vicinity
APPLICATION OF THE MODEL TO THE DUWAMISH RIVER ESTUARY 19

to

_l

^20

z

z"l8

a. 16 iLU

o 14

en 6

Freshwater

Observed, 3-foot(1-metre)depth 

Computed, top sublayer _____

i i i i I i i i i I i i i i I i i

10 IS 

JULY 1967

20 25

FIGURE 12. Dissolved-oxygen concentrations in Duwamish River 

estuary at 16th Avenue South Bridge at times of high and low 

tides during July 1967.

20

18

O 16

4 LU

££'4

S^ 

o ._

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en

en 4

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Observed, 3-foot( 1-metre)depth 

Computed, top sublayer

Spokone Street 

Bridge

10 IS 20 

JULY 1967

25

FIGURE 13. Dissolved oxygen concentrations in Duwamish River 

estuary at Spokane Street Bridge at times of high and low tides 

during July 1967.

of the wedge toe, the differences between the observed 

and computed data at that station are measures of the

Observed, 3-foot(l-metre)depth 

Computed, top sublayer

10 IS

AUGUST 1967

25 3O

FIGURE 14. Dissolved oxygen concentrations in Duwamish River 

estuary at 16th Avenue South Bridge at times of high and low 

tides during August 1967.

*-. 6

Observed, 3-foot(l-metre)depth 

Computed, top sublayer

Spokane Street 

Bridge

10 IS 20 

AUGUST 1967

25

FIGURE 15. Dissolved oxygen concentrations in Duwamish River 

estuary at Spokane Street Bridge at times of high and low tides 

during August 1967.

errors in the boundary conditions at the wedge toe. The 

differences between the observed and computed data at 

Spokane Street Bridge at low tide are measures of the 

errors of the upper layer models. The concentrations at 

Spokane Street Bridge at high tide are strongly dependent on the concentrations in Elliott Bay.

In each graph, the concentrations at 16th Avenue
20 NUMERICAL MODEL OF THE SALT-WEDGE REACH OF THE DUWAMISH RIVER ESTUARY

Observed, 3-foot(l-metre)depth 

Computed, top sublayer

15 20 

JULY I96B

FIGURE 16. Water temperatures and tide stages in Duwamish 

River estuary at times of high and low tides during July 1968.

16th Avenue 

South Bridge Observed, 3-foot(l-mefre)depth 

Computed, top sublayer

15 20 

AUGUST I96S

FIGURE 17. Water temperatures and tide stages in Duwamish 

River estuary at times of high and low tides during August 1968.

LU

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LU 

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Freshwater 1

7

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South Bridge ~ _ A^ - jfc ^ ~ -"^^

' **"" * ~ ~* *

Computed top sublayer 

Computed wedge

Observed:

3-foot(l-metre) depth A

3-feet(l-metre) above bed ,

" First Avenue /

South Bridge A ^ A-'' 

_ £ ^- -^ ^- «=^*i^--«r«^p--s=--^p--^vr:^^. -^

-

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Bridge ./ 

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1 5 " 10 15 20 25 " 31

4U

20

0

60

40

20

0

JULY I96B

FIGURE 18. Chlorophyll a concentrations in Duwamish River CBtuary computed at times of high and low tides and observed about 

twice a day two times per week during July 1968.

LU

cc 

cc

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i p. n Computed wedge - 

A\ \ \ \ \ A . . . Observed: 

filljiM 16th Avenue - 

' L' 1' V I»A o xu o _, 3 -foot (hnetre) depth A 

; v 1; " » South Bridge _ . ;. ' . u _, _. 

' ' *AVA 3-feet(l-metre) above bed 

$.--' -'~~~'^,*^^.^f^^_^.^^f_^_^^^ ^^

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f ' A » Soutn Bridge

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\j ihk Spokane Street

- il A\ \ \ *S Bridge

IL u Jijlji Wi ^i ^CT^^hKl-'T-i- wr-J-Mj-r^t-j-^-^^-igJ

tv 

20

0

60

40

20

0

10 15 20 

AUGUST 1968

25

FIGURE 19. Chlorophyll a concentrations in Duwamish River estuary computed at times of high and low tides and observed about 

twice a day two times per week during August 1968.

South and Spokane Street Bridges follow the trends in 

the concentrations in the freshwater inflow. Variations 

within a day, due to the effects of tides or solarradiation intensity, are apparent in both the computed 

and observed data.

TEMPERATURE

The computed temperatures as shown in figures 7, 8,
APPLICATION OF THE MODEL TO THE DUWAMISH RIVER ESTUARY 21

Observed, 3-foot(l-metre)depth 

Computed, top sublayer

15 20 

JULY 1968

FIGURE 20. Dissolved oxygen concentrations in Duwamish River 

estuary at times of high and low tides during July 1968.

16, 17, and 22 usually agree with the observed temperatures within 2°C, and the daily averages usually 

agree within about 1°C. High temperatures typically 

are associated with the less saline water at low tides, 

and low temperatures with the more saline water at 

high tides. The association is expected because the 

seawater in summer is usually cooler than the river 

water. However, for some unknown reason, high temperatures occasionally are associated with high tide as 

observed at Spokane Street Bridge in August 1967 (fig. 

8); the water there is usually a few degrees cooler than 

it is at 16th Avenue South Bridge. The temperatures in 

the wedge, which are not shown in these figures, usually did not vary spacially by more than 1°C. During 

the summer months, the wedge temperatures were 

usually between 10° and 14°C.

CHLOROPHYLL a

Computed and observed chlorophyll a concentrations 

are compared in figures 9, 10, 11, 18, and 19. Three 

phytoplankton blooms were observed during these 

periods, one each in July and August 1967 and one in 

late July-early August 1968. High DO concentrations, 

often in excess of the saturation concentrations, are 

associated with each of the blooms (figs. 12-15, 20-21). 

During blooms, the chlorophyll a concentrations were

Observed, 3-foot(l-metre)depth 

Computed, top sublayer

16th Avenue 

South Bridge

Spokone Street 

* Bridge

' V VV,A.

VN-\~'

I i

15 20 

AUGUST 1968

25

FIGURE 21. Dissolved-oxygen concentrations in Duwamish River 

estuary at times of low and high tides during August 1968.

15

UJ LJ

O 111 < u- 5 

Z 0 

-5

2 16

3 14

5 16

Spokane Street 

Bridge

Observed, 3-foot (1-metre) depth 

Computed, top sublayer

10 15 20 

SEPTEMBER 1971

25 30

FIGURE 22. Water temperatures and tide stages in Duwamish 

River estuary at times of high and low tides during September 

1971.
22 NUMERICAL MODEL OF THE SALT-WEDGE REACH OF THE DUWAMISH RIVER ESTUARY

to 8

Freshwater

16th Avenue 

South Bridge

Observed, 3-foot(1-metre)depth 

Computed, top sublayer

_ Spokane Street

W$£^^

10 15 20 

SEPTEMBER 1971

25 30

FIGURE 23. Dissolved oxygen concentrations in Duwamish River 

estuary at times of high and low tides during September 1971.

much higher in the upper layer than in the wedge. The 

agreement between the observed and computed 

chlorophyll a concentration data is poorer than that for 

the temperature data. The larger errors in the phytoplankton model mostly are due to oversimplified 

mathematical descriptions (equations 26 through 31) 

of the biological processes and somewhat to the inaccuracies introduced by the insufficient data used to 

define the boundary conditions for chlorophyll a.

