- Email: [email protected]
Longitudinal estuarine diffusion in San Francisco Bay, California
Water Research Pergamon Press 1969. Vol. 3, pp. 1-20. Printed in Great Britain
LONGITUDINAL ESTUARINE DIFFUSION IN SAN FRANCISCO BAY, CALIFORNIA B. GLENNE a n d R. E. SELLECK Oregon State University, Corvallis, Oregon, U.S.A. and University of California, Berkeley, U.S.A.
(Received 2 August 1968) Abstract--The concept of one-dimensional diffusion is discussed and applied to the San Francisco Bay, California, to evaluate the degrce of mixing and dispersion existing within the estuarine system. The intensity of mixing and dispersion is expressed in terms of an overall, longitudinal diffusion coefficient E which can be varied throughout the system. A method for computing E from observations of water quality constituents in an estuary is presented and discussed. Concentration distributions of chlorosity and silica derived from field observations taken over four years of continuous sampling of the Bay waters are used to evaluate the diffusion coefficient within the two arms of the Bay system. The results derived from the two differont concentration distributions are found to be comparable, giving at least a partial verification of the proposed procedures. The relationships between the observed coefficients and the hydrologic, hydrographic, and tidal wave characteristics of the Bay system are examined to ascertain if the diffusion coefficient can be predicted directly without making resourse to water quality observations. The only relationship investigated which appears to have merit is E = c o n s t . (~ u ' D ) 3/* where ~ is the mean advective velocity in the estuary, u'= the mean tidal velocity, and D the hydraulic channel depth. The equation is applicable only to those regions of an estuary where the advective velocity is known to have a significant effect on the value of E. Procedures which may be used to predict the concentrations of a water quality constituent in the Bay system--once the diffusion coefficients are known--are discussed briefly. An example of a steady state computation is presented and discussed.
INTRODUCTION IN THE last decade the concept of eddy diffusion has advanced our knowledge of mixing hays and
estuaries. Basically, the diffusion approach offers a continuous mathematical model described by a non-linear, partial differential equation. Solutions are generally obtained by a finite-difference technique with a digital computer manipulating the arithmetic. The purpose of this paper is to state and refine the diffusion concept by: quantifying the longitudinal mixing process in the San Francisco Bay, California, by evaluating and using diffusion coefficients; presenting equations for predicting steady and unsteady longitudinal concentration profiles of water quality parameters in natural estuaries. The investigation was performed in conjunction with a comprehensive study of San Francisco Bay which was conducted at the University of California, Berkeley, and is described by STORRS et al. (1964). Having completely different tidal and advective characteristics, the northern and southern branches of the San Francisco Bay system serve well to illustrate the application of the described procedure. The physical characteristics of the Bay system are summarized in Fro. 1 and TAnLV.S1 and 2. A remark is pertinent about the so-called one-dimensionality of the procedure. Only after the estuary has been divided into small cross-sectional volumes are the cross-sectional areas and advective flows assumed constant for each small volume. This permits a reasonable mathematical approximation of the existing natural conditions. 1
2
B. GLENNE and R. E. SELLECK REVIEW
The general three-dimensional convective diffusion equation of continuity was stated early by classical hydrodynamicists. A three-dimensional equation is not easily applied to natural estuaries, however, because of the irregular physical boundaries of such systems.
/
N
J-~L -~m =.
SAN PASLO BAY
SACRAMENTO
~ICHMONO
PACIFIC OCEAN OAKLAND
SOUTHERN S~N " FRANCISCO BAY
,.~4NMATEO
%~_~'
'?
,~ ....
FIG. 1. Location map.
ARONS and STOMMEL(1951) were among the first to propose solving pollution problems in estuaries with a one-dimensional, longitudinal convective diffusion equation. From observed steady state salt distributions, STOVa,mL (1953) presented a method using finite differences to determine the magnitude of the diffusion coetticients in the equation. PRrrCHAP,D (1952) deliberated along the same lines and
Longitudinal Estuarine Diffusion in San Francisco Bay, California
3
applied the method to the salt balance in the St. James estuary in Chesapeake Bay. KENT (1958) considered the diffusion coefficient to vary with time as well as position. He used a steady state procedure to compute values of the eddy diffusion coefficient in the Delaware Estuary. OICUBO(1965) studied the validity of the mathematical derivation of the equations used in the previous work. SELLECK and PEARSON (1961) used an Orzan tracer to investigate the diffusion effected in the landward end of the South San Francisco Bay. The longitudinal diffusion coefficient was found to have
TABLE 1. PHYSICAL CHARACTERISTICS OF SOUTHERN SAN FRANCISCO BAY
Name of sampling station
Section No.
CB-6 1 CB-5 2 CB-4 3 CB-2 4 LB-2 5 LB-3 6 LB-6 7 SB-3A 8 SB-1 9 SB-4 10 SB-5 11 SB-6 12 SB-7 13 SB-8 14 SB-9 15
Distance from Golden Gate
Length of section
Crosssection area at MTL
Hydraulic depth
Mean tidal range
(ft)
(ft)
(ft 2)
(ft)
(ft)
0 8255 16510 24420 32320 42160 52000 62650 73300 83800 94300 108100 121900 129700 137500 143350 149200 159475 169750 177775 185800 191050 196300 200050 203800 208450 213100 215800 218500 226750
938000
~ 177
4.0
,,~79
4.1
16510 15810 19680 21300 21000 27600 15600 11700 20550 16050 10500 7500 9300 5400 16500
825000 850000 915000 890000 920000 1030000 820000 653000 333000 359000 292000 286300 232000 165500 1011200 126200 92000 65710 37000 33070 19400 19350 10300 8240 6900 3200
4.3 38.7 4.5 24.4 4.8 16.9 5.2 12.2 5.6 14.0 5.9 13.7 6.1 11.4 6.4 11.2 6.7 6.8 6.9 6.8 7.1 7.6 7.3 5.1 7.4 4.5
values ranging from 460 to 1380 ft2/sec. The California State Department of Water Resources performed diffusion studies in the northern branch of the Bay system. OWEN (1964) found diffusion coefficients ranging from 200 to 14,000 ft2/sec. F r o m hydraulic model studies IPPEN and HAgI~r,IAN (1966) proposed that the actual value of the diffusion coefficient departs from an idealized value in a manner which is proportional to the energy dissipated in the estuary. They described the ratio of the
4
B. GLENNE and R. E. SELLECK
actual diffusion coefficient to the idealized value as a function of a stratification parameter, or in mathematical terms --~ = c o n s t . \ 2 P , F 2 J
(11
TABLE 2. PHYSICALCHARACTERISTICSOF NORTHERN SAN FRANCISCOBAY
Name of sampling Section station No.
CB-6 16 NB-3/4 17 NB-5/6 18 NB-9 19 SP-3/2 20 SP-4/5 21 SP-8/7 22 SP-9 23 SU-1 24 SU-2 25 SU-3 26 SU-4 27 SU-5 28 SU-6A 29/31 SU-9 32 SU-10 33
Distance from Golden Gate
Length of section
Crosssection area at MTL
Hydraulic depth
Mean tidal range
fit)
(ft)
(ft 2)
(ft)
fit)
0 11850 23700 29800 35900 45700 55500 61650 67800 79100 90400 103250 116100 125500 135000 151300 167600 174650 181700 186450 191200 198250 205300 213100 220900 229500 238100 247950 257800 268750 279700 289850
938000
~ 177
4.0
23700 12200 19600 12300 22600 25700 18900 32600 14100 9500 14100 15600 17200 26000 21900 20300
1165000 1361000 762000 684000 507000 504000 490000 646000 573000 580000 265000 245000 141000 186000 175000 162000 194000 216000 202000 383000 216000 217000 135000 104000 128000 122200 43000 71700 71400 73900
~89 31.3
3.9 4.2
26.3 4.0 30.1 4.1 12.7 4.4 9.6 4.4 29.9 4.3 37.4 4.1 21.0 14.4 3.9 10.4 13.9 31.0 3.3. 20.5 3.1 24.9 19.0
in which E = diffusion coefficient, b = a characteristic length in the mixing process, ~ = mean advcctive velocity, Q = advectiveflow, T = tidal period, Pt = tidal volume, F = Froude No. = u,~'x/(gD), u'~ = mean tidal velocity, D = hydraulic channel depth and n = exponent to be determined from data. In the Rotterdam estuary HARLEMANand ABRAHAM(1966) used the tidal excursion as a measure of the characteristic length b.
