A MODEL OF THE VERTICAL DISTRIBUTION OF PELAGIC EGGS.
A COMPUTER REALIZATION.
Trond Westgård
I n s t i t u t e of Marine Research P.O.Box 1870
5024 Bergen, Norway
ABSTRACT
The p r e s e n t e d mathematical model and i t ' s r e a l i z a t i o n i n an i n t e r - a c t i v e computer program d e s c r i b e s t h e v e r t i c a l d i s t r i b u t i o n of p e l a g i c f i s h and plankton eggs. The water column has a d e n s i t y g r a d i e n t and t h e v e r t i c a l d i f f u s i o n c o e f f i c i e n t i s a f u n c t i o n of t h e wind blowing a t t h e s u r f a c e . The d e n s i t y and t u r b u l e n c e c o e f f i c i e n t p r o f i l e s of t h e water column a r e defined t o be i n v a r i a n t with time.
The model i s consequently a b l e t o d e s c r i b e only t h e s h o r t term f l u c t u a t i o n s of t h e c o n c e n t r a t i o n of eggs. The eggs may be d i s t r i b u t e d i n s e v e r a l s i z e - and buoyancy-groups. The t o t a l number of eggs i n t h e water column i s c o n s t a n t d u r i n g one s e s s i o n . Despite t h e crude formulation of t h e model i t i s u s e f u l f o r a number of purposes of which t h e main a r e t h e e v a l u a t i o n of sampling d e s i g n s f o r p l a n k t o n i c eggs and t h e e s t a b l i s h m e n t of t h e coincidence between contaminants and b i o l o g i c a l r e s o u r c e s i n a v u l n e r a b l e l i f e s t a g e .
INTRODUCTION
The p r e s e n t model i s mainly t h e model of t h e v e r t i c a l d i s t r i b u t i o n of f i s h eggs i n t h e homogenous wind mixed upper l a y e r s of t h e ocean a s d e s c r i b e d by Sundby (1983). The p r e s e n t work s o l v e s t h e e q u a t i o n s numerically i n o p p o s i t i o n t o t h e a n a l y t i c a l s o l u t i o n t h a t Sundby p r e s e n t e d , consequently i t has been p o s s i b l e t o i n c l u d e a d e n s i t y g r a d i e n t i n t h e watercolumn. The numerical s o l u t i o n a l s o makes i t p o s s i b l e t o i n s p e c t t h e t r a n s i e n t s o l u t i o n of t h e e q u a t i o n s . T h i s means t h a t i t models t h e time from t h e eggs a r e i n t r o d u c e d i n t h e watercolumn a t c e r t a i n depths u n t i l t h e i r v e r t i c a l c o n c e n t r a t i o n p r o f i l e reach an e q u i l i b r i u m . P o s s i b l e a p p l i c a t i o n a r e a s o f t h e model a r e :
-
Design of e f f i c i e n t sampling s t r a t e g i e s of p l a n k t o n i c eggs under d i f f e r e n t weather, s a l i n i t y and temperature c o n d i t i o n s .-
Estimation of t h e amount of eggs o u t s i d e t h e p a r t of t h e water column t h a t i s sampled.-
Evaluation of t h e v e r t i c a l coincidence of p o l l u t a n t s and t h e spawning products of commercial f i s h s p e c i e s and important plankton s p e c i e s .With small m o d i f i c a t i o n s t h e model could a l s o be used f o r t h e d e s c r i p t i o n of t h e v e r t i c a l d i s t r i b u t i o n of o i l d r o p s and a i r b u b b l e s i n t h e water column.
The paper i n c l u d e s an u s e r guide t o t h e computer program. The u s e r may then e a s i l y compare t h e model's r e s u l t with o b s e r v a t i o n s made i n f i e l d i n v e s t i g a t i o n s . Since i t i s b e l i e v e d t h a t an experimental computer program l i k e t h e p r e s e n t could e a s i l y be modified t o s o l v e r e l a t e d problems t h e computer code i s included i n Appendix 1.
The p r e s e n t work i s a p a r t of t h e HELP-program a t t h e I n s t i t u t e of Marine Research, Bergen. A r e s e a r c h program which main g o a l i s t o a s s e s s t h e p o s s i b l e e f f e c t s of o i l s p i l l s on v u l n e r a b l e r e s o u r c e s of commercially important f i s h s p e c i e s along t h e c o a s t of Norway caused by t h e i n c r e a s e d o i l a c t i v i t y i n Norwegian waters.
THE GOVERNING EQUATIONS
The v e r t i c a l d i s t r i b u t i o n of eggs i s dependent on t h e d e n s i t y p r o - f i l e of t h e water column, t h e v e r t i c a l eddy t u r b u l e n c e c o e f f i c i e n t , t h e v e r t i c a l c u r r e n t and t h e d e n s i t y and diameter of t h e s p h e r e s . When t h e v e r t i c a l c u r r e n t i s n e g l i g i b l e , a l l h o r i z o n t a l g r a d i e n t s a r e z e r o and t h e v e r t i c a l p h y s i c a l p r o p e r t i e s of t h e water column a r e supposed t o be i n v a r i a b l e with t i m e t h e p r o c e s s i s d e s c r i b e d by t h e following e q u a t i o n s :
1 2
3
gd d e ( z ) v ( z i l when d<
D(z) ( 1 A )2 1
w(z) =
-
mk1$d + k l q ( z ) ) I A Q ( z )
I 3
s i g n ( A e ( z 1 ) v ( z ) ~ when d>
D(z) (18)K(z=O) =
k61
' 6 2 ~ ~ f o r O<
W< 13
m / s ( 6 )a o )
= ~ ( w ( z ) ~ ( z , t ) )a t a
zEquation
(7)
has t h e c o n s t r a i n t s :f ( z , t ) = O ; O z
<
H ;anlo,t)
= = O O ( ta~ a
zThe d e f i n i t i o n s of t h e v a r i a b l e s a r e :
Unit z = depth ( z e r o a t s u r f a c e p o s i t i v e downwards) ( m )
- 1
w(z) = ascendingldescending v e l o c i t y of a s p h e r e a t depth z (ms )d = diameter of a s p h e r e ( m )
D(z) = maximum s p h e r e diameter f o r ~ t o k e ' s e q u a t i o n t o apply ( m )
@ e = d e n s i t y of a s p h e r e (k&3)
e S ( z ) = d e n s i t y a t depth z
(&G3)
2 -1 v ( z ) = kinematic molecular v i s c o s i t y a t d e p t h z ( m s
1
S ( z ) = s a l i n i t y a t depth z T ( z ) = temperature a t depth z
K(z) = eddy t u r b u l e n c e c o e f f i c i e n t a t depth z 2
-1
( m sW = wind v e l o c i t y a t t h e s u r f a c e
(mi1)
n ( z , t ) = c o n c e n t r a t i o n of s p h e r e s a t depth z a t t i m e t (mm3) f ( z , t ) = b i r t h l d e a t h of s p h e r e s a t depth z a t t i m e t ( 2 s - l )
g = a c c e l e r a t i o n of g r a v i t y ( m c Z )
ki j = n e c e s s a r y c o n s t a n t s H = t h e bottom depth
Table 1. Values of necessary c o n s t a n t s .