The phytoplankton blooms observed at 16th Avenue 

South, First Avenue South, and Spokane Street 

Bridges are simulated by the computed data. However, 

these blooms were not initiated within the modeled 

region of the estuary; the blooms were inputs to the 

model in the freshwater inflow and were maintained 

and advected through the estuary.

Chlorophyll a concentration in the upper layer also 

was modeled without phytoplankton growth in the estuary downstream of the wedge toe. Results of this 

simulation (not shown in the figures) indicated that 

chlorophyll a and DO concentrations in the upper layer 

were too low relative to the observed concentrations. 

Thus, phytoplankton in the estuary downstream of the 

wedge toe must grow sufficiently to balance the diluting effect of the water entrained from the wedge.

Data for samples taken at East Marginal Way 

Bridge at low tide when the water there was fresh and 

at Renton Junction where the water is always fresh 

rarely show chlorophyll a concentrations greater than 

Wfjig/l or supersaturated DO concentrations. However, 

the chlorophyll a concentrations in the freshwater inflow, which are artificial concentrations computed from 

observed data in the vicinity of the wedge toe, generally were much higher than 10 fjig/l during phytoplankton blooms, and the computed DO concentrations 

in the freshwater inflow often were above saturation 

levels (figs. 12, 14, 20, 21) during blooms. These differences between the actual freshwater concentrations 

and the artificial freshwater input concentrations 

suggest that the brackish-water reach upstream from 

the wedge toe is an incubator for the phytoplankton 

that bloom in the estuary. Thus, the biological processes in this brackish-water reach are important in determining the phytoplankton populations in the upper 

layer of the salt-wedge reach of the estuary.

The agreement between computed and observed 

chlorophyll a concentrations is better for the 1968 

bloom than for either of the blooms in 1967. In 1968, 

the observed bloom at both First Avenue South and at 

Spokane Street Bridges lasted as long as at 16th Avenue South Bridge, and the magnitudes of the observed 

chlorophyll a concentrations were about the same at 

the three stations. The input of high chlorophyll a concentrations in the freshwater inflow to the estuary during the relatively long 1967 period of blooms generally 

is confirmed by the observed concentrations at 16th 

Avenue South Bridge. However, during 1967, observed 

data show that blooms did not persist as long and 

chlorophyll a concentrations were not as high at First 

Avenue South and Spokane Street Bridges as at 16th 

Avenue South Bridge. For example, the August 1967 

bloom is barely discernible in the observed data for 

Spokane Street Bridge (fig. 11). Because the computed 

chlorophyll a concentrations at the two downstream 

locations closely follow the concentrations at 16th Avenue South Bridge, the computed concentrations at 

Spokane Street and First Avenue South Bridges are in 

error for the 1967 blooms.

The reasons for the lack of downstream persistence 

of the phytoplankton blooms in 1967 as compared to 

1968 are not know. Possible explanations include the 

following: (1) the species that bloomed in the 2 years 

could have been different, and those that bloomed in 

1967 perhaps could not grow well in the more highly 

saline water of the downstream part of the estuary; (2) 

herbivores that graze on phytoplankton could have 

been more numerous in the estuary in 1967 than in 

1968; and (3) a toxic substance that inhibits phytoplankton growth could have been present in the lower 

estuary during 1967 but not in 1968.

The few available data on the phytoplankton species 

present in the estuary during blooms appear in table 

12. Data for the 1968 bloom exist only for the station at 

First Avenue South Bridge and only for August 5. On 

that day, the most abundant taxa were oval flagellates 

that probably were of marine origin. Oval flagellates 

were also the most abundant taxa at First Avenue
APPLICATION OF THE MODEL TO THE DUWAMISH RIVER ESTUARY 23

South Bridge during the bloom on August 24,1967, but 

not 2 days earlier on August 22, 1967. On this earlier 

date Cyclotella sp., which is of freshwater origin, was 

most abundant.

The available data on the herbivores in estuary 

samples on a few days during the blooms of August 

1967 and August 1968 (table 13) show similar concentrations of herbivores during both blooms. The phytoplankton loss rate caused by grazing can be estimated 

by multiplying the filtering rate of an individual organism by the concentration of the grazing organisms. 

Hutchinson (1967, p. 528) reports filtering rates observed by L. A. Erman for the rotifer Brachionus calcyciflorus of 0.002 to 0.014 ml per hour per organism. 

Thus, the loss rate due to herbivore grazing is approximately

(0.005 ml hr-1 organisms" 1 ) (10 organisms ml- 1 ) = 

0.05 hr-i.

Because this value is one-fourth the maximum 

phytoplankton growth rate used in the model (0.2 hr- 1 ) 

and is probably nearly as large as the daily average 

phytoplankton growth rate, grazing by herbivores 

could be an important mechanism for controlling the 

phytoplankton populations in the estuary. However, 

because so few data exist on the concentration of herbivores in the water flowing into the estuary, modeling 

herbivore populations and their interactions with the 

phytoplankton in the Duwamish River estuary is not 

feasible at present.

No data from the Duwamish River estuary are 

available to support or refute the possibility of toxicants affecting the growth of phytoplankton.

DISSOLVED OXYGEN

Observed and computed DO concentrations are compared in figures 12-15, 20, 21, and 23. The data show 

daily peaks in the DO concentrations during phytoplankton blooms at both 16th Avenue South and 

Spokane Street Bridges. During the 1967 period of 

blooms, the period of computed high DO concentrations 

lasted a few days longer than the period of high observed concentrations, as did the period of the computed high chlorophyll a concentrations. Except during 

blooms the computed and observed DO concentrations 

in the upper layer usually agree within about 2 mg/1. 

The daily means often agree within 1 mg/1.

The wedge DO concentrations (not shown) computed 

with the present model are nearly identical with those 

computed previously by Stoner, Haushild, and McConnell (1974) with the earlier version of the model, and 

the computed and observed concentrations usually 

agree within 2 mg/1.

PREDICTION OF FUTURE 

DISSOLVED-OXYGEN CONCENTRATIONS

The model was used to estimate changes in DO concentrations in the estuary in response to an increase in 

effluent discharge from RTF, with June-September 

1971 used as the test period. One set of computations 

was made using historical input data as described in 

the preceding section. Another set of computations was 

made using much of this same data but with some 

changes to account for the effect of an increased flow 

from RTF. Four variables changed in the predictions 

were the flow rate and BOD of the RTF effluent, the 

DO concentration of the freshwater inflow, and the 

oxygen consumption rate in the wedge. The changes 

were estimates based on the designed capacity of RTF 

and on information provided by Metro personnel, who 

had given consideration to the available preliminary 

estimates furnished by their consultants.

For the predictions, the RTF effluent discharge was 

increased to 223 ft3/s (6.31 m3/s) as compared to an 

average of about 37 ft3/s (1.05 m3/s) during JuneSeptember 1971. The BOD of the future RTF effluent 

discharge was assumed to be 5 mg/1 as compared to the 

daily mean of about 3 mg/1 (range 1-11 mg/1) during 

June-September 1971. All freshwater DO concentrations, DOf, were decreased by 2 mg/1 to account for 

increased oxygen consumption between RTF and the 

wedge toe and an increased quantity of effluent with a 

low DO concentration in the freshwater inflow.

