Transverse Diffusion of 

Solutes in Natural Streams 

OEC 1 4 1972 

GEOLOGICAL SURVEY PROFESSIONAL PAPER 582-C 

Transverse Diffusion of 

Solutes in Natural Streams 

By NOBUHIRO YOTSUKURA and ERNEST D. COBB 

DISPERSION IN SURFACE WATER 

GEOLOGICAL SURVEY PROFESSIONAL PAPER 582-C 

UNITED STATES GOVERNMENT PRINTING OFFICE. WASHINGTON : 1972 
UNITED STATES DEPARTMENT OF THE INTERIOR 

ROGERS C. B. MORTON, Secretary 

GEOLOGICAL SURVEY 

V. E. McKelvey, Director 

Library of Congress catalog-card No. 72-600198 

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

Washington, D.C. 20402- Price 55 cents (paper cover) 

Stock Number 2401-2147 
CONTENTS 

Page 

Abstract___________________________________________________________________ C1 

Introduction_______________________________________________________________ 1 

Acknowledgments____________________________________________________________ 2 

Equation of transverse diffusion______________________________________________ 2 

Analytical solutions_________________________________________________________ 4 

Procedure for verification by stream data______________________________________ 6 

South River test____________________________________________________________ 6 

Atrisco Feeder Canal test____________________________________________________ 9 

Bernardo Conveyance Channel test___________________________________________ 12 

Missouri River test_________________________________________________________ 13 

Applications_______________________________________________________________ 15 

Conclusions________________________________________________________________ 19 

References_________________________________________________________________ 19 

ILLUSTRATIONS 

Page 

FIGURE 1. Sketch of an infinitesimal stream tube in a natural stream_______________________________________________ C2 

2. Effects of variable diffusion factor on the solutions to diffusion equation _______________________ ------------ 4 

3-5. Comparison of observed and theoretical concentration distribution: 

3. South River test L ___ _ __ _ _ _ _ _ _ _ _ _ __ __ _ __ ___ _ ____ __ __ __ _ _ ___ __ __ ____ __ __ __ __ __ ___ _ _ ____ __ __ __ 7 

4. South River test 2--------------------------------------------------------------------------- 8 

5. South River test 3_ ___ _ _ _ _ _ _ _ _ _ _ __ __ __ __ __ _ _ __ _ _ __ _ __ _ __ ___ _ _ _ _ _ _ __ _ _ ___ ____ __ __ _ ___ ____ ___ _ _ 8 

6. Comparison of concentration-transverse distance curve and concentration-cumulative discharge curve, South 

River test 2----------------------------------------------------------------------------------- 9 

7-10. Comparison of observed and theoretical concentration distribution: 

7. Atrisco Feeder Canal test L _ _ __ _ _ _ __ __ __ _ _ __ __ __ __ __ __ _ _ _ _ __ _ __ _ _ _ _ _ _ _ _ __________ _ _ _ __ _ _ _ __ _ _ 10 

8. Atrisco Feeder Canal test 2------------------------------------------------------------------- 11 

9. Atrisco Feeder Canal test 3 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 11 

10. Bernardo Conveyance Channel________________________________________________________________ 13 

11. Relation of relative cumulative discharge to relative transverse distance, Missouri River near Blair, Nebr _ _ _ _ _ 14 

12. Comparison of observed and theoretical concentration distribution, Missouri River near Blair, Nebr__________ 14 

13-14. Theoretical relation of degree of mixing, Pm: 

13. To distance parameter, a, and source location q/, for point source injections_______________________ 16 

14. To distance parameter, a, and source location, qat' and q82 ', for line source injections________________ 17 

TABLES 

TABLE 1-4. Computation of best-fit transverse diffusion coefficients: Page 

1. South River_________________________________________________________________________________ C9 

2. Atrisco Feeder Canal_________________________________________________________________________ 12 

3. Bernardo Conveyance Channel________________________________________________________________ 12 

4. Missouri River near Blair, N ebr_ _______________________________________________________ ------- 15 

5. Cross-sectional form parameter, UD2juy2, of natural streams___________________________________________ 18 

6. Comparison of estimated and observed mixing distances, Water Research Association data ____ ------------- 18 

III 

DISPERSION IN SURFACE WATER 

TRANSVERSE DIFFUSION OF SOLUTES IN NATURAL STREAMS 

By N OBUHIRO YoTSUKURA and ERNEST D. CoBB 

ABSTRACT 

An analytical model is developed for the transverse diffusion 

of solutes from steady sources placed in a natural stream with 

steady discharge. 

In theoretical derivations, the steady-state convective diffusion 

equation for a natural stream is convertEd into an approximate 

equation suitable for analytical treatment by the introduction of 

cumulative partial discharge as an independent variable. Analytical solutions for the latter equation are obtained for point and 

line source injections of solutes. 

The validity of the model is tested using data from four 

natural streams, three of which are alined straight, while one 

has mild curves. The verification was satisfactory for the three 

straight channels, and the constant in Elder's formula for the 

transverse diffusion coefficient ranged from 0.2 to 0.3. In the 

curved channel, the model appears workable, but an increase 

of the diffusion coefficient in the downstream direction is observed. 

The applications to stream problems are discussed in the final 

section. The model may be employed for the description and 

prediction of thermal and other waste disposal problems, simulation studies by tracers, and the determination of transverse 

diffusion coefficient. Another use of the model, the estimation 

of mixing distances, is explained and demonstrated using the 

data of the Water Research Association, England. 

INTRODUCTION 

The transverse diffusion (mixing) of soluble material 

in natural streams is an important mechanism of transport in the disposal and dispersal of industrial and 

municipal wastes in streams. Thanks to the frequent 

reporting of mass media, the public is now familiar 

with infrared photography of polluted streams in which 

a plume of hot waste water is often shown hugging a 

bank for miles downstream without substantial mixing 

with the rest of the flow. A careful observer traveling 

by airplane will sometimes notice the same phenomenon 

dowstream from the junction of a clear main stream 

and a sediment-laden tributary. Observations on the 

behavior of pollutants and tracers have established 

that, while v-ertical mixing is completed in a relatively 

short downstream distance, a much longer distance is 

required for the completion of transverse mixing in 

natural water courses. 

Despite most waste releases being in the form of a 

point source, which induces highly nonuniform distribution of waste concentration in a cross section, the 

design of an outfall and the evaluation of downstream 

pollution have often been !>ased on the vague concept 

that the waste is somehow dispersed uniformly throughout an arbitrary volume of receiving water. There appears to be no standard to evaluate the merit of point 

release against line release or that of bank-side release 

against midstream release. Many ordinances for thermal 

loading of streams refer to the so-called mixing zone 

without any quantitative definition of the zone. Again 

there is no accepted method of estimating the distance 

and degree of mixing under various loading conditions. 

There obviously is a need to establish a workable 

model of transverse diffusion in natural streams to 

provide answers to the above problems, as well as other 

questions pertaining to efficient pollution control. 

Knowledge of the transverse diffusion coefficient is a 

prerequisite in assessing the total dispersive capacity of 

a stream. As the recent study of Fischer (1967a) has 

shown, longitudinal dispersion in open channel flow is 

due to the combined action of variable longitudinal 

convection along the transverse axis and turbulent 

diffusion in the transverse direction. Whether the longitudinal dispersion is described in terms of the effective 

dispersion coefficient or by means of numerical simulation, knowing the transverse diffusion coefficient is 

vital. 

A review of the literature on transverse diffusion 

is helpful to delineate further the objectives of the 

present study. A basic source of information on transverse diffusion is Elder (1959), who found from his 

flume experiment that the transverse distribution of 

solute in two-dimensional flow is Gaussian and that the 

Cl 
C2 DISPERSION IN SURFACE WATER 

transverse diffusion coefficient, Ez, is given by 

(1) 

where D is the depth of flow and U * is the average shear 

velocity. The constant, {3, was given by Elder as 0.23. 

Orlob (1959), Sayre and Chamberlain (1964), Sayre and 

Chang (1968), Engelund (1969), and Prych (1970) 

investigated the diffusion of plastic particles floating 

on the surface of a two-dimensional flume flow and 

obtained the value of {3 ranging from 0.17 to 0.26. 

