Chapter I
The Mathematical Model
by
Jacques C.J. NIHOUL
Foreword
This chapter is a synopsis of the mathematical model which has heen
the framework of the theoretical and experimental research conducted in
the last three years in the scope of the Belgian National Program on the
Environment — Sea Project'.
The successive steps in the construction of the Model have been
described in several progress reports and published papers and the details are not reproduced here. A list of references is attached.
A detailed description of the modern concepts and techniques of
marine modelling can be found in }'^odelling
of Mavine Systems,
edited by
Jacques C. J. Nihoul, Elsevier Publ. Amsterdam, 19T^«
The ordering of this chapter closely follows that of the book ajid
the same notations are used.
1. Sponsored by the Ministry for Science Policy, Belgium.
10 -
53° N
-i5^°N
^1
j .
1 . 1 .
-
11 -
1.- Demarcation of the system
Support
The support of the model is the Southern Bight of the
and the Scheldt Estuary as shown in figure 1.1.
The array of broken l i n e s , dots and squares shows the
measurements are made several times a year. Data concerning
part of the Bight are provided by British colleagues in the
general program of exchanges and i n t e r c a l i b r a t i o n s .
North Sea
area where
the westera
scope of a
The area in front of the Belgian coast and the Scheldt Estuary are
covered by a finer grid of measurements not reproduced on the figure.
The hydrodynamical part of the model i s extended t o the whole
North Sea whenever necessary t o provide appropriate Northern boundaiy
conditions for the Southern Bight.
Saope
The primary objective of the model is the study of the space and
time characteristics of water motion, surface elevation, temperature,
salinity,turbidity, oxygen, nutrients, trace metals, chlorinated hydrocarbons and living species \
The situation is summarized in figure 1.2.
Reduction of scope — Compartments
Attention is restricted to the most important compartments :
i) dissolved substances,
ii) suspensions,
iii) plankton — phytoplankton,
zooplankton,
iv) fish — pelagic,
benthic,
v) shellfish and molluscs,
vi) bottom sediments.
1. Petroleum products are examined in a separate program.
- 12 -
l\ Nutrients
Plankton
Biomass
Trace
Metals
fig. 1.2.
-
13 -
v i i ) macro- and meio-benthos ,
v i i i ) b a c t e r i a — faecal,
marine h e t e r o t r o p h i c ,
benthic.
The s t a t e variables include the specific massief each compartment
and the aggregate concentrations of the selected chemicals in compartments (i) to ( v i ) .
§êiê£ÎS§_î}yïriSSΧ • -^ » ^^ > Si .
§ÊiS5!Ëê^_^ʧ}!Z_5ê"t§lË : Zn ,
Cd ,
§êl^ÇΧ^_£hlorinateà_hjrdrocarbons
Pb ,
:
Cn ,
Fe ,
Mn ,
Mg .
pp' DDT , DDD , DDE , a l d r i n e ,
d i a l d r i n e , endrine, l i n d r a n e , heptachlore, epoxid heptachlore, PCB .
It niust L^e emphasized
these
v a r i a b l e s , not every
present
state, experimental
trations
hierarchy
most
that, although
data
them
are
in all coir,, ar tmpn t s at ail
difficulties
a
here
one of
but
of
..rgently
it i s , in most
priorities
needed
in
the m a t h e m a t i c a l
is actually
not
always
sampling
analysed
available
work
of
and
covers
on all c h e m i c a l
s t a t i o n s . This
cases, a consequence
the e x p e r i m e n t a l
model
experimentally.
may
be due
the n e c e s s i t y
focusing
of
all
the
concen-
to
attention
In
technical
accepting
to what
is
(see chapter 2 ) .