Stoner, Haushild, and McConnell (1974) estimate 

that oxygen in the wedge might be consumed at the 

rate of 0.27 mg/1 per hour for a future RTF effluent 

discharge of 223 ft3/s (6.31 m3/s). For this study their 

consumption rate was increased to 0.34 mg/1 per hour, 

a 25-percent increase, to account for the increased entrainment rate from the wedge as was described earlier.

Results for both sets of computations for the month 

of September appear in figure 24. The freshwater DO 

concentrations shown are all 2 mg/1 less than those 

determined from observed concentrations in September 1971 (fig. 23). The differences between DO concentrations computed with and without increased RTF 

effluent discharge for Septermber 1971 vary little with 

time but are less at Spokane Street Bridge than at 16th 

Avenue South Bridge. Spatial and temporal variations 

of differences for June-August 1971 (not shown in the 

figures) were similar to those for September.

The estimated monthly average decreases in DO 

concentrations in the Duwamish River estuary during 

June-September 1971 are given in table 8. These decreases are all less than the constant decrease of 2 mg/1 

assumed for the concentration in the freshwater inflow 

to the estuary, DO/-. Most of the computed difference is
24 NUMERICAL MODEL OF THE SALT-WEDGE REACH OF THE DUWAMISH RIVER ESTUARY

co 8

1 1 1

Freshwoter I

I6th Avenue South Bri |\ .,197 \

Spokone Street 

Bridge

1971 

Future

x1 1 1 1

10 15 20 

SEPTEMBER 1971

25 30

FIGURE 24. Dissolved-oxygen concentrations in the top sublayer of 

the Duwamish River estuary model at high and low tides during 

September 1971, computed for RTF effluent discharges for 1971 

(monthly average = 35.7 cubic feet per second or 1.01 cubic meters 

per second) and for the future (223 cubic feet per second or 6.31 

cubic meters per second).

caused by the decreased DO of the freshwater inflow 

and to the increased oxygen consumption in the wedge 

water which eventually is entrained into the upper 

layer. The increased consumption of oxygen by the increased BOD in the upper layer accounts for less than 

10 percent of the computed difference.

At 16th Avenue South Bridge in the upstream part 

of the estuary, the decreases in DO concentrations of 

the top and middle sublayers are influenced mostly by 

the decrease in DOy-, whereas the decreases in DO concentrations in the bottom sublayer respond more to the 

decreases in wedge DO concentrations. The average 

differences at 16th Avenue South Bridge (table 8) were 

computed using data at times of the high and low tides 

(usually a total of four times per day). At Spokane

TABLE 8. Estimated decreases in the monthly averages of computed 

DO concentrations in the Duwamish River estuary during JuneSeptember 1971 for an increase in the RTF effluent discharge to the 

probable future maximum

Station

16th Avenue South 

Bridge, averages 

for high and low 

tides.

Spokane Street 

Bridge, averages 

for low tides 

only.

Middle sublayer __ 

Bottom sublayer _ 

Wedge

Middle sublayer __ 

Bottom sublayer __ 

Wedge

June

1.9

1.7 

1.2

.7

1.6 

1.1 

.5 

.2

Decreases (mg/1) 

July Aug.

1.8 

1.7 

1.2

.8

1.6 

1.3

.7 

.3

1.7 

1.5 

1.3 

1.0

1.5 

1.2 

1.0 

.2

Sept.

1.6 

1.4 

1.2 

1.1

1.3 

1.1 

.7 

.2

Street Bridge near the mouth of the estuary, computed 

DO concentrations in the wedge and the sublayers at 

high tide are strongly dependent on the DO concentrations in Elliott Bay. Therefore, the average differences 

at Spokane Street Bridge (table 8) were computed 

using only data at times of the low tides.

Because of a lack of chlorophyll a data for 1971, 

chlorophyll a concentrations were not computed for 

1971 or for the future when the RTF effluent discharge 

rate increases. However, because the nutrient concentrations are believed to have been sufficiently high in 

1971 so as not to limit the growth of phytoplankton 

(see section "Supplemental Information"), the increased RTF effluent should not affect the phytoplankton concentrations in the estuary or the amount 

of oxygen produced or consumed by phytoplankton.

The predictions which were made using estimates of 

the probable future values of the input data and 

parameters indicate that the decreases in DO concentrations in the upper layer of the estuary will not be 

more than the decreases in DO concentration of the 

freshwater input at the wedge toe. This indication 

agrees with the relatively unimportant influence of 

BOD on DO concentrations in the sublayers noted in 

the modeling of the historical data. Lastly, the predictions were made using preliminary estimates of the 

effect of future conditions on model inputs and 

parameters; the confidence in the predictions will increase as these estimates improve.

SUMMARY AND CONCLUSIONS

A numerical model of a salt-wedge estuary developed by Fischer (1974) has been expanded and used 

to calculate the distributions of salinity, temperature, 

chlorophyll a concentrations, BOD, and DO concentrations in the Duwamish River estuary, King County, 

Wash. The section of the estuary included in the model 

extends from the estuary mouth to the toe of the saltwater wedge. The location of the wedge toe, and hence 

the upstream boundary of the model, is a function of 

river discharge and tide stage.

In the model, the estuary is divided vertically into 

the wedge and the upper layer, the latter in turn being 

divided into three sublayers. Longitudinally, the estuary is divided into about 35 segments; laterally, the 

estuary is assumed to be homogeneous. The watertransport processes modeled were longitudinal advection and dispersion in the wedge, entrainment from the 

wedge to the upper layer, longitudinal and vertical advection in the upper layer, and vertical diffusion in the 

upper layer. These transport processes were computed 

using conservation of volume equations, observed data, 

and predicted tide stages.

Additional processes in the model for computing
REFERENCES 25

temperature are heat addition to the water by solar 

radiation and heat transfer at the surface by long-wave 

radiation, evaporation, and conduction.

Phytoplankton are modeled as a chlorophyll a concentration. The growth of phytoplankton is computed 

as a function of temperature and light intensity; computed respiration is a function of temperature only. 

Observed nutrient concentrations are believed to be 

sufficiently high so that nutrients are not limiting the 

phytoplankton growth in the estuary. Therefore, nutrients and their effects on phytoplankton were not 

modeled. The few data on herbivores suggest that their 

numbers can be sufficiently high to affect the phytoplankton population; however, the data are insufficient 

to allow the modeling of herbivore populations and 

their effects on the phytoplankton.

Additional processes in the model for computing DO 

concentrations in the upper layer are oxygen transfer 

at the surface, consumption of oxygen by BOD, and the 

production and consumption of oxygen by phytoplankton during photosynthesis and respiration. In the 

wedge the model uses a constant oxygen-consumption 

rate. The BOD in the upper layer of the estuary is low, 

and its effect on the DO concentration is small. The 

production of oxygen by photosynthesis in the upper 

layer is sufficient to cause supersaturation levels of DO 

during phytoplankton blooms but is relatively unimportant at other times. The effect of phytoplankton 

respiration on DO concentrations in the Duwamish 

River estuary is small.

The model was calibrated with data for the summer 

months of 1968 and verified with data for the summer 

months of 1967, 1969, and 1971. At any given time the 

observed and computed temperatures in the upper 

layer at two stations in the estuary agree within about 

2°C, and the daily average temperatures usually agree 

within 1°C. The computed chlorophyll a concentrations 

increased and decreased with the observed data during 

a phytoplankton bloom in 1968, but the computed 

blooms in 1967 persisted farther downstream and 

lasted a few days longer than did the observed blooms. 