According to Prych's (1970) summary of the depthaveraged transverse diffusion coefficient, the values of 

{3 in a laboratory flume range from 0.11 (Sullivan, 1968), 

to 0.18 (Sayre and Chang, 1968). Prych also shows that 

the correct value of {3 for Elder's data is 0.16 rather than 

0.23 and is in good agreement with his and Okoye's 

measured value of 0.14. Fischer (1967a) confirmed that 

the transverse solute distribution in a three-dimensional 

flume flow can be predicted satisfactorily by the use of 

{3=0.23. Sayre and Chang (1968) concluded that the 

Fickian diffusion theory including the reflectionsuperposition technique is satisfactory in describing the 

transverse diffusion of solute in a rectangular flume. 

Recently Fischer (1969) proposed a method of estimating the transverse diffusion coefficient in stream 

bends and showed that t.he secondary currents could 

increase the constant of equation 1 as much as tenfold. 

In comparison with laboratory flume studies, information on diffusion in natural streams is rather limited. 

Glover (1964) obtained {3=0.72 in a reach of the Columbia River by using a mixture of radioisotopes as a 

tracer. He then used the reflection-superposition principle to predict the diffusion at a large distance from 

the source and obtained fair agreement with observed 

data. Fischer (1967b) reported a {3 value of 0.24 measured in a straight reach of the Atrisco Feeder Canal, an 

irrigation canal 2 feet deep and 60 feet wide. Lately, 

Yotsukura, Fischer, and Sayre (1970) reported the 

value of Ez=0.6DU* as observed in the Missouri River. 

In contrast to the moment method used by Glover and 

Fischer, Yotsukura, Fischer, and Sayre used a numerical 

simulation of the diffusion equation to determine the 

transverse diffusion coefficient. 

Two points about transverse diffusion are apparent 

from the above review. The first is that the form of the 

transverse diffusion coefficient is now well established 

by equation 1, even though the value of the constant, 

~' may vary considerably between laboratory flumes 

and natural streams. The second is that, as long as 

there is no appreciable convective transport in the 

transverse direction, application of the Fickian theory 

with much basic information compiled, it appears feasible and appropriate to integrate it into a workable 

model of transverse diffusion which is directly applicable 

to waste disposal problems. 

This report presents a comprehensive study on the 

steady-state transverse diffusion of solute in natural 

streams. The theoretical part of the study derives an 

appropriate diffusion equation and then red~ces itt? a 

form suitable for analytical treatment by IntroduCing 

the cumulative partial discharge, q, as the independent 

variable replacing the transverse distance, z. The transformation provides a convenient means of accounting 

for the solute mass balance and of adapting the analytical solutions as adequate approximations to stream 

conditions with variable velocity u(z). A set of analytical 

solutions for a point source and a line source are obtained by the method of images. The theoretical development is then compared with diffusion data from a 

number of streams, and the results are presented as a 

table of computed transverse diffusion coefficient and 

as a graphic comparison of observed versus theoretical 

concentration distribution. Several applications of the 

theory are presented in the final section. 

ACKNOWLEDGMENTS 

The dye diffusion tests in Atrisco Feeder Canal and 

, Bernardo Conveyance Channel were conducted under 

the direction of James K. Culbertson. H. B. Fischer, 

D. R. Dawdy, and C. F. Nordin participated in the 

tests. M. A. Kennan rendered valuable assistance 

throughout the present study. Contributions of the 

above-mentioned U.S. Geological Survey personnel are 

gratefully acknowledged. 

EQUATION OF TRANSVERSE DIFFUSION 

Consider a uniform section of a steady natural stream 

as sketched in figure 1. The coordinate x is measured in 

the longitudinal (downstream) direction, y is directe.d 

vertically downward from the water surface, and z IS 

is satisfactory for transverse diffusion problems. Now FIGURE 1.-An infinitesimal stream tube in a natural stream 
TRANSVERSE DIFFUSION OF SOLUTES IN NATURAL STREAMS C3 

measured transversely from the left bank along the 

water surface. It is assumed that the section is far 

enough downstream from the solute injection site that 

the vertical distribution of solute concentration is 

already uniform. The problem is thus restricted to two 

space dimensions and the vertical variation of such 

quantities as velocity, diffusion coefficients, and solute 

concentration is neglected through the use of depthaveraged values. Density of water is assumed to be 

homogeneous through the system. The symbol y is used 

to denote the flow depth measured from the water 

surface to the channel bed and is thus a function of z. 

The transport (convective diffusion) equation of a 

solute is derived for an elementary volume yt::.zt::.x containing a solute concentration, c, by noting that the 

mass continuity of solute within the volume must be 

satisfied at any time. The net inflow rate of solute 

due to the convective velocity u(z) into the volume 

yt::.zt::.x is given by 

The net inflow rate due to transverse diffusion, for 

which the diffusion coefficient is designated by Ez(z), 

is expressed by 

The time rate of increase of solute mass within the 

volume is given by 

a(cyt::.xt::.z) 

-at-· 

The sum of inflow rates of solute through all boundaries 

is now equated to the time rate of change in order to 

satisfy the mass continuity, and the following equation 

of solute transport is obtained 

(2) 

In this derivation the convective velocity is assumed to 

act only in the longitudinal direction so the volume 

yt::.zt::.x forms a stream tube. Also the turbulent diffusion 

in the longitudinal direction is neglected because the 

transport due to this diffusion is very small relative 

to the transport by the convection and transverse 

diffusion. 

The equation of steady-state transverse diftusion is a 

special case of equation 2 when it is assumed that the 

solute is injected into the stream continuously at a 

uniform rate. Since the solute concentration c at any 

downstream location will give a value which is independent of time, the time derivative term in equation 2 is 

dropped. Furthermore, since u, y, and Ez are assumed to 

be functions of z alone, equation 2 is simplified to the 

form 

(3) 

The boundary conditions at the banks are given by 

ac 

EzYaz =0 at z=O and B (4) 

where B is the channel width. Because of the steady 

state of solute distribution, the inflow rate of solute at 

the injection site must be equal to the outflow rate at a 

downstream cross section or 

M='Y J: cuydz, (5) 

where M is the time rate of solute inflow by weight and 

'Y is the specific weight of water. 

When u andy vary transversely as in typical natural 

streams, it is impossible to obtain closed-form analytical solutions to equations 3, 4, and 5. The only 

feasible method of solving them directly is to use a 

finite-difference technique with a digital computer. In 

various applications, however, it is desirable to have 

analytical solutions which can be used for the approximation of real problems, because direct numerical 

simulation which normally requires time-consuming 

preparation of input data may not be justified in terms 

of time and cost. 

A new independent variable q will be defined by 

(6) 

It is seen that q is the cumulative discharge measured 

from the left bank and becomes Q at z=B, where Q is 

the total discharge. By introducing q in place of z, 

equations 3, 4, and 5 are transformed to 

OC 0 ( 20C) ox= oq EzUY oq (7) 

oc Ezuy2

0q =0 at q=O and Q (8) 

M="fiQcdq· (9) 

Equation 7 may "be considered as the diffusion equation 

with convective velocity of unity and transverse diffusion coefficient of EzUY 2

• Even though equation 7 appears more suitable for analytical treatments than 
C4 DISPERSION IN SURFACE WATER 

equation 3, such analytical solutions have not been 1 

obtained and the solution must still be sought through 

numerical techniques. 

Fortunately there is strong empirical evidence that 

the solution to equation 7 having a variable coefficient 

e,uy2 (henceforth called diffusion factor to differentiate 

it from the diffusion coefficient e 2) can be approximated 

sufficiently by the solution of another equation having 

a constant factor EzUY2

, or 

de -- d2c 

dX =ezUY2 (jq2' (10) 

where e 2uy2 is the constant diffusion factor defined by 

- 1.J:Q E2Uy2

=Q 0 E2Uy2dq. (11) 

Some examples of numerical solution to equation 7 are 

shown in figure 2 to illustrate the point. In this hypothetical calculation, Q was assumed to be 100 cfs (cubic 

feet per second) and a line source was located between 

0.4 and 0.5 relative cumulative discharge. Three schemes 

of variation of the diffusion factor, e 2uy2

, were employed 

as shown at the bottom of figure 2. The concentration 

distributions at several downstream cross sections for 

the three different schemes are shown on the upper 

part of figure 2, indicating that they are indeed similar 

to each other despite such large variations in e 2uy2

• All 

schemes have EzUY2 equal to 0.1. 