Equation (1) i s given i n Sundby(1983). Equation ( 1 A ) i s t h e well-known s t 0 k e . s e q u a t i o n f o r t h e s i n k l r i s e v e l o c i t y of a s o l i d s p h e r e i n a continous f l u i d phase. This e q u a t i o n i s v a l i d up t o a Reynold number of 0 . 5 according t o Sundby, The i n t e r m e d i a t e r e g i o n b e f o r e t h e motion becomes completely t u r b u l e n t i s d e s c r i b e d by e q u a t i o n ( 1 B ) . The maximum diameter f o r which t h e Stoke e q u a t i o n a p p l i e s given i n ( 2 ) i s d e r i v e d from (1A) by s u b s i t u t i n g w(z) i n t h e l e f t handside of ( 1 A ) with :
Where R e i s ~ e ~ n o l d ' s number f o r t h e motion of t h e s p h e r e . The c r i t i c a l v a l u e of Re i s chosen t o be 0 . 5 h e r e and t h i s v a l u e i s used when
(8)
i s s u b s t i t u t e d i n t o ( 1 A ) . Equation ( 1 B ) i s v a l i d upto R e v a l u e s of a t l e a s t 5 . 0 and consequently cover t h e a c t u a l speed range f o r p l a n k t o n i c eggs (Sundby,1983).Equation
( 4 )
i s t h e I n t e r n a t i o n a l Equation of S t a t e of Sea Water (IES 1980) a s g i v e n i n Fofonoff and M i l l a r d ,1983.
S i n c e t h e p l a n k t o n i c eggs i s mostly a t moderate depths t h e water i s taken t o be incompressible, i . e . t h e s e c a n t bulk modulus term i n IES 1980 i s ignored.
The kinematic molecular viscosity of sea water is a functbon of the salinity and temperature of which temperature is the most important.
Sundby (pers.comm.) using data from Krummel and Ruppin (1905) arrived at equation (5) which is strictly valid for a salinity of 30 0100.
The eddy turbulence coefficient given in (6) is valid for the upper wind mixed layers of an homogenous water column. The expression was derived by Sundby (1983) from field data on vertical distribution of fish eggs using the 12 hour mean wind velocity before the sampling of the actual profile.
The last equation (7) is the well-known advection/diffusion equation for a scalar concentration given in standard text books like Pond and Pickard (1983). It should be noted that the advection term has a positive sign and not a negative, this is because w(z) is positive upwards i.e. in the negative z-direction.
Sundby (1983) gives the steady state solution of (7) when K(z) and w(z) are both constant. He also includes the distribution of eggbouyancy and eggdiameter groups, i.e. the distribution of ascending/descending velocities of the eggs, which was anticipated to be a truncated normal distribution.
Here the equations (1)-(7) are assumed to be valid for one specific set of values of the diameter and buoyancy and the computer program allows the user to define a reasonable number of such sets that best fits his actual empirical distribution of egg diameter and buoyancy.
It is emphasized that
R ( z )and w(z) are functions of depth in the present work.
NUMERICAL SOLUTION
Equation (7) is solved by setting up a staggered grid as shown
inFig.1. To ease the implementation of the scheme the concentration of spheres has two dummy grid cells, one just over the surface and one just below the bottom. The content of these cells are always zero.
Each of the layers are dZ m thick. The concentration of spheres is
measured in the middle of each layer. The vertical velocity and eddy
turbulence coefficient are measured between the layers and both are
set to zero at the surface and at the bottom to satisfy the
c o n s t r a i n t s of
( 7 )
i . e . t h e numerical e q u i v a l e n t s t o t h e c o l u t i o n domain' s boundary condi t o n s.
When K ( z ) and w(z) a r e c o n s t a n t s t h e numerical s o l u t i o n of ( 7 ) i s w e l l d e s c r i b e d and could be found i n f o r example A m e s (1977) and Roache ( 1 9 7 2 ) . I n o u r c a s e K(z) and w(z) a r e f u n c t i o n s of d e p t h , z . S i n c e t h e problem i s d e f i n e d i n only one space dimension t h e t i m e s t e p and g r i d s i z e could be made small and an e x p l i c i t scheme has been chosen.
F i g u r e 1. The s t a g g e r e d g r i d used i n t h e computations.
When w i s a f u n c t i o n of z , t h e second o r d e r upwind d i f f e r e n c i n g method i s e f f i c i e n t f o r t h e buoyancy f l u x term term i n
( 7 ) ,
t h e method h a s a l s o been c a l l e d t h e donor c e l l method (Roache, 1 9 7 2 ) . Thenumerical r e p r e s e n t a t i o n of t h e buoyancy term, advn, i s then :
Wlnl
-
advn = W2 "2 dT
dZ Where
n ( i Z ) f o r w1
>
0 n1 = [n ( i Z + l ) f o r w
<
O1
n2 =
[
n ( i z - l ) f o r w2>
0 n ( i Z ) f o r w2<
OThe l o c a l change i n n ( i Z ) due t o d i f f u s i o n , d i f n , i s e v a l u a t e d i n a s o r t of two s t e p Euler scheme (Dag S l a g s t a d , per;. comm. ) t h a t i s simple and y e t e f f i c i e n t .
K ( i Z ) ( n ( i Z + l ) - n ( i Z ) )
-
K ( i Z - l ) ( n ( i Z ) - n ( i Z - 1 ) ld i f n = dT
(13)
dz2
It i s n o t i c e d t h a t when K(z) i s c o n s t a n t t h e scheme d e g e n e r a t e s t o t h e u s u a l c e n t e r e d second d i f f e r e n c e method f o r t h e d i f f u s i o n term.
I n t h e c a s e when w(z) and K(z) a r e c o n s t a n t s Roache
9. G.
g i v e s an a c c u r a t e d e s c r i p t i o n of t h e numerical d i f f u s i o n f o r t h e p r e s e n t e d numerical s o l u t i o n of( 7 ) .
I n t h a t c a s e t h e p h y s i c a l d i f f u s i o n c o e f f i c i e n t could be modified and t h e s t e a d y s t a t e s o l u t i o n w i l l e x a c t l y match t h e a n a l y t i c a l s o l u t i o n . The e r r o r of t h e d i f f u s i o n c o e f f i c i e n t i n t h e t r a n s i e n t s o l u t i o n could a l s o be e s t i m a t e d i n t h a t c a s e . I n t h e p r e s e n t paper t h e p h y s i c a l c o e f f i c i e n t of d i f f u s i o n K ( z ) has been modified by an approximate numerical d i f f u s i o n term and t h i s v a l u e has been used t o modify t h e K(z) v a l u e i n each of t h e nZ l a y e r s . For f u r t h e r d e t a i l s t h e i n t e r e s t e d r e a d e r a r e r e f e r r e d t o t h e computer code included i n t h e Appendix 1.A USER G U I D E TO THE PROGRAM
The program i s implemented on a ND-500 computer i n Fortran-77 and t h e r e s u l t i s d i s p l a y e d e i t h e r on a g r a p h i c a l s c r e e n o r a pen p l o t t e r u s i n g t h e GPGS-F l i b r a r y (Anon.,1984). The l i s t of t h e program i s given i n Appendix 1.