Longitudinal Estuarine Diffusion in San Francisco Bay, California
5
Some additional concepts of longitudinal diffusion are furnished by various other investigations not directly concerned with the problems of estuarine mixing. KOLMOC-OROrr (1941) hypothesized that in isotropic turbulence the diffusion coefficient can be described by E = c o n s t . G 1/3 L41a
(2)
in which G ffi mean rate of energy dissipation per unit mass of fluid and L = mean size of eddies participating in the diffusion process. In open channel flow G can be approximated through Chezy's formula. Equation (2) may then be expressed as
/L,t\I/3
in which R = hydraulic radius of the channel. Assumptions made at times are that the mean size of the eddies is directly proportional to the hydraulic radius a n d that for wide channels the hydraulic radius is approximately equal to the channel depth. Equation (3) then becomes E = const,
o
(4)
TAYLOR (1954) proposed a formula for the prediction of longitudinal dispersion coefficients for uniform turbulent flow in straight closed conduits. ELDER (1959) expanded this formula to open channel flow to obtain E = 5.9 D ~/(~o/P)
(5)
in which 7o = wall shear stress and p = density o f the fluid. Equations (4) and (5) are obviously similar. AL-SAFFAR (1964) found the mean diffusion coefficients in laboratory flumes to vary approximately as the 0.85 power of the channel velocity (expressed as a Reynolds number). ORLOn (1959) related the lateral eddy diffusion coefficient to a channel scale. THEORETICAL
CONSIDERATIONS
General equations Gt~NNE (1966) has shown that the one-dimensional convective diffusion equation of continuity can be written
;
Ss+ A ?5
(6)
in which x = longitudinal distance, A = cross-sectional area, ~ -- mean constituent concentration, t = time ordinate and S, -- net rate of constituent mass being created or destroyed per unit length of the estuary. Equation (6) is a linear, first-order non-homogenous partial differential equation in E, which can be solved by ordinary methods. Integrating seaward from the landward termination of the system gives
A E ~x = A • 5 -
Ss dx +
A ~-t dx
(7)
20 The landward terminal of the system is the point where the tidal influence or cross-sectional area becomes insignificant, or where the constituent concentration has such a low level so as not to influence significantly the magnitude of the integral terms on the right hand side of equation (7). The terms AE(O5/Ox) and Afi~ represent the time average diffusive and advective transport rates of the constituent mass, respectively, across the most seaward cross-section of the reach (see Fla. 2). The first integral term o n the right hand side of the equation is the summation of sinks and sources of constituent mass entering or leaving the reach. A constant of integration may be included in this term if a source of constituent mass exists at the landward termination of the estuary. The last term o n the right hand side of equation (7) represents the non-steady depletion or accumulation of constituent
6
B. GLENNE and R. E. SELLECK
mass within the landward reach. Replacing the two integral terms with the shorter expressions XS, and ZF, equation (7) becomes AE
(8)
"=- = ox
o
o
With the aid of numerical methods, equation (8) can be used to obtain estimates of the diffusion coefficient E when the constituent concentrations and advective flow conditions are known. The limitations on equation (8) are described in detail by GLENNE (1966). Some of these are:
,ucA INCREMENTAL
CONTROL
VOLUME
TIME AXIS
j+l
--
O i-I
I
SPACE AXIS
i÷l
REACH ~
F i I
I
~ - - - - ~ - SECTION
', LANDWAROEND OFESTUARY
L
I
~ . ~ - ' ~ , ' . v , , . . -
~,..
FIG. 2. Sketch of reach and control volume. the constituent in question must react according to the diffusion mixing concept (Fick's First Law); diffusion in the estuary must essentially be one-dimensional; stratification in the estuary should be small enough to make the constituent gradient truly representative; second order changes of a, ~ and A with distance are insignificant when compared with first-order changes;
Longitudinal Estuarine Diffusion in San Francisco Bay, California
7
the parameters E, A, ti and e are average values taken over at least one, and preferably several, complete tidal cycles; the values of ti and c are also averaged over the entire cross-sectional area; A is essentially unchanged from one tidal cycle to the next. For a non-conservative water quality constituent, equation (8) takes the form X
AE "~x ~ = A~-~
X
(Ss-kV6)+ ~o
(9)
in which k = first-order rate constant and V = volumes of the sections making up the reach. Equation (6) may be expressed in simpler forms by assuming either separately or successively that: the system is in steady state; no external sources or sinks exist for the system; A and t~ are constants with respect to x; E is a constant with respect to x. the constituent under study is conservative (k = 0).
Evaluating the diffusion coefficients With known constituent concentrations, sources, and sinks, equation (8) can be used to evaluate the diffusion coefficient. [Equation (6) is not as well suited for this purpose because of the derivatives of E contained therein.] Letting the subscript i denote the ith station on the longitudinal axis and the subscript j the jth time trace (see FI~. 2), equation (8) can be written in a central finite difference form as follows:
Af,~E,,j -~x ,,j = A"ju"Jei'J-X(S')"J+~o _
~x/i,y
i
,-1,j Axi_,,~
(10)
1
4Ax,. y
[(e'+l'J+x-6'-"J+')+(e'+"J-l-e'-"i-x)]
(lOa)
c-~,j = { (e~, j+ t - c~, j - 1)
(lOb)
(0-~t9 ,-1.j = At,,j1 (e,_l,j+ ~-e,_,,j_,)
(lOc)
i-2
(Ss),, s = ~ (Ss),, j +-~ ['(Q,e,),_ 2, j + (Q,e,),, j] - 1 e,_ 1, j [(QL),- 2, j + (QL)~, j] (10d) 0
0
in which Q, = flow rate of source discharging into a section and QL = flOWrate of sink withdrawing water from the section. The evaluation of Ei.j from equation (10) requires a knowledge of the hydrographical characteristics and the advective flows, as well as the constituent profiles. The summation process commences with a small landward reach and progresses in the seaward direction with the cumulative addition of finite estuarine sections. The process increases in accuracy with decreasing lengths of estuarine segments. A matrix in time and space of observed diffusion coefficients results. For the San Francisco Bay study the lengths of the sections were dictated by the locations of the water quality sampling stations, and the time intervals by the lapses in time between consecutive sampling cruises.
Steady state constituent concentration computations Equation (8) is also convenient for computing steady state constituent concentrations when E is known. Dropping the unsteady term of the equation and employing central finite differences on the distance axis gives
A~Ei( Ci+l--Ci~
1.)=
Aifii ( ( i + 1 +2' ~ -
1 ) _ ~ o (S~) ~
(11)
8
B. GLENNE and R. E. SELLECK i
i-2
Z (s~),= o
(s,), + 2 (s,), o
i-2
= ~ (Ss)i+(Qsgs)i-(QI.id~_l)-¼ki(Ai+Ai_l)(Ax~+Axi_2)6i_ 1 0
(lla)
Solving for ~ ~ equation (11) takes the form
2Ax~ ['A~E~ . Aifi i c] +2 -- A, ( 2 E , - ~,Ax,) LTx[ ci-~ + ~
~
]
e,_l -~o (S,),
(12)
one initial condition is required to solve equation (12). Generally this is a known or assumed value of ~J~_~at some point in the estuary.
General constituent concentration computations Equation (6) is best used to determine unsteady state constituent concentrations when E is known, i.e.,
0
OC
~-~(Auc)
]..,
-[Ss]i,j+[A
O~
Employing a forward finite difference approach and referring to FIG, 2, the individual terms of the above equation may be approximated by the expressions
(S,)i, j = (q,)i, j (es)i, y - (qL)i, j el, j -- ki, j Ai, j e~, j
(13a) (13b)
0 ( A ~ e ) ~,~
2 (xi, j - x i_ l, j)
'
'
(13d)
(A~()t- 2, j = As- 1, j ui- 2, y Oi- 2, j (A~d)~+t,l = (A~,ig~,Id~,j)-F x t ' j - x t - l ' J ( A i + l , j ~ i + l , j d i + l , j - A t , Xi+ l, j-- Xi, j
j~i, jg~,j) (13e)
(AE-ff-x)i_½, i=½(A,,yEi, j+A,_l,jEi_l,j,i\xi, i_xi_l,y,]
(A~ o~
oe
(1311)
Ox)i+½,j=(AE),+½,j~f+½,j
(a~)
(AE)i+½, j = A i j E i j 4 xt'J--X~-l'] ( A i + l , y E i + l , y - A l , jEl, j) • , 2(Xi+l,y--Xt, j
~X i+½, j
=
(13g)
(13i)
(~,, ,- ~,_,, 4:) 2(x,, ,-x,_,.,)[~,+~,,-~,., .:,,-~,_I.:](13j) .
kXi, j--Xi-1,
(xi+x,j-x~-t,j)kXi+l,y-xi,
j
xi,.l-xi-1,
in which q, = flow rate per foot Of estuary length of a source discharging into the estuary, ~, = constituent concentration of the source, and qL = flOW rate per foot of estuary length of a sink with-
Longitudinal Estuarine Diffusion in San Francisco Bay, California
DIFFUSION COEFFICIENTS, ~ o
~l ~
ft 2 / s e c
~o
~ o
~ ~ ~ I--E.
9
~ ~.c,.o_0.~..0 , , ,,,
.. ,,, ORt~SE
CTION
3
• --
SECTION
4
•
SECTION
5
SECTION
6
•
•
n
SAN
MATEO
BRIDGE
SECTION 7
8r SECTtON 8
SECTION 9
- -
DUMSN~TON
BRIDGE
SEC3"t0N I0
ETI(~I
•
II
o--
omole
" 0 --SECTION 12 ~ ALVISO ° S L O ~ l ~ 0 0 SECTION t 3 ~ - = m m m SECTION 1 4
~I-~
S. P.R.R.
Q ellDql) o •
~ •
eo
-
•
-
•
s
IJl
•
B:~IDGE eeoee,,o*
~
oe
• • •
I
l
•
•
SECTION IS
I
•
J
,
i
-,,
I ill
-,,','= -,
I
,
FIG. 3. Diffusion coc(~cicnts in southern reach from chloride tracer.
