MATHEMATICAL MODELING OF THE DISTRIBUTION OF FISH EGGS FROM SPAWNING REGIONS OF A RIVER
GOUR-TSYH YEH * and CHI FU YEH
Stone and Webster Engineering Corporation, Boston, Massachusetts 02107 (U.S.A.)
(Received 26 February 1979)
ABSTRACT
Yeh, G.-T. and Yeh, C.F., 1980. Mathematical modeling of the distribution of fish eggs from spawning regions of a river. Ecol. Modelling, 8: 97--107.
A deterministic mathematical model, based on the conservation of numbers, is pro- posed for predicting the distribution of fish eggs after they drift away from the spawning regions. In the model, the effects of egg settling, diffusion capability of the water body, river currents and river boundaries are all included. Close form solution is obtained for the governing partial differential equation. In deriving the solution, egg-flux across river banks, water surface or bottom is suppressed. Application of the model is illustrated by an example. The result should provide the biological community with a quick and easy way of predicting the location of fish eggs in a river.
INTRODUCTION
The passing of non-screenable planktonic organisms, including fish eggs and larva, through condenser cooling water systems of power plants has recently been of concern in environmental impact evaluations. Entrainment of these ichthyo- plankton is a far more serious problem and less easily solved than those of entrapment or impingement of larger fishes, because the latter problems can be minimized by improved engineering. At present, numerous studies are in progress at power stations throughout the country to investigate the effects of entrainment on ultimate populations of fish over a broad geographical area. Coutant (1970, 1971), Marcy (1975) and Van Winkle (1977) have provided a comprehensive review of entrainment studies. Jensen (1978) has reported recent developments in techniques to protect aquatic organisms at the water intakes of steam-electric power plants. Entrainment by power plants can be much reduced by siting the plants in nonproductive areas. Thus, there has been emphasis on studying the distribution of fish eggs by means of field samples. However, the results often conflict because of the patchiness of distribution, efficiency and selectivity of sampling gear and water movement.
* Now at Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, U.S.A.
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Several problems are encountered in carrying out these measurements and analyzing the results (Copeland et al., 1976; Jude, 1976). It is therefore the intent of this paper to develop a mathematical model for predicting the dis- tance that ichthyoplankton travel after they are spawned, fertilized and hatched, with specific reference to fish eggs.
Mathematical models based on the principle of conservation of mass have been widely adapted to predict the distribution of dye in a river (Yeh, 1973, Yeh and Tsai, 1976). These dye-dispersion models, validated by field measure- ments, have been used with confidence by the engineering communi ty to pre- dict the dispersion of the effluent discharges from power plants. The distribu- tion of fish eggs should, in many respects, resemble those of effluent dis- charges but there are significant differences, the most important being the settling of fish eggs. This sinking phenomenon is not considered when dye modeling, thus inhibiting the direct employment of dye-dispersion models to study the fish egg distribution. The model proposed in this paper will include the effects of the settling of fish eggs, the diffusion capability of the water body, and the effects of river currents and boundaries.
MODEL FORMULATION
The governing equation that describes the distribution of fish eggs in a turbulent river can be derived in the form
an an an+__a (E an ) a ( a_~) a (E a n ) ~=--u~x--ws a--z ax x ~ x +~y Ey +~z ~'~z +S (I)
where n is the egg abundance, i.e., the number of eggs per unit volume of water; t is the time; x, y and z are the longitudinal, lateral and vertical coordi- nates, respectively; Ex, Ey and Ez are the dispersion coefficients in the x-, y- and z-directions; u is the river current; w s is the settling velocity of the egg; and S is the source function, representing the number of eggs per unit volume per unit time produced in the spawning region. The source function, S, can be represented mathematically by
S =N forO<-.x<LandBl<y<.B2andHl<,.z<.H2andO<,.t<T
S = 0 otherwise (2)
where T is the spawning duration and (0, L)X(B1, B2)X(H1, H~) is the spawning region.
Eq. (1) is based on the conservation of the number of eggs in an infinitely small volume. The term on the left-hand side is the rate of change of eggs with t ime in the small element. The first term on the right-hand side repre- sents the net rate at which eggs are brought into the small volume by the river current. The second term represents the number of eggs brought into the small volume by the settling velocity of the eggs. The third through fifth terms represent the net flux of eggs due to the turbulent action of the water
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body. These three terms represent the effects of the diffusive capability of the river on the egg abundance. It is noted that Eq. (1) is identical to the well- known advective turbulent diffusion equation (Yeh, 1973), except that an extra term, wsan/az, is added to account for the settling of the eggs.