The u s e r i s p r e s e n t e d with t h e following d i a l o g u e :
Give number of eggdiameter groups :
2
Give number of eggbuoyancy groups :
2
Give number of depth l a y e r s :
9
Give name of f i l e with d a t a : ( y o u r - u s e r - i d ) y o u r : d a t a Give maximum depth i n meters : 100.0
Give wind speed i n m / s :
s.0
Give t h e number of days t o run t h e model :
7.0
Give hours between showtimes : 0.5
The maximum number of egg diameter and egg buoyancy groups i s
9
s o t h e maximum number of egg groups i s81.
T h i s should be s u f f i c i e n t f o r most purposes. The maximum number of l a y e r s t h a t t h e u s e r has measured o r computed t h e i n i t i a l d i s t r i b u t i o n of e g g s , s a l i n i t y , temperature and t u r b u l e n c e i n t h e water column i s 200. The d a t a f i l e i s made by t h e u s e r u s i n g an e d i t o r .The l a y o u t of t h e d a t a f i l e be a s follows :
5 . 2
l
The temperature i n each of t h e l a y e r s ( o C )
. l
The s a l i n i t y i n each of t h e l a y e r s (0100).
The i n i t i a l c o n c e n t r a t i o n of eggs i n each of t h e l a y e r s ( n o . / m 3 )::: l
I n t h e sample d i a l o g u e above t h e u s e r s p e c i f i e s 50 l a y e r s i n t h e d a t a f i l e and g i v e s a bottomdepth of 100 meters. The depth of each l a y e r , dZ, i s then 2 m . The temperature, s a l i n i t y and c o n c e n t r a t i o n d a t a should be s t r a i g h t forward t o s p e c i f y . The r e l a t i v e v a l u e s of t h e eddy t u r b u l e n c e c o e f f i c i e n t compared t o t h e eddy t u r b u l e n c e computed f o r t h e upper l a y e r u s i n g
( 6 )
i s n o t easy t o compute u n l e s s t h e whole water column i s homogenous o r t h e r e i s a s t r o n g pycnocline and t h e eggs i s above t h a t l e v e l . I n t h e s e c a s e s t h e v a l u e s should be s e t t o 1 . 0 , i n t h e c a s e of a s t r o n g pycnocline t h e bottom d e p t h i s s e t t o t h e l e v e l of t h e pycnocline. I n o t h e r c a s e s only g e n e r a l advice i s p o s s i b l e . The r e a d e r i s adviced t o c o n s u l t t h e papers of Gargett( 1 9 8 4 ) , S t i g e b r a n d t ( 1 9 8 5 ) , Omstedt (1985) and Rodi (1980). Gargett g i v e s some g u i d e l i n e s on how t o compute K ( z ) v a l u e s based upon t h e s t a b i l i t y of t h e water column below t h e wind mixed l a y e r s i n t h e oceans. I n t h e l a s t t h r e e papers more complex models of t h e t u r b u l e n c e p r o c e s s i s p r e s e n t e d .
1 . 0 - 1 . 0
The l a s t p a r t of t h e d a t a f i l e s p e c i f i e s t h e p r o p e r t i e s of t h e eggs. I t i s t h e s a l i n i t y a t which t h e egg have n e u t r a l buoyancy t h a t i s given i n t h e f i l e . The reason i s t h a t t h i s i s t h e v a l u e measured i n l a b o r a t o r y experiments and t h a t t h e eggs have t h e i n t e r n a l p r e s s u r e and temperature a s t h e surrounding water (Sundnes e t . a l . , 1965)
C.-- T h i s nurnber a u s t always be 1 . 0 I ?
The r e l a t i v e s i z e of t h e eddy t u r b u l e n c e i n each l a y e r compared t o t h e wind induced v a l u e i n t h e t o p l a y e r
I n t h e next q u e s t i o n s t h e u s e r g i v e s t h e wind speed a t t h e s u r f a c e t h a t he wants t o u s e i n t h e sirnulation, t h e number of days t o run t h e model and how o f t e n he wants t h e r e s u l t t o be d i s p l a y e d a t t h e s c r e e n . The l a s t q u e s t i o n t h e u s e r must d e a l with i s :
0 . 1
-
1.5E-03 33.2 0 . 1 The f i r s t number i s t h e egg diameter i n m.
1.5E-03 33.4 0 . 4 T h i s diameter i s r e p e a t e d a s many times a s i t 1 . 6 ~ - 0 3 33.2 0.4 i s buoyancy groups, then t h e n e x t diameter group 1 . 6 ~ - 0 3 33.4 0 . 1 s t a r t s . The l a s t column g i v e s t h e f r a c t i o n t h a t
t h e diameter/buoyancy group i s of t h e t o t a l .
Do you want a hardcopy (Y/N) ?
X
If
t h e answer i s Y ( y e s ) a f i g u r e s i m i l a r t o F i g , 2 i s drawn on t h e pen p l o t t e r . The t o t a l number of eggs p r e s e n t i n t h e water column i s computed every time t h e f i g u r e i s d i s p l a y e d t o g i v e t h e u s e r a crude check on t h e accuracy of t h e computations. The number of eggs should be c o n s t a n t s i n c e they n o t a r e s u b j e c t t o d e a t h and no new eggs a r e i n t r o d u c e d .EXAMPLES
The p r e s e n t model have s o f a r n o t been compared with a c t u a l f i e l d d a t a except f o r t h e s t a e d y s t a t e s o l u t i o n s shown by Sundby ( 1 9 8 3 ) . For i l l u s t r a t i v e purposes I have made two examples showing what t h e eggs p r o p e r t i e s and d i f f e r e n t p h y s i c a l conditons l e a d s t o q u i t e d i f f e r e n t v e r t i c a l c o n c e n t r a t i o n p r o f i l e s of eggs. I n both c a s e s t h e i n i t i a l c o n c e n t r a t i o n p r o f i l e of eggs i s symmetricly d i s t r i b u t e d around 50 m from 20 t o 80 m.