It is apparent from the above development that the 

purpose of the empirical verification to be described 

later is to determine to what degree a solution to 

equation 10 approximates prototype diffusion rn a 

natural stream. 

ANALYTICAL SOLUTIONS 

The analytical solutions are sought for the following 

set of equations 

(10) 

--ac 

EzUY2 dq =0 at q=O and Q (12) 

M=-r J: cdq. (9) 

The form of these equations is identical to conventional 

equations of diffusion with constant diffusion coefficient. 

A well-known solution to equation 10 is 

(13) 

2 

. :!;: * * • Scheme A 

:r 

+Scheme 8 

X =2000 ft -¥ X Scheme C 

.ol' f- . f. 

10 . ~ 0 i I.JJ 

f-

::> 

_, 2 0 

en 

LL. 

0 . z -* t * ~ 0 ~ 1 -¥ X f= 

<( t X =4000 ft X a:: + f- . z 

I.JJ 

(..) 

z 

0 0 (..) 

I.JJ 

> 

f= 2 <( 

_, 

I.JJ 

a:: 

¥ ¥ ¥ ¥ ¥ -¥ X 

X =6000 ft 

t t X .... 

X 

0 

0 0.2 0.4 0.6 0.8 1.0 

"' >-. I I I I :;:, 

• Scheme A ~0.2 ~ X +Scheme 8-

X X x Scheme C 0::: 

0 

f- + + + (..) ¥ -¥ <( 

LL. 0.1 - . + + . - . 

z X X 0 

U5 

::> + + LL. 

LL. 

C5 X I I I I X 

0 

0 0.2 0.4 0.6 0.8 1.0 

RELATIVE CUMULATIVE DISCHARGE (q/Q) 

FIGURE 2.-Effects of variable diffusion factor on the solutions 

to diffusion equation 

where K1 and K2 are the constants of integration. Since 

equation 10 is a linear partial differential equation, any 

number of superposition of equation 13 is also a solution. If the solute is assumed to be injected as point 

source at a location, x=O and q=q8 , the solution satisfying the boundary condition, equation 12, and the 

continuity condition, equation 9, can be constructed 

by the method of images (Glover, 1964). According to 

this method, satisfying the boundary equation 12 is 

equivalent to reflecting the solute concentration at the 

boundary as if it were from a mirror image source with 

respect to the boundary. 

The appropriate solution is thus obtained by adding 

an infinite number of solutions similar to equation 13, 
TRANSVERSE DIFFUSION OF SOLUTES IN NATURAL STREAMS C5 

each solution representing the concentration from an 

actual source and successive image sources which come 

into play whenever the solute is reflected at the boundary. The series solution is written as 

c(x,q)= Kl ± e- -- 4~~uY2x-- [ { 

(2nQ+q.-q) 2 

"\147rEzUY2X n=O 

(14) 

In order to satisfy equation 9, equation 14 is integrated 

from q=O to Q and the result is equated to Mf'Y· The 

constant K1 is fixed in this manner as 

M Kl=--· 

'Y 

If the average concentration is defined by 

- 1 fQ M 

c=Q)o cdq= 'YQ' 

(15) 

(16) 

it will be more convenient to express the concentration 

cas a relative value to;. Also q and qs may be expressed 

as fractions of Q. 

Equation 14 is thus reduced to the nondimensional 

form 

af en { -~(2n+q',-q')Z c'(a,q')=- ~ e 2 

.J2; n=O 

where 

+e -~'Cln+o'.+o'l'} 

+ ± {e -~(2n-q',-q')2 

n=l 

a2 } +e 2 --(2n-q',+q')2 J 

I C c =;;;; 

c 

(17) 

(18) 

(19) 

(20) 

and 

(21) 

It is useful to note that, as a consequence of normalization of variables and parameters, ff,c' dq' = 1. Equation 

17 is similar to the solution obtained by Sayre and 

Chang (1968) through the time integration of an unsteady solution to the Fickian diffusion equation, 

which includes a time derivative term as well as a 

diffusion term in the longitudinal direction. Also, 

equation 21 is the nondimensional parameter used by 

Sayre (1965). Sayre's developments, however, are for a 

rectangular channel flow and use a different set of 

variables and constants. 

When the solute is injected as a line source, the new 

solution may be constructed by the superposition of the 

point source solution, equation 17. Let the line source 

be located between q' s1 and q' s2 (q' s2>q' si) and assume 

that it consists of m point sources equally spaced at 

q' s~, q' s1 +Aq' s, q' s1 +2Aq' s, ... , q' s1 + (m-2)Aq' s, and 

q' s2· If the solute injection rate isM for all sources combined, each point source will produce the concentration 

distribution which is 1/m times equation 17, which, 

however, is adjusted for each source location. The 

superposition of all concentration distribution from the 

m sources will yield, for example, an expression corresponding to the first term of equation 17 such as 

a ± {:i:_!_e -~(q'11 +(i-1)..:1q',+2n-q')2} • 

.J2~=o i=lm 

B b . . 1 Aq~ d . . 

y su stitutmg , , +A , an Increasing m to 

m qs2-qsl qs 

infinity and reducing Aq' s to zero at the same time, the 

above term is reduced to a limit form 

(22) 

By repeating a similar superposition and limit 

process to all other terms appearing in equation 17, 

the solution for a line source is finally obtained as 
C6 DISPERSION IN SURFACE WATER 

When diffusion is complete and the concentration, c, is 

uniform across the channel, P m attains the value of 

unity. The definition is compatible with the principle of 

mass continuity and is easily calculated from the plot 

of nondimensional concentration c(C against nondimensional cumulative discharge q/Q. Since the transverse distribution of solute is analytically determined 

as a function of a, q;t' q' s2' and q', the cross-sectional 

degree of mixing, P m' can likewise be determined 

analytically. This relation between P m and a, q;t' and 

q; 2 is discussed later in connection with the estimation 

of mixing distances. 

PROCEDURE FOR VERIFICATION BY STREAM 

(23) DATA 

where erf designates the error function defined by 

2 rq erf q= .J;Jo e-P2

dp, 

and q;2 and q;1 are two end points of the line source 

expressed as fractions of total discharge Q. Equation 23 

is again similar to the solution obtained by Sayre (1970) 

by the convolution of the point source solution with a 

line input function. It is also noted that equation 23 

is normalized and !a1 c' dq' = 1· 

Parameters representing the progress of transverse 

diffusion are used frequently in the following sections. 

One is the parameter a defined in equation 21. Even 

though this parameter is obtained by converting to 

nondimensional terms the parts of equation 14, its 

meaning is clear if it is noted that 2 E2'1ty2 x is approximately equal to the variance, u:, of the transverse 

solute distribution which is obtained in a channel 

without transverse boundaries and that equation 21 

is reduced accordingly to 

(21) 

As the distance x increases, the time xj U increases and 

so does u:. Therefore a decreases as the distance 

increases and the diffusion becomes more complete 

as the result of solute reflection at boundaries. 

The degree of mixing, P m' originally defined as 

percent mixing by E. D. Cobb and J. F. Bailey (written 

commun., 1965), may be written as 

where 

- 1 rQ c=Q )o cd.q 

(24) 

(16) 

The procedure for testing the theoretical model by 

stream data consisted of (1) seeking and obtaining the 

best matching analytical solution to cross sectional 

data in terms of transverse solute distribution and 

degree of mixing, P m' (2) computing EzUY2 from the 

equation 

(21) 

(3) examining the consistency of computed EzUY2 for 

all cross sections used in a particular test, (4) computing the average diffusion coefficient E z by 

--2 

E =EzUY 

z uy2 (25) 

where the averaging means discharge weighting as 

defined in equation 11, and finally, (5) comparing the 

values of E z with Elder's formula as well as with values 

determined by other methods independent of the 

present model. The results are presented as a table of 

computed values of EzUY2 and Ez, and a graphic comparison of observed versus theoretical solute concentration distributions. 

SOUTH RIVER TEST 

The first verification study was made on the South 

River just upstream from the town of Waynesboro, V a. 

Three separate tests were made at this site. The reach 

was 2,300 feet long with a few very slight bends. The 

upper 1,000 feet of the reach was characterized by a 

large pool followed by two small riffies immediately 

downstream from the pool. The streambed was primarily 

sand and gravel with a few cobbles and larger rocks. 