,_
10
B. GLENNE and R. E. SELLECK
DIFFUSION o =
COEFFICENTS,
sq f t / s e c
o
I
u I I I.,t~S/~I
SECTION
5
SECTION
4
SECTION
5
SECTION
6
0
o
I
I
i
l
I
F R A N C I S C O - OAKLAND
I
OAf
• --
•
I
I
I
I
BRIDGE
•
•
•
•
P, 1:) o
o
u~ c) m
~SAN SECTION
7
SECTION
8
SECTION
9
MATEO - - H A Y W A R O
BRIOGE
o--e
• m
G'3 o ill
•
•
--
CUMOARTON
SECTION
I0 o--o--l--o--o
SECTION
II
•
o--o•
O - - O
O - - O - - O
BRIDGE o--o
•
o
~
o
o
-
• • o - - o - - o - - o • •
-
o
-
-
o
•
•
o --SECTION
12
SECTION
13
SECTION
14
SECTION
15
0 0
I
]
f ~ll
~'~'o~G~" o - - o
j~-SPRR
. •
.
e~ooeo
I~rlDGEe o e m O I
i
.
.
.
oo
,
•
,De
,
I
.
.
•
•
e•
• - -
• - -
J I ill
.
.
.
•
•
Ioo•
•
J
FIG. 4. Diffusion coefficients in southern reach from silica tracer.
I
Longitudinal Estuarine Diffusion in San Francisco Bay, California
01FFUSiON
COEFFICIENT,
(fit/see)
8 I
I
I
I
II
i
I
I
8
I
I
I
I I [
I
-SECTION
I7
S~C T I O N
18
i°
11
I
ANC~EL
I
ISLANO
° I
RICHI~I~ RAFIEL.
-- SAN B l ~ OC.~
<< m m
•
SECTION
•
0
0
20
ee
O •
e
mo
,.¢~3 4DO - -
CO --
PC. SAN P E O R O
--
CA.qQUIN EZ STRAITS 0RIDGE
--
BIENICIA - MARTINEZ 8RID(
--
PtTTS8 URG
OQ O O 3 0
:0
Z
--
SECTION
21
SECTION
22
SECTION
23
SECTION
24
SECTION
25
SECTtON
27
SECTK~I
28
z
o
•
IND
-
-
e
e - -
•
•
eo
~ ¢=
o e J l ~
O
•
•
Io
OCP
o
• oloeo--
o ooo
o oe o
~
o o
O O
m
o
¢)
o o
o~
o
o
o o
•
~ •
o ee
oo
--
i
- -
I
~
I
t
i ill
i
i
t
~
I
ttll
t
Fzo. 5. Diffusion coefficients in northern reach from chloride tracer.
I
I
I
12
B. GLENNE and R. E. SELLECK
drawing water from the estuary. To use this method for the prediction of constituent concentrations, equation (13) needs to be solved for cl,j+ ~, or
_
/
tj+z--t jr/~
6~
6
1
The boundary conditions are C~,o, the initial constituent distribution and two end conditions established by equation (8). DIFFUSION
COEFFICIENTS,
ft/sec
o
SECT~N
t7
SECTION
IB
S ( C TION
19
SIECTION
20
$ ( C TION
21
SECTION
22
0
0
•
'1
--
ANGEL I S L A N 0
o
R~:HMONO - S A N R A F ~ L IBRIOGE
--
OT. $ ~
PI[0RO
;2
"*o
o•
o
o
e--era
o - -
-- CARQUINEZ $ 1 1 ~ I T S 8~10~E
SECTHON
23
SECTION
24
SECTION
E5
ECTION
26
SECTION
27
•
*
o•
o
~
¢ml
*
a
--
~CT~N
oJ
o
-
• •
-- *
* -
-
*
~
**
m
o
o
*o
o•
o
~
-
*o
oool~
o
~
m
~
o
•
•e
m
-
-
o
.
BENICIA - Me,R'rlNEZ 8 ~ O ( o
a
o - - m
o
o
28 --
I
l
I
I
I
l
I
J
r
,
i
I
~TTS~JRG
ill
FIG. 6. Diffusion coefficients in northern reach from silica tracer. To insure stability of the iterative solution, the time interval has to be kept within a prescribed limit. Using a numerical analysis of parabolic partial differential equations, it can be shown that the stability criterion is approximately
Atj/(Axi) 2 <~ 1/2Ei
(15)
5.0
O I0.0 O / 3.: (1)
)f-
15.0
z0.0
"
A
Z~
--
L~
~ 0
0
OBSERVED 15 ~ 62 ,, ........ o " 31 JUL 62 r-t__ II JUL 62 ESTIMATEDAOVECTIVEFLOWS (BY DWR AT CHIPPS ISLAND ) 4,260 cu ft/sec AUG 62
A
AA
I
~
~
0
I
13 13 \ 13~.~ I::]
t"1 N
~ ~1
[]
E]
--
50~ 000
THEORETICALPROFILEOBTAINEDFROMEQUATION (12) " (INITIALCON01TIONUSED c=IT"og/IAT23'7OOf'}
I
2OO,000 DISTANCE FROM GOLDEN GATE ,ft
0 °
/
I
Flo. 7. Example of a steady state integration.
I00,000
~
I
rJl
~o
E
r~ 0 0~ .~" ,.. r-L
14
B. GLENNE and R. E. SELLECK CALCULATIONS
FOR SAN FRANCISCO
BAY
Equation (10) was used to evaluate the diffusion coefficients in the Bay system. This requires a knowledge of the concentrations of either a natural or artificially induced water tracer. Two natural tracers---chloride with a seaward origin and silica with a landward origin--were used to make independent evaluations of the coefficient. The constituent data were collected in the Bay during an extensive sampling program lasting from 1959 to 1964. The period between sampling cruises averaged 56 days and each station was sampled on three different tidal stages and at two or more depths. The distance between the sampling stations varied from 2700 ft to 16,300 ft as shown in TABLES 1 and 2. The raw data were then normalized to obtain representative concentration profiles. This process is described in detail by GLENNE (1966). I000
I
0 n,I-
600
h0 i (_9
400
--
0 It
I
I
X
25
+
24
0
25
rl
26
0
27
I
I
'"1
Z~
28
0 []
o 6
A 200
n
I
SECTION
8O0
x
0
--
,,
z~
X
t,LL ZLL UJ h'
_~o ~z too
XO
A
I00
80 --
#
,¢
0
o~ 60
-X
:~Z Lt. LLI
40
*
%
I
I
I
I00
200
400
600
O n 0 A
-XX
20
--
IC I0
I
I
I
I
20
40
60
80
I 8 0 0 I000
A D V E C T I V E V E L O C I T Y IN P E R C E N T O F G E O M E T R I C M E A N A D V E C T I V E V E L O C I T Y IN 1 9 6 1 - 6 2 W A T E R Y E A R
FIG. 8. Relationship between the diffusion coefficient and advective velocity in Suisun Bay. Although each observed concentration profile did not extend throughout the entire estuary because of the nature of the sampling program employed, it was possible to calculate four separate matrices of diffusion coefficients in time and space. The results are presented in FIGS. 3-6. FiGtn~ES 3-6 do not show all the diffusion coefficients evaluated, but rather the coefficients passing two acceptance criteria. Coefficients were rejected if: (i) the observed constituent concentration profile gradients were less than 10 - s g/1 of chloride per foot or 10 - s mg/1 of silica per foot. (ii) the overall value of the three terms on the right hand side of equation (10) was less by one order of magnitude than the value o f the largest o f the three terms considered individually.
Longitudinal Estuarine Diffusion in San Francisco Bay, California
15
These criteria were found necessary because relatively slight inaccuracies in the data may create large errors in the estimates of the coefficient for the regions where the criteria are not fulfilled. FIGURES 3-6 indicate that the computed diffusion coefficients are essentially the same for both the silica and chloride tracers. The variations in the computed values appear to be larger, however, when using silica tracer. This may result from the relatively large inaccuracies involved in estimating the sinks and sources of silica in the system. For example, a significant uptake of silica by diatoms may occur during the summer season. Equation (12) was used to calculate a steady state chloride concentration profile in the northern reach of the Bay system. The average hydrologic and hydrographic conditions of August 1962
I000
[
800
600
--
(D
@ FUJ 5; o (.9 LL 0
[
I
}
400
X
19
+
20
0
21
0
22
I
I
o [3 0
200
0 0 +
la
n,- I_ bJ Z n LIJ
_z,3
I
(3[:]
L)
t.--LL
I
SECTION
0
I00
8o I --
"
60
0
0
0
X []
÷
X
0 4-
0 "IX
121
O - - - - - X ÷
rl
0~1-
÷ X
÷ l
4-
0 {g
¥
0
0 X
40 x ~Z u. ILl
20
io
--
. . . . . I0
I __ 20
I
I
40
60
] 8 0 IO0
I
I
200
400
[ 600
I 800 I000
A D V E C T I V E V E L O C I T Y I N PER C E N T O F G E O M E T R I C MEAN ADVECTIVE VELOCITY IN 1961-62 WATER YEAR
Fie. 9. Relationship between the diffusion coefficient and advective velocity in San Pablo Bay. were used in the computation. The flow from the Sacramento-San Joaquin River Delta was supplied by the California State Department of Water Resources (DWR). The boundary conditions used was a chloride concentration of 17.00 g/1 at the seaward entrance to the northern reach. The results are shown in Fie. 7. Also plotted in Fie. 7 are the chloride concentrations observed in the reach during July and August of 1962. The agreement between the observed and computed concentrations of chlorosity as shown in FIe. 7 appears reasonably good. Subsequent investigations have shown, however, that the chlorosity distribution in the Bay system is seldom in steady state, and the high degree of correlation between the observed and computed values exists partly because the boundary condition was determined from the observed values. A digital computer program was also written to calculate unsteady state
B. GLENNE and R. E. SSLLECK
16
constituent concentrations using equation (14). The results of those calculations will be discussed in detail in a subsequent paper.