I n t h e f i r s t example t h e r e i s a thermocline a t t h e d e p t h o f 50 m and a
o o
bottom depth of 100 m. The temperature i s
7
C above 50 m and 5 C below. The s a l i n i t y i s34
0100 above 50 m and 35 0100 below. The wind speed i s 5.0 m / s . The turbulence c o e f f i c i e n t below t h e thermocline i s supposed t o be 1/10 of t h e v a l u e i n t h e wind mixed l a y e r . Two groups of eggs a r e given one with a diameter of1.5
mm and n e u t r a l buoyancy a t 33.0 0/00 and one with a diameter of1.3
mmm and n e u t r a l buoyancy a t 36.0 0/00. The r e s u l t i s shown i n Fig. 2 a f t e r t h e model has been running f o r twelve hours. It i s seen t h a t t h e heavy eggs c o n c e n t r a t e s q u i t e n e a r t h e bottom, while t h e l i g h t e r eggs with p r o p e r t i e s n e a r e . g . cod eggs have a pure p e l a g i c d i s t r i b u t i o n . The i n i t i a l c o n c e n t r a t i o n of eggs i s given by a dashed l i n e i n F i g . 2.T i m e in h o u r a ' 1 2 . 0 2 N u m b e r af c q g r . I I 0 0 E t 0 2 o
1 0 0
1
I I l I l I l I0 . 0 O. 5 I . O 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0
C o n c a n t r o t i o n ( n o . / m 3 )
Figure 2. Sample o u t p u t of t h e model showing a c o n c e n t r a t i o n p r o f i l e of f i s h eggs a f t e r
12
hours ( s o l i d l i n e ) . The i n i t i a l con- c e n t r a t i o n i s shown w i t h a dashed l i n e .T i m e in h o u r r ' 1 2 . 1 0 N u m b e r of e q q r . I . I O O E t 0 2
-
E O-
.c a CiC o n c e n t r a t i o n ( o o . / m J )
Figure
3.
Sample o u t p u t of t h e model showing a c o n c e n t r a t i o n p r o f i l e of copepod eggs a f t e r 12 hours ( s o l i d l i n e ) . The i n i t i a l c o n c e n t r a t i o n i s shown w i t h a dashed l i n e .F i g ,
3
shows a simulated p r o f i l e of copepod eggs. Eggs of t h e copepod Calanus finmarchicus a r e extremely heavy compared t o most f i s h eggs ( S a l z e n , 1956) and they a r e seen ( F i g .3)
t o g e t a c o n c e n t r a t i o n n e a r t h e bottom i n a homogenous 100 m wind mixed water column. The s a l i n i t y and temperature were s p e c i f i e d t o be35
0100 and5
o C r e s p e c t i v e l y . The wind speed were s e t t o 2 . 0 m / s . Note t h a t d e s p i t e t h e f a c t t h a t t h e copepod eggs a r e s o heavy compared t o f i s h eggs they s t a y remarkably high i n t h e water column. The reason i s t h a t t h e eggs have a diameter of only135
pm. I f t h e eggs had been 500 pm most of t h e eggs would have been i n t h e lower5
m of t h e water column a f t e r 12 hours.
DISCUSSION
When t h e aim i s t o model t h e v e r t i c a l d i s t r i b u t i o n of p l a n k t o n i c eggs over a p e r i o d of 1-2 days t h e p r e s e n t e d model i s w e l l s u i t e d . When t h e model i s used i n t h e homogenous wind mixed upper l a y e r t h e r e s u l t s corresponds w e l l with f i e l d o b s e r v a t i o n of f i s h eggs (Sundby,
1983).
A t t h e moment when t h i s r e p o r t was w r i t t e n t h e r e was n o t a v a i l a b l e s u i t a b l e f i e l d d a t a t o e v a l u a t e t h e model's a b i l i t y t o d e s c r i b e t h e v e r t i c a l c o n c e n t r a t i o n of eggs i n a s t r a t i f i e d water column. I t i s s t r o n g l y b e l i e v e d , however, t h a t t h e model g i v e s a good d e s c r i p t i o n a s l o n g a s t h e r e l a t i v e s t r e n g t h of t h e v e r t i c a l d i s t r i b u t i o n of t u r b u l e n c e i s s p e c i f i e d c o r r e c t . Even i f t h i s d i s t r i b u t i o n d i v e r g e from t h e r e a l one t h e model g i v e s r e a s o n a b l e r e s u l t s . The d a t a f o r t h e example shown i n F i g . 2 was a l t e r e d s o t h a t t h e r e l a t i v e s t r e n g t h of t h e t u r b u l e n c e below t h e thermocline was p u t t o 0 , 5 i n t s t e a d of 0 . 1 . There were no d r a s t i c changes. The main e f f e c t s were t h a t t h e l i g h t e r egg group g o t a deeper d i s t r i b u t i o n while t h e heavy group r a i s e d some- what ,
An important parameter i n t h e model i s t h e ~ e ~ n o l d ' s number when t h e d e s c r i p t i o n of t h e r a i s e l s i n k v e l o c i t y of eggs s w i t c h e s from e q u a t i o n ( 1 A ) t o ( 1 B ) . Davis, 1972, s t a t e s t h a t e q u a t i o n ( 1 A ) i s v a l i d upto a Reynold number of 1 . 0 and i n p r a c t i c e even could be extended t o Reynold numbers a s high a s 50.0. I have had no o p p u r t u n i t y t o check t h e v a l i d i t y of t h i s f o r p e l a g i c eggs and have adopted t h e assumption
of Sundby,
op. G..
If
t h e o b j e c t i s t o d e s c r i b e t h e v e r t i c a l d i s t r i b u t i o n of f i s h eggs over extended t i m e p e r i o d s s e v e r a l o t h e r a s p e c t s must be included i n t h e model e q u a t i o n s . Below t h e most important f a c t o r s t h a t must be included i n such a model a r e d e s c r i b e d .The spawning and h a t c h i n g of eggs must be i n c l u d e d and t h e s t a t e of t h e eggs must be expanded with t h e development s t a g e of t h e eggs t o allow f o r t h e f a c t t h a t eggs when they develop might change t h e i r buoyancy (Solemdal, 1967,1971)
.
I f t h e h o r i z o n t a l g r a d i e n t s i s f a r from homogenous and t h e r e i s a v e r t i c a l s h e a r i n t h e c u r r e n t s t h e p r e s e n t e d model i s n o t s u i t a b l e . The model a l s o l a c k s t h e a b i l i t y t o model t h e e f f e c t of s h i f t i n g wind c o n d i t i o n s . More s o p h i s t i c a t e d models of t h e e f f e c t of wind on t h e t u r b u l e n c e i n t h e upper l a y e r s a r e then needed. Such a model should a l s o i n c l u d e a i r - t e m p e r a t u r e , -humidity and - p r e s s u r e i n a d d i t i o n t o wind speed.
A promising example of a b e t t e r model of t h e t u r b u l e n c e k-€ model (Rodi, 1980, Omstedt,
1985).
That model makes i t p o s s i b l e t o i n c l u d e e f f e c t s l i k e t h e deepening o f t h e wind mixed l a y e r and t h e development of a s e a s o n a l thermocline a s opposed t o t h e p r e s e n t model where t h e d e n s i t y p r o f i l e i n t h e water column i s f i x e d . The b e n e f i t of such a model however becomes only manifest when t h e r e is spawning over extended time p e r i o d s .When t h e r e s u l t s of t h e p r e s e n t model a r e compared t o f i e l d d a t a I s u s p e c t t h a t i n a r e a s with s t r o n g h o r i z o n t a l and v e r t i c a l s h e a r s i n t h e c u r r e n t s t h e r e s u l t s w i l l o f t e n be d i s a p p o i n t i n g . T h i s i s because advection p r o c e s s e s h e r e p l a y s an important r o l e . The o n l y way t o t o make p r o g r e s s i n such a r e a s i s t o d e f i n e d e t a i l e d 3D-models.
