DIFFUSION
OF
10MENTUT4 AND SALIHITY
IN
LAKE MARACAIBO,
VENIEZUELA
by
JOHN ROSS YEARSLEY
S.B., Massachusetts Institute of Technology
(1958)
M.S.,
University of Washington
(1966)
SUBMITTED IN PARTIAL FULFILLM1ENT OF THE REQUIREENTS FOR
THE DEGREE OF MASTER OF SCIENCE
at the
MASSACHUSETTS INSITUI'E OF TECHNOLOGY
May 1968
Signature
of
Auth6r.
.
.
. ,
,
"
.-.
%-,, -.
.
.
.
..
[Department oi Geologya Geophysics
A
(May 23, 1968)
Certified by . . . . . .
. . -,...
....
*
C*/
Accepted by.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Thesis Supervisor
.
, .
.
.
.
.
.
. .
.
Chairman, Departmental Cormittee
on Graduate Studonts
Lindgren
WIT
(iqOM 1.\
6
.t9.
MITRAJinoAmlE
(ii)
ABSTRACT
DIFFUSION OF MOM101,
UM AND SAL-ITfCY
IN
LAKE MARACAIBO, VENEZUELA
by
John Ross Yearsley
Submitted to the Department of Geology and Geophysics
on May 23, 1968 in partial fulfillment of the requirements
for the degree of Master of Science
A theoretical model of the time-dependent motion in a
rotating stratified fluid, driven by a wind stress applied at
the surface, is developed. Theoretical expressions for the
radial, azimuthal and vertical velocities, and salinity are
obtained.
The steady-state velocity and salinity profiles predicted
by the theory are compared to the results of a field survey
made in Lake Maracaibo by Redfield et al (1955). No data is
available from Lake Mbaracaibo to verify the validity of the
time-dependent portion of the theory, which predicts that
the velocity and salinity fields in the Lake require approximately
19 days to reach equilibrium a-fter a change in the applied wind
stress.
Thesis Supervisor: Arthur T. Ippen
Title: Ford Professor of Engineering
(iii)
ACOTOWLEDGEIMITS
Financial support for this research was provided by a
Ford Foundation grant to the Inter-American Project in the
Department of Civil Eigineering, M.I.T.
Professor A. T. Ippen, who supervised the thesis, read the
manuscript critically and offered numerous helpful suggestions
and criticisms.
Drs. Ralph H. Cross and Claes Rooth also read the manuscript
and provided constructive criticism.
(iv)
TABLE OF CONTENTS
Page
Abstract . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements.
.
.
.
.
.
.
.
Table of Contents . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
. ............
I. INTRODUCTION . . . . . . . . .
. . . .
. .
1.2 Description of the region . . . . . .
1
1
.
.
iii
iv
. . . .
1.1 Description of the problem . . . . . . .
i
.
1
. .
3
1.4 Development of the navigation channel. . . . .
4
1.5 Hydrologic conditions. . ..
1.3 Economic importance of the region. . . .
.
1.6 Wind patterns over the Lake. .
.
.
.
.
.
.
5
.
.
.
.
. .
9
9
1.7 Tidal characteristics.
.
.
.
.
.
.
.
.
1.8 Salinity intrusion .
.
.
.
.
.
.
.
.
.
1.9 Flushing time of the Lake. .
.
.
.
1.10 Previous investigatidns . . . .&.
..
.
. .
.. .
2.1
Equations of motion. .
.
.
2.2 Important assumptions. .
.
.
.
.
-
..
...
2.3 Scaling the equations. . .
. . . .
2.4 Perturbations expansion.
. .
.
23
23
. .. .
1.11 Scope of the present investigation. .
II. DEVELOPMEN'IT OF THE MATHEMATICAL MODEL .
11
.
.
. . .
30
.
32
.
.
32
.-..
34
.
......
. ..
.
.
35
. . .
40
.
.
(v)
Page
2.5 Interior expansion .
.
.
.
.
2.6-Boundary layer expansion
.
.
.
2.7 Boundary conditions. .
.
.
.
..
2.8 Solutions. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
42
.
.
.
.
45
........
.
.
.
.
.
.
III. APPLICATION OF THE THEORY TO LAKE MARACAIBO.
.
.
. . .
3.1 Estimating the size of important constants
.
3.2 Transient response .
.
.... .. .
.
.
.
.
.