The solution of Eq. (1) with Eq. (2) requires the specification of initial and boundary conditions. These are obtained from the physical conditions as follows:
n = na at t = 0 (3)
n is bounded as x -~ +oo (4)
an Ey ~ - = 0 a t y = 0 (5)
an Zy ~-~ = 0 at y = B (6)
an Ez - ~ = w~n at z = 0 (7)
and
an Ez ~z = 0 at z = H (8)
where H and B are the depth and width of the river, respectively, and na is the initial egg abundance. Conditions (5), (6) and (8) state that flux of the eggs through the river banks and water bo t tom is prohibited. Condit ion (4) states that at infinity the abundance of eggs approaches a complete mixing value. Condition (3) implies that the river has constant abundance initially. Condition (7) states that at the water surface the flux due to the vertical fluctuation is balanced by settling of the eggs.
The solution of Eq. (1), subject to boundary conditions (3) through (8) with the source function given by Eq. (2), is
+ ~ N B 2 - B 1 ~__ wsz i=1 4 B B c~i(z) ~i(H1" H2) exp \ Yj(x, t, L, T) (9) 2 Ez]
where @j, ~/, and F u are functions defined by Eqs. (A3), (A7) and (A6), respectively, and Y; is defined similarly to F u. The detailed derivation of Eq. (9) is given in the Appendix. It is the tool obtained in this paper for pre-
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dicting the distribution of the fish eggs. It is used to calculate the abundance of eggs at any point in the river at any time released from a spawning region at a uniform source rate, N.
APPLICATIONS
The application of this model to practical problems depends on the proper choice of dispersion coefficients, Ex, Ey and E~, and the settling velocity, w~-. The settling velocity depends on many factors, of which the most important are: size and shape of the fish eggs, density of water and egg, and Reynolds number. Determination of the settling velocity has been extensively studied (Graf and Acaroglu, 1966; Watson, 1969).
Previous studies on the dispersion coefficients have centered on the disper- sion of patches of material, such as fluorescent dye (Jobson and Sayre, 1970; Miller and Richardson, 1974). When considering the dispersion of particulate material such as fish eggs, however, the concern is with the transport of vari- ous particle sizes characterized by different settling velocities. It is natural to ask if dispersion coefficients determined from dye studies can be used in the dispersion of fish eggs. To answer this question, extensive field or laboratory experiments on the dispersion of fish eggs are needed. Equation (9) may be used to obtain estimates of Ex, Ey and Ez from experiments. A continuous volume source of fish eggs would be discharged into the water. Egg abundance, spatial coordinates and times would be recorded. After this, an opt imum search scheme (Green, 1970) would be used to determine the proper values for these coefficients.
Lacking experimental data on Ex, Ey and E~ for fish egg distribution in a river, representative values for these coefficients obtained from dye studies on the Hudson River in New York are used in Eq. (9) and hypothetical results are calculated. For a river 1000 m wide and 16 m deep, typical values have been chosen: u = 0.3 m s - 1 , Ex = 50 m 2 s -1, Ey = 6 m 2 S - 1 and E~ = 0.002 m 2 s -1 .
To assess the effects of source location and settling velocity on the distribu- tions, three different source locations and three settling velocities are con- sidered. The results are shown in Figs. 1 through 6. Reasonable qualitative results are obtained. Figures 1, 2 and 3 show the abundance distribution from spawning regions located in three different depth levels. Each Figure describes isopleths of various egg abundances for three settling velocities and indicates that the area enclosed by the same isopleths is larger when the settling velocity is small. This is understandable since the rate at which eggs are removed from the water body decreases with decreasing settling velocity if everything else remains the same.
Figures 4, 5 and 6 show the distribution of egg abundance from three dif- ferent settling velocities, ws = 0.0, 1.0 and 2.0 cm s -1, respectively. Each Figure depicts the isopleth from three source levels of depth. Figure 4 demonstrates that symmetrical distribution is obtained if the source is located at the center
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&
- 1 5 0 0.0
4.o
8.0
12.0
16.0 - 50
O. 0 150 300 450 600 750 900 1050
I ) \' i = " ' / " L ' , , v ,
i i / t 1"0 1/0 ' / ' 0 \
~,,, ; i / \ ' , ~ , ," / 1.~i
. -~--- ..~-__ i . - " 0 . 5 / i
, ' ~ . . " <~'~. : . . . . . . . i " ,/
N O. 0 150 300 450 600 750 900 1050
1200 0.0
4.0
8.0
_ 12.0
16.0 1200
Longitudinal Distance
Fig. 1. P r e d i c t e d i s o - a b u n d a n c e l ine o n t h e p l a n e y = 0 r e s u l t i n g f r o m a v o l u m e s o u r c e x s = - 5 t o 5 m , y s = 2 9 5 t o 3 0 5 m , z s = 0 t o 4 m f o r w s = 0 . 0 c m s - 1 ( ) , w s = 1 .0 c m s - 1 ( - - . - - ~ a n d Ws = 2 .0 c m s - 1 ( . . . . ), r e s p e c t i v e l y . U n i t s ( m ) fo r b o t h a x e s in t h i s a n d s u c c e e d i n g f igu res .