The specific masses of compartments ( i ) and ( i i ) may be called
" s a l i n i t y " and " t u r b i d i t y " respectively; the d i s t i n c t i o n between s a l i n i t y
and t u r b i d i t y being, in a sense, a r b i t r a r y as the experimentalist w i l l
c a l l dissolved everything which goes through a f i l t e r of pre-decided
fineness. This must be borne in mdnd also while i n t e r p r e t i n g the budgets
of n u t r i e n t s and pollutants in the two compartments.
Reduction
of support — Space
averaging
All over the support except in the Scheldt Estuary the turbvilent
mixing ensures a uniform density.
While separate three-dimensional and combined depth-averaged and
width averaged two-dimensional models are being developed for the Scheldt
Estuary, the model for the Southern Bight i s fxirther simplified by a s suming constant mass density and considering only average properties over
the t o t a l depth.
1.
Mass p e r
unit
volume.
- Il; -
The mechanical variables are then reduced to the two horizontsil
components of the depth-averaged velocity vector and the surface elevation.
For the study of specific chemical and ecologiceil i n t e r a c t i o n s , the
support may be divided in a limited number of niahes where s i m i l a r conditions p r e v a i l . In a f i r s t approach, one may study the dynamics of the
aggregate properties of the niche obtained by further i n t e g r a t i n g
(averaging) over the horizontal dimensions of the niche. Niche (or box)
models of t h i s sort have been developed and tested against the experimental observations made in the Ostend "Bassin de Chasse"^ a closed sea
basin at the coast which has been extensively studied in the p a s t .
2.- State variables and control parameters
The s t a t e variables are defined by the scope of the system as
s t a t e d above.
As a r e s u l t of turbulence and other e r r a t i c or rapidly o s c i l l a t i n g
motions of the sea, each variable shows fluctuations around a mean value.
Only these mean values are significant for the model.
The evolution equations are written for the mean variables and only
the general effect of the fluctuations (through non-linear terms in the
basic equations) i s taken i n t o account.
The mean v a r i a b l e s , i.e. the smooth running fijnctions of space and
time, obtained by f i l t e r i n g out the fluctuations are denoted by special
symbols. A bar over the symbol indicates the average over depth of the
mean v a r i a b l e .
In addition to the mechanical variables and temperature, the other
s t a t e variables represent e s s e n t i a l l y specific masses or concentrations.
(The specific biomass of phytoplankton, the specific mass of copper in
s o l u t i o n , suspension or plankton, e t c . ) These "concentration" variables
are noted p^ (smooth running part r „ ) . For convenience p^ (r^) w i l l
be referred t o as the specific mass of "constituent a"; the word cons t i t u e n t being used in a conventional sense as i t may denote a whole
aggregate ( a l l dissolved substances . . . ) or the aggregate content of a
- 15 -
compartment in a specific chemical (concentration of mercury in pelagic
fish . . . ) •
The mean (smooth running) variables are noted as follows :
depth- ave rage d
Velocity vector
U
IT
(three dimenaional) (two dimensional)
Flow rate vector
U = H ÏÏ
Surface elevation
Ç
Total height of water
(where h i s the depth)
Temperature
Specific mass of
a
H = h + Ç
e
ê
r»
Ta
"In addition t o the s t a t e v a r i a b l e s , different kinds of parameters
appear inevitably in the mathematical description of the system. These
may be called aontvol parameters as they influence the evolution of the
system (hence appear in the evolution equations) but are not predicted
by the model i t s e l f (no specific evolution equation i s written for them).
The f i r s t kind of control parameters one thinks of are the guidanae
parameters which are at the disposal of man to manage the marine system
according t o some optimal design.
Most of the parameters, however, which control the evolution of the
system cannot be chosen to conform with man's concern. They are imposed
by Nature. These parameters a r i s e from the i n i t i a l demarcation of the
system, the necessity of r e s t r i c t i n g the s t a t e variables and formulating
the laws of t h e i r evolution in a simple and t r a c t a b l e way. They r e f l e c t
a l l the aspects of the n a t u r a l system, of which the model does not take
charge; usually because the additional equations required for t h e i r p r e diction would jeopardize the simulation by t h e i r d i f f i c u l t y , t h e i r dubiousness or simply by increasing the size of the system beyond the
computer's a b i l i t y .