Despite t h e shortcomings of t h e p r e s e n t model i t i s an u s e f u l to01 f o r a r e s e a r c h e r who want t o g a i n i n s i g h t i n t h e v e r t i c a l mixing process of p e l a g i c f i s h and plankton eggs. The r e s e a r c h e r may experiment with t h e p r o p e r t i e s of eggs and p h y s i c a l v a r i a b l e s and observe t h e e f f e c t s d i r e c t l y a s a "movie" on h i s g r a p h i c a l s c r e e n . Such experiments w i l l l e a d t o b e t t e r i n s i g h t and i n e v i t a b l y t o b e t t e r f i e l d sampling s t r a t e g i e s .
REFEREWCES
Anon. 1984. GPGS-F u s e r . s guide. 6 t h e d i t i o n . T a p i r , Trondheim,
Ames, F.A. 1977. Numerical methods f o r p a r t i a l d i f f e r e n t i a l e q u a t i o n s . Second e d i t i o n . Academic P r e s s .
Davis, J . T . 1972. Turbulence Phenomena. An i n t r o d u c t i o n t o t h e eddy t r a n s f e r of momentum, mass, and h e a t , p a r t i c u l a r l y a t
i n t e r f a c e s . Academic P r e s s .
Fofonoff, N.P. and M i l l a r d , R . C . 1983. Algorithms f o r computation of fundamental p r o p e r t i e s of seawater.
Unesco Tech. Paper i n Mar. S c i . , No. 44,
53
pp.G a r g e t t , A.E. 1984. V e r t i c a l eddy d i f f u s i t i v i t y i n t h e ocean i n t e r i o r . J . Mar. R e s . (42):359-393
Krummel, 0 . and Ruppin,E. 1905. Wissensch. Meeresuntersuchungen d e r K i e l e r Kommission, Vol. 9 , p . 29. ( K i e l )
Omstedt, A . 1985. Modelling f r a z i l i c e and g r e a s e i c e formation i n t h e upper l a y e r s of t h e ocean.
Cold Regions Science and Technology, 11:87-98
Pond, S. and P i c k a r d , G.L.
1983.
I n t r o d u c t o r y dynamical oceanography.2nd e d i t i o n . Pergamon P r e s s .
Roache, P . J . 1972. Computational f l u i d dynamics.
Hermosa P u b l i s h e r s . Albaquerque.
Rodi, W . 1980. Turbulence models and t h e i r a p p l i c a t i o n i n h y d r a u l i c s
-
a s t a t e of t h e a r t review.
I n t . Assoc. Hydrol. Res., S e c t . Fundam. Div.2: Exp. Math. F l u i d Dyn.
,
Delf t , The Netherlands, pp.
1-104.S t i g e b r a n d t , A .
1985.
A model f o r t h e s e a s o n a l pycnocline i n r o t a t i n g systems with a p p l i c a t i o n t o t h e B a l t i c Proper.J . Phys
.
Oceanogr. , 15
: 1392-1404Solemdal, P. 1967, The e f f e c t of s a l i n i t y on buoyancy, s i z e and development of f lounder eggs, S a r s i a , 29 : 431-442
Solemdal, P. 1971. Prespawning f l o u n d e r s t r a n s f e r r e d t o d i f f e r e n t s a l i n i t i e s and t h e e f f e c t s on t h e i r eggs. " V i e e t Milieu", Tro- isieme Symposium Europeen de B i o l o g i e Marine. pp 409-423.
Sundby, S.
1983.
A one-dimensional model f o r t h e v e r t i c a l d i s t r i b u t i o n of p e l a g i c f i s h eggs i n t h e mixed l a y e r .Deep Sea Research Vol. 30
( 6 A )
: 645-661.Sundnes, G . , Leivestad, H. and I v e r s e n , 0. 1965. Buoyancy determi- n a t i o n of eggs from t h e cod (Gadus morhua L . )
J . Cons. perm. i n t . Explor. Mer, 29:249-252
APPENDIX
1. Soureeeode of the program,C ' * ( T r o n d ) M o d e l - 2 : s y m b * * C
C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c C
C D i f f u s i o n / a d v e c t i o n m o d e l f o r the v e r t i c a l d i s t r i b u t i o n o f p e l a g i c C s p h e r e s / e g g s i n the w i n d m i x e d l a y e r s o f the o c e a n .
C
C R i s e / s i n k v e l o c i t y of e g g s a n d the e d d y d i f f u s i v i t y c o e f f i c i e n t in C the s u r f a c e l a y e r is c o m p u t e d a c c o r d i n g to f o r m u l a e s b y S v e i n S u n d b y C g i v e n i n : C D e e p S e a R e s e a r c h 3 0 ( 6 ) 6 4 5 : 6 6 1 , 1 9 8 3 .
C
C All u n i t s i n t h i s p r o g r a m a r e S I - u n i t s i . e . : k g , m a n d s C
C T h e p r o g r a m is w r i t t e n i n F O R T R A N - 7 7 w i t h a f e w N D - e x t e n s i o n s . 1 . e . C
C - T h e D o for . . . . E n d d o c o n s t r u c t is u s e d C - S o m e v a r i a b l e n a m e s a r e m o r e t h a n 6 c h a r a c t e r s
C - B o t h u p p e r a n d l o w e r c a s e l e t t e r s a r e u s e d i n the c o d e C
C T r o n d W e s t g å r d . I n s t i t u t e o f M a r i n e R e s e a r c h . B e r g e n . N o r w a y . C
C 0 2 . 1 2 . 1 9 8 7 C
P R O G R A M M O D E L Z C
C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c
C
C D E F I N I T I O N S A N D D E C L A R A T I O N S :
C
c - - -
C S P A C E D I M E N S I O N :
C
C n Z m a x = M a x i m u m n u m b e r o f g r i d s t e p s i n the m o d e l i n Z - d i r e c t i o n P a r a m e t e r ( n Z m a x = 2 0 0 )
C n Z = N u m b e r o f g r i d s t e p s i n the m o d e l i n the Z - d i r e c t i o n I n t e g e r n Z
C Z = D e p t h in m
R e a l Z
C d Z = D i s t a n c e b e t w e e n g r i d p o i n t s i n the Z - d i r e c t i o n i n m . R e a l d Z
C i Z = I n d e x of g r i d p o i n t in the Z - d i r e c t i o n . I n t e g e r i Z
C p r Z = P h y s i c a l r a n g e in the Z - d i r e c t i o n i n m R e a l p r Z
c - - -
C C O N C E N T R A T I O N O F E G G S / S P H E R E s :
C
C n D m a x = M a x i m u m n u m b e r o f egg d i a m e t e r g r o u p s P a r a m e t e r ( n D m a x = ? )
C n D = A c t u a l n u m b e r o f egg d i a m e t e r g r o u p s i n the r u n I n t e g e r n D
C iD = I n d e x of egg d i a m e t e r g r o u p I n t e g e r iD
C n B m a x = M a x i m u m n u m b e r of egg d e n s i t y g r o u p s P a r a m e t e r ( n B m a x = ? )
C n B = A c t u a l n u m b e r o f egg d e n s i t y g r o u p s i n the r u n I n t e g e r n B
C i B = I n d e x o f egg d e n s i t y g r o u p I n t e g e r iB
C Nall(iZ) = N u m b e r o f e g g s a t time z e r o i n e a c h d e p t h l e v e l i n all C e g g g r o u p s in d i a m e t e r a n d b u o y a n c y . R e a d f r o m f i l e .