DISCUSSION Inspection of FIGS. 3-6 shows that several deductions can be made regarding the behavior o f diffusion coefficients in the Bay system. In the South Bay, which is a well-mixed estuary having relatively small advective flow, the diffusion coefficients are smaller than those observed in the northern arm o f the system. The diffusion coefficients appear to decrease in value in embayments and increase in or near the straits o f Carquinez and San Pablo. As a check, the computations were repeated using 1000
E
T
800 _
SECTION X
15
600
+
II
--
0
~2
[3
13
0
14
400
I-b~ :E
i
j
i
1
1
1
I
I
4-
,9,
(.9 h 0 FZ ~J (J r~ i_
1
"t" b
200
0 0
n
too
h' Z
+ I00
0
+
O ~ Q 0 ~ + ~
o~ ~-
+ ÷
÷o
80 60
X X X
UJO
o~
40
:DZ tL,~[
20--
10 I0
I
I
I
I
20
40
60
80
I00
I
I
200
400
6 0 0 8 0 0 I000
ADVECTIVE V E L O C I T Y IN PER C E N T OF G E O M E T R I C MEAN A O V E C T I V E V E L O C I T Y IN I 9 6 1 - 6 2 WATER YEAR
FIG. 10. Relationship between the diffusion coefficient and advective velocity in the southern reach. a reduced or "effective" cross-sectional area of the main channel in San Pablo Bay. However, because the reductions in cross-sectional areas were at most 15 per cent, this proposed alternative did not significantly change the value of the diffusion coefficient. In the South Bay an increase in the diffusion coefficient is found near the landward end of the estuary. This is most likely due to the added mixing caused by the advective flow in the region where the value of f~/u'm becomes significant. In attempting to reduce the variations in the observed diffusion coefficients, the coefficients were plotted against the corresponding net advective flows. The results are shown in FIGS. 8-10. In the northern reach the advective flows are expressed in percent of the in sit, geometric mean o f the advective flows for the 1961-62 water year. In the South Bay the geometric mean was computed for the measurements made during the six cruise periods conducted in 1961-52. The diffusion coefficients
E
'I0°
O'--b ,o
I00
,
X
X
X
T",
,~
X
X
I0'
X
X
x
X
•
X
,
x
0
o
,
X
,
A oo
,
~
•
~
,
z~
o *
i
~ ~
•
o
o
I0z
•
"T~--T~
~
•
•
•
,
, ,
Z~
STRATIFICATION
,
0
,
,
0
I0 ~ INDEX
O'
0
~-'rrTT ,
, ~ - ~ )
O
O0
/gOO
O
,
\
0
O 0
I ~-
0
•
0
r~ - F T - T ~ F T ~
•
O
-
•
I
&& •
;
•
I ,
r
•
I
•
•
•
+
io"
+
I I I II
FIG. l 1. Relationship between E/~b and the stratification index in the northern reach.
X
X
o
, , , Tl-
÷
I
I
-F
~¢TIO# S[CT~
X 0 0
I
I
23
T9
I
]
+#
I
I
+
+
[
g~CTION SECTIme 27 4- SECTION 2e
I I
I
I II
+ +-
io*
I I I I'~
0
@
I:I
18
B. GLENNE a n d R, E. SELLECK
fill
I I
I
I
I
[III
I I
Ii. I
I
÷.
÷ ÷ ÷ -
~_g~g_g_~
4.4 4-
* ,m
4 4
•
4
• 4
o
0
0
O O 4
4
0 O0
0 0
O
-
.~
0 0 0
0
.~ c3 13o <3 0<1, ~ I : I D CI
o
o
o
o
'o
o
o o
o
°~
x
x
13° 0
o x
o
O
x
X X
lllll
I
I
I
f
IIIII ~O
I
I
I
I
No_-
Longitudinal Estuarine Diffusion in San Francisco Bay, California
19
are expressed as percentages of a mean diffusion coefficient found at each section, with the mean diffusion coefficient being computed from tho~arithmetic mean of the chloride concentrations observed in the above-mentioned time periods. As evidenced by the figures, there is a definite dependency of the diffusion coefficients on the advective flows in the northern reach. This has also been noted by OWEN (1964). Lines fitted through the points shown in FIGS. 8 and 9 indicate that E varies approximately as the 3/4 power of a. In the South Bay no such dependency is apparent. The probable reason for this is that the tidal mixing greatly over-shadows the advective mixing in most of the southern reach. It is evident that some of the variations evidenced in FIGS. 8-10 result from "inaccuracies" in the measurements. This is especially true of the results depicted for the southern reach. Without extensive additional data analysis, however, it is difficult to quantitate the magnitude of these inaccuracies. Equation (1), with b = u'=(T/2), was used to obtain the results shown in FIG. 11. A useful relationship does not appear to exist between the observed diffusion coefficients and those predicted by equation (1). In FIG. 12 the same approach was used, but the term b was made equivalent to the hydraulic depth. This appears to be the only relationship valid for the northern reach of the San Francisco Bay. Letting n = - 1/4 gives the relationship E = const. (aura'D) a/4
(16)
CONCLUSIONS Using Fick's First Law and a mass continuity concept, the mixing or diffusion coefficients in estuaries can be evaluated from a knowledge of the constituent concentrations, sources, and sinks, as well as the general advection through the system. The following observations itppear to be valid regarding the nature of the diffusion coefficients determined for the San Francisco Bay: the values of the diffusion coefficients obtained using silica as the tracer generally coincide with the corresponding values obtained using chloride as the tracer; the diffusion coefficients exhibit considerable variations with time as well as distance; in northern San Franciso Bay where the advection is significant, the diffusion coefficients are found to vary approximately as the 3/4 power of the advective velocity; of the semi-empirical methods examined for predicting directly values of the diffusion coefficients only equation (16) shows promise of giving satisfactory results. The one-dimensional diffusion model, in general, presents a feasible method for solving steady and non-steady state transport problems of pollutants and other constituents in waterways and estuaries. The steady state solution can be calculated by hand, whereas the non-steady state solution is best handled by a computer. Solutions to transport problems in estuaries commonly depend on the information available regarding the magnitude of the diffusion coefficient. It is hoped that the many estuary investigations now being conducted will contribute significantly to such information.
Acknowledgements
This study was sponsored by the State Water Quality Control Board, Sacramento, California, under Standard Agreement No. 12-24, dated July 1, 1960, with the Regents of the University of California, and by the Research Grants Division (WP-649) of the National Institutes of Health of the U.S. Department of Health, Education and Welfare. The investigation was carried out at the Sanitary Engineering Research Laboratory of the University of California, Berkeley, under the supervision of Professor ERMANA. PEARSON.The authors wish to thank Professor ROBERT L. WmGEL, University of California, and Professor DONALD R. F. HA~J,EMAN, Massachusetts Institute of Technology, for advice and support. REFERENCES AL-SAI~AR A. M. (1964) Eddy diffusion and Mass transfer in open channel flow. Ph.D. Dissertation, Dept. of Civil Eng., Univ. of California, Berkeley. ARONS A. B. and S T O O L H. (1951) A mixing length theory of tidal flushing. Trans. Am. geophys. Un. 32, 3, 419-421.
20
B. GLENNEand R. E. SELLECK
ELDERJ. W. (1959) The dispersion of marked fluid in turbulent shear flow. J. Fluid Mech. 5, 544-560. GLm~r~ B. (1966) Diffusive Processes in Estuaries Sam Eng. Res. Lab. Publ. No. 66-6, Univ. of California, Berkeley. H,~.moa~ D. R. F. and AaR~tAM G. (1966) One-Dimensional Analysis of Salinity Intrusion in the Rotterdam Waterway Delft Hydraul. Lab. Publ. No. 44, Netherlands. IPI'EN A. T. (1966) Estuary and Coastline Hydrodynamics McGraw-Hill, New York. K~NT R. E. (1958) Turbulent Diffusion in a Sectionally Homogeneous Estuary Tech. Rep. No. 16, Chesapeake Bay Inst., John Hopkins Univ. Kot~oooRor~ A. (1941) The local structure of turbulence in incompressible viscous fluids for very large Reynolds numbers. Vol. 30--301. OKtrau A. (1965) Equations describing the diffusion of an introduced pollutant in a one-dimensional estuary. In Studies in Oceanography pp. 216--226. Univ. of Washington Press, Seattle. O ~ o a G. T. (1959) Eddy diffusion in homogeneous turbulence. 3". HydrauL Div. Am. Soc. cir. Engrs 85, HY9, 75-101. OWEN L. W. (1964) The Sacramento-San Joaquln Delta--an analysis of its hydrology and salinity intrusion problems. Paper presented at Specialty Conf., Div. Irrigation and Drainage, Am. Soc. cir. Engrs, El Paso, Texas. Pm'rcrtAl~D D. W. (1952) Salinity distribution and circulation in the Chesapeake Bay estuarine system. J. mar. Res. 11, 2, 106--123. SELLECKR. E. and I ~ S O N E. A. (I 961) Tracer Studies and Pollution Analysi,~ of Estuaries California State Water Pollution Control Board. Publ. No. 23, Sacramento. STO~mL H. (1953) Computation of pollution in a vertically mixed estuary. Sewage ind. Wastes 25, 9, 1065-1071. STORRS P. N., SELLECKR. E. and PEARSONE. A. (1964) The San Francisco Bay water pollution investigation. Paper presented at Fourth Water and Waste Conf., Univ. of Texas, Austin. TAYLORG. I. (1954) The dispersion of matter in turbulent flow through a pipe. Proc. R. Soc. Series A, 223, 1155, 446-468.