There were a number of water-plant clumps in the 

reach. The stream width varied from about 40 to 80 

feet with a mean width of about 60 feet. The average 

depth was from 1.0 to 1.5 feet. The flow during the 

study varied from 50 to 58 cfs. The average velocity in 
TRANSVERSE DIFFUSION OF SOLUTES IN NATURAL STREAMS C7 

the upper 1,200 feet of the reach was about 0.8 fps 

(feet per second), while that in the lower 1,100 feet 

was about 0.6 fps. 

The solute used for the study was a fluorescent dye, 

Rhodamine WT. For the first test, the dye was injected 

continuously at the center of the stream by an airpowered, spring-diaphragm-regulated constant-rate injection unit. Once the dye entered the stream, a cross 

current in the channel caused the dye to flow toward 

the right bank. The stream was sampled at four downstream cross sections located at 800, 1,200, 1,600, and 

2,000 feet from the injection site. Ten evenly spaced 

points were sampled at each cross section; three successive samples were obtained at each point. 

The dye was injected for the second and third tests 

at a site 1,000 feet downstream from the first injection 

site to avoid the effect of the pool and riffles mentioned 

earlier. The same constant-rate-injection unit was used. 

In the second test, the dye, injected at the center of 

the channel, moved toward the left bank. In the third 

test the dye was injected at one-quarter of the width 

of the stream from each bank, the flow rate of the dye 

solution being approximately split between the two 

points. Samples for the second and third tests were 

obtained at cross sections located 400, 600, 1,000, and 

1,290 feet downstream from the injection site. Sampling 

was done in a manner similar to sampling in the first 

test. Velocity and depth data were obtained at each 

sampling point by the standard current-meter method 

of the U.S. Geological Survey. 

The concentrations of Rhodamine WT in the samples 

were determined by a Turner model 111 fluorometer 

after the samples were stored a few days to allow for 

the settling of sediment and adjusting to room temperature. All concentrations were temperature corrected as 

necessary and adjusted for effective background concentrations. The average concentration of the three 

samples obtained at each point was considered representative of that point. The degree of mixing P m and 

the cross-sectional average concentration c were computed at each cross section by equations 24 and 16. 

All concentration data are presented graphically in 

figures 3, 4, and 5. Since the relative concentration c' 

is computed by discharge weighting, the area underneath each distribution curve is unity and is directly 

comparable with theoretical solutions. To find the bestfit solution, however, a theoretical source location must 

be matched to the actual source location. Since the 

data included only geometric information of the source 

and not qs', the matching was accomplished empirically. 

Each observed distribution, c' vs q', was compared 

with several theoretical solutions, c' vs q', having the 

same degree of mixing, P m' as the data but having 

different source locations, q/. A representative qs' was 

selected as giving the overall best-fit among all cross 

sections for a particular run. On the basis of this common source location, the best-fit theoretical solution for 

each cross section was taken as having the same degree 

of mixing as the data. These theoretical solutions are 

superimposed on the data in figures 3, 4, and 5. The 

agreement between the theoretical curves and the data 

points is satisfactory in most cross sections. 

1.3 .-------.----r--,-----.----.--~-y-------,.--,..----, 

1.1 

1.0 

0.8 

1.2 

1.1 

~ '<- 1.0 

z 

0 

f= 0.9 <( 

c:: 

ft5 0.8 

~ .. 0 () 1.2 

LLJ 

> 

~ 1.1 ...J 

LLJ 

c:: 1.0 

0.9 

0.8 

1.2 

1.1 

1.0 

0.9 

• Observed 

-Theoretical 

X =800ft 

x= 1200 ft 

x=1600ft 

• 

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 

RELATIVE CUMULATIVE DISCHARGE (q!Q) 

FIGURE 3.-Comparison of observed and theoretical 

concentration distribution, South River test 1 

The data plotted by discharge weighting as used in 

this report present a much clearer picture of diffusion 

than do the data plotted by conventional distance 

weighting (c' vs z'=z/B). The data of test 2 are presented as examples in figure 6. The cross-sectional 
C8 DISPERSION IN SURFACE WATER 

• Observed 

-Theoretical 

il, 

~1 

z 

0 

~ <( 

0:: 

I-

-z 0 w 

() 2 z 

0 

() 

w 

> 

~ <( 

....J 1 w 

0:: 

o~~--~--~--~---L--~---L--~--.....1-~ 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 

RELATIVE CUMULATIVE DISCHARGE' (q!Q) 

FIGURE 4.-Comparison of observed and theoretical 

concentration distribution, South River test 2 

average c based on distance weighting fluctuates a great 

deal from section to section, while that based on discharge weighting remains fairly steady. The degree of 

mixing also tends to be less consistent when it is based 

on the c' vs z' plot than when it is computed from c' vs 

q' plot. Furthermore, the transverse location of the 

peak concentration in the c' vs z' plot fluctuates between 

z' =0.2 to 0.5, while in the c' vs q' plot it remains steady 

at about q' =0.35. It is obvious from the velocity data 

that the meandering of high velocity threads caused 

irregular shifting of the peak in the c' vs z' plot while 

it produced no similar effect on the c' vs q' plot. These 

and other similar plottings confirmed that the c' vs q' 

plot that Cobb and Bailey proposed is sound from the 

concept of mass continuity and is also appropriate for 

the nondimensional presentation of natural stream 

data. Since the location of the peak concentration 

normally corresponds to the location of the source, the 

0 

2 

~1 

~ 

z 

0 

~ <( 

0:: 0 Iz 

w 

() 

6 2 () 

w 

> 

~ <( 

....J 

~ 1 

0 

2 

• Observed 

-Theoretical 

o~~--~--~---L---L--~---L--~--.....1-~ 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 

RELATIVE CUMULATIVE DISCHARGE (q!Q) 

FIGURE 5.-Comparison of observed and theoretical 

concentration distribution, South River test 3 

representative source location determined from the plot 

of c' vs q' was considered to be the most reliable. 

After the best-fit theoretical solution was chosen for 

a particular cross-section, the value of diffusion factor 

e2uy2 was computed by equation 21 from known values 

Q, x, and a, the last being determined from the bestfit solution. Before computing the average diffusion 

coefficient Ez by equation 25, the effective value of 

·uy2 for the cross section was determined by weighting 

each upstream cross-sectional uy2 by appropriate 

longitudinal distance. Such weighting was necessary 

because the stream was not uniform and the observed 

diffusion pattern represented the sum effe.£!_ of the 

nonuniformity. The cross-sectional value of uy2 varied 

from 0.9 ft3 per sec at the middle of the reach to 4.0 
TRANSVERSE DIFFUSION OF SOLUTES IN NATURAL STREAMS C9 

2 

0 

2 

lv 

~ 

- 1 z 

0 

;:::::: 

<X: 

a:: 

I-

~ 0 u a 2 u 

w 

> 

;:::::: 

::5 

~ 1 

0 

2 

\ 

\ 

~ ....... 

............... -.. 

o~~--~--~--~--~--~--~--~~--~ 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 

TRANSVERSE COORDINATE (ZfB or q/Q) 

FIGURE 6.·-Comparison of concentration-transverse distance 

curve and concentration-cumulative discharge curve, South 

River test 2-

toward the end of the 2,300-foot long reach. The values 

of the pertinent parameters as well as the calculated 

Ez are tabulated in table 1. The consistency of Ez is 

satisfactory except in the first cross section of both 

test 1 and 2, where the values of P m show some anomaly. 

With an average Ez for tests 2 and 3 of 0.05 ft2 per 

sec, the calculated constant of Elder's formula is 

about 0.3 based on the estimate that u* is 0.13 fps 

and Dis 1.3 feet in this particular reach. On the other 

hand, Ez for test 1 is about three times the value of 

Ez for tests 2 and 3. Undoubtedly this represents a 

composite effect of high diffusion in the pool and 

riffles of the upper reach and less intense mixing in 

the lower reach. Since the theory does not accommodate 

such abrupt change of diffusion characteristic in the 

reach, test 1 should not be considered part of the 

verification. Nevertheless, the information is of considerable interest. Examination of figures 4 and 5 

and table 1 shows that the verification of the theory 

by this set of tests is satisfactory. 