Although they are rarely known beforehand and must b e , in most
c a s e s , determined approximately by separate models, experimental data
or sideways t h e o r e t i c a l r e f l e c t i o n , the control parameters which resxilt
- 16 -
from the closure of the system must be regarded, in the language of the
theory of control, as fixed and d i s t i n c t , therefore, from the guidance
parameters mentioned above.
The separation between s t a t e variables and control parameters i s ,
of course, more or less a r b i t r a r y and function of the model's capability
and ambition.
For instance, a l l models of primary productivity ( s t a t e variables :
nutrients and plankton) are controlled by the incident l i g h t . In a f i r s t
s t a g e , the incident l i g h t may be taken as a fixed control parameter and
be given an empirical value. The model can be refined and give the i n cident l i g h t at every depth as a function of the i n t e n s i t y of l i g h t at
the sea surface, using the transparency of water as a new control parameter. In an even more perfect version of the model, the transparency
of water can be included in the s t a t e variables and inferred from the
t u r b i d i t y which i t s e l f can be predicted by the model.
The chemical reaction rates may be regarded as control parameters
hopefiilly determined by chemical k i n e t i c s , i.e. by laboratory experiments
or by some fundamental molecular theory conducted in p a r a l l e l with the
model but not part of i t .
The dynamics of translocations
( t r a n s f e r of a chemical element from
one compartment t o another) must be given appropriate mathematical form.
This cannot be done in general without introducing several control parameters the values of which can only be ascertained experimentally." *
To ascertain the value of the control parameters and determine the
laws of i n t e r a c t i o n , complementary variables are measured or calculated.
These include, for instance, transparency of water, primary production,
diversity and s t a b i l i t y indexes, a n t i b i o t i c e f f e c t s , e t c .
1. The text between quotation marks is reproduced from Modelling of Marine Systems
edited by Jacques C.J. Nihoul, Amsterdam, Elsevier Publ., 1974.
-
17 -
3 . - Evolution equations
The assumption of uniform sea water density which i s j u s t i f i e d by
experiment (apart from the Scheldt Estuary where s l i g h t l y different
models are being developed) allows a decoupling between the mechanical
variables and the others. The hydrodynamic equations can be solved independently of the other evolution equations and the values of the flow
velocity determined by the former s u b s t i t u t e d in the l a t t e r which in
turn can be solved knowing the i n t e r a c t i o n s between the c o n s t i t u e n t s .
The hydrodynamic models which one can develop in t h i s way differ
whether one i s interested in unsteady sea motions produced by t i d e s and
storm surges or in the residual — "steady" — circulation which r e s u l t s
from the average of the actual flow over a time sufficiently long to
cancel out t i d a l o s c i l l a t i o n s and t r a n s i t o r y wind currents.
The evolution equations for the s t a t e variables Ta are coupled
through the terms expressing the i n t e r a c t i o n s of the constituent a
with other constituents 6 , y , ... They also depend on the water
motion determined by the hydrodynamic models.
In a f i r s t approach, i t i s rewarding t o separate the two effects
and study f i r s t the dispersion (by c u r r e n t s , sedimentation and turbulence) of a "passive" constituent i.e. one which does not have s i g n i f i cant interactions with others. Then, t o i n v e s t i g a t e the i n t e r a c t i o n s
{e.g. the path of a pollutant i n the food chain), one can develop niche
models concerned with mean concentrations over some reasonably homogeneous regions of space. Niche models are not affected by the d e t a i l e d
hydrodynamics of the sea, only i t s effects on imputs and outputs at the
niche's frontiers remain t o be known.
Although passive dispersion models can only give conservative e s timates of the d i s t r i b u t i o n of chemicals and species in the sea and
niche models can only give an average knowledge of the i n t e r a c t i o n s ,
they provide nevertheless a f i r s t valuable insight into the mechanisms
of the marine system.