LONGITUDINAL ESTUARINE DIFFUSION IN SAN FRANCISCO BAY, CALIFORNIA B. GLENNE a n d R. E. SELLECK Oregon State University, Corvallis, Oregon, U.S.A. and University of California, Berkeley, U.S.A.
(Received 2 August 1968) Abstract--The concept of one-dimensional diffusion is discussed and applied to the San Francisco Bay, California, to evaluate the degrce of mixing and dispersion existing within the estuarine system. The intensity of mixing and dispersion is expressed in terms of an overall, longitudinal diffusion coefficient E which can be varied throughout the system. A method for computing E from observations of water quality constituents in an estuary is presented and discussed. Concentration distributions of chlorosity and silica derived from field observations taken over four years of continuous sampling of the Bay waters are used to evaluate the diffusion coefficient within the two arms of the Bay system. The results derived from the two differont concentration distributions are found to be comparable, giving at least a partial verification of the proposed procedures. The relationships between the observed coefficients and the hydrologic, hydrographic, and tidal wave characteristics of the Bay system are examined to ascertain if the diffusion coefficient can be predicted directly without making resourse to water quality observations. The only relationship investigated which appears to have merit is E = c o n s t . (~ u ' D ) 3/* where ~ is the mean advective velocity in the estuary, u'= the mean tidal velocity, and D the hydraulic channel depth. The equation is applicable only to those regions of an estuary where the advective velocity is known to have a significant effect on the value of E. Procedures which may be used to predict the concentrations of a water quality constituent in the Bay system--once the diffusion coefficients are known--are discussed briefly. An example of a steady state computation is presented and discussed.
INTRODUCTION IN THE last decade the concept of eddy diffusion has advanced our knowledge of mixing hays and
estuaries. Basically, the diffusion approach offers a continuous mathematical model described by a non-linear, partial differential equation. Solutions are generally obtained by a finite-difference technique with a digital computer manipulating the arithmetic. The purpose of this paper is to state and refine the diffusion concept by: quantifying the longitudinal mixing process in the San Francisco Bay, California, by evaluating and using diffusion coefficients; presenting equations for predicting steady and unsteady longitudinal concentration profiles of water quality parameters in natural estuaries. The investigation was performed in conjunction with a comprehensive study of San Francisco Bay which was conducted at the University of California, Berkeley, and is described by STORRS et al. (1964). Having completely different tidal and advective characteristics, the northern and southern branches of the San Francisco Bay system serve well to illustrate the application of the described procedure. The physical characteristics of the Bay system are summarized in Fro. 1 and TAnLV.S1 and 2. A remark is pertinent about the so-called one-dimensionality of the procedure. Only after the estuary has been divided into small cross-sectional volumes are the cross-sectional areas and advective flows assumed constant for each small volume. This permits a reasonable mathematical approximation of the existing natural conditions. 1
2
B. GLENNE and R. E. SELLECK REVIEW
The general three-dimensional convective diffusion equation of continuity was stated early by classical hydrodynamicists. A three-dimensional equation is not easily applied to natural estuaries, however, because of the irregular physical boundaries of such systems.
/
N
J-~L -~m =.
SAN PASLO BAY
SACRAMENTO
~ICHMONO
PACIFIC OCEAN OAKLAND
SOUTHERN S~N " FRANCISCO BAY
,.~4NMATEO
%~_~'
'?
,~ ....
FIG. 1. Location map.
ARONS and STOMMEL(1951) were among the first to propose solving pollution problems in estuaries with a one-dimensional, longitudinal convective diffusion equation. From observed steady state salt distributions, STOVa,mL (1953) presented a method using finite differences to determine the magnitude of the diffusion coetticients in the equation. PRrrCHAP,D (1952) deliberated along the same lines and
Longitudinal Estuarine Diffusion in San Francisco Bay, California
3
applied the method to the salt balance in the St. James estuary in Chesapeake Bay. KENT (1958) considered the diffusion coefficient to vary with time as well as position. He used a steady state procedure to compute values of the eddy diffusion coefficient in the Delaware Estuary. OICUBO(1965) studied the validity of the mathematical derivation of the equations used in the previous work. SELLECK and PEARSON (1961) used an Orzan tracer to investigate the diffusion effected in the landward end of the South San Francisco Bay. The longitudinal diffusion coefficient was found to have
TABLE 1. PHYSICAL CHARACTERISTICS OF SOUTHERN SAN FRANCISCO BAY
Name of sampling station
Section No.
CB-6 1 CB-5 2 CB-4 3 CB-2 4 LB-2 5 LB-3 6 LB-6 7 SB-3A 8 SB-1 9 SB-4 10 SB-5 11 SB-6 12 SB-7 13 SB-8 14 SB-9 15
Distance from Golden Gate
Length of section
Crosssection area at MTL
Hydraulic depth
Mean tidal range
(ft)
(ft)
(ft 2)
(ft)
(ft)
0 8255 16510 24420 32320 42160 52000 62650 73300 83800 94300 108100 121900 129700 137500 143350 149200 159475 169750 177775 185800 191050 196300 200050 203800 208450 213100 215800 218500 226750
938000
~ 177
4.0
,,~79
4.1
16510 15810 19680 21300 21000 27600 15600 11700 20550 16050 10500 7500 9300 5400 16500
825000 850000 915000 890000 920000 1030000 820000 653000 333000 359000 292000 286300 232000 165500 1011200 126200 92000 65710 37000 33070 19400 19350 10300 8240 6900 3200
4.3 38.7 4.5 24.4 4.8 16.9 5.2 12.2 5.6 14.0 5.9 13.7 6.1 11.4 6.4 11.2 6.7 6.8 6.9 6.8 7.1 7.6 7.3 5.1 7.4 4.5
values ranging from 460 to 1380 ft2/sec. The California State Department of Water Resources performed diffusion studies in the northern branch of the Bay system. OWEN (1964) found diffusion coefficients ranging from 200 to 14,000 ft2/sec. F r o m hydraulic model studies IPPEN and HAgI~r,IAN (1966) proposed that the actual value of the diffusion coefficient departs from an idealized value in a manner which is proportional to the energy dissipated in the estuary. They described the ratio of the
4
B. GLENNE and R. E. SELLECK
actual diffusion coefficient to the idealized value as a function of a stratification parameter, or in mathematical terms --~ = c o n s t . \ 2 P , F 2 J
(11
TABLE 2. PHYSICALCHARACTERISTICSOF NORTHERN SAN FRANCISCOBAY
Name of sampling Section station No.
CB-6 16 NB-3/4 17 NB-5/6 18 NB-9 19 SP-3/2 20 SP-4/5 21 SP-8/7 22 SP-9 23 SU-1 24 SU-2 25 SU-3 26 SU-4 27 SU-5 28 SU-6A 29/31 SU-9 32 SU-10 33
Distance from Golden Gate
Length of section
Crosssection area at MTL
Hydraulic depth
Mean tidal range
fit)
(ft)
(ft 2)
(ft)
fit)
0 11850 23700 29800 35900 45700 55500 61650 67800 79100 90400 103250 116100 125500 135000 151300 167600 174650 181700 186450 191200 198250 205300 213100 220900 229500 238100 247950 257800 268750 279700 289850
938000
~ 177
4.0
23700 12200 19600 12300 22600 25700 18900 32600 14100 9500 14100 15600 17200 26000 21900 20300
1165000 1361000 762000 684000 507000 504000 490000 646000 573000 580000 265000 245000 141000 186000 175000 162000 194000 216000 202000 383000 216000 217000 135000 104000 128000 122200 43000 71700 71400 73900
~89 31.3
3.9 4.2
26.3 4.0 30.1 4.1 12.7 4.4 9.6 4.4 29.9 4.3 37.4 4.1 21.0 14.4 3.9 10.4 13.9 31.0 3.3. 20.5 3.1 24.9 19.0
in which E = diffusion coefficient, b = a characteristic length in the mixing process, ~ = mean advcctive velocity, Q = advectiveflow, T = tidal period, Pt = tidal volume, F = Froude No. = u,~'x/(gD), u'~ = mean tidal velocity, D = hydraulic channel depth and n = exponent to be determined from data. In the Rotterdam estuary HARLEMANand ABRAHAM(1966) used the tidal excursion as a measure of the characteristic length b.