80
4.2 Inadequacies of the theory.
.
Summary.
.
.
.
.
-
-
.
.
.
.
.
.
.
.
.
.
.
.
88
.
.
.
.
.
.
.
.
.
95
. . . . . ..
.
.
5.2 Recommendations for further work
BIBLIOGRAPHY.
LIST
.
.
.
.
.
.
SYrBOLS.
.
.
.
..
OF FIGURES
.
.
.
...
.
.
.
.
VII. LIST OF
IX.
72
80
CONCLUSIONS.
VIII.
56
4.1 Comparison of theory and data. . . . . . . . .
5.1
VI.
46
56
IV. DISCUSSION . . . . . . . . . . . . . . . . . . . .
V.
43
LIST OF TABLES.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
95
.
.
.
.
.
.
.
97
..
.
.
95
.
.
...
.
.
.
.
.
. ,
.
99
.
.
102
. . .0. . . . . . .
104
.
.
.
.
I. INTRODUCTION
SDescription of the problem.-- Lake laracaibo is a large
lake in northwestern Venezuela which is connected to the Gulf of
Venezuela by the Straits of Maracaibo and Tablazo Bay (see
In recent years a general increase in the salinity
Figure 1).
of the Lake, as well as a major sedimentation problem in the
navigation channel connecting the Lake to the Gulf, has been
observed.
Because the Lake is a major national resource for
Venezuela, research was begun 1963 to investigate these two
problems. The resulting project has been a cooperative effort
between M.I.T., the University of Zulia in Maracaibo, and the
Instituto Nacional de Canalizaciones (I.N.C.), with M.I.T.
conducting the basic research and acting as consultant to the
I.N.C. in planning field investigations, analysis of field data
and the construction of a hydraulic model of the Straits and
Tablazo Bay.
The University has been concerned primarily with
measuring the fresh-water discharge from the Lake and the I.N.C.
has been responsible for collecting the field data.
1.2 Description of the region.-- Important physical dimensions
of the region, as given by Partheniades (1966), are as follows.
The Lake has an oval shape with a north-south dinension of
approximately 150 kiloneters and east-west dimension of 110
kilometers. The Lake bottom is relatively flat with a maximum
-2-
1*00'
11000'
Son Carlos Gulf of Venezuela
Zapara
Straits of Maracaibo
-1030'
10*00'
Lake Maracaibo
930'
9030'
DEPTHS IN
METERS
72*00'
FIGURE I. LAKE MARACAIBO.
71*30'
From Redfield et al (1955)
7100'
M3-
depth of 35 meters and an average depth of about 23 meters.
The area of the Lake is about 10,000,square kilometers. The
Straits of 11racaibo and Tablazo Bay both have an area of
approximately 1100 square kilometers. The Straits are about
40 kilometers long and vary in width from 18 kilometers at the
southern end to 7 kilometers at the northern end.
A channel
runs through the Straits and is about 1000 meters wide and
12 to 18 meters deep.
The distance from the northern end of
the Straits to the Gulf of Venezuela is about 24 kilometers.
Tablazo Bay is extremely shallow, with depths varying between
1 and
5
meters, except in some natural channels where they may
reach 7 meters.
Tablazo Bay is separated from the Gulf of
Venezuela by a series of shifting islands and bars, the largest
of which is the island of Zapara.
The major exchange of water
between the Gulf of Venezuela and the Tablazo Bay-Straits of
Maracaibo-Lake 4aracaibo system occurs through a natural
channel about 2 kilometers wide between the island of Zapara
and the penninsula of San Carlos.
The Maracaibo Basin is separated from the rest of Venezuela
on the east and south, and Colombia on the west, by the high
mountain ranges of the Andes, whose altitudes reach a maximum
of about
5000
meters to the south of the Lake.
1.3 Economic importance of the region.-- Because of large
petroleum reserves located beneath the Lake and the area
-14surrounding the Lake, the production of crude oil has been
extremely important in the economy of Venezuela.
However, the
Lake also supports a fisheries industry and agriculture may
become necessary in future years to support an increase in the
population of Venezuela.
Since an increase in Lake salinity
may have an adverse effect on the fisheries and agricultural
industries in the region it is important to understand the
processes which control the diffusion of salinity in the Lake.
1.4 Development of the navigation channel.-- As mentioned in the
previous section, the production of crude oil in Lake Maracaibo
has provided a significant contribution to the economy of
Venezuela.