R e a l Nall(nZmax)
C N F r a c t i o n = F r a c t i o n o f e g g g r o u p iD/iB a t t i m e z e r o . R e a d f r o m f i l e . R e a l N F r a c t i o n
C N ( i Z , i D , i B ) = C o n c e n t r a t i o n o f e g g s p r . m 3 i n l a y e r i Z , i Z = 1 , n Z C a t t i m e s t e p i T i n d i a m e t e r g r o u p iD a n d b u o y a n c y g r o u p iB
R e a l N ( O : n Z m a x + l , l : n D m a x , l : n B m a x )
C N p l ( i Z , i D , i B ) = V a l u e o f N ( i Z , i D , i B ) a t the n e x t t i m e s t e p R e a l N p l ( O : n Z m a x + l , l : n D m a x , 1 : n B m a x )
C a d v N = C h a n g e i n N ( i Z , i D , i B ) d u r i n g o n e t i m e s t e p d u e to a d v e c t i o n
R e a l a d v N
C d i f N = C h a n g e i n N ( i Z , i D , I B ) d u r i n g o n e t i m e s t e p d u e to d i f f u s i o n
R e a l d i f N
C b t h N = C h a n g e i n N ( i Z , i D , i B ) d u r i n g o n e t i m e s t e p d u e to C l o c a l g e n e r a t i o n (birth).
R e a l b t h N
C d t h N = C h a n g e i n N ( i Z , i D , i B ) d u r i n g o n e t i m e s t e p d u e to C l o c a l d e g e n e r a t i o n (death).
R e a l d t h N
C t o t N = T o t a l a m o u n t o f N p r e s e n t a t t i m e i T + d T R e a l t o t N
C P R O P E R T I E S O F T H E E G G S / S P H E R E S :
C
C E g g V l c = T e r m i n a l r i s e / s i n k v e l o c i t y o f a n e g g d u e to b u o y a n c y
C e f f e c t s i n m/s
R e a l E g g V l c
C E g g S a l ( i D , i B ) = T h e s a l i n i t y w h e r e the egg h a s n e u t r a l b u o y a n c y R e a l E g g S a l ( l : n D m a x , l : n B m a x )
C E g g D i a ( i D , i B ) = T h e d i a m e t e r o f a n egg in m R e a l E g g D i a ( l : n D m a x , l : n B m a x )
C E g g V m e a n ( i D , i B ) = T h e m e a n r i s e / s i n k v e l o c i t y o f the e g g s / s p h e r e s C d i s t r i b u t e d in the i Z , i D , i B s p a c e a t t i m e z e r o .
R e a l E g g V m e a n ( l : n D m a x , l : n B m a x )
C E g g V m a l l = T h e m e a n r i s e / s i n k o f the a b s o l u t e v a l u e o f the v e l o c i t i e s
C o f a l l e g g s / s p h e r e s a t time z R e a l E g g V m a l l
C E g g V m a x = T h e m a x . r i s e / s i n k v e l o c i t y o f the e g g s / s p h e r e s d i s t r i b u t e d
C i n the i Z , i D , i B s p a c e a t time z e r o . R e a l E g g V m a x
C R e y n o l d = R e y n o l d ' s n u m b e r for the egg's a d v e c t i o n R e a l R e y n o l d
c - - -
C T I M E :
C
C n T = N u m b e r o f t i m e s t e p s the m o d e l s h o u l d r u n I n t e g e r n T
C d T = T i m e s t e p i n s R e a l d T
C i T = T i m e s t e p n u m b e r I n t e g e r i T
C p r T = T o t a l time i n s e c o n d s the m o d e l i s a l l o w e d to r u n R e a l p r T
c - - -
C W A T E R A N D A T M O S P H E R E :
C
C Temp(iZ) = T e m p e r a t u r e i n s e a w a t e r i n d e g r e e s C e l s i u s .
C a t d e p t h i Z
R e a l T e m p ( 1 : n Z m a x )
C S e a S a l ( i Z ) = S a l i n i t y o f s e a w a t e r i n 0 / 0 0 a t d e p t h i Z R e a l S e a S a l ( 1 : n Z m a x )
C D e n s O ( S e a S a 1 , T e m p ) = T h e d e n s i t y o f s e a w a t e r i n k g / m 3 a t a t m o s p h e r i c C p r e s s u r e . T h e I n t e r n a t i o n a l E q u a t i o n o f S t a t e o f s e a w a t e r .
R e a l D e n s 0
C K i n M V = K i n e m a t i c m o l e c u l a r v i s c o s i t y i n m2/s R e a l K i n M V
C W i n d S p d = W i n d s p e e d a t the s u r f a c e i n m/s R e a l W i n d S p d
C KV(iZ) = V e r t i c a l t u r b u l e n c e c o e f f i c i e n t i n m 2 / s R e a l KV(0:nZmax)
C K V C = V e r t i c a l t u r b u l e n c e c o e f f i c i e n t i n the u p p e r l a y e r s R e a l K V C
C K F r a c t i o n ( i Z ) = T h e r e l a t i v e v a l u e o f K V C for KV(iZ) R e a l K F r a c t i o n ( n Z m a x )
c - - -
C M I S C E L L A N E O U S :
C
C K V N = A p p r o x i m a t e n u m e r i c d i f f u s i o n c o e f f i c i e n t i n m 2 / s R e a l K V N
C E r r o r = P e r c e n t a g e e r r o r i n the t r a n s i e n t d i f f u s i o n c o e f f i c i e n t . R e a l E r r o r
C D a t a F i l e = N a m e o f f i l e w h e r e the s a l i n i t y . t e m p e r a t u r e a n d e g g d a t a C a t t i m e z e r o i s r e a d f r o m .
C h a r a c t e r D a t a F i l e * 3 2
C S c r e e n = T h e G P G S - F n u m b e r o f the g r a p h i c a l s c r e e n u s e d I n t e g e r S c r e e n
C P l o t t e r = T h e G P G S - F n u m b e r o f the g r a p h i c a l p l o t t e r u s e d I n t e g e r P l o t t e r
C S h o w T i m e = N u m b e r o f h o u r s b e t w e e n e a c h g r a p h i c a l o u t p u t R e a l S h o w T i m e
C I S h w T m e = N u m b e r o f t i m e s t e p s b e t w e e n e a c h o u t p u t I n t e g e r I S h w T m e
C H o u r s = T h e time in h o u r s w h e n t h e r e is S h o w T i m e R e a l H o u r s
C D a y s = T h e n u m b e r s o f d a y s the u s e r w a n t s to r u n the m o d e l R e a l D a y s
C ~ 1 . ~ = 2 N e c e s s a r y c o n s t a n t s i n the c a l c u l a t i o n s . R e a l c l , c 2
C
C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c C
C I N I T I A L I Z A T I O N :
C
C S c r e e n a n d P l o t t e r is the G P G S - F I D o f the T a n d b e r g g r a p h i c a l s c r e e n C a n d the H P - 7 5 5 0 p e n p l o t t e r .