Of course, the model can combine dispersion and interactions and
provide the detailed prediction of the s t a t e of the system at a l l points
and time.
- 18
However the gigantic amount of computer work which is required to
solve large systems of coupled partial differential equations commends
that the full scale simulation be restricted to dramatic cases where estimates are insufficient or to special problems (like the dimpings) where,
in the area of interest, only a limited number of constituents are involved in a significant way.
The general dispersion-interaction model is described in [Nihoul
(1973a)] and subsequent papers. Different simplified forms of this model,
pertinent to special studies, are given in the following and illustrated
by examples of application.
4.- Tides and storm surges model
i)
State_yariables
Mean depth-averaged h o r i z o n t a l velocity vector
H
Water height
H
(The surface elevation
ç
i s given by
H= h + ç
where
h
i s the water
depth.)
ii)
Extemal_forces_ger_unit_mass_of_sea_water
Tide-generating force
E
Gradient of atmospheric pressure
V(—)
Wind s t r e s s
Ts
Relation of wind s t r e s s t o wind velocity
at reference height
iii)
V
s
= — v llvl
H
Evolution_eq.uations
3H
3^ + V . ( H U )
=0
• | ï + U . V U + f e , A U = g - V(-â- + gç) + a v^
dt
-^
p
- | u
llull + l
V llv
- 19 where the
where
e^
and e^
V = e
axes are h o r i z o n t a l , the
3
1 3x
. _
+ e2
e ^ - a x i s v e r t i c a l and
3
dXr
iV) Control_garameters
f
:
Coriolis parameter (twice the v e r t i c a l component of the angul a r velocity of the earth)
p
:
specific mass of the sea water
a
:
horizontal effective viscosity
D :
bottom f r i c t i o n
C :
atmospheric drag coefficient
h
depth
:
coefficient
The time variation of the velocity vector
a
3
Y
5
:
advection
U
is the result of
U . VU
rotation
f e^ ^ u produced by the Coriolis effect in
axes fixed on the r o t a t i n g earth
. .
2
mixing by shear effect and turbulence
a Vu
friction on the bottom
- — ÏÏ ||1J||
n
e
acceleration by agents of three different types
e.1
the wind s t r e s s on the sea surface
the wind velocity
e.2
related to
V at some reference height
n
the gradient of the atmospheric pressure and of the surface
elevation
e.3
— V ||V
,Pa
• V(^+
gç)
the external force ^ . The type of external force one has
in mind here i s e s s e n t i a l l y the tide-generating force which
i s generally assumed to derive from a p o t e n t i a l ,
{i.e.
V A g = O). S can then be combined with the pressure and
surface elevation gradients.
- 20 -
5 . - nodel of r e s i d u a l
i)
circulation
State_yariables
Stream function
ij;
The two components of t h e r e s i d u a l flow r a t e v e c t o r
U
= - ^
U
= ^ ^ .
0,1
3x2
0,2
ii)
UQ a r e given by
3x^
External_forces_2er_imit_mass_of_sea_water
Residual s t r e s s
where
(Tg)Q
ô
i s t h e r e s i d u a l wind s t r e s s and
(Tt)o
the residual t i d a l
s t r e s s [Nihoul (197^)3
(T,)o
where
ç^
= [gç^
Vç^
+ V.