The blooms are initiated in the brackish waters upstream from the wedge toe and are advected into the 

modeled reach through the upstream boundary. Thus, 

the biological processes in the brackish reach upstream 

from the wedge toe (outside the model boundaries) are 

important in determining the phytoplankton populations within the modeled reach of the estuary.

Except during bloom periods the computed and observed DO concentrations agreed within about 2 mg/1. 

During blooms both the computed and observed DO 

concentrations reached supersaturated levels; however, the error in the computations during the blooms 

is larger than during periods without blooms.

The effect on the DO concentration of increasing the 

discharge of treated RTF sewage effluent to the proposed future maximum (223 fts/s or 6.3 m3/s) was estimated with the model. When using the summer data of 

1971 as a base, the computed monthly average DO concentrations in the estuary decreased by 2 mg/1 or less. 

Because the concentrations of nutrients in the 

Duwamish estuary are presently believed to be sufficiently high so as not to limit the growth of phytoplankton, the increase in nutrient concentrations 

caused by the increased amounts of sewage in the estuary should not affect the phytoplankton growth 

rates.

REFERENCES

American Public Health Association, American Water Works Association, and Water Pollution Control Federation, 1971, Standard methods for the examination of water and wastewater [l3th 

ed.]: Washington, B.C., 874 p.

Barlow, J. P., Lorenzen, C. J., and Myren, R. T., 1963, Eutrophication of a tidal estuary: Limnology and Oceanography, v. 8, no. 2, 

p. 251-262.

Bella, D. A., and Grenney, W. J., 1970, Finite difference convection 

errors: Am. Soc. Civil Engineers Proc., Jour. Sanitary Eng. Div., 

v. 96, no. SA6, p. 1361-1375.

Chen, C. W., and Orlob, G. T., 1972, Ecologic simulation for aquatic 

environments, final report: Walnut Creek, California, WTater 

Resources Engineers, Inc., 156 p.

Cupp, E. E., 1943, Marine plankton diatoms of the west coast of 

North America: Scripps Inst. Oceanography Tech. Ser. Bull. 5 

(D, 238 p.

Dawson, W. A., and Tilley, L. J., 1972, Measurement of salt-wedge 

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26 NUMERICAL MODEL OF THE SALT-WEDGE REACH OF THE DUWAMISH RIVER ESTUARY

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SUPPLEMENTAL INFORMATION
28 NUMERICAL MODEL OF THE SALT-WEDGE REACH OF THE DUWAMISH RIVER ESTUARY

PHYTOPLANKTON

GENERAL

Slack, Averett, Greeson, and Lipscomb (1973) define phytoplankton as the community of suspended or floating plant organisms that 

drift passively with water currents. The Duwamish River estuary 

contains phytoplankton species of both freshwater and marine origin; the former enter the estuary with the water from the GreenDuwamish River, and the latter enter from Elliott Bay. Only suspended phytoplankton are discussed in this report; phytoplankton 

that float on the surface are not of importance in the Duwamish 

River estuary at present.

During their stay in the estuary, phytoplankters may grow, produce oxygen by photosynthesis, decrease in mass and consume oxygen by respiration, be transported by currents, settle, and be eaten 

by grazers. The eventual fate of the phytoplankton not eaten-by 

grazers in the estuary is either being transported into Elliott Bay or 

becoming a part of the streambed sediment.

Phytoplankton growth and photosynthesis occur mainly in the 

upper layer of the estuary. Although the top part of the saltwater 

wedge is within the photic zone, light in most of the wedge is insufficient to support active phytoplankton photosynthesis (Welch, 1969). 

The growth of phytoplankton and the production of oxygen by photosynthesis are the important biological processes in the model. Although modeled, the use of oxygen and the decrease in biomass attributed to respiration are less important biological processes in the 

estuary. Grazing was not modeled. Phytoplankton photosynthesis 

increases DO concentrations in the estuary significantly during 

blooms but has a negligible effect on DO concentration during 

nonbloom periods. A phytoplankton bloom, as defined in this report, 

is either 0.5 million or more cells per litre (as denned by Slack and 

others, 1973) or a chlorophyll a concentration in excess of 4 jU.g/1 (as 

defined by Welch, 1969) during a continuous period longer than a 

day.

SAMPLE COLLECTION, PREPARATION, 

AND ANALYSIS

The concentrations of chlorophyll a and other pigments were determined for water samples collected in the Green-Duwamish River 

and in Elliott Bay. The plant-pigment data were supplemented by 

phytoplankton cell counts and taxonomic identification for selected 

samples.

The river samples were collected during 1967-69, usually at six 

stations starting at Spokane Street Bridge and ending at the Renton 

Junction monitor (fig. 1). The Elliott Bay samples were collected by 

Metro personnel (under the direction of R. I. Matsuda) at four stations during the April-June periods of 1967-69. Bay samples were 

collected from depths of about 3 ft (1 m) and river samples were 

collected from about 3 ft (1 m) below the water surface (surface 

samples), or about 3 ft (1 m) above the streambed (bottom samples). 

Samples at most stations were collected with the Emsworth version 

of a 4-litre Van Dorn sampler. However, many river samples were 

obtained from the pumped stream supplying the automatic waterquality monitors.

Immediately after sample collection, a 125 ml aliquot was preserved with Lugol's solution for cell counting and identification. 

Phytoplankton were identified and counted by W. A. Dawson and L. 

J. Tilley using the inverted-microscope technique (Slack and others, 

1973).

For analysis of plant pigments, volumes between 0.5 and 4.0 litres 

were filtered through 0.45-jum membrane filters. The river samples 

were filtered at about one-half atmosphere of positive pressure 

within 1 hour of the time of collection. The bay samples were subjected to slightly greater delay before filtration in a vacuum apparatus at no more than one-fifth atmosphere negative pressure. The

folded filters containing the phytoplankton were stored in desiccators at 5°C. Acetone extracts from the filter and samples were 

analyzed for plant pigments by the method of Richards with 

Thompson (1952), as modified by Strickland and Parsons (1968).

The work of Ward and Whipple (1959) was the most useful reference for identification of freshwater organisms. Marine organisms 

were identified with the aid of the works of Cupp (1943), Wailes 

(1939), Hustedt (1930), Schiller (1930), Leegaard (1915), and Kofoid 

and Campbell (1929).

ABUNDANT PHYTOPLANKTON TAXA

The most abundant organisms found in Elliott Bay samples (table 

9) were Skeletonema costatum and Thalassiosira sp. The samples 

listed in table 9 were selected for taxonomic analysis to represent 

times of peak chlorophyll a concentrations in the surface waters of 

Elliott Bay near Seattle during the April-June periods of 1967-69 (R. 

I. Matsuda, Metro, oral commun., 1971). Although the data in table 9 

are for spring months, the two dominant genera probably enter the 

Duwamish River estuary with the bay water not only in the spring 

but in summer and early fall as well. Welch (1969) identified 

Skeletonema as the most abundant taxa in samples from 16th Avenue South Bridge during a bloom in early August 1965 and speculated that an abundant diatom might be Thalassiosira in a 

later bloom during the same month. Skeletonema was one of the 

abundant organisms in samples from some estuary stations during 

blooms in August 1967 (table 12).