C
S c r e e n = 63
P l o t t e r = 5 8 8 0 D o f o r i Z = l , n Z m a x , l
Temp(iZ) = 0 . 0
S e a S a l ( i Z ) = 0 . 0 E n d d o
D o f o r i Z = O , n Z m a x , l
KV(iZ) = 0 . 0 E n d d o
D o for i Z = O , n Z m a x + l , l D o f o r iD = I , n D m a x , l
D o for iB = 1 , n B m a x . l N ( i Z , i D , i B ) = 0 . 0 N p l ( i Z , i D , i B ) = 0 . 0 E n d d o
E n d d o E n d d o C
W r i t e (*,'("$Give n u m b e r o f e g g d i a m e t e r g r o u p s : " ) ' ) R e a d ( * , * ) n D
W r i t e (*,'('.$Give n u m b e r o f e g g b u o y a n c y g r o u p s : " ) . ) R e a d ( * , * ) n B
W r i t e ( * , ' ("$Give n u m b e r o f d e p t h l a y e r s : ' , ) . ) R e a d ( * , * ) n Z
I f ( n D . g t . n D m a x . o r . n B . g t . n B m a x . o r . n Z . g t . n Z m a x ) S t o p ' I n v a l i d v a l u
&e o f n D , n B or n Z '
C
W r i t e (*,'("$Give n a m e o f f i l e w i t h d a t a : " ) ' ) R e a d ( * , * ) D a t a F i l e
O p e n ( 1 0 . f i l e = D a t a ~ i l e , a c c e s s = ~ r s e q ~ , s t a t u s = ' o l d ' , e r r = 8 8 8 8 8 ) R e a d ( l O , * . e r r = 9 9 9 9 9 ) ( T e m p ( i Z ) . i Z = l , n Z )
R e a d ( l O , * , e r r = 9 9 9 9 9 ) ( S e a S a l ( i Z ) , i Z = l , n Z ) R e a d ( l O , * , e r r = 9 9 9 9 9 ) ( N a l l ( i Z ) , i Z = i . n Z ) R e a d ( l O , * , e r r = 9 9 9 9 9 ) ( K F r a c t i o n ( i Z ) , i Z = l , n Z ) D o f o r iD = l , n D , l
D o f o r iB = l , n B , 1 R e a d
E g g D i a ( i D , i B ) , E g g S a l ( i D , i B ) , N F r a c t i o n D o for iZ = l , n Z , l
N ( i Z , i D , i B ) = Nall(iZ)*NFraction E n d d o
E n d d o E n d d o
W r i t e ( * , ' ("$Give m a x i m u m d e p t h i n m e t e r s R e a d ( * , * ) p r Z
d Z = p r Z / n Z
W r i t e (*,'("$Give w i n d s p e e d in m/s : " ) ' ) R e a d ( * . * ) W i n d S p d
K V C = 7 6 . i E - 0 4 + 2 . 2 6 ~ - 0 4 W i n d S p d W i n d S p d
W r i t e (*,'("$Give the n u m b e r o f d a y s to r u n the m o d e l : " ) ' ) R e a d ( * , * ) D a y s
W r i t e (*,'("$Give h o u r s b e t w e e n s h o w t i m e s : " ) ' ) R e a d ( * , * ) S h o w T i m e
pr^ = ~ a y s * 2 4 * 6 0 * 6 0
E g g V m a x = - 1 0 0 0 0 0 . 0 E g g V m a l l = 0 . 0
D o f o r iD = l , n D , 1 D o f o r iB = l , n B , l
E g g V m e a n ( i D , i B ) = 0 . 0 D o for i Z = l , n Z , l
C a 1 1 S p e e d ( S e a S a l ( i Z ) , T e m p ( i Z ) , E g g D i a ( i D , i B )
& E g g S a l ( i D , i B ) , E g g V l c , K i n M V , R e y n o l d ) I f ( R e y n o l d . g t . 5 . 0 ) t h e n
W r i t e ( * , * ) ' W a r n i n g ! ! R e y n o l d g r e a t e r t h a n 5 . 0 '
P a u s e . P u s h CR to c o n t i n u e ' E n d i f
I f ( E g g V l c . g t . E g g V m a x ) E g g V m a x = E g g V l c E g g V m e a n ( i D , i B ) = E g g V m e a n ( i D , i B ) + E g g V l c E g g V m a l l = E g g V m a l l + Abs(EggV1c)
E n d d o
E g g V m e a n ( i D , i B ) = EggVrnean(iD,iB)/nZ E n d d o
E n d d o
E g g V m a l l = E g g V m a l l / ( n D * n B * n Z ) C
C A c o r r e c t i o n o f the p h y s i c a l d i f f u s i o n is g i v e n to g e t the a p p r o x i m a t e
C s t e a d y s t a t e s o l u t i o n . A t r a n s i e n t n u m e r i c a l d i f f u s i o n i s i m p o s s i b l e C to a v o i d .
C
K V N = 0 . 5 * A b s ( E g g V m a l l ) * d Z
I f ( ( K V C - K V N ) . I t . 0 . 0 0 1 ) Stop' N u m e r i c d i f f u s i o n to b i g . D o for i Z = 1 , n Z - l , l
KV(iZ) = K V C - K V N
I f ( K V ( i Z ) . l t . O ) KV(iZ) = 0 . 0
KV(iZ) = KV(iZ)*KFraction(iZ)
E n d d o C
C A d v e c t i o n / d i f f u s i o n or p u r e d i f f u s i o n . dT is p u t to a q u a r t e r o f t h e C n e c e s s a r y time s t e p for the u p w i n d d i f f e r e n c i n g m e t h o d :
C I d e a l l y K V N s h o u l d be d e l e t e d . C
C d T = 0 . 2 5 * ( 1 . / ( 2 . * ( K V C - K V N ) / ( d Z e d Z ) + (Abs(EggVmax)/dZ))) dT = 0 . 5 * ( 1 . / ( 2 . * ( K V C - K V N ) / ( d Z * d Z ) + (Abs(EggVrnax)/dZ))) C
C T h e a p p r o x i m a t e e f f e c t i v e t r a n s i e n t d i f f u s i o n c o e f f i c i e n t K V N T is : C K V N T = 0 . 5 * A b s ( E g g V m a l l ) * d Z * ( l - C ) ; w h e r e c = A b s ( E g g V m a l l ) * d T / d Z C W h e n K V C is c o r r e c t e d by K V N to g e t the c o r r e c t s t e a d y s t a t e the C a p p r o x i m a t e p e r c e n t e r r o r in the e f f e c t i v e t r a n s i e n t d i f f u s i o n C c o e f f i c i e n t , E r r o r , is :
C
E r r o r = l O O . * ( K V N * A b s ( E g g V m a l l ) * d T / d Z ) / K V C
W r i t e ( * , * ) ' A p p r o x i m a t e p e r c e n t a g e e r r o r i n t r a n s i e n t KV :
' , E r r o r
P a u s e ' P u s h CR to c o n t i n u e '
C
n T = p r T / d T
I S h w T m e = N i n t ( ( S h o w T i m e * 6 0 . * 6 0 . ) / d T ) I f ( I S h w T m e . e q . 0) I S h w T m e = l
c l = d T / d Z
c 2 = dT/(dZ*dZ)
C
C E v a l u a t i o n w i t h T I M E s t a r t s h e r e C
D o f o r iT = 1 , n T , 1 C
C E v a l u a t i o n o v e r E G G G R O U P S s t a r t s h e r e ! ! ! ! ! ! ! ! ! ! ! C
D o f o r iD = 1 , n D , 1 D O for i B = 1 , n B , 1
C
C E v a l u a t i o n w i t h S P A C E i n Z - d i r e c t i o n s t r a t s h e r e ! ! ! ! ! ! ! ! ! ! !