ATRISCO FEEDER CANAL TEST 

The second set of data used for verification was from 

diffusion tests on the Atrisco Feeder Canal near 

Bernalillo, about 15 miles north of Albuquerque, 

N. Mex. Part of the data was published and discussed 

by Fischer (1967b) and Yotsukura (1968). The present 

analysis, however, will examine all of the data. The 

study reach on this canal was much more uniform than 

that of the South River and had a straight alinement 

for about 12,000 feet. The stream cross section had a 

nearly rectangular shape for the entire length. The bed 

consisted of sand dunes, approximately 10 feet long and 

0.8 to 1 foot high. The stream width varied from about 

51 to 65 feet with an average width of about 60 feet. The 

average depth in the reach was about 2.2 feet, the mean 

velocity was 2.2 fps, and the discharge during the study 

varied from 255 cfs to 269 cfs. 

TABLE !.-Computation of best-fit transverse diffusion coefficients, South River 

Test Date z', q', Q (cfs) X (ft) Pm a ";.Uyi ;;yr E. 

(fV per sec2) (ft3 per sec) (ft2 per sec) 

1 June 8, 1966 0. 5 0. 75 58 800 0. 934 1. 60 0. 821 3. 3 0.249 

1,200 . 936 1. 58 . 561 3.4 . 165 

1, 600 . 944 1. 53 . 449 3.0 . 150 

2,000 . 949 1. 50 . 374 2. 7 . 139 

2 June 9, 1966 0. 5 0.35 54 400 . 781 3. 63 . 277 2. 9 . 096 

600 . 735 4.40 . 126 2.4 . 053 

1,000 . 771 3. 80 . 101 2.0 . 051 

1, 290 . 829 2.97 . 128 2. 2 . 058 

3 June 9, 1966 0. 25 0. 10 50 400 . 825 4. 62 . 146 2.9 . 050 

and and 600 . 872 3.98 . 132 2.4 . 055 

0. 75 0.85 1,000 . 921 3. 37 . 110 2. 0 . 055 

1,290 . 944 3.09 . 101 2. 2 . 046 

4-66-743 0-72~2 
ClO DISPE'RSION IN SURFACE WATER 

Rhodamine WT dye was injected at a constant rate 

into the stream by a Mariotte constant-head tank. For 

the first and second tests, the dye was injected at the 

center of the stream; whereas, for the third test, it was 

injected at a point 1 foot from the left bank. The stream 

was sampled at eight to ten cross sections, located 

between 200 and 6,500 feet downstream from the injection site. All samples in a cross section were obtained 

simultaneously by submerging a set of 24 milliliter vials 

suspended from a nylon cord stretched across the 

channel. The vials were spaced at a uniform interval of 

2 feet. Velocity data were obtained at every sampling 

cross section within 1,200 feet from the injection site, 

but only at a few sections further downstream. Depth 

data were obtained at each sampling section. A more 

detailed description of the study reach and the test 

procedure was given by Fischer (1967b). 

The concentrations of the samples were determined 

in a manner similar to that for the South River data. 

Since only one sample was obtained at each point, the 

concentration of that sample was considered steadystate concentration at that point. The procedure of 

data analysis was the same as for the South River data. 

The results of tests are given in figures 7, 8, and 9, 

and table 2. Several cross-sectional data from test 3 

(side injection test) are not presented because the 

deviations of the computed c for these sections were 

more than 10 percent of the average c of all cross sections and were considered excessive for solute mass 

continuity. The error in c for the side injection test 

tends to be large relative to the error inc for the midstream injection, because the high solute concentration 

which contributes heavily to c, is in the region of slow 

bank-side flow where velocity and its measurement 

tend to be erratic. The source location was determined 

as in the South River tests. For tests 1 and 2, however, 

velocity measurements made within 50 feet of the injection site indicate that, at the center of the channel 

where the injection nozzle was located, the relative 

cumulative discharge from the left bank, q/, was about 

0.45, which agrees well with the value determined from 

the concentration data. 

The agreement of observed and theoretical solute 

distributions appears quite satisfactory. The theoretical 

distributions are based on the best matching values 

of q' s and a (P m) tabulated in table 2. Especially 

noticeable is the result that the solute distributions 

at distances less than 1,000 feet from the injection site 

show typical Gaussian pattern. Comparison of such 

distributions with the typically skewed distributions 

of the plot, c' vs z', again demonstrated that the 

present analytical approach using the cumulative 

discharge q' as the transverse variable is indeed compatible with the conditions in natural streams. In this 

connection, the equation relating the variance of the 

TU 

.:;-... 

z 

0 

t= 

<( 

a:: 

1-

z 

UJ 

u 

z 

0 

u 

UJ 

> 

t= 

<( 

_J 

UJ 

a:: 

2 

0 

2 

0 

X =800ft 

X =3200 ft 

i i • • • •x~~=== •I• J 

1.0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 

RELATIVE CUMULATIVE DISCHARGE. (q!Q) 

FIGURE 7.-Comparison of observed and theoretical 

concentration distribution, Atrisco Feeder Canal test 1 
TRANSVERSE DIFFUSION OF SOLUTES IN NATURAL STREAMS Cll 

distribution with the transverse diffusion factor is 

written in the present theory as 

(26) 

where u 2 q is the variance of the plot of c vs q. Equation 

26 is derived from equation 10, as in Sayre and Chang 

(1968) from the rectangular-channel equation of con-

~ ;:;..: 

z 

0 

i= 

<( 

a:: 

Iz 

w 

(.) 

z 

0 

(.) 

w 

> 

i= 

<( 

_J 

w 

a:: 

2.---.---.---~---r---.--~--~~------.-~ 

0 

0 

8 e I • ••• 

. ..... I W W J 

:r 

• • • • • X =3600 ft 

• • • • • 

1r······~······~····· X =6500 ft 

0 I I I I I I 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 

RELATIVE CUMULATIVE DISCHARGE (qiQ) 

FIGURE B.-Comparison of observed and theoretical concentration distribution, Atrisco Feeder Canal test 2 

ventional notation. Equation 26 however is useful 

only for the region where the solute has not reached 

either bank. 

z 

0 

i= 

<( 

a:: 

Iz 

w 

(.) 

z 

0 

(.) 

w 

> 

i= 

<( 

_J 

w 

a:: 

7,---,---.---.---.---.---~---.---,---.--~ 

6 

5 

4 

3 

2 

2 

0 

4 

3 

2 

0 

2 

X =400ft 

• • Observed 

- Theoretical 

0

2

1 

~ 0 - x-6500 ft l 

[ ~ .. ~----.........~. ·, ·~·~· ~0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 

RELATIVE CUMULATIVE DISCHARGE (q/Q) 

FIGURE 9.-Comparison of observed and theoretical concentration distribution, Atrisco Feeder Canal test 3 
C12 DISPERSION IN SURFACE WATER 

TABLE 2.-Computation of best-fit transverse diffusion coefficients, Atrisco Feeder Canal 

Test Date z', q', Q (cfs) 

1 June 21, 1966 0. 5 0.4 269 

2 June 22, 1966 0. 5 0.45 258 

3 June 23, 1966 0. 0 0. 0 263 

As for the magnitude of the computed values of Ez, 

the consistency throughout the tests is remarkable. A 

few discrepancies, as in section 2400 of test 1 and 

section 3600 of test 2, are due to the apparent discrepancy of data as shown by the degree of mixing. 

The values of Ez in parentheses were computed by less 

representative values of uy2 than others, for there 

were not enough velocity-distribution data for these 

distant cross sections. Another problem noticed in 

the analysis is that the degree of mixing, P m' becomes 

rather sensitive to random data errors when P m exceeds 

about 0.95. Thus the values of P m and a for high mixing 

values are not as reliable as when values of P m are less 

than 0.95. With these considerations taken into account, 

the average transverse diffusion coefficient for the tests 

is estimated to be 0.1 square foot per second which 

agrees well with Fischer's (1967b) value of 0.11 ft2

/per 

sec. Fischer computed the coefficient on the basis of 

the variance of the transverse distribution of solute 

mass-flow rate, cuy, with respect to distance z. With Ez 

being equal to 0.1, the constant of Elder's formula is 

0.22 for an averageD of 2.2 feet and U* of 0.204 fps. 

The verification of the theory is again satisfactory. 