The list of abundant freshwater organisms in samples from the 

Green-Duwamish River during March-August 1967 (table 10) shows 

a diversity of dominant genera. Several genera appear in both listings of abundant taxa in samples from the two stations in the freshwater part of the river (tables 10, 11). Many organisms found in the 

freshwater samples from the river must have been periphyton which 

had been dislodged upstream and subsequently carried downstream 

by the river currents.

The abundant organisms in samples from six stations on the estuary and river during 2 days of an August 1967 bloom and from one 

estuary station during 1 day of an August 1968 bloom are listed in 

table 12. The diversity in the abundant river taxa is again evident by 

the data for samples from the Renton Junction monitor station and 

East Marginal Way Bridge. The marine diatom, Skeletonema costatum, was one of the abundant species only in the samples collected 

from the two farthest downstream stations on August 22, 1967.

Cyclotella sp. and oval flagellates, along with "coccoids," dominate 

the abundance listing for the estuary during the August 1967 

blooms. Cyclotella sp., a freshwater diatom, was not numerous 

enough to be listed among the abundant freshwater organisms (tables 10, 11); it ranked about ninth in abundance of the freshwater 

autotrophic organisms found in the Green-Duwamish River. The 

oval flagellates were probably marine organisms from Elliott Bay; 

oval flagellates were the 20th most abundant autotrophic organism 

found in the bay samples of the April-June periods of 1967-69. However, because the taxonomic analyses of the estuary samples may not 

have distinguished between freshwater and marine oval flagellates, 

the probable origin of these microorganisms is designated in table 12 

as "saltwater(?)" to indicate some uncertainty. Regardless of their 

origin, both Cyclotella sp. and oval flagellates flourished in brackish 

estuary water having a wide range in salinity.

The longitudinal distributions of Cyclotella sp. (fig. 25) indicate a 

peak concentration in the brackish surface water between Boeing 

Bridge (mile 6.5 or kilometre 10.5) and First Avenue South Bridge 

(mile 3.4 or kilometre 5.5). Downstream of this reach, the consistently lower concentrations at Spokane Street Bridge indicate that 

the prevailing high salinity there may limit the growth of this diatom. Upstream of the peak-concentration reach, the concentration
SUPPLEMENTAL INFORMATION 29

TABLE 9. Abundant taxa in samples collected at four stations in 

Elliott Bay near Seattle during 1967-69

[Phytoplankton samples collected by R. I. Matsuda, Municipality of Metropolitan Seattle, 

and counted and identified by W. A. Dawson and L. J. Tilley]

Chlorophyll a Concentration

tration Percentage Presence 

Date (jug/1) Taxon Cells/ml of total of bloom

1967

May 17 4.1 Skeletonema costatum _ ._ _ 1,500 74 Yes

Thalassiosira sp _ 430 22 No 

1968 

Apr. 29 9.8 Skeletonema costatum ______ 610 41 Yes 

Thalassiosira sp __ _ 460 31 No

Apr. 29 3.3 Skeletonema costatum - 850 75 Yes

Flagellate, oval _ ____ __ 170 15 No

May 27 11.4 Skeletonema costatum ______ 7,700 88 Yes

May 27 9.2 Skeletonema costatum _ 2,400 68 Yes

June 3 11.7 _ _ do 1 600 66 Yes

Skeletonema costatum ______ 360 15 No 

1969 

May 14 4.6 __ _ _ ___ _ do 12,000 98 Yes

May 14 3.6 ___ ______ do ____ ______ 12,000 98 Yes

May 21 2.3 __ _ ______ do ____ ____ _ 4,700 91 Yes

Chaetoceros sp 370 7 No

May 21 3.5 Skeletonema costatum ______ 4,700 86 Yes 

Chaetoceros sp 660 12 Yes

TABLE 10. Abundant taxa in 1967 freshwater samples from two 

stations on the Green-Duwamish River

[Taxa counted and identified by W. A. Dawson and L. J. Tilley]

Chlorophyll a Concentration

Time tration Percentage Presence 

Date (P.s.t.) (fig/1) Taxon Cells/ml of totals of bloom

Renton Junction Monitor (mi 13.1, km 21.1

Mar. 14 1020 0.8 Unidentified blue-green 2,500 50 Yes 

algae. 

Apr. 18 1035 3.8 Hannaea sp _ _ _ _ 34 27 No

May 12 1200 2.5 Pennate diatoms _ 90 17 No

24 1425 3.6 Unidentified blue-green algae 7,800 71 Yes 

June 8 1245 1.8 Chrysococcus sp _ 350 23 No

27 0805 3.6 "Coccoids" and clusters __ 2,300 40 Yes 

July 5 0820 4.0 Oscillatoria sp _ __ - 1,700 34 Yes

11 0800 3.7 "Coccoids" and clusters __ 1,500 29 Yes 

25 1245 4.5 Flagellate spp __________ 2,200 31 Yes

Aug. 22 1630 2.7 "Coccoids," solitary _ __ 5,600 61 Yes

East Marginal Way Bridge 1 (mi 7.8, km 12.6)

Mar. 14 1040 0.5 Crenothnx sp ____ ___ ___ 700 87 Yes

May 12 1245 2.5 Synedra sp __ _ ____ _ 54 13 No

24 1020 2.2 Oscillatoria sp _ __ _ _ 190 36 No

June 20 0830 2.2 _ _ ___ ____ do __ __ __ 930 38 Yes

July 5 0845 3.3 "Coccoids" and clusters __ 840 25 Yes 

25 1225 6.0 Flagellate sp ____________ 3,900 36 Yes

'Samples obtained during low tides when river did not contain saltwater here.

of Cyclotella sp. is lower at East Marginal Way Bridge, mile 7.8 or 

km 12.6 (concentration being tide-dependent here), and decreases 

rapidly thereafter with distance upstream. 

Longitudinal distributions of oval flagellates (fig. 26) indicate peak 

concentrations downstream from Boeing Bridge (mile 6.5 or 

kilometre 10.5) with the concentration at Spokane Street Bridge 

(mile 1.2 or kilometre 1.9) either higher or only slightly less than 

concentrations at adjacent upstream stations. The rapid decrease in

TABLE 11. Five most abundant taxa in freshwater samples obtained 

at two stations on the Green-Duwamish River during MarchAugust 1967

[Taxa counted, identified, and ranked by W. A. Dawson and L. J. Tilley]

Taxon and sampling station

Abundance Renton Junction Monitor East Marginal Way Bridge 

ranking (mi 13.1, km 21.1) (mi 7.8, km 12.6)

1 "Coccoids" and clusters _ Oscillatoria sp.

concentration of oval flagellates with distance upstream from Boeing 

Bridge suggests that they are marine organisms reacting to the increasingly fresher water. In general, the concentration of "coccoids" 

decreases downstream and, therefore, with increasing salinity

(fig. 27).

PRODUCTION AND CONSUMPTION OF OXYGEN 

BY PHYTOPLANKTON

When blooming, the phytoplankton in the upper layer of the 

Duwamish River estuary produce more oxygen by photosynthesis 

than they consume by respiration, frequently resulting in DO supersaturation. DO supersaturation has not been observed in the wedge. 