The oil is shipped to refineries in other parts of
th.e world in large ships which enter the Lake by way of a
navigation channel.
The channel begins in the Gulf of Venezuela,
passes through the natural channel between Zapara and San Carlos,
and continues through Tablazo Bay and the Straits of Maracaibo
into the Lake.
Prior to 1938 the natural channel through this region was
used, but continual shifting in its depth and alignment prompted
the oil companies using the channel to begin a dredging program.
Under this program the channel depth was maintained at 20 feet
until 1947 when the Venezuelan Government recommended that the
depth be increased to 35 feet.
In 1952 the I.N.C. was formed
and given the responsibility of maintaining the channel at
recommended depth.
depth to
45
In 1957 the I.N.C. decided to increase the
feet to accomodate the deep-draught tankers which
were coming into service. This work was begun in 1960 and
completed in 1963. The dredging operations are presently
performed by the dredge "Zulia" which maintains a channel
800 to 1000 feet wide and
45
feet deep.
1.5 Hydrologic conditions.-- The hydrologic balance of the
Maracaibo Basin is determined by computing: (1) the average
runoff from the upland areas of the basin; (2) the average
evaporation and transpiration of Lake Maracaibo and associated
marshland; and (3) the average precipitation onto the marsh
and water surface of the Lake.
The hydrologic balance is then
given by:
S =
- E + Pr
(1.5.1)
where S is the hydrologic balance, R is the average runoff,
E is the evaporation and transpiration, and Pr is the precipitation over the Lake, all quantities having the units of
cubic meters per second.
The most comprehensive study of hydrologic conditions in
the Maracaibo Basin was done by Corona (1964), who analyzed
-6-
rainfall and evaporation data for the period 1944-1960.
Monthly averages of the hydrologic balance, S, were determined
and then multiplied by the number of seconds per month, giving
an estimate of the net monthly hydrologic balance:
Ba = S x (60 x 60 x 24 x 30)
(1.5.2)
where Ba is the net monthly hydrologic balance in cubic meters.
Ramon Cadenas (personal communication) has applied the same
methods as Corona to data covering the period 1964-1967.
The average net monthly hydrologic balance, obtained by
averaging the results of the above investigators over the
period 1944-1960, 1964-1967, is shown in Figure 2. The net
monthly hydrologic budget of the Lake for individual years,
during the same period, is shown in Figure 3.
The long-term average hydrologic balance (Figure 2)
indicates that the dry season extends through the months of
January, February, Iarch and April, while the wettest period
occurs during October and November.
It should be pointed out that there is considerable
spatial variation of rainfall within the Maracaibo Basin. The
Catatuibo sub-basin, in the extreme southwestern portion of the
Maracaibo Basin, has a mean monthly maximun rainfall of
500
millimeters in the month of October, whereas the same mean
0
40
0
8
4-
0D
C
.0
Jan.
FIGURE 2.
March
May
July
Sept.
Average net monthly hydrologic balance.
and Cadenos (unpublished data).
Oct.
Jan.
From Corona (1964)
.
8-
E
Z
.D
U
4
0
a)
cO
1945
FIGURE
1950
3.
1955
1960
Net monthly hydrologic balance for the period 1944-1960, 1964-1967.
Corona 0964) and Codenas (unpublished data).
1965
From
1967
-9-
monthly rainfall over the southern half of Lake Maracaibo is
about 200 millimeters.
1.6 Wind patterns over the Lake.-- Lake YMaracaibo is located
in the belt of the Northeast Trade Winds and because of this
the winds over the northern half of the Lake are from the
northeast about 70% of the time.
As shown by Redfield et al (1955)
the wind over the northern half of the Lake is predominantly
from the northeast during the months January-April, July and
December, being somewhat variable during the remaining portions
of the year. The average monthly wind speed varies between
6 and 9 meters per second.
Over the southern half of the Lake the wind regime is not
well known.
Redfield et al (1955) report that during their
survey the wind was predominantly from the west, and local
inhabitants indicated that this was the case for most of the
year.
The high mountain ranges to the west and south of the
Lake apparently deflect the Trade Winds in such a way that
the wind over the south end of the Lake is predominantly
westerly.