Longitudinal Estuarine Diffusion in San Francisco Bay, California
5
Some additional concepts of longitudinal diffusion are furnished by various other investigations not directly concerned with the problems of estuarine mixing. KOLMOC-OROrr (1941) hypothesized that in isotropic turbulence the diffusion coefficient can be described by E = c o n s t . G 1/3 L41a
(2)
in which G ffi mean rate of energy dissipation per unit mass of fluid and L = mean size of eddies participating in the diffusion process. In open channel flow G can be approximated through Chezy's formula. Equation (2) may then be expressed as
/L,t\I/3
in which R = hydraulic radius of the channel. Assumptions made at times are that the mean size of the eddies is directly proportional to the hydraulic radius a n d that for wide channels the hydraulic radius is approximately equal to the channel depth. Equation (3) then becomes E = const,
o
(4)
TAYLOR (1954) proposed a formula for the prediction of longitudinal dispersion coefficients for uniform turbulent flow in straight closed conduits. ELDER (1959) expanded this formula to open channel flow to obtain E = 5.9 D ~/(~o/P)
(5)
in which 7o = wall shear stress and p = density o f the fluid. Equations (4) and (5) are obviously similar. AL-SAFFAR (1964) found the mean diffusion coefficients in laboratory flumes to vary approximately as the 0.85 power of the channel velocity (expressed as a Reynolds number). ORLOn (1959) related the lateral eddy diffusion coefficient to a channel scale. THEORETICAL
CONSIDERATIONS
General equations Gt~NNE (1966) has shown that the one-dimensional convective diffusion equation of continuity can be written
;
Ss+ A ?5
(6)
in which x = longitudinal distance, A = cross-sectional area, ~ -- mean constituent concentration, t = time ordinate and S, -- net rate of constituent mass being created or destroyed per unit length of the estuary. Equation (6) is a linear, first-order non-homogenous partial differential equation in E, which can be solved by ordinary methods. Integrating seaward from the landward termination of the system gives
A E ~x = A • 5 -
Ss dx +
A ~-t dx
(7)
20 The landward terminal of the system is the point where the tidal influence or cross-sectional area becomes insignificant, or where the constituent concentration has such a low level so as not to influence significantly the magnitude of the integral terms on the right hand side of equation (7). The terms AE(O5/Ox) and Afi~ represent the time average diffusive and advective transport rates of the constituent mass, respectively, across the most seaward cross-section of the reach (see Fla. 2). The first integral term o n the right hand side of the equation is the summation of sinks and sources of constituent mass entering or leaving the reach. A constant of integration may be included in this term if a source of constituent mass exists at the landward termination of the estuary. The last term o n the right hand side of equation (7) represents the non-steady depletion or accumulation of constituent
6
B. GLENNE and R. E. SELLECK
mass within the landward reach. Replacing the two integral terms with the shorter expressions XS, and ZF, equation (7) becomes AE
(8)
"=- = ox
o
o
With the aid of numerical methods, equation (8) can be used to obtain estimates of the diffusion coefficient E when the constituent concentrations and advective flow conditions are known. The limitations on equation (8) are described in detail by GLENNE (1966). Some of these are:
,ucA INCREMENTAL
CONTROL
VOLUME
TIME AXIS
j+l
--
O i-I
I
SPACE AXIS
i÷l
REACH ~
F i I
I
~ - - - - ~ - SECTION
', LANDWAROEND OFESTUARY
L
I
~ . ~ - ' ~ , ' . v , , . . -
~,..
FIG. 2. Sketch of reach and control volume. the constituent in question must react according to the diffusion mixing concept (Fick's First Law); diffusion in the estuary must essentially be one-dimensional; stratification in the estuary should be small enough to make the constituent gradient truly representative; second order changes of a, ~ and A with distance are insignificant when compared with first-order changes;
Longitudinal Estuarine Diffusion in San Francisco Bay, California
7
the parameters E, A, ti and e are average values taken over at least one, and preferably several, complete tidal cycles; the values of ti and c are also averaged over the entire cross-sectional area; A is essentially unchanged from one tidal cycle to the next. For a non-conservative water quality constituent, equation (8) takes the form X
AE "~x ~ = A~-~
X
(Ss-kV6)+ ~o
(9)
in which k = first-order rate constant and V = volumes of the sections making up the reach. Equation (6) may be expressed in simpler forms by assuming either separately or successively that: the system is in steady state; no external sources or sinks exist for the system; A and t~ are constants with respect to x; E is a constant with respect to x. the constituent under study is conservative (k = 0).
Evaluating the diffusion coefficients With known constituent concentrations, sources, and sinks, equation (8) can be used to evaluate the diffusion coefficient. [Equation (6) is not as well suited for this purpose because of the derivatives of E contained therein.] Letting the subscript i denote the ith station on the longitudinal axis and the subscript j the jth time trace (see FI~. 2), equation (8) can be written in a central finite difference form as follows:
Af,~E,,j -~x ,,j = A"ju"Jei'J-X(S')"J+~o _
~x/i,y
i
,-1,j Axi_,,~
(10)
1
4Ax,. y
[(e'+l'J+x-6'-"J+')+(e'+"J-l-e'-"i-x)]
(lOa)
c-~,j = { (e~, j+ t - c~, j - 1)
(lOb)
(0-~t9 ,-1.j = At,,j1 (e,_l,j+ ~-e,_,,j_,)
(lOc)
i-2
(Ss),, s = ~ (Ss),, j +-~ ['(Q,e,),_ 2, j + (Q,e,),, j] - 1 e,_ 1, j [(QL),- 2, j + (QL)~, j] (10d) 0
0
in which Q, = flow rate of source discharging into a section and QL = flOWrate of sink withdrawing water from the section. The evaluation of Ei.j from equation (10) requires a knowledge of the hydrographical characteristics and the advective flows, as well as the constituent profiles. The summation process commences with a small landward reach and progresses in the seaward direction with the cumulative addition of finite estuarine sections. The process increases in accuracy with decreasing lengths of estuarine segments. A matrix in time and space of observed diffusion coefficients results. For the San Francisco Bay study the lengths of the sections were dictated by the locations of the water quality sampling stations, and the time intervals by the lapses in time between consecutive sampling cruises.
Steady state constituent concentration computations Equation (8) is also convenient for computing steady state constituent concentrations when E is known. Dropping the unsteady term of the equation and employing central finite differences on the distance axis gives
A~Ei( Ci+l--Ci~
1.)=
Aifii ( ( i + 1 +2' ~ -
1 ) _ ~ o (S~) ~
(11)
8
B. GLENNE and R. E. SELLECK i
i-2
Z (s~),= o
(s,), + 2 (s,), o
i-2
= ~ (Ss)i+(Qsgs)i-(QI.id~_l)-¼ki(Ai+Ai_l)(Ax~+Axi_2)6i_ 1 0
(lla)
Solving for ~ ~ equation (11) takes the form
2Ax~ ['A~E~ . Aifi i c] +2 -- A, ( 2 E , - ~,Ax,) LTx[ ci-~ + ~
~
]
e,_l -~o (S,),
(12)
one initial condition is required to solve equation (12). Generally this is a known or assumed value of ~J~_~at some point in the estuary.
General constituent concentration computations Equation (6) is best used to determine unsteady state constituent concentrations when E is known, i.e.,
0
OC
~-~(Auc)
]..,
-[Ss]i,j+[A
O~
Employing a forward finite difference approach and referring to FIG, 2, the individual terms of the above equation may be approximated by the expressions
(S,)i, j = (q,)i, j (es)i, y - (qL)i, j el, j -- ki, j Ai, j e~, j
(13a) (13b)
0 ( A ~ e ) ~,~
2 (xi, j - x i_ l, j)
'
'
(13d)
(A~()t- 2, j = As- 1, j ui- 2, y Oi- 2, j (A~d)~+t,l = (A~,ig~,Id~,j)-F x t ' j - x t - l ' J ( A i + l , j ~ i + l , j d i + l , j - A t , Xi+ l, j-- Xi, j
j~i, jg~,j) (13e)
(AE-ff-x)i_½, i=½(A,,yEi, j+A,_l,jEi_l,j,i\xi, i_xi_l,y,]
(A~ o~
oe
(1311)
Ox)i+½,j=(AE),+½,j~f+½,j
(a~)
(AE)i+½, j = A i j E i j 4 xt'J--X~-l'] ( A i + l , y E i + l , y - A l , jEl, j) • , 2(Xi+l,y--Xt, j
~X i+½, j
=
(13g)
(13i)
(~,, ,- ~,_,, 4:) 2(x,, ,-x,_,.,)[~,+~,,-~,., .:,,-~,_I.:](13j) .
kXi, j--Xi-1,
(xi+x,j-x~-t,j)kXi+l,y-xi,
j
xi,.l-xi-1,
in which q, = flow rate per foot Of estuary length of a source discharging into the estuary, ~, = constituent concentration of the source, and qL = flOW rate per foot of estuary length of a sink with-
Longitudinal Estuarine Diffusion in San Francisco Bay, California
DIFFUSION COEFFICIENTS, ~ o
~l ~
ft 2 / s e c
~o
~ o
~ ~ ~ I--E.
9
~ ~.c,.o_0.~..0 , , ,,,
.. ,,, ORt~SE
CTION
3
• --
SECTION
4
•
SECTION
5
SECTION
6
•
•
n
SAN
MATEO
BRIDGE
SECTION 7
8r SECTtON 8
SECTION 9
- -
DUMSN~TON
BRIDGE
SEC3"t0N I0
ETI(~I
•
II
o--
omole
" 0 --SECTION 12 ~ ALVISO ° S L O ~ l ~ 0 0 SECTION t 3 ~ - = m m m SECTION 1 4
~I-~
S. P.R.R.
Q ellDql) o •
~ •
eo
-
•
-
•
s
IJl
•
B:~IDGE eeoee,,o*
~
oe
• • •
I
l
•
•
SECTION IS
I
•
J
,
i
-,,
I ill
-,,','= -,
I
,
FIG. 3. Diffusion coc(~cicnts in southern reach from chloride tracer.
,_
10
B. GLENNE and R. E. SELLECK
DIFFUSION o =
COEFFICENTS,
sq f t / s e c
o
I
u I I I.,t~S/~I
SECTION
5
SECTION
4
SECTION
5
SECTION
6
0
o
I
I
i
l
I
F R A N C I S C O - OAKLAND
I
OAf
• --
•
I
I
I
I
BRIDGE
•
•
•
•
P, 1:) o
o
u~ c) m
~SAN SECTION
7
SECTION
8
SECTION
9
MATEO - - H A Y W A R O
BRIOGE
o--e
• m
G'3 o ill
•
•
--
CUMOARTON
SECTION
I0 o--o--l--o--o
SECTION
II
•
o--o•
O - - O
O - - O - - O
BRIDGE o--o
•
o
~
o
o
-
• • o - - o - - o - - o • •
-
o
-
-
o
•
•
o --SECTION
12
SECTION
13
SECTION
14
SECTION
15
0 0
I
]
f ~ll
~'~'o~G~" o - - o
j~-SPRR
. •
.
e~ooeo
I~rlDGEe o e m O I
i
.