In the model the production of oxygen during photosynthesis is 

assumed to be proportional to the increase in chlorophyll a concentration caused by phytoplankton growth. Similarly, the consumption 

of oxygen during respiration is assumed to be proportional to the 

decrease in chlorophyll a concentration. The coefficient of proportionality was assumed to be the same for both processes, which implies a photosynthesis-to-respiration ratio of unity for stable populations. The coefficient of proportionality relating the increase of DO to 

chlorophyll a, which is denoted by DOc in the main body of this 

report, was estimated from data obtained in the Duwamish River 

and estuary during phytoplankton blooms. The procedure used is as 

follows: 

1. The salinity, DO concentration, and the chlorophyll a concentration, Sf,, DOb, and CPf,, were observed at a point in the river or 

estuary where there was a bloom. The salinity and DO concentration were observed at two other points, one upstream from 

the first where there was no bloom, Su and DOU, and the other in 

the wedge, Sw and DOj,,. The chlorophyll a concentrations at the 

second two points usually were not observed but were known to 

be a small fraction of the concentration at the first point. 

2. The relative volumes of water from the upstream point, Vu , and 

from the wedge point, Vw , required to produce a mixture of 

salinity, Sf,, was computed as

V

Ow~ob j -IT 1 TT

u - o o and VW -1-VU .

ow ou

3. The DO concentration of this mixture, DOOT , was computed as 

DOm = Vu DOU + Vw DOW .

4. The excess DO concentration, defined as ADO=DO6 -DOm, was 

plotted as a function of the chlorophyll a concentration CPf, (fig. 

28). 

5. If one assumes that the excess DO concentration is the net increase in oxygen concentration due only to phytosynthesis and 

respiration of the phytoplankton during the production of all 

the chlorophyll a at the bloom point, then, the slope of the line 

fit to the data in figure 28 is an average measure of the desired 

coefficient of proportionality, DOc- The available data from the
30 NUMERICAL MODEL OF THE SALT-WEDGE REACH OF THE DUWAMISH RIVER ESTUARY 

TABLE 12. Abundant taxa during two blooms in the Green-Duwamish River

[Phytoplankton counted and identified by W. A. Dawson]

Date Station

1967 

Aug. 22 Renton 

Junction 

Monitor. 

East Marginal Way 

Bridge.

Boeing 

Bridge.

16th 

Avenue 

South 

Bridge.

First 

Avenue 

South 

Bridge.

Spokane 

Street 

Bridge.

Aug. 24 Renton 

Junction 

Monitor.

East Marginal Way 

Bridge.

Boeing 

Bridge.

16th 

Avenue 

South 

Bridge.

First 

Avenue 

South 

Bridge.

Time 

(P.s.t.) 

and tide 

stage

1530 

Midflood

0930 

Late ebb

1520 

Midflood 

0915 

Late ebb 

1425 

Midflood

0845 

Midebb

1400 

Early 

flood 

1405 

Early 

flood 

0825 

Midebb

1340 

Early 

flood 

1345 

Early 

flood 

0735 

Early 

ebb 

1305 

Early 

ebb

0755 

Early 

ebb 

1310 

Early 

flood 

0930 

Early 

ebb 

1540 

Early 

flood

0190 

Early 

ebb 

1525 

Early 

ebb

0830 

Start 

of ebb 

1440 

Start 

of flood

0805 

Low high

0815 

Low high 

1415 

Low low

1425 

Low low

0740 

Low high

0750 

Low high 

1345 

Low low

1355 

Low low

Depth 

location1

do ____.

______ do ____

______ do ____

______ do ____

--- do ____

_ __. do 

______ do ____

______ do __

__ __ do 

 __ do .___

-____do ____

_ __ do ____

 do

 do _ _

_ dc . _

Salinity 

(ppt)

16.7

12 9

9ft ^

1 q A

27.4

0

0

.2

27.7

7.4

16.7

11.5

OO 1

Chlorophyll a 

concentration

(ft/1)

2.6 

10.5

11.3 

41.3 

21.9

5.6

37.8 

3.5 

6.6

15.6 

1.2 

2.0 

5.5

1.8 

.9 

6.7 

9.6

21.9

8.4

47.8 

13.2

33.7

3.9 

58.9

5.6 

10.0

2.6 

66.8

1.6

Taxon

("Coccoids" and clusters) __ __

("Coccoids" and clusters) _

("Coccoids" and clusters) __ 

("Coccoids" and clusters) __ __

("Coccoids" and clusters) ____

Flagellate, oval 5-7 fam __ __ 

("Coccoids" and clusters) __ __

("Coccoids" and clusters) ____

("Coccoids" and clusters) ____

("Coccoids" and clusters) ____

Skeletonema costatum _ __

("Coccoids" and clusters) -___ 

Flagellate, oval 6-8 fim __--

("Coccoids" and clusters) ____

("Coccoids and clusters) ____

("Coccoids" and clusters) _-__

("Coccoids" and clusters) _-__

("Coccoids" and clusters) __

("Coccoids" and clusters) __ _

("Coccoids" and clusters) __ __ 

Flagellate, oval 5-7 fam __ __

Flagellate, oval 5-7 fam __ __

("Coccoids" and clusters) ____

Flagellate, oval 5-7,um _ 

("Coccoids" and clusters) _ _

Concentration

Percentage Probable Presence 

Cells/ml of total2 origin of bloom

710 

510 

(6,700) 

5,300 

410 

(2,800) 

1,600 

980 

31,000 

(1,400) 

33,000 

1,200 

(3,900) 

13,000 

(3,700)

57,000 

510 

(1,800) 

690 

110 

(1,200) 

13,000 

1,000 

(2,400)

39,000 

(2,000)

550 

(55)

1,400 

1,200

2,000 

1,100 

550 

(310) 

200 

97 

89 

61 

23

1,100 

(990)

3,400 

(12,000)

11,000 

8,900 

(5,800) 

2,600 

2,100 

470 

(4,000) 

80,000 

2,200 

(5,400) 

4,600 

2,400 

710 

(3,300) 

77,000 

150 

110 

(42) 

1,700 

(190) 

66,000 

3,000 

(360) 

1,100 

250 

(91)

12,000 

(140)

840 

(25) 

110,000 

1,000 

(510) 

37 

(15)

42 

30 

(398) 

88 

7 

(47) 

56 

35 

98 

(4) 

93 

3 

(11) 

98 

(3)

98 

1 

(3) 

71 

11 

(123) 

92 

7 

(18)

99

(5)

77 

(8)

54 

46

50 

26 

14 

(8) 

39 

19 

18 

51 

19

91

(85)

90 

(321)

53

44 

(28) 

46 

38 

8 

(72) 

97 

3 

(7) 

61 

32 

10 

(45) 

99 

<1 

<1 

«D 

87 

(1) 

94 

4 

(1) 

61 

14 

(5)

99

«1)

82 

(2) 

99 

<1 

«1) 

83 

(34)

do

 _ do -

_______ do - 

_ _____ do - -

________ do ----- 

_______ do 

_ _ do -----

 _ _ do

do

 ___ do -

__-___. do -

__ __ do - -

 -- do - 

__-_ do -----

__ ___ do ____.

--___ _ do __ _ ...

 __ do _ __ _

__ __ __ do ______

 __- do ________

--__- do

________ do 

________ do - _ 

________ do _ _

Freshwater

do ________

Saltwater(?) . __ _

Saltwater(?) __ ._

do ________

________ do ________

 do __-__-._

_ _ do ________

__ ___ do ________

_ _____ do ________

________ do ________

________ do ____ 

 do _.__ __

___ ___ do ___ ___

________ do ... _ __

do ___ __

Saltwater(^)

_ _____ do _____ _

. do ________

____ __ do ________

__ _____ do __ _ ___

Salt wa ter(? )

Freshwater

Saltwater(?) _______

__ ___ do _ _ _

Saitwaterf?) _ _ _

Freshwater .