cl l i
-150 O. 0 150 300 450 600 750 900 1050 1200 0.0 I I 1 t . I " - I I i I 1 I O . q
0 . 3 . / "
4.o
8.0
12.0
/-/
).-/
\'
' I i
16.0 I I | / I I 1 -150 O.0 150 300 450 600 750
I 1 . 4 / 1.o/ ~ \ . . . . . ' / / , \ ,,
, p .
0.5
t I
I i 900
I 1050
4.o
8.0
12.0
16.0 1200
Longitud2nal Distance
Fig. 2. S a m e as Fig. 1 e x c e p t t h a t z s = 6 to 10 m .
102
J::l
12.0
-150 0 150 300 450 600 750 900 1050
0.o ' ' ' ' ' ' I
4.0
-
-,,1<o 0 \
16.0 I I \ \ I \ , \ ! l ' , 1 I -150 o.0 150 300 450 600 750 900 lO50
0.3
1200
0.0
8.0
..~ 12.0
16.0 1200
Longitudinal Distance
Fig . 3. S a m e as F ig . 1 e x c e p t t h a t z s = 1 2 t o 1 6 m .
J:l &
- 1 5 0 0.0
4.o
8.0
12.0
16.0
-150
0 150 300 450 600 750 900 1050
7 i , I i S I I I O. /
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/ ~ . ~ 1.0/
0.3 ~.
i i I - , I I 1 0 150 300 450 600 750 900 1050
12OO 0.0
4.0
8.0
12.0
16.0
1200
Longitudinal Distance
Fig . 4. P r e d i c t e d i s o - a b u n d a n c e l i n e o n t h e p l a n e y = 0 f o r w s = 0 . 0 c m s - 1 r e s u l t i n g f r o m a v o l u m e s o u r c e l o c a t e d at x s = - - 5 t o 5 m , Ys = 2 9 5 t o 3 0 5 m , a n d z s ffi 0 t o 4 m ( . . . . ), z s = 6 t o 1 0 m ( ) a n d z s = 1 2 t o 1 6 m ( - - . - - . ) , r e s p e c t i v e l y .
103
1&
-150 0 150 300 450 600 750 900 0 . 0 [ I j , ~ , , . - - ~ " I ~1 I ",..[ \ I
Fig. 5. S a m e as in Fig. 4 e x c e p t t h a t w s = 1 .0 c m s - 1 .
-150 0 150 300 450 60O 750 90O
0.0 1 I I ~ I ~ , I I \ ~ I
I 1 40 ~'~ \ \ I \ ,, I o . 5 1 \
4.0 ',3 ~ ~ ~ z ~ ; \ ,
• I
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4.O
8.0
-- 12.0
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1050 1200
Longitudinal Distance
Fig. 6. S a m e as Fig. 4 e x c e p t t h a t w s = 2 .0 c m s - 1 .
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of the river. It also points out that if the sources are located at an equal dis- tance from the top and bot tom, they have identical distributions with respect to the top and bottom, respectively. This results from the fact that the egg is assumed to be neutrally buoyant with respect to the water. This symmetrical and reciprocal property is no longer preserved when eggs have a density differ- ent from the water, as can be seen from Figs. 5 and 6, which present distorted isopleths toward the downstream and towards the bot tom of the river.
CONCLUSION
An analytical model based on the conservation of numbers is proposed to predict fish egg abundance from spawning areas in the rivers. The predictive equation is derived. The model includes the effects of river geometry, fish egg settling and water dispersibility. The sensitivity of model parameters is analyzed. Reasonably good quantitative and qualitative results are obtained. The verification and validation of the model remain to be done.
A P P E N D I X So lu t ion of Eqs. (1) t h r o u g h (8)
The solution of Eq. (1) subject to the initial and boundary conditions (3) through (8) can be obtained by means of Green's theorem (Yeh and Tsai, 1976)
T f . . n = f f f S(xo, Yo, Zo, to)exp[ws(z--Zo)/2Ez -- w~(t--to)/4Ez]
0 - -~ 0 0
G(x, y, z, t; x o, Yo, Zo, to) dzo dYo dxo dto + na (A1)
where G is the Green's function of Eq. (1) and can be verified having the explicit expression of
G = 1 exp[-- { ( x - - x ° ) - - u ( t - - t ° ) } u I + 2 E cos~-~--][DrY~ x/ 4~rEx( t _ to ) (4Ex(t -- to)) B i=1
(A2)
X cos~--B-- ~ exp[_--~-~] E , ( t - - t o ! ¢~(z)¢j(zo)exp[--X~E~(t--to)])
respectively. When the settling velocity of the eggs approches zero, a term of 1/H must be added to the third term of Eq. (A2).