(H"^UU)]O
denotes t h e s u r f a c e e l e v a t i o n produced by t i d e s and t r a n s i t o r y
wind f o r c e s .
iii)
Stead2;_state_residual_e2.uation
dX^
dXj
h
dX^
3X2
^^1
^
^^2
_, 3h = ^ «^3 + i l ^ ^
where
«2
%^ and
^2
a r e t h e two h o r i z o n t a l components of
i s t h e v e r t i c a l conrponent of
VAô .
i v ) Çontrol_£arameters
K:
:
bottom f r i c t i o n c o e f f i c i e n t
f
:
Coriolis parameter
h
:
depth.
for r e s i d u a l flow
ô
3h
- ^
„
^2
and where
- 21 -
The space distribution of the stream function
ip is the result of
a
effects
:
B :
combination of bottom slope and Coriolis
combination of bottom slope and bottom friction
effects
_ â l rd±_ 9h_ _!_ ^±_ 9h -•
h '-9x^ 9x^
9x2 9X2
Y
: combination of bottom slope and residual stress effects
1 3X2
^ 3^1
residual stress forcing
h
(JO,
.
6.- Passive dispersion models
i ) State_variables
Depth-averaged concentration of any passive constituent
or depth-averaged temperature
(c = r„
or
a
—
c = e)
Depth-averaged horizontal velocity vector
(given by separate hydrodynamic model)
Water height
(given by separate hydrodynamic model)
—.
„
i i ) Inputs — Outputs
Total input in a water column of unit base
(including volume sources, surface and bottom
fluxes. If these result in a net output, A
is negative.)
HA
- 22
i i i ) E v o l u t i o n _ e a u a t i o n [Nihoul (l9T3h)]
1 ^ + ÏÏ . V7 + H""" V .
(YP
aXi
^
^ r ^ c ÏÏ) = A +
H"^
V . [y. ^ ÏÏ (ÏÏ . VÏÏ)]
"U.
Il
+ V . K Vc
iv)
Çontrol_£arameters
Oj
:
migration (sedimentation or ascension)
a, = 0
velocity
f o r t e m p e r a t u r e and n e u t r a l l y buoyant con-
stituents
K
M
'
:
h o r i z o n t a l eddy
:
shear effects coefficients
,
diffusivity
[Nihoul ( 1 9 7 1 ) , ( 1 9 7 2 ) ,
( I 9 7 3 h ) , (197^+)].
v) I n t e r p r e t a t i o n of t h e e v o l u t i o n e q u a t i o n
The e v o l u t i o n i n time of t h e depth-averaged v a r i a b l e
c" i s t h e
r e s u l t of
a
:
a d v e c t i o n by t h e depth-ave raged v e l o c i t y
ÏÏ.
V'c
3
:
s h e a r e f f e c t c o r r e c t i o n t o t h e advection t a k i n g i n t o account
t h a t , as a r e s \ i l t of m i g r a t i o n , t h e maxim\am of
c
may occur
i n a region of t h e w a t e r column where t h e a c t u a l h o r i z o n t a l
velocity is significantly different
from
ÏÏ
[Nihoul (1973b),
(I97i+)]
H
V.
(Y,
'2
—=:^ c U)
u
s h e a r e f f e c t d i s p e r s i o n [Nihoul ( l 9 7 l ) , ( 1 9 7 2 ) , ( 1 9 7 3 b ) ,
(197^)]
H ' V.
[Y^ ^ Ï Ï
(IT. TcV
tiirbulent dispersion
V . K Vc
e x t e r n a l inputs (or outputs)
A
- 23 -
7 . - Niche i n t e r a c t i o n s
model
i ) State variables
Averages over the whole niche's space of
a l l i n t e r a c t i n g chemical and ecological
s t a t e variables ra or 6
i i ) Inputs — Outputs
Total input (output i f negative) in the
niche (including volume sources and fluxes
or flows in and out of the niche at the
boundaries)
i i i ) Evolution_equations
ds
~ ^o
dt
"*" l a ' * ' ^ 1 ' ^ 2 > ' ' * ' ^ n ^
la represents the r a t e of production (or destruction) of s^
by chemical, biochemical or ecological i n t e r a c t i o n s . In general I^
i s a function of time axid a l l i n t e r a c t i n g variables s^ . I„ depends
on the p a r t i c u l a r i n t e r a c t i o n s involved. In many cases, i t can be
simply approximated by combinations (in sums and products) of simple
laws such t h a t
(constant)
a.
k
b.