·----------------- X (ft) Pm a E,uy2 uy;· E, (ft3 per sec2) (ft3 per sec) (ft2 per sec) 

400 0.493 8. 3 1. 312 14. 0 0. 094 

600 0. 575 6. 8 1. 304 13.4 0.097 

800 0.629 6. 0 1. 256 12. 9 0. 097 

1,000 0. 662 5. 5 1. 196 12.4 0. 097 

2,400 0.875 3. 0 1. 675 (11.3) (0. 148) 3, 200 0.879 2. 9 1. 344 (11. 7) (0. 115) 4, 000 0. 916 2.4 1. 570 ( 12. 6) (0. 125) 

1,600 0. 775 4. 36 1. 094 10. 9 0. 100 

2,000 0. 846 3. 69 1. 222 11. 1 0. 110 

2,400 0. 872 3.45 1. 165 11. 3 0. 103 

2,800 0. 910 3. 09 1. 245 11. 4 0. 109 

3, 200 0. 927 2. 88 1. 255 11. 7 0. 107 

3, 600 0. 958 2. 30 1. 748 12. 2 0. 143 

4,000 0. 957 2. 32 1. 547 12.6 0. 123 

6, 500 0. 9Rl 1. 72 1. 730 ( 13. 0) (0. 133) 

400 0. 297 8. 00 1. 350 13. 1 0. 103 

600 0. 333 6. 90 1. 210 12. 7 0.095 

1, 600 0.479 4. 30 1. 169 11. 9 0. 098 

3, 200 0. 629 3. 00 1. 200 (11. 7) (0. 103) 4,000 0. 689 2. 60 1. 279 (12. 6) ( 0. 102) 

6, 500 0. 815 2.00 1. 330 ( 13. 0) (0. 102) 

BERNARDO CONVEYANCE CHANNEL TEST 

A dye test was conducted on the Bernardo Conveyance Channel after those on the Atrisco Feeder 

Canal to ascertain that the constant of the transverse 

diffusion coefficient remains the same for both dune 

bed and plain bed regime. The conveyance channel is 

about 55 miles south of Albuquerque, N. Mex. An 

extensive study of sediment transport in the channel 

was carried out and documented in detail by Scott 

(1968) and Culbertson and Scott (1970). The dye diffusion test was conducted like that in the Atrisco Feeder 

Canal. A straight reach of about 8,000 feet, south of 

U.S. Highway 60, was used. The channel was of typical 

plain-bed configuration with a discharge of 627 cfs. The 

average width was 66 feet, the depth 2.3 feet, the 

velocity 4.1 fps, and the shear velocity was 0.203 fps. 

at the time of the test. 

The results of the test are shown in figure 10 and 

table 3 in the same manner as the preceding results. The 

data at sections 500, 1500, and 2000 were deleted 

because the computed c differed from the average of 

all cross sections by more than 10 percent. The agreement of observed versus theoretical distribution is 

TABLE 3.-Computation of best-fit transverse diffusion coefficients, Bernardo Conveyance Channel 

Date z', q'. Q (cfs) X (ft) Pm a E,uy2 (ft5 per sect) uy2 (fta per sec) E. (ft2 per sec) 

Feb. 14 1967 0. 5 0. 45 627 1,000 0. 507 8. 0 3. 07 26 0. 118 

2, 500 . 789 4. 2 4.46 -------------- 0. 172 

3, 500 . 80R 4. 0 3. 51 -------------- 0. 135 

4, 500 . 869 3.4 3. 78 ------- 0. 145 

5, 500 . 918 3. 0 3. 97 -------------- 1. 153 

6, 500 . 950 2. 5 4. 84 -------------- 0. 186 
TRANSVERSE DIFFUSION OF SOLUTES IN NATURAL STREAMS C13 

not as good as for the two preceding tests, and neither 

is the value of E z as consistent as in the other tests. 

One reason for the random scatter of data is the continual shift in the bottom configuration and the subsequent change in the velocity distribution. The 

average Ez for this test was estimated to be 0.14 

ft 2 per sec. and the constant, {1, was about 0.3. The 

:@ 

~ 

z 

0 

i= 

<( 

a:: 

1-

z 

UJ 

() 

z 

0 

() 

UJ 

> 

i= 

<( 

_J 

UJ 

a:: 

4.--.---,---,--~--~--~--~~--~--~ 

2 

0 

2 

0 

1.5 

0 

1.5 

X = 1000 ft 

X =2500 ft 

• • Observed 

-Theoretical 

I 

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 

RELATIVE CUMULATIVE DISCHARGE (q/Q) 

FIGURE 10.-Comparison of observed and theoretical 

concentration distribution, Berna.rdo Conveyance Channel 

variation of 11.y2 was from 25 to 28 ft3 per sec., and a 

single common value of 26 was used for aH sections. 

In spite of some discrepancies, the data appear to 

support the theoretical development and indicate that 

the constant of Elder's formula does not vary significantly between the plain-bed and the dune-bed regime. 

MISSOURI RIVER TEST 

A transverse diffusion test was conducted in 1967 as 

a part of a comprehensive measurement of mixing 

characteristics of the Missouri River (Yotsukura and 

others, 1970). In contrast to the three preceding tests 

which were done in essentially straight channels, the 

Missouri River reach below Blair, Nebr., had two mild 

alternating curves within its 6-mile stretch. At the time 

of the test, the river discharge was 34,100 cfs, the 

average depth was 9 feet, and the width ranged from 

500 to 700 feet. The channel bed, which consisted predominantly of coarse sands, was of plain-bed configuration and the average velocity was about 5.7 fps, while 

the shear velocity was estimated to be 0.24 fps. The 

tracer, Rhodamine BA dye, was injected continuously 

at a uniform rate from a Mariotte tank installed at the 

Blair Highway bridge for about 4 hours. The steadystate concentration distribution was sampled at 10 

cross sections between 1,700 and 33,000 feet downstream 

from the injection site. For more details on channel 

characteristics and test procedures, see Yotsukura, 

Fischer, and Sayre, 1970. 

In a study of the convective diffusion process in a 

curved reach, detailed information on the shifting 

pattern of the velocity-depth distribution is essential. 

In the present test, however, only two measurements 

were made on the tranverse distribution of velocity 

and depth; at 1,000 feet upstream from the Blair 

Highway Bridge (river mile 648.3) and at 17,500 feet 

downstream from the bridge (river mile 645). The assumption was thus made that the velocity data above 

the bridge were applicable to sampling sections 1 

through 4, while the downstream velocity data were 

applicable to sections 5 through 7. A mirror image of 

the downstream data was assumed to exist at sections 8, 

9, and 10. The two measured discharge distributions 

are presented in figure 11. The applicability of these 

discharge distributions was partly checked by computin the average dye concentration c, because the wrong 

combination of solute concentration and partial discharge distributions would produce an abnormal c 

value. Conducting such a test was crucial in the area of 

cross sections 5 through 8, because these were in the 

region where the thalweg shifts from the right- to the 

left-bank side, while the river alternates its direction 

of curvature. 
C14 DISPERSION IN SURFACE WATER 

0.9 

a 

;;:- 0.8 -

l.LJ 

~ 0.7 <( 

I 

(.) 

~ 0.6 Cl 

l.LJ 

~ 0.5 

::5 

~ 0.4 :::> 

(.) 

~ 0.3 

;::: 

<( 

_J 

l.LJ 

0:: 

0.1 0.2 0.3 0.4 0.5 0.6 0.8 

RELATIVE TRANSVERSE DISTANCE (z/8) 

FIGURE H.-Relation of relative cumulative discharge to 

relative transverse distance, Missouri River near Blair, 

Nebr. 

The results of the analysis are shown in figure 12 

and table 4. The data at cross sections 1, 2, and 3 were 

deleted because the computed c exceeded the acceptable 

range of error ( ± 15 percent of the average). The 

agreement between observed and computed solute distribution is not as satisfactory as the agreement for 

the preceding data fron1 straight channels. The location 

of the observed peak concentration for sections 5, 6, 

and 7 is different from what is expected, probably 

because of the discharge distribution used in the computation. For example, the discharge distribution 

employed for section 7, though adequate for the computation of c, concentrated too much discharge on the 

right-bank side and pushed the observed peak location 

too much to the left on the c' vs q' plot. The reverse 

situation occurred at section 6. 

The computed values of e 2uy2 and Ez appear to increase downstream. This seems natural because, as the 

solute passes through river bends, its mixing is assisted 

more and more by the increased transverse convection 

of secondary currents. The effect is manifested as a 

greater-than-normal increase in the degree of mixing, 

a decrease in a and an increase in e zUY2 according to 

equation 21. The value of uy2=912 ft3 per sec used in 

the computation of E z is the mean of uy2= 1290 ft3 per 

sec at the upstream cross section and 530 ft3 per sec at 

the downstream section. If uy2 truly decreased in the 

downstream direction, the value of E z would increase 

more pronouncedly than is shown in table 4. 