Substituting Eq. (2) into Eq. (A1) and making the use of Eqs. (A2) through (A5), one obtains the solution of Eq. (9). It can be shown that Fij and @j(H1, //2) in Eq. (9) are given by
Fij(x, t; L, T) = - - ~ . e x p ( - - a i j b i j ~ x t ) . [exp(aubijx/~ ) Ot ij b ij [
+ ~ -- exp(b~/4), erf aij~/T4 V'-4-E~t
x )11 V~4-Ex(t--T) 4 - - bo/2 + - - " e x p ( - - a i i b i j ~ ( t -- T)) •
( b i j (x - -L) ].erf(ai, x / t _ T + x - - L " e x p ( a i i b i j ~ ) [exp(-x/4Ex(t-- T)] V ~ E x ( t - - T))
- - erf ( ~ i j ~ x -- eifl2 Jl -- • exp[--aiieijx/-4-Nx(t-- T)I N/r~x t OL ijC ij
- - L c i j ( x - - L ) i er f ( t~i jx /r~_T X --'T;) "{exp(t~iJciix/T--~-T'[exp( ' ~/-~x~-~-T)/" x/4Ex(t
106
-- exp(-- c i r x x
Ierf(au t~/~-~T - x - - L - - c u / 2 ) --erf (air~/-f--T-- ~ / 4 E . ( t -- T)
_ x _ cu/2~-lt (A6) ]J} ~ 4 E x ( t - T)
and
2 2 2 "~ aj ~/(H1, H2) =kr A - - ~ - - .-~.2 4E2k2 + Ws
# [[4E~X~ + w s • s in(ki l l2)-- 2 COS(~k r H2 )
_ 2 4EzXrW s 1 Wsg-~ _ 1 4 E 2 z k 2 - - w s • exp 2 ~-z 2j [4E 2k2 + w 2 " sin(krH1) 4E2k2 + w2
• cos(kjH1) } • e x p I - - l w s ~2
respectively. In Eq. (A6), Olir , bi j , and cir, are represented by the following equations:
air = x /u2 /4Ex + (i2~2/B2)Ey + k~Ez + w2/4Ez (A8)
bi j U oQj - - - + ( A 9 )
2E x E x
and
u -- air (AIO) cir = 2 Ex Ex
The formula for Yj is identical to that for Fij except that au, bu and c u have to be replaced by at, br and cr, which are defined by
~i = ~/u2/4Ex + k2E~ + w2/4Ez ( A l l )
u + ~J (A12) bj = 2Ex E~
and u
cj = 2E x Ex (A13)
respectively.
107
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Copeland, B.J., Miller, J.M., Watson, W., Hodson, R., Birkhead, W.B. and Schneider, J., 1976. Meroplankton: problems of sampling and analysis of entrainment. In: L.D. Jensen (Editor), Third National Workshop on Entrainment and Impingement. Ecological Analysts, Melville, New York, N.Y., 199--137.
Coutant, C.C., 1970. Biological aspects of thermal pollution I. Entrainment and discharge canal effects. CRC Crit. Rev. Environ. Control, 1(3): 341--381.
Coutant, C.C., 1971. Effects on organisms of entrainment in cooling water. Steps toward predictability. Nucl. Saf., 12: 600--607.
Graf, W.H. and Acaroglu, E.R., 1966. Settling velocities of natural grains. Int. Assoc. Sci. Hydrol., 11: 27--43.
Green, R., 1970. Optimization by the pattern search method. Research Paper No. 7, Division of Water Control Planning, Tennessee Valley Authori ty, Knoxville, Tennessee.
Jensen, L.D. (Editor), 1978. Fourth National Workship on Entrainment and Impingement: Section V -- Engineering Aspects of Entrainment and Impingement. Ecological Analysts, Melville, New York, N.Y., 424 pp.
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Miller, A.C. and Richardson, E.V., 1974. Diffusion and Dispersion in Open Channel. J. Hydr. Div., ASCE, 100(HY1): 159--171.
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Van Winkle, W. (Editor), 1977. Assessing the Effects of Power Plant Induced Mortality on Fish Populations. Pergamon, New York, N.Y., 380 pp.
Watson, R.L., 1969. Modified Rubey's law accurately predicts sediment settling velocities. Water Resour. Res., 5: 1147--1150.
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