^a(3
c.
^o/S7 ^0
d.
^ 1 Sa - k ^
e.
where the
a
U
'
k^ ,
(linear)
^P
(bilinear)
S7
2
S^
^0
k , -H S ,
k^^
and k^^^
(logistic)
(idchaeles-Mentem-Monod)
are functions of time and control para-
meters .
iv) Çontrol_parameters
Several control parameters influence the i n t e r a c t i o n laws and appear
in p a r t i c u l a r in the expressions of the coefficients k^ , k^^ , . . .
- 2U -
In some cases, i t i s simpler t o consider these coefficients as r e s u l t i n g
control parameters to be determined experimentally.
v)
Istergretation_of_the_eyolution_eguation
The niche-averaged value of the s t a t e variables
r„
or
6
changes
in time as a r e s u l t of
a
:
input or outputs in or out of the niche
S^
3
:
chemical, biochemical or ecological interactions
Ij, .
8.- Examples of applications
8 . 1 . - Tides_aiid_storm_surges_moàel
The model has been applied with success by Ronday (1973) to the
calculation of t i d e s in the Korth Sea and in the Southern Bight.
Figures 1.3 and 1.U show a comparison between lines of equal phases
and amplitudes according t o observation and according t o simulation.
8 . 2 . - Residual_çirçulation_model
The model has been applied with success by Ronday (1972), Runfola
and Adam (1972), Nihoul and Ronday (197^) t o the calciilation of the r e sidual c i r c u l a t i o n in the North Sea and in the Southern Bight.
Figure 1.5 shows the residual flows of water masses in the North
Sea estimated from observation.
Figure 1.6 shows the calculated stream lines in the North Sea with
the assumption of constant depth.
Figiire 1.7 shows the calciilated stream lines in the North Sea
taking the depth variations i n t o account and demonstrating the influence
of the bottom slope on the residual c i r c u l a t i o n .
Figure 1.8 shows the calculated stream lines in the Southern Bight
when the t i d a l s t r e s s i s neglected.
25 -
Jeo"
50°
fiL,.
1.3.-
LiTS
'if
^qual
b'trvatiors
tiaal
(after
ti'^ast'^
frot-lfrd-
ani
ar 1
anol'tuclfs
)o
i'
f-e" " ^ - f -
J>;CH , 1 9 2 J .
Sf'a
aerc-dinj
to
- 26 -
fig.
1.4.- Lines of
equal
tiddl
the m a t h e n a t i c a l
phases
model
and
(after
amplitudes
Ronday,
in
1973).
the ISiorth S< a a c c o r d i n g
to
- 27 -
J
fiy.
1.5.-
Water
rrassPs
in
the
North
sea
accordinj
to
Laevastu
(1963).
150'
- 28 ;\
\'
\
\
- -
" /
/
- \
///
fig. 1.6.
Residual streamlines
V " const
in the North
Sea calculated, assuming constant depth (after
Ronday, 1 9 7 2 ) . The actual values of ii
are
10
times the indicated figures.
- 29 -
fia.
1.7.
rteoi-ual
'jtreamiines
1^1= c c n ^ t
ir t h e \ o r t h
Sea ( a f t t r R o n d a y , 1 y ' ' 2 ) . Thi> a c t u a l v a l u e s
cf
1^ a r e
10
tirn s t h t i n d i c a t e d f i q u r o s .
- 30 -
fig. 1.8.- Residual circulation in the Southern Bight «ithout tidal stress.
Streamlines
^ - const
(in 10^* m ^ / s ) .
(After Nihoul and Ronday, 1974),
fig. 1.9.- Residual circulation in the Southern Bight with the tidal stress.
Strealines
V» - const
(in 10^ m ^ / s ) .
(After Nihoul and Ronday, 1 9 7 4 ) .