Yes 

Yes 

Yes 

Yes 

No 

Yes 

Yes 

Yes 

Yes 

Yes 

Yes 

Yes 

Yes 

Yes 

Yes

Yes 

Yes 

Yes 

Yes 

No 

Yes 

Yes 

Yes 

Yes

Yes 

Yes

Yes

No

Yes 

Yes

Yes 

Yes 

Yes 

No 

No 

No 

No 

No 

No

Yes 

Yes

Yes 

Yes

Yes 

Yes 

Yes 

Yes 

Yes 

No 

Yes 

Yes 

Yes 

Yes 

Yes 

Yes 

Yes 

Yes 

Yes 

No 

No 

No 

Yes 

No 

Yes 

Yes 

No 

Yes 

No 

No

Yes 

No

Yes 

No 

Yes 

Yes 

Yes 

No 

No
SUPPLEMENTAL INFORMATION 

TABLE 12. Abundant taxa during two blooms in the Green-Duwamish River Continued

31

Date Station

Spokane 

Street 

Bridge.

1968 

Aug 5 First 

Avenue 

South 

Bridge.

Time 

(P.s.t.) Depth 

and tide locastage tion1

0645 Surface

Low high

Low high

Low low

Low low 

0940 Surface _______

Low low 

1830 __ do _

Early 

ebb

Chlorophyll a 

concenSalinity tration 

(ppt) (/u/1) Taxon

("Coccoids" and clusters) ____

27.3 26.0 Flagellate, oval 6-10 /urn _ _

Concentration

Percentage 

Cells/ml of total2

25,000 

300

36 

23 

(6) 

18,000 

1,800 

(15,000) 

88 

83 

31

19,000 

720

16,000 

480

97 

1

40 

26

(7) 

90 

9 

(74) 

41 

38 

14

95 

4

96 

3

Probable 

origin

Saltwater(?) . __

Freshwater _

Saltwater(?) _ __

Saltwater _ . ._

Saltwater*?) -__

Freshwater __ _

________ do ____

Saltwatert?) ____

________ do ____

Saltwater(?) _

Saltwater(?) _

Presence 

of bloom

 _ Yes

____ No

____ No

 . No

____ No

____ Yes

__._ Yes

____ Yes

____ No

____ No

____ No

 . Yes

.___ Yes

.... Yes

____ No

'Surface and bottom indicate about 3 feet (1 m) below water surface and about 3 feet (1 m) above streambed, respectively.

"Total is sum of concentrations for the various taxa counted but does not include concentration of "coccoids" and "coccoid" clusters.

GREEN-DUWAMISH RIVER KILOMETRE 

5 10 15 20

100,000

10,000

1000

100

August 22, 1967 

Flood tide

August 22, 1967 

Ebb tide

August 24, 1967 

Low high and 

early ebb tide

August 24, 1967 

Low low and 

early flood tide'

5 10 

GREEN-DUWAMISH RIVER MILE

15

Duwamish River estuary yield DOc=0.125 mg oxygen per microgram chlorophyll a.

ESTIMATED INFLUENCE OF NUTRIENTS ON 

PHYTOPLANKTON GROWTH

The effluent from RTP considerably increases concentrations of 

ammonium and total and soluble phosphate in the Duwamish River 

estuary (Welch, 1969, fig. 8; Tilley and Dawson, 1971, fig. 3) but does 

not greatly increase the concentration of nitrate (Welch, 1969, fig. 8; 

Santos and Stoner, 1972, p. 62-63). The nutrients contributed by the 

RTP effluent have the potential of increasing the phytoplankton 

biomass in the estuary. Welch (1969) reported that the addition of 

RTP effluent to Duwamish River estuary samples increased the 

population of a green-algae population (Scenedesmus sp.) when the 

samples were incubated in flasks under uniform light of about 7,000 

lux and a temperature of 20°-21°C, without mixing for 5 or 6 days. In 

one series of tests, the maximum assimilated carbon-14 (a measure 

of the biomass produced) in samples containing 5-, 10- and 25- 

percent RTP effluent was about 14, 30, and 36 percent greater than 

in samples without effluent. Welch (1969) also stated that the observed data for chlorophyll a concentration in the estuary showed no 

significant increase in the estuary's phytoplankton biomass following a 46-percent increase in effluent discharge between 1965 and 

1966. Later data suggest that chlorophyll a concentrations (figs. 

9-11,18,19) and cell concentrations (table 12) in the 1967-68 blooms 

were at about the same level as they were in the 1965-66 blooms 

(Welch, 1969, figs. 3, 5). As Welch also noted, it is impossible to make 

definitive comparisons between maximum biomass of blooms because of the difficulty of sampling the maximum biomass for any one 

bloom.

Computed Michaelis-Menton factors were used in estimating the 

effects of nutrients on growth rates of phytoplankton in the 

Duwamish River estuary. These and similar factors, like FL or FT, 

can be included in equation 29 for reducing the growth rate because 

of deficiency of a nutrient or the effects of any other parameter. The

FIGURE 25. Longitudinal distributions of Cyclotella sp. during a 

bloom in 1967 and data at two points during a bloom in 1968.
32 NUMERICAL MODEL OF THE SALT-WEDGE REACH OF THE DUWAMISH RIVER ESTUARY

100,000

GREEN-DUWAMISH RIVER KILOMETRE 

5 10 15 20

UJ 10,000

1000

100

2 >.

 August 5, 1968 \ 

O: Low low tide '. 

High high tide 

August 24, 1967 

Low low and 

early flood tide

August 24, 1967

. Low high and 

' early ebb tide

August 22, 1967 

xFlood tide

5 '10 

GREEN-DUWAMISH RIVER MILE

FIGURE 26. longitudinal distributions of oval flagellates during a 

bloom in 1967 and data at two points during a bloom in 1968.

Michaelis-Menton factor, FN, for a particular nutrient is expressed 

in the form

(50)

C+C,

where C is the concentration of the particular nutrient and C ^ is C 

when FN = %.

Existing literature does not provide values of C^ for all phytoplankton taxa found in the Duwamish River estuary. However, 

likely ranges of CM> for the types and sizes of phytoplankton in the 

estuary may be 0.01 to 0.1, 0.002 to 0.02, and 0.001 to 0.05 mg/1 for 

nitrate nitrogen, ammonium nitrogen, and phosphate phosphorus, 

respectively. Especially helpful in providing these data were the 

reports by Eppley, Rogers, and McCarthy (1969), Maclsaac and Dugdale (1969), and the Lake Tahoe Area Council (1968-70). The midpoints of the likely ranges of Cy2 for nitrate and ammonium are 0.055 

and 0.011 mg/1, respectively; Maclssac and Dugdale (1969) reported 

respective values of 0.062 and 0.023 mg/1 for natural marine communities growing in eutrophic conditions.

GREEN-DUWAMISH RIVER KILOMETRE 

5 10 15 20

IJ 10,000

1000

100

August 22, 1967 

Ebb tide

August 22, 1967 

Flood tide

August

Low low and 

early flood tide

/\

August 24, 1967 

Low high and 

early ebb tide

5 10 

GREEN-DUWAMISH RIVER MILE

FIGURE 27. Longitudinal distributions of "coccoids" plus "coccoid" 

clusters during a bloom in 1967.