.
.
oo
,
•
,De
,
I
.
.
•
•
e•
• - -
• - -
J I ill
.
.
.
•
•
Ioo•
•
J
FIG. 4. Diffusion coefficients in southern reach from silica tracer.
I
Longitudinal Estuarine Diffusion in San Francisco Bay, California
01FFUSiON
COEFFICIENT,
(fit/see)
8 I
I
I
I
II
i
I
I
8
I
I
I
I I [
I
-SECTION
I7
S~C T I O N
18
i°
11
I
ANC~EL
I
ISLANO
° I
RICHI~I~ RAFIEL.
-- SAN B l ~ OC.~
<< m m
•
SECTION
•
0
0
20
ee
O •
e
mo
,.¢~3 4DO - -
CO --
PC. SAN P E O R O
--
CA.qQUIN EZ STRAITS 0RIDGE
--
BIENICIA - MARTINEZ 8RID(
--
PtTTS8 URG
OQ O O 3 0
:0
Z
--
SECTION
21
SECTION
22
SECTION
23
SECTION
24
SECTION
25
SECTtON
27
SECTK~I
28
z
o
•
IND
-
-
e
e - -
•
•
eo
~ ¢=
o e J l ~
O
•
•
Io
OCP
o
• oloeo--
o ooo
o oe o
~
o o
O O
m
o
¢)
o o
o~
o
o
o o
•
~ •
o ee
oo
--
i
- -
I
~
I
t
i ill
i
i
t
~
I
ttll
t
Fzo. 5. Diffusion coefficients in northern reach from chloride tracer.
I
I
I
12
B. GLENNE and R. E. SELLECK
drawing water from the estuary. To use this method for the prediction of constituent concentrations, equation (13) needs to be solved for cl,j+ ~, or
_
/
tj+z--t jr/~
6~
6
1
The boundary conditions are C~,o, the initial constituent distribution and two end conditions established by equation (8). DIFFUSION
COEFFICIENTS,
ft/sec
o
SECT~N
t7
SECTION
IB
S ( C TION
19
SIECTION
20
$ ( C TION
21
SECTION
22
0
0
•
'1
--
ANGEL I S L A N 0
o
R~:HMONO - S A N R A F ~ L IBRIOGE
--
OT. $ ~
PI[0RO
;2
"*o
o•
o
o
e--era
o - -
-- CARQUINEZ $ 1 1 ~ I T S 8~10~E
SECTHON
23
SECTION
24
SECTION
E5
ECTION
26
SECTION
27
•
*
o•
o
~
¢ml
*
a
--
~CT~N
oJ
o
-
• •
-- *
* -
-
*
~
**
m
o
o
*o
o•
o
~
-
*o
oool~
o
~
m
~
o
•
•e
m
-
-
o
.
BENICIA - Me,R'rlNEZ 8 ~ O ( o
a
o - - m
o
o
28 --
I
l
I
I
I
l
I
J
r
,
i
I
~TTS~JRG
ill
FIG. 6. Diffusion coefficients in northern reach from silica tracer. To insure stability of the iterative solution, the time interval has to be kept within a prescribed limit. Using a numerical analysis of parabolic partial differential equations, it can be shown that the stability criterion is approximately
Atj/(Axi) 2 <~ 1/2Ei
(15)
5.0
O I0.0 O / 3.: (1)
)f-
15.0
z0.0
"
A
Z~
--
L~
~ 0
0
OBSERVED 15 ~ 62 ,, ........ o " 31 JUL 62 r-t__ II JUL 62 ESTIMATEDAOVECTIVEFLOWS (BY DWR AT CHIPPS ISLAND ) 4,260 cu ft/sec AUG 62
A
AA
I
~
~
0
I
13 13 \ 13~.~ I::]
t"1 N
~ ~1
[]
E]
--
50~ 000
THEORETICALPROFILEOBTAINEDFROMEQUATION (12) " (INITIALCON01TIONUSED c=IT"og/IAT23'7OOf'}
I
2OO,000 DISTANCE FROM GOLDEN GATE ,ft
0 °
/
I
Flo. 7. Example of a steady state integration.
I00,000
~
I
rJl
~o
E
r~ 0 0~ .~" ,.. r-L
14
B. GLENNE and R. E. SELLECK CALCULATIONS
FOR SAN FRANCISCO
BAY
Equation (10) was used to evaluate the diffusion coefficients in the Bay system. This requires a knowledge of the concentrations of either a natural or artificially induced water tracer. Two natural tracers---chloride with a seaward origin and silica with a landward origin--were used to make independent evaluations of the coefficient. The constituent data were collected in the Bay during an extensive sampling program lasting from 1959 to 1964. The period between sampling cruises averaged 56 days and each station was sampled on three different tidal stages and at two or more depths. The distance between the sampling stations varied from 2700 ft to 16,300 ft as shown in TABLES 1 and 2. The raw data were then normalized to obtain representative concentration profiles. This process is described in detail by GLENNE (1966). I000
I
0 n,I-
600
h0 i (_9
400
--
0 It
I
I
X
25
+
24
0
25
rl
26
0
27
I
I
'"1
Z~
28
0 []
o 6
A 200
n
I
SECTION
8O0
x
0
--
,,
z~
X
t,LL ZLL UJ h'
_~o ~z too
XO
A
I00
80 --
#
,¢
0
o~ 60
-X
:~Z Lt. LLI
40
*
%
I
I
I
I00
200
400
600
O n 0 A
-XX
20
--
IC I0
I
I
I
I
20
40
60
80
I 8 0 0 I000
A D V E C T I V E V E L O C I T Y IN P E R C E N T O F G E O M E T R I C M E A N A D V E C T I V E V E L O C I T Y IN 1 9 6 1 - 6 2 W A T E R Y E A R
FIG. 8. Relationship between the diffusion coefficient and advective velocity in Suisun Bay. Although each observed concentration profile did not extend throughout the entire estuary because of the nature of the sampling program employed, it was possible to calculate four separate matrices of diffusion coefficients in time and space. The results are presented in FIGS. 3-6. FiGtn~ES 3-6 do not show all the diffusion coefficients evaluated, but rather the coefficients passing two acceptance criteria. Coefficients were rejected if: (i) the observed constituent concentration profile gradients were less than 10 - s g/1 of chloride per foot or 10 - s mg/1 of silica per foot. (ii) the overall value of the three terms on the right hand side of equation (10) was less by one order of magnitude than the value o f the largest o f the three terms considered individually.
Longitudinal Estuarine Diffusion in San Francisco Bay, California
15
These criteria were found necessary because relatively slight inaccuracies in the data may create large errors in the estimates of the coefficient for the regions where the criteria are not fulfilled. FIGURES 3-6 indicate that the computed diffusion coefficients are essentially the same for both the silica and chloride tracers. The variations in the computed values appear to be larger, however, when using silica tracer. This may result from the relatively large inaccuracies involved in estimating the sinks and sources of silica in the system. For example, a significant uptake of silica by diatoms may occur during the summer season. Equation (12) was used to calculate a steady state chloride concentration profile in the northern reach of the Bay system. The average hydrologic and hydrographic conditions of August 1962
I000
[
800
600
--
(D
@ FUJ 5; o (.9 LL 0
[
I
}
400
X
19
+
20
0
21
0
22
I
I
o [3 0
200
0 0 +
la
n,- I_ bJ Z n LIJ
_z,3
I
(3[:]
L)
t.--LL
I
SECTION
0
I00
8o I --
"
60
0
0
0
X []
÷
X
0 4-
0 "IX
121
O - - - - - X ÷
rl
0~1-
÷ X
÷ l
4-
0 {g
¥
0
0 X
40 x ~Z u. ILl
20
io
--
. . . . . I0
I __ 20
I
I
40
60
] 8 0 IO0
I
I
200
400
[ 600
I 800 I000
A D V E C T I V E V E L O C I T Y I N PER C E N T O F G E O M E T R I C MEAN ADVECTIVE VELOCITY IN 1961-62 WATER YEAR
Fie. 9. Relationship between the diffusion coefficient and advective velocity in San Pablo Bay. were used in the computation. The flow from the Sacramento-San Joaquin River Delta was supplied by the California State Department of Water Resources (DWR). The boundary conditions used was a chloride concentration of 17.00 g/1 at the seaward entrance to the northern reach. The results are shown in Fie. 7. Also plotted in Fie. 7 are the chloride concentrations observed in the reach during July and August of 1962. The agreement between the observed and computed concentrations of chlorosity as shown in FIe. 7 appears reasonably good. Subsequent investigations have shown, however, that the chlorosity distribution in the Bay system is seldom in steady state, and the high degree of correlation between the observed and computed values exists partly because the boundary condition was determined from the observed values. A digital computer program was also written to calculate unsteady state
B. GLENNE and R. E. SSLLECK
16
constituent concentrations using equation (14). The results of those calculations will be discussed in detail in a subsequent paper.