C
C U p w i n d d i f f e r e n c i n g .
C T h e e v a l u a t i o n o f t h e d o - l o o p d i r e c t i o n w a s a l s o n e c e s s a r y at l e a s t C o n the N D - 5 0 0 c o m p u t e r to get s t a b i l i t y .
C
I f ( E g g V m e a n ( i D , i B ) . l e . 0 ) t h e n C
D o f o r i Z = l , n Z , l C
I f ( i Z . e q . 1) t h e n E g g V 1 = 0 . 0
S S = (SeaSal(l)+SeaSal(2))/2 T T = (Temp(l)+Temp(2))/2
C a 1 1 S p e e d ( S S , T T , E g g D i a ( i D , i B ) , E g g S a l ( i D , i B ) , E g g V 2 , K i n M V , R e y n o l d )
E l s e i f ( i Z . e q . n Z ) t h e n
S S = ( S e a S a l ( n Z - l ) + S e a S a l ( n Z ) ) / 2
T T = (Temp(nZ-l)+Temp(nZ))/2
C a 1 1 S p e e d ( S S , T T , E g g D i a ( i D , i B ) , E g g S a l ( i D , i B ) , E g g V l , K i n M V , R e y n o l d )
E g g V 2 = 0 . 0 E l s e
S S = ( S e a S a l ( i Z - l ) + S e a S a l ( i Z ) ) / 2
T T = (Temp(iZ-l)+Temp(iZ))/2
C a 1 1 S p e e d ( S S , T T , E g g D i a ( i D , i B ) , E g g S a l ( i D , i B ) , E g g V 1 , K i n M V . R e y n o l d )
S S = ( S e a S a l ( i Z ) + S e a S a l ( i Z + 1 ) ) / 2 T T = ( T e m p ( i Z ) + T e r n p ( i Z + 1 ) ) / 2
C a l l S p e e d ( S S , T T , E g g D i a ( i D , i B ) , E g g S a l ( i D , i B ) , E g g V 2 , K i n M V . R e y n o l d )
E n d i f C
a d v N = c l * ( E g g V 2 * N ( i Z , i D , i B ) - E g g V l * N ( i Z - l , i D , i B ) ) C
C C e n t r a l d i f f e r e n c i n g in Z f o r d i f f u s i o n ( D a g S l a g s t a d p e r s . c o m m ) : C
d i f N = c2*( KV(iZ) ( N ( i Z + l . i D , i B ) - N ( i Z , i D , i B ) ) KV(iZ-1) * ( N ( i Z , i D , i B ) - N ( i Z - l , i D , i B ) ) ) b t h N = + 0 . 0
d t h N = - 0 . 0
N p l ( i Z , i D , i B ) = N ( i Z , i D , i B ) + a d v N + d i f N + b t h N + d t h N E n d d o
E l s e
D o f o r i Z = n Z , l , - l I f ( i Z . e q . l) then
E g g V 1 = 0 . 0
C a 1 1 S p e e d ( S e a S a l ( 2 ) , T e m p ( 2 ) , E g g D i a ( i D , i B ) ,
& E g g S a l ( i D , i B ) , E g g V 2 , K i n M V , R e y n o l d )
E l s e i f ( i Z . e q . n Z ) then
C a 1 1 Cp e e d ( S e a S a l ( n Z - l ) , T e m p ( n Z - l ) , E g g D i a ( i D , i B ) ,
& E g g S a l ( i D , i B ) , E g g V l , K i n M V , R e y n o l d )
E g g V 2 = 0 . 0 E l s e
S S = ( S e a S a l ( i Z - l ) + S e a S a l ( i Z ) ) / 2 T T = (Temp(iZ-l)+Temp(iZ))/Z
Gall S p e e d ( S S , T T , E g g D i a ( i D , i B ) , E g g S a l ( i D , i B ) , E g g V l , K i n M V , R e y n o l d )
S S = ( S e a S a l ( i Z ) + S e a S a l ( i Z + l ) ) / Z T T = (Temp(iZ)+Temp(iZ+l))/Z
C a 1 1 S p e e d ( S S , T T , E g g D i a ( i D , i B ) , E g g S a l ( i D , i B ) , E g g V Z , K i n M V , R e y n o l d )
E n d i f
S h o w R e s u l t s KV(iZ-1) * ( N ( i Z . i D , i B ) - N ( i Z - l , i D , i B ) ) ) b t h N = + 0 . 0
d t h N = - 0 . 0
N p l ( i Z , i D , i B ) = N ( i Z , i D , i B ) + a d v N + d i f N + b t h N + d t h N E n d d o
E n d i f E n d d o E n d d o C
C P r e s e n t a t i o n i f S h o w T i m e : C
I f ( i T . e q . 1 . o r . m o d ( i T , I S h w T m e ) . e q . O) t h e n H o u r s = d T * ( i T - 1 . ) / 3 6 0 0 .
t o t N = O .O
D o for i Z = l , n Z , l Nall(iZ) = 0 . 0 D o f o r iD = l , n D , l
D o for i B = l , n B , l
Nall(iZ) = Nall(iZ) + N ( i Z , i D , i B ) * d Z t o t N = t o t N + N ( i Z , i D , i B ) * d Z
E n d d o E n d d o E n d d o C a l l
( N a l l ( l ) , n Z , H o u r s , t o t N , p r Z , S c r e e n , P l o t t e r ) E n d i f
C
C M a k e s r e a d y for a n e w time s t e p : C
D o f o r i Z = l , n Z D o for i D = l , n D
D o f o r i B = l , n B
N ( i Z , i D , i B ) = N p l ( i Z , i D , i B ) E n d d o
E n d d o E n d d o C
E n d d o C
S t o p ' N o r m a l t e r m i n a t i o n o f the s e s s i o n . G o o d B y e ! ! ! '
88888 S t o p E r r o r w h e n o p e n i n g d a t a f i l e '
99999 S t o p ' E r r o r w h e n r e a d i n g o n g i v e n d a t a f i l e . C
E n d C
C * * S h o w R e s u l t s * * C
S u b r o u t i n e S h o w R e s u l t s ( A r r a y , N o I n A r r a y . H o u r s , C h e c k S u m m L e n g t h ,
& S c r e e n , P l o t t e r )