Figure 29 shows the Michaelis-Menton factors for nitrate + nitrite 

and ammonium nitrogen and for phosphate phosphorus that were 

computed using equation 50 and midpoint values of the likely ranges 

of Cfc. Nutrient-concentration data were available for 3-ft (1-m) 

depth samples collected semimonthly and monthly during most of 

the probable bloom periods, July-early September of 1967 and 

1970-71 at East Marginal Way, 16th Avenue South, and Spokane 

Street Bridges. Analyses of these data indicate that (1) concentrations of ammonium nitrogen, NH/t-N; nitrate + nitrite nitrogen 

(NO3+NO2)-N; and total phosphate, PO4-P, equaled or exceeded 

0.25 mg/1 in 79, 92, and 93 percent of the samples, respectively; and 

(2) variations in nutrient concentrations were unrelated to 

chlorophyll a concentrations. For a concentration of 0.2 mg/1, the 

curves in figure 29 give factors ranging from 0.79 to 0.95 times 

maximum growth. Two inferences important to the modeling and the 

management of the Duwamish River estuary may be implied from 

the foregoing analysis:

1. Given the present phytoplankton inflow to the Duwamish River 

estuary, present concentrations of nutrients in the estuary usually are sufficiently high to be considered nonlimiting to the 

growth of the phytoplankton;

2. Therefore, if the phytoplankton inflow to the estuary does not 

increase, future increases in nutrient concentrations in the estuary from additional RTF effluent or from other sources probably will not greatly increase the phytoplankton biomass in the 

estuary.
io

o 6

to 2* 

SUPPLEMENTAL INFORMATION

1.0 r

33

I6th Avenue South 

Bridge

1 

 / Spokane Street

0.125 / ®~~~~ Bridge

V 

0 /

0 a / o

A&

A

®

East Marginal Way 

*"" Br'dge

/V

0 20 40 60 80 100 

CHLOROPHYLL a CONCENTRATION, IN MICROGRAMS PER LITRE

FIGURE 28. Relation between concentrations of excess dissolved 

oxygen and chlorophyll a in some 3 ft (1-m) deep samples from 

three stations during 1967-68. Excess dissolved-oxygen concentration is the difference between sample concentration and a computed concentration for wedge and river water mixed in proportion 

to sample salinity.

The study of the effects of nutrients on phytoplankton growth rates 

in the Duwamish River estuary is continuing.

HERBIVORES

Herbivores are organisms that obtain their nourishment by consuming plants. Both freshwater and marine species were found in 

water samples taken from the Duwamish River estuary during 

periods of phytoplankton blooms in 1967 and 1968. Concentrations 

ranged from 0 to 110 herbivores/ml (table 13). These data suggest 

that concentration of herbivores and number of taxa tend to be 

higher in the water sampled from the two farthest downstream stations. Out of the seven freshwater taxa identified, rotifers, Didinium 

nasutum, Vorticella sp., and other ciliates were most often present in 

the samples. Twelve marine taxa were identified, of which holotrich 

spp.,Laboea conica, and other heterotrich spp. occurred most often in 

the samples. The freshwater taxa were predominant in samples from 

the upstream stations, whereas marine taxa were predominant in 

samples from the downstream stations.

0 0.2 0.4 0.6 0.8

NUTRIENT CONCENTRATION, IN MICROGRAMS PER LITRE

l.O

FIGURE 29. Relations between Michaelis-Menton factors and concentrations of nutrients.

TABLE 13. Concentrations of herbivores in water samples from the 

Green-Duwamish River for 3 days during phytoplankton blooms in 

1967 and 1968

Date

Time

(P.s.t.)

Sampling 

depth designation1 Taxa present2

Concentration 

(herbivores/ml)

Renton Junction Monitor

1967

Aug. 22 

24 

24

1530 Surface _______ F6, F7

1930 ______ do__________ F6

1540 ______ do____ F3,F6

1.1

.2

6.8

East Marginal Way Bridge

Aug. 22 0930 Surface ________ F4, F6

22 1520 ______ do _ F6, F7

24 0910 ______ do________ F4, F6, F7

24 1525 ______ do _______ Fl, F4, F6, S12

Boeing Bridge

Aug. 22 0915 Surface __________ Fl, S12

22 1425 ____ do________ F4, F6

24 0830 ______ do______ S4, S10, Sll, S12

24 1440 ______ do ______

.4 

.3

3.2

0

16th Avenue South Bridge

Aug. 22 0645 Surface __________ F2, Sll, S12

22 1400 ______ do__________ Fl, F6, S12

22 1405 Bottom____________ Fl, F6, F7, S12

24 0805 Surface __________ S4, S12

24 0815 Bottom__________ F6, S8, S12

24 1415 Surface __________ Fl, S12

24 1425 Bottom____________ F5, S8, S12

2.8 

.6 

3.4 

1.8 

4.0
34 NUMERICAL MODEL OF THE SALT-WEDGE REACH OF THE DUWAMISH RIVER ESTUARY

TABLE 13. Concentrations of herbivores in water samples from the 

Green-Duwamish River for 3 days during phytoplankton blooms in 

1967 and 1968 Continued

Date

Time

(P.s.t.)

Sampling 

depth designation1 Taxa present2

Concentration 

(herbivores/ml)

First Avenue South Bridge

Aug. 22

22

22

24

24

24

24

0825

1340

1345

0740

0750

1345

1355

______ do 

. Fl, S4, Sll, S12

. Fl, F4, Sll, S12

. . S7, S12

.____ S4

. - S4, S8, S12

.____ F1.F6, S12

. _ _ Sll, S12

5.4

2.8

1.7

13

2.1

4.3

.7

Spokane Street Bridge

. 22

22

22

22

24

24

24

24

0735

0755

1305

1310

0645

0705

1310

1320

Bottom

Bottom_ _ .

Bottom

Surface _ _

.. F2, F7, SI, S6, Sll, S12

______ F6, S5, S8, S12

______ S3, S4, S12

______ F3, F6, SI, S5, S6

. __ S2, S4, S7, S12

_ . F3, S2, S4, S7, S12

______ F4, S12

______ S9

3.6

16

8.0

3.1

110

2.1

12

6.8

First Avenue South Bridge

1968

Aug. 5 0940 Surface __________ F7, S5, S12 8.9

5 1830 ___ do_ ___ F4, F6, S5, S12 37

'Surface and bottom indicate samples from about 3 ft (1 m) below the water surface and 

about 3 ft (1 m) above the streambed, respectively.

2Herbivores counted and identified by W. A. Dawson are referred to by F (freshwater 

taxa) or S (saltwater taxa) and a number to indicate a specific taxa:Fl-Didinium nasutum, 

F2-Nemata, F3-Rhizopoda and Actinopoda, F4-Vorticella sp., FS-Filamentous bacteria 

(decomposers), F6-Other ciliates, and F7-Rotifers; Sl-Appendicularian ("Dikopleura"), S2- 

Gyrodinium spirale, S3-Laboea strobila, S4-Laboea conica, S5-Laboea sp., S6-Nauplius 

larvae, S7-Noctiluca scintillans, S8-Parundella sp., S9-Strombidium sp., SlO-Copepodid 

larvae, Sll-Holotrich spp., and S12-Other heterotrich spp.

S-U.S. GOVERNMENT PRINTING OFFICE: 1976 - 690-036/94