DISCUSSION Inspection of FIGS. 3-6 shows that several deductions can be made regarding the behavior o f diffusion coefficients in the Bay system. In the South Bay, which is a well-mixed estuary having relatively small advective flow, the diffusion coefficients are smaller than those observed in the northern arm o f the system. The diffusion coefficients appear to decrease in value in embayments and increase in or near the straits o f Carquinez and San Pablo. As a check, the computations were repeated using 1000
E
T
800 _
SECTION X
15
600
+
II
--
0
~2
[3
13
0
14
400
I-b~ :E
i
j
i
1
1
1
I
I
4-
,9,
(.9 h 0 FZ ~J (J r~ i_
1
"t" b
200
0 0
n
too
h' Z
+ I00
0
+
O ~ Q 0 ~ + ~
o~ ~-
+ ÷
÷o
80 60
X X X
UJO
o~
40
:DZ tL,~[
20--
10 I0
I
I
I
I
20
40
60
80
I00
I
I
200
400
6 0 0 8 0 0 I000
ADVECTIVE V E L O C I T Y IN PER C E N T OF G E O M E T R I C MEAN A O V E C T I V E V E L O C I T Y IN I 9 6 1 - 6 2 WATER YEAR
FIG. 10. Relationship between the diffusion coefficient and advective velocity in the southern reach. a reduced or "effective" cross-sectional area of the main channel in San Pablo Bay. However, because the reductions in cross-sectional areas were at most 15 per cent, this proposed alternative did not significantly change the value of the diffusion coefficient. In the South Bay an increase in the diffusion coefficient is found near the landward end of the estuary. This is most likely due to the added mixing caused by the advective flow in the region where the value of f~/u'm becomes significant. In attempting to reduce the variations in the observed diffusion coefficients, the coefficients were plotted against the corresponding net advective flows. The results are shown in FIGS. 8-10. In the northern reach the advective flows are expressed in percent of the in sit, geometric mean o f the advective flows for the 1961-62 water year. In the South Bay the geometric mean was computed for the measurements made during the six cruise periods conducted in 1961-52. The diffusion coefficients
E
'I0°
O'--b ,o
I00
,
X
X
X
T",
,~
X
X
I0'
X
X
x
X
•
X
,
x
0
o
,
X
,
A oo
,
~
•
~
,
z~
o *
i
~ ~
•
o
o
I0z
•
"T~--T~
~
•
•
•
,
, ,
Z~
STRATIFICATION
,
0
,
,
0
I0 ~ INDEX
O'
0
~-'rrTT ,
, ~ - ~ )
O
O0
/gOO
O
,
\
0
O 0
I ~-
0
•
0
r~ - F T - T ~ F T ~
•
O
-
•
I
&& •
;
•
I ,
r
•
I
•
•
•
+
io"
+
I I I II
FIG. l 1. Relationship between E/~b and the stratification index in the northern reach.
X
X
o
, , , Tl-
÷
I
I
-F
~¢TIO# S[CT~
X 0 0
I
I
23
T9
I
]
+#
I
I
+
+
[
g~CTION SECTIme 27 4- SECTION 2e
I I
I
I II
+ +-
io*
I I I I'~
0
@
I:I
18
B. GLENNE a n d R, E. SELLECK
fill
I I
I
I
I
[III
I I
Ii. I
I
÷.
÷ ÷ ÷ -
~_g~g_g_~
4.4 4-
* ,m
4 4
•
4
• 4
o
0
0
O O 4
4
0 O0
0 0
O
-
.~
0 0 0
0
.~ c3 13o <3 0<1, ~ I : I D CI
o
o
o
o
'o
o
o o
o
°~
x
x
13° 0
o x
o
O
x
X X
lllll
I
I
I
f
IIIII ~O
I
I
I
I
No_-
Longitudinal Estuarine Diffusion in San Francisco Bay, California
19
are expressed as percentages of a mean diffusion coefficient found at each section, with the mean diffusion coefficient being computed from tho~arithmetic mean of the chloride concentrations observed in the above-mentioned time periods. As evidenced by the figures, there is a definite dependency of the diffusion coefficients on the advective flows in the northern reach. This has also been noted by OWEN (1964). Lines fitted through the points shown in FIGS. 8 and 9 indicate that E varies approximately as the 3/4 power of a. In the South Bay no such dependency is apparent. The probable reason for this is that the tidal mixing greatly over-shadows the advective mixing in most of the southern reach. It is evident that some of the variations evidenced in FIGS. 8-10 result from "inaccuracies" in the measurements. This is especially true of the results depicted for the southern reach. Without extensive additional data analysis, however, it is difficult to quantitate the magnitude of these inaccuracies. Equation (1), with b = u'=(T/2), was used to obtain the results shown in FIG. 11. A useful relationship does not appear to exist between the observed diffusion coefficients and those predicted by equation (1). In FIG. 12 the same approach was used, but the term b was made equivalent to the hydraulic depth. This appears to be the only relationship valid for the northern reach of the San Francisco Bay. Letting n = - 1/4 gives the relationship E = const. (aura'D) a/4
(16)
CONCLUSIONS Using Fick's First Law and a mass continuity concept, the mixing or diffusion coefficients in estuaries can be evaluated from a knowledge of the constituent concentrations, sources, and sinks, as well as the general advection through the system. The following observations itppear to be valid regarding the nature of the diffusion coefficients determined for the San Francisco Bay: the values of the diffusion coefficients obtained using silica as the tracer generally coincide with the corresponding values obtained using chloride as the tracer; the diffusion coefficients exhibit considerable variations with time as well as distance; in northern San Franciso Bay where the advection is significant, the diffusion coefficients are found to vary approximately as the 3/4 power of the advective velocity; of the semi-empirical methods examined for predicting directly values of the diffusion coefficients only equation (16) shows promise of giving satisfactory results. The one-dimensional diffusion model, in general, presents a feasible method for solving steady and non-steady state transport problems of pollutants and other constituents in waterways and estuaries. The steady state solution can be calculated by hand, whereas the non-steady state solution is best handled by a computer. Solutions to transport problems in estuaries commonly depend on the information available regarding the magnitude of the diffusion coefficient. It is hoped that the many estuary investigations now being conducted will contribute significantly to such information.
Acknowledgements
This study was sponsored by the State Water Quality Control Board, Sacramento, California, under Standard Agreement No. 12-24, dated July 1, 1960, with the Regents of the University of California, and by the Research Grants Division (WP-649) of the National Institutes of Health of the U.S. Department of Health, Education and Welfare. The investigation was carried out at the Sanitary Engineering Research Laboratory of the University of California, Berkeley, under the supervision of Professor ERMANA. PEARSON.The authors wish to thank Professor ROBERT L. WmGEL, University of California, and Professor DONALD R. F. HA~J,EMAN, Massachusetts Institute of Technology, for advice and support. REFERENCES AL-SAI~AR A. M. (1964) Eddy diffusion and Mass transfer in open channel flow. Ph.D. Dissertation, Dept. of Civil Eng., Univ. of California, Berkeley. ARONS A. B. and S T O O L H. (1951) A mixing length theory of tidal flushing. Trans. Am. geophys. Un. 32, 3, 419-421.
20
B. GLENNEand R. E. SELLECK
ELDERJ. W. (1959) The dispersion of marked fluid in turbulent shear flow. J. Fluid Mech. 5, 544-560. GLm~r~ B. (1966) Diffusive Processes in Estuaries Sam Eng. Res. Lab. Publ. No. 66-6, Univ. of California, Berkeley. H,~.moa~ D. R. F. and AaR~tAM G. (1966) One-Dimensional Analysis of Salinity Intrusion in the Rotterdam Waterway Delft Hydraul. Lab. Publ. No. 44, Netherlands. IPI'EN A. T. (1966) Estuary and Coastline Hydrodynamics McGraw-Hill, New York. K~NT R. E. (1958) Turbulent Diffusion in a Sectionally Homogeneous Estuary Tech. Rep. No. 16, Chesapeake Bay Inst., John Hopkins Univ. Kot~oooRor~ A. (1941) The local structure of turbulence in incompressible viscous fluids for very large Reynolds numbers. Vol. 30--301. OKtrau A. (1965) Equations describing the diffusion of an introduced pollutant in a one-dimensional estuary. In Studies in Oceanography pp. 216--226. Univ. of Washington Press, Seattle. O ~ o a G. T. (1959) Eddy diffusion in homogeneous turbulence. 3". HydrauL Div. Am. Soc. cir. Engrs 85, HY9, 75-101. OWEN L. W. (1964) The Sacramento-San Joaquln Delta--an analysis of its hydrology and salinity intrusion problems. Paper presented at Specialty Conf., Div. Irrigation and Drainage, Am. Soc. cir. Engrs, El Paso, Texas. Pm'rcrtAl~D D. W. (1952) Salinity distribution and circulation in the Chesapeake Bay estuarine system. J. mar. Res. 11, 2, 106--123. SELLECKR. E. and I ~ S O N E. A. (I 961) Tracer Studies and Pollution Analysi,~ of Estuaries California State Water Pollution Control Board. Publ. No. 23, Sacramento. STO~mL H. (1953) Computation of pollution in a vertically mixed estuary. Sewage ind. Wastes 25, 9, 1065-1071. STORRS P. N., SELLECKR. E. and PEARSONE. A. (1964) The San Francisco Bay water pollution investigation. Paper presented at Fourth Water and Waste Conf., Univ. of Texas, Austin. TAYLORG. I. (1954) The dispersion of matter in turbulent flow through a pipe. Proc. R. Soc. Series A, 223, 1155, 446-468.