4·~~--~--~--~---.---.---r---,--,---, 

3 A • Observed 

2 

2 

IU 3 

~ 

z 2 

0 

;::: 

<( 

0:: 1 Iz 

l.LJ 

(.) 0 z 

0 A (.) 

l.LJ 2 > 

;::: 

<( 

w::l 1 0:: 

0 

2 

• 

x=22,410 ft 

0 

2 

• 

x=27,690 ft 

0 

FIGURE 12.-Comparison of observed and theoretical concentration distribution, Missouri River near Blair, Nebr. 

The agreement between the data and theory is fair. 

On the other hand, the transverse diffusion coefficient 

ranging from 0. 7 to 1.4 ft2 per sec is in good agreement 

with 1.3 ft2 per sec of an earlier study (Yotsukura and 

others, 1970), considering that the latter figure was 

estimated from a direct-simulation method employing 

somewhat different velocity distributions than the 

present analysis. Since the average depth was 9 feet and 

the average shear velocity was 0.24 fps, the above range 

of the diffusion coefficient corresponds to a range of 

Elder's constant between 0.3 and 0.65. The applicability 
TRANSVERSE DIFFUSION OF SOLUTES IN NATURAL STREAMS C15 

TABLE 4.-Computat1'on of besf-fit transver.c;e diffusion coefficif>nts, M1'ssouri R1'ver near Blair, Nebr. 

Date z', q'. Q X 

(cfs) (ft) 

Nov. 17, 1967 0.4 0. 4 34, 100 8, 730 

11. 8.50 

15,490 

18, 720 

22,410 

27,690 

32, 970 

of the present theory to a mildly curved stream appears 

promising; however, the complete verification requires 

more detailed data than were available in this test. 

APPLICATIONS 

The applications of the theory are in two categories. 

The first describes and predicts the transverse distribution of solute or waste concentration and requires 

detailed information on the transverse distribution of 

convective velocity and depth, as well as information on 

the conditions of the solute injection scheme. The 

monitoring of thermal and other waste discharges, the 

simulation of waste effluent by tracers, and the determination of the transverse diffusion coefficient are 

thrHe major investigations in the category. The second 

category is the description and prediction of overall 

efficiency of mixing without reference to actual solute 

concentration. To solve these problems, detailed flow 

information may not be available, and the accuracy 

requirement for prediction is often less than for problems in the first category. The need to estimate downstream distance for complete or partial mixing is a 

typical problem in the second category. The following 

pages present a summary of the study and applications 

of the theory in these categories. 

It is appropriate to review the assumptions involved 

in the theory which was developed to describe the 

steady-state transverse diffusion of a solute in a homogeneous steady streamflow. It is assumed that the 

solute is injected into the stream continuously at a 

uniform rate. By the use of the partial discharge q as 

an independent variable in place of the transverse 

distance z, the theory is adapted to the stream condition that velocity and depth vary from point to point 

in a cross section. No assumptions are made as to the 

cross-sectional shape or the velocity distribution. Even 

though one of the conditions of the theory is the presence 

of a prismatic channel, a mild degree of nonuniformity 

may be accommodated. On the other hand, the theory 

is not applicable to a nonuniformity which produces 

significant transverse convective velocity, as this upsets 

the orderly alinement of the stream tubes and overshadows the gradient-type transverse diffusion caused 

by turbulence. Thus the theory is not applicable for a 

reach where severe bends and (or) abrupt cross-sectional 

Pm a E,uy2 uy2 E 

(fts per sec2) (ft3 per sec) (ft2 per sec) 

0. 422 10.0 666. 0 912 0. 73 . 5.56 7. 2 946.4 1. 04 . 589 6. 5 888.4 0. 97 . 666 5. 5 1,026. 7 1. 13 . 699 5. 1 997.5 1. 09 . 785 4. 3 1, 135. 6 1. 25 . 825 3. 7 1, 288. 1 1. 41 

changes occur. Also, if the solute release is by high 

velocity jet, the theory cannot be applied for the 

vicinity of the release site. It is also important to note 

that the transverse diffusion coefficient, Ez or Ez, 

is a strictly empirical coefficient, even though its range 

is well established. The present theory determines from 

data but cannot predict the values of Ez or Ez in natural 

streams. 

In dealing with problems in the first category, the 

determination of the source condition should conform 

to the above-mentioned assumptions. In other words, 

the waste discharge rate !::t.q8 should be added to the 

ambient stream discharge to determine the total discharge Q. The location of the line source (qsl and qs2, 

where q82 -qsl =!::t.q8) or the point source (qs) should be 

fixed in terms of cumulative partial discharge. In many 

tracer studies, the tracer discharge rate is insignificant 

relative to the stream discharge. Nevertheless, if the 

injection is a line source, the location should be given 

by qsl and q82 so that !::t.qs represents the portion of the 

stream discharge affected by the tracer. 

The calibration of a stream reach, with respect to its 

diffusion coefficient, may be carried out in a manner 

similar to the preceding verification studies. Briefly, 

given a steady condition of stream and solute discharges, 

one can make a survey of the transverse distribution 

of solute concentration, velocity, and depth at several 

downstream cross sections. One can compare the observed nondimensional distribution, c' vs q', with the 

theoretical distribution, obtained using the given 

source condition, until the best match is obtained. 

One can then compute the value of EzUY2 for the section 

by equation 21 using figures for Q, x, and a determined 

by the above comparison. The average value of Eztty2 

for the reach should be obtained after the computation 

is done for all secticns. The survey should be carried 

out for at least three strean1 discharges (high, medium, 

and low), and the value of EzU.Y2 may be correlated to 

stream discharge. 

The prediction of solute concentration reverses 

the above process, namely, finding a from Q, x, EzUY2

, 

fixing the source condition, and then translating an 

appropriate analytical solution into an actual concentration distribution. 
C16 DISPERSION IN SURFACE WATER 

The computation of the transverse diffusion coefficient, Ez, is not necessary for calibration and prediction. However, since the data of the transverse distribution of velocity and depth are parts of the survey, 

Ez can easily be computed by equation 25. 

To estimate the mixing distance or efficiency, the 

simplest scheme uses the degree of mixing, P m' as 

defined previously. The analytical relation between 

P m and a was calculated for typical point- and linesource arrangements and is pre sen ted in figures 13 and 

14. Along with the distance equation, 

(21) 

these figures can be used to estimate a relation between the distance x and the degree of mixing P m· 

Noteworthy is the fact that the required distance x 

E 

~ 

0.9 

(.!) 0.7 

z 

x 

~ 

LL. 

0 

LLJ 

::::! 0.6 (.!) 

LLJ 

0 

0.5 

0.4 

for bank side injection (q;=O) is four times as long as 

distance for midstream injection (q;=0.5) to obtain 

the same degree of mixing. The premise can also be 

stated figuratively that a near-bank injection is equivalent to a midstream injection in the same channel with 

a discharge twice as large as the actual. Another premise 

of practical interest applies to multipoint sources. 

Assume that the channel is divided into N stream tubes 

of equal discharges, and the source is at the center of 

each stream tube. For this arrangement of N sources, 

the distance x necessary for a certain mixing level can 

be reduced to 1/N2 times the distance for a single midstream source. 

Equation 21 can be further simplified as needed. Let 

the channel depth, D, be defined by 

D~J.B ydz, 

B (27) 

DISTANCE PARAMETER (a) 

FIGURE 13.-Theoretical relation of degree of mixing, P m. to distance parameter, a, and source location, q' ,, for point source injection 
TRANSVERSE DIFFUSION OF SOLUTES IN NATURAL STREAMS C17 

where B is the width at the water surface. Also the 

average U is defined 

foBuydz Q 

U-- -=--· 

fo B BD ydz 0 

(28) 

With these average quantities and equation 1 for the 

transverse diffusion coefficient, equation 21 is transformed to 

1 UD2 U B2 

X=--·--·-·-· 

2a2{3 'UY2 u* D (29) 

To use equation 29, two expressions, UD2juy 2 and 

1/2a 2{3, need to be evaluated. The nondimensional ratio 

UD 2/uy 2 accounts for the deviation of flow and geometry of a natural stream from that of an idealized 

rectangular channel. The data tabulated in table 5 

were obtained from the streams used for the vertification, and the ratio UD 2

/uy 2 ranges from 0.3 to 0.9. 