31 -
Figure 1.9 shows the calculated streamlines in the Southern Bight
taking the t i d a l s t r e s s (calculated from the t i d a l model) i n t o account
and demonstrating i t s cogent influence on the residual c i r c u l a t i o n .
8.3•- Passive_disgersion_model
The model has been applied with success by Nihoul (1972) and by
Adam and Runfola (1972) t o the determination of the dispersion p a t t e r n
subsequent t o a dye release or a dumping.
/I N
1 km
fig. 1.10.
Simulation of a dye-release experiment in the North Sea
[after Adam and R u n f o U a (1972)].
Position : 51"20' N , 1°34' E .
Curves : 1/50 of initial central concentration
72 h - 96 h - 108 h after release.
48 h -
- 32 -
Figure 1.10 illustrates the influence of the shear effect on the
anisotropy of the patch of dye.
51°55i
51°50'
2°35
fig. 1.11.
Comparison between observed and predicted shape of a patch of rhodamine B
68 hours after release.
The experimental curve is the irregular curve drawn by Talbot (1970) [The
broken piecewise straight lines are the ship's trajectories]. The theoretical curve is the regular ellipse predicted by the simplified model
[Nihoul (1972)].
Figure 1.11 snows a comparison "between the o"bserved and predicted
shape of a patch of rhodaunine B, sixty-eight hours after r e l e a s e . The
t h e o r e t i c a l curve is calculated using a simplified version of the model
which f i t s the best e l l i p s e to the t i d a l velocity vector diagram.
- 33 -
8. U. - Ni che_interactions_mod.el
The model has been a^oplied with success by P i c h o t and Adam t o t h e
s t u d y of chemical, b i o c h e m i c a l and e c o l o g i c a l i n t e r a c t i o n s i n t h e Ostend
Bassin
de Chasse
Figures 1.12,
1.13 and l . l U show t h e e v o l u t i o n of s i x
different
forms of phosphorus :
X.
d i s s o l v e d phosphate i n s e a w a t e r ;
Xg
r
-
X3
d i s s o l v e d phosphate i n i n t e r s t i t i a l w a t e r ;
-
X4
P
in non-living matter in suspension;
i n bottom s e d i m e n t s ;
X5 P
in plankton;
Xg
in benthos.
P
They i l l u s t r a t e t h e cogent i n f l u e n c e of t h e n o n - l i n e a r terms i n
t h e i n t e r a c t i o n laws on t h e e x i s t e n c e of a s t e a d y s t a t e and t h e time
n e c e s s a r y t o reach i t .
References
NIHOUL, J . C . J . , (1971). P r o c . Worth Sea Science Conf., Aviemore, S c o t l a n d ,
1, 89.
(1972). Bull.
Soa. So. Lg^ 10^ 521.
(1973a). Mem. Soc.
So. Lg, 4^ 115.
(1973b). P r o c . Second lUTAM-IUGG Symposium on T u r b u l e n t
Diffusion i n Environmental P o l l u t i o n , C h a r l o t t e s v i l l e , U . S . A . , 1973.
(197^). Modelling
of Marine Systems^
Elsevier Publ.,
Amsterdam.
RONDAY, F . C . , (1972). Modèle mathématique pour l ' é t u d e de l a c i r c u l a t i o n
r é s i d u e l l e dans l a mer du Nord, Marine Science Branch.j
Manuscript
Report Series,
N° 2 7 , Ottawa.
(1973). Modèle mathématique pour l ' é t u d e de l a c i r c u l a t i o n
due à l a marée en mer du Nord, Marine Science Branch.,
Manuscript
Report Series,
N° 2 9 , Ottawa.
3k
10
10
'<5
10
^6
10
fig. 1.12.
Evolution of six different forms of P in a closed sea basin assuming completely
linear interactions [after Adam (1973)].
40
120
-I
J
40
120
80
40
I
I
80
I
1
L_
120
fig. 1.13.
Evolution of six different forms of P in a closed sea basin assuming quadraticbilinear interactions Lafter Adam (1973)].
- 36 -
xi
40
80
120
40
80
80
120
40
80
120
-J
40
.
L_
120
•^6
-1
40
80
120
40
u_
60
120
fig. 1.14.
tvolution of six different forms of P in a closed sea basin assuming strongly
non-linear interactions [after Adam (1973)].
- 37 -
Programme n a t i o n a l s u r l ' e n v i r o n n e m e n t physique e t b i o l o g i q u e — P r o j e t
mer. P r o g r e s s Reports
IJ 1 NIHOUL, Mathematical Model for t h e Study of Sea P o l l u t i o n .
N2
NIHOUL, A p p l i c a t i o n of Averaging Techniques t o t h e Mathematical
Modelling of Sea P o l l u t i o n .
W3
NIHOUL, Mathematical Model, P r o c . ICES, Meeting on P o l l u t i o n i n t h e
North Sea, Lowestoft, March 2 5 - 2 6 , 1971.
N h
RONDAY, Etude de l a d i s p e r s i o n d'un p o l l u a n t en mer du Nord.
N 5
NIHOUL, Shear Effect and Eddy Diffusion i n t h e Southern North Sea.
N 6
NIHOUL, Ecosystemology a p p l i e d t o Sea P o l l u t i o n .
N 7
FRANKIGNOUL, P r e l i m i n a r y r e p o r t on A i r - S e a I n t e r a c t i o n .
N 9
ADAM-RUNFOLA, Numerical R e s o l u t i o n of Diffusion
Equation.
N 10 NIHOUL, Mathematical Models.
N 11 NIHOUL, A simple Model of Entrainment and Turbulent Diffusion
a Density I n t e r f a c e i n t h e Ocean.
across
N 12 FRANKIGNOUL-STRAIT5 Coherence and v e r t i c a l Propagation of Highfrequency Ener[p/ i n t h e Deep Sea.
N 13 RONDAY, Etude de l a c i r c u l a t i o n r é s i d u e l l e en mer du Nord au mois
de j a n v i e r .
N 14 ADAl^l-RUNFOLA, Numerical Methods for t h e Computation of Shear E f f e c t
Diffusion.
N 15 RUNFOLA-ADAM, R e s i d u a l and Wind-driven C i r c u l a t i o n i n t h e Southern
Bight.
N 16 DUBOIS, A new S t a t i s t i c s of Ecosystems.
N 17 RONDAY, Modèle mathématique pour l ' é t u d e de l a c i r c u l a t i o n due à l a
marée en m.er du Nord.
N 19 PICHOT-HECQ, E x p l o i t a t i o n des données au p o i n t f i x e
lations croisées.
M06 p a r c o r r é -
N 20 NIHOUL, Diffusion of T u r b i d i t y by Shear Effect and Turb\ilence i n
t h e Southern Bight of t h e North Sea.
N 21 DUBOIS, D i v e r s i t y of s p a c e - t i m e dependent f l u c t u a t i n g
of S p e c i e s .
Distributions
- 38 -
N 22 I
II
DROISSART-SMITZ, A p p l i c a t i o n s du c a l c u l a t e u r h y b r i d e aux
problèmes é c o l o g i q u e s .
LEROY, Modèles n o n - l i n é a i r e s d ' i n t e r a c t i o n s
écologiques.
H 23
GENET, Développement économique optimal de r é g i o n s
a f f e c t é e s p a r une p o l l u t i o n t r a n s f r o n t i è r e .
frontalières
N 2I+
DUBOIS, The Behaviour of t h e Index of F l u c t u a t i o n s
V o l t e r r a - L o t k a Model.
D
N 25
NIKOUL, I n t e r a c t i o n s ax t h e Sea Boundaries as a Handicap t o
Modelling.
N 26
LAMBERMONT-LEBON, On a new D e r i v a t i o n of t h e Erosion Flux induced
by a Tiirbulent Flow.
in the