As for the other expression 1/2a 2{3, {3 ranges from 0.2 to 

0.3 for a straight channel but can go up to 0.6 for channels with mild bends. The value of a should be chosen 

from figures 13 and 14 if the conditions are known for 

Pm and q;. 

The choice of a value for P m for so-called complete 

mixing depends on the nature of a particular study. 

For discharge measurements by dye dilution methods 

a mixing distance based on the value of P m of 98 percent is recommended because this mixing insures that 

the theoretical maximum deviation of concentration 

from the mean is about 3 percent. For less accurate 

predictions, a P m of 95 percent is adequate. In this 

connection, figures 13 and 14 indicate that the mixing 

distance is very sensitive to P m for high mixing values. 

For example, the distance required to improve the 

1.0~-------=~------~--------.--------,--------.--------,,--------.--------,--------, 

E 

~ 

0.8 

(.!l 0.7 

z 

x 

~ 

LJ.. 

0 

UJ 

~ 0.6 (.!l 

UJ 

0 

0.5 

DISTANCE PARAMETER (a) 

FIGURE 14.-Theoretical relation of degree of mixing, P m' to distance parameter, a, and source location, q' •1 and q' d, for line source 

injections 
C18 DISPERSION IN SURFACE WATER 

TABLE 5.-Cross-sectional form parameter, UD2fuy2, of natural streams 

Stream 

South River; test L ____________ _ 

South River: tests 2 and 3 _______ _ Atrisco Feeder CanaL ___________ _ 

Bernardo Conveyance ChanneL __ _ Missouri River near Blair ________ _ 

B 

(ft) 

60 

60 

60 

66 

600 

D 

(ft) 

1. 21 

1. 44 

2. 0 

2.3 

10. 0 

mixing from 95 to 99.5 percent may be as much as the 

distance required to attain the initial mixing of 95 

percent. On the other hand, such a theoretical result 

is difficult to verify by stream data, because as noted 

previously field measurements of P m often cannot attain 

sufficient accuracy. 

The application of equation 29 has been tested with 

the data collected by Barsby, Delannoy, and Watt 

(1967) on mainly shallow mountain streams of straight 

alinement. The tracer was salt, the concentration of 

which was measured by conductance measurement. 

No data for transverse distribution of velocity and 

depth were collected. The comparison of observed and 

estimated m1x1ng distance 1s presented in table 6. 

The values for Q, B, D, U, and P m. are from the above 

report and U* is computed by -JgDS. In the estimate, it 

u 

(ft per sec) 

0. 8 

.6 

2. 2 

4. 1 

5. 7 

Q 

(cfs) 

58 

52 

263 

627 

34,100 

UIJ2 

(fta per sec) 

1. 17 

1. 24 

8.8 

21. 7 

570 

uy2 

(ft3 per sec) 

2. 7-3.4 

2. 0-2. 9 

11-14 

26 

912 

0.43-0.35 

. 62- . 43 . 80- . 63 . 83 

. 62 

was assumed that 13=0.23 and UD2fuy 2= 1, while a was 

obtained from figure 13, in which the value of P m was 

known. 

The agreement between observed and estimated 

distance is fairly good for the Alwen River. For the 

Brenig, Wye, and Devon Rivers, the estimated x's 

are much larger than the observed values. This discrepancy is explained reasonably well by the possibility 

of UD2

/uy2 being less than unity in these rivers. The 

discrepancy for the Whitewater River is probably due 

to this being the only stream with significant bends, 

thus, higher 13 values. On the other hand, the proposed 

equation does not apply at all to the Hambleden Brook. 

The foregoing analysis, in comparison with the verification studies, demonstrates that mixing distance cannot be predicted with a high degree of accuracy unless 

TABLE 6.-C omparison of estimated and observed mixing distances, Water Research Association data 

Test River Q (cfs) B (ft) D (ft) U (fps) 

1 Brenig ____________ _ 12. 1 21.4 0.8 0. 7 

2 Alwen ____________ _ 8. 2 13. 5 0. 57 1. 07 

3 Alwen ____________ _ 5. 3 12. 7 0. 5 0.84 

4 Alwen ____________ _ 5. 3 13.4 0.47 0.84 

5 Alwen _____________ _ 25.5 37. 0 0. 64 1.1 

6a VVye ______________ _ 59.0 20.4 1. 15 2. 57 

7 a Devon ____________ _ 170.0 40. 1 2. 1 2. 04 

8 VVhi tewa ter _______ _ 17. 5 26. 6 1. 03 0. 64 

9 Hambleden Brook __ _ 4. 6 7. 2 0. 83 0. 77 

u. (fps) 

0. 66 

o. 33 

0.44 

0.30 

0.48 

0. 27 

0.49 

0. 27 

0.46 

Pm 

0. 923 

. 989 

. 520 

. 830 

. 907 

. 942 

. 773 

. 957 

. 996 

. 780 

. 860 

. 944 

. 966 

. 605 

. 751 

. 773 

. 845 

. 986 

. 820 

. 881 

. 972 

. 991 

. 987 

. 993 

. 996 

. 941 

. 970 

. 995 

a 

3.06 

2. 18 

7.80 

3. 87 

3. 20 

2.86 

4.36 

2. 71 

1. 95 

4. 31 

3. 61 

2. 84 

2. 60 

6.29 

4. 57 

4.36 

3. 74 

2.27 

3. 96 

3.44 

2. 51 

2. 15 

2. 25 

2.08 

1. 95 

2.87 

2. 55 

2.00 

Estimated X (ft) 

141 

278 

37 

150 

220 

275 

70 

183 

352 

125 

178 

28R 

344 

270 

510 

394 

535 

1,450 

440 

585 

1, 100 

1, 500 

700 

810 

905 

28 

35 

57 

Observed :t (ft) 

100 

200 

50 

100 

200 

270 

60 

160 

310 

50 

100 

200 

270 

200 

400 

100 

230 

770 

100 

200 

500 

800 

100 

300 

800 

70 

130 

250 
TRANSVERSE DIFFUSION OF SOLUTES IN NATURAL STREAMS C19 

detailed information of channel geometry and velocity 

is available. Nonetheless, equation 29 is useful if the 

limitations of the formula are fully understood. 

When the assumptions are that a=2.3, /3=0.23, and 

UD2juy2=0.6 for complete mixing resulting from a 

midstream injection, equation 29 is reduced to the 

simplest form 

U B2 

x=0.25 u*. n· (30) 

Equation 30 is comparable to the formula derived by 

G. M. Rimmar (British Standards Institution, 1964) 

B2 

x=0.130z(0.70z+11) gD' (31) 

where Oz is the Chezy coefficient of channel roughness. 

Equation 31 is written for the foot-pound-second 

system and is equivalent to assuming, in equation 29, 

th t 2 UD2j-2-1 d f.l __ {g__ h R a a= , uy- , an fJ- (0.70z+11)' w ere fJ 

ranges from 0.11 to 0.37, as Oz varies from 60 to 6. 

Both equations 30 and 31 are valid only for conditions 

of eomplete mixing of solutes by midstream injection 

in a straight channel. 

CONCLUSIONS 

The preceding section summarized the present study 

concerning application to practical problems. This section is devoted only to the essential conclusions derived 

frmn the field verifications of the proposed model. 

1. The proposed model enables steady-state transverse 

diffusion of solutes in natural streams to be described satisfactorily by analytical solutions of the 

diffusion equation. It is satisfactory insofar as 

present knowledge on transverse diffusion coefficients prescribes the use of a cross-sectional 

average value, Ez, rather than use of local values, 

Ez(z), in diffusion models. 

2. The study established that the cumulative partial 

discharge, q, rather than the transverse distance, 

z, is the prime variable to be considered in transverse diffusion processes. 

3. The model enables the reduction of transverse 

diffusion data in terms of nondimensional variables 

consistent with the mass balance of solutes in 

natural streams. 

4. Calculated diffusion coefficients by the model are in 

good agreement with Elder's formula (/3=0.23) 

and are consistent within straight shallow reaches 

of a natural stream. 

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U.S. GOVERNMENT PRINTING OFFICE: 1972 0-466-743