1.7 Tidal characteristics.-- As indicated in an earlier section
the Tablazo Bay-Straits of IHaracaibo-Lake Maracaibo system
communicates writh the Gulf of Venezuela through the narrow
-10-
channels between the islands and sand bars along the northern
boundary of Tablazo Bay.
The most important of these is the
2 kilometer-wide channel between the island of Zapara and the
peninsula of San Carlos.
Two secondary entrances exist at the
northeastern boundary of the Bay between the islands of
Canonera, Canonerita, and the mainland on the east.
The total
width of these secondary channels is about 3 kilometers, but
their depth is very small, ranging from practically zero at low
tides to 3 to
5
feet at high tides.
Therefore, the main tidal
exchange occurs through the natural channel between Zapara and
San Carlos, and only a small amount occurs through CanoneraCanonerita.
The tidal range at Zapara is about 3 feet, the predominant
constituent being the lunar semi-diurnal (N2) harmonic.
Frictional effects reduce the amplitude of the tidal wave as
it enters the Bay and the Straits of Maracaibo.
At Maracaibo,
approximately midway between the southern and northern limits of
the Straits, the tidal range of the lunar semi-diurnal
constituent is only 1 foot.
The tidal range in the Lake is further reduced by friction,
and, more important, as a result of the considerable change in
width between the Straits and the Lake.
Harmonic analysis of
tidal records from the Lake, made by Redfield et al (1955), give
a tidal range of
0.04
feet for the lunar semi-diurnal
-11-
constituent in the Lake.
Tidal velocities at the northern entrance to Tablazo Bay,
as given by Partheniades (1966), are of the order of 1.0 meters
per second, whereas measurements in the Lake by Redfield
indicate that tidal velocities there are of the order of
0.05-0.1 meters per second.
1.8 Salinit
intrusion.-- Salinity surveys in the Lake have
shown that the water of the Lake can, in a general manner,
be characterized by two different parts:
(1) The epilimnion, or upper layer, in which the chlorides
vary only slightly from place to place and with depth, and
which occupies more than 705o of the total Lake volume.
(2)The hypolimnion, or lower layer, in which salt
concentrations are generally higher and increase with depth.
Its shape is conical and is located appromxiately in the center
of the Lake.
Chemical analysis of the Lake water indicates that
practically all the salinity in the Lake is due to the entrance
of Gulf water.
Due to its higher density this water enters
the Lake along the bottom.
Long-term records of Lake chlorinity in the epilimnion
have been made by the Creole Oil Company and the I.N.C. in
unpublished form.
Since the epilimnion comprises the major
-12-
volume of the Lake and because its chloride content is relatively
uniform, these records provide a means of determining salinity
changes in the Lake.
Figure h shows the chloride content of the
Lake epilimnion as a function of time for the period 1944-1966.
From Figure 4 it can be seen that prior to 1958 the Lake
chlorinity remained at about 0.700 parts per thousand (ppt)
with the exception of the period 1948-1950 when the chlorinity
rose to 1.200 ppt.
Between 1958 and 1966, where the records
end, the Lake chlorinity increased from 0.700 ppt to about
3.500 ppt. The present Lake chlorinity is probably in the
neighborhood of 3.500-4.000 ppt, although no reliable data
is available to substantiate this.
Comparison of Figures 3 and 4 suggest that these increases
are associated with two phenomena, abnormal hydrologic balance
and deepening of the navigation channel. During the period
1948-1949.the channel depth remained at 20 feet, but the
hydrologic balance was below normal, from which one would
infer that the resulting chlorinity increase was caused by
the low hydrologic balance.
This is supported by the fact that
in 1950-1951 the hydrologic balance was above average and a
corresponding decrease in Lake chlorinity was observed.
Beginning in mid-1957 and continuing into 1960, where,
unfortunately, the hydrologic record ends, there was a period
of extreme drought.
Increase in the Lake chlorinity began in
late 1958 and was, no doubt, associated with the drought.
2.0006
06
a.
C
-
1.00
1945
FIGURE
1950
4.
1955
1960
1965
1967
Chlorinity In the epilimnion of Lake Maracoibo during 1944-1960, 1964-1967.
From unpublished data of the Creole Oil Co. and the I.N.C.
-114-
However, in 1960 the channel depth was increased to 45 feet
and this could also have contributed to the change in Lake
chlorinity. That the increase in channel depth may be
responsible, in part, for the chlorinity increase is suggested
by the fact that the hydrologic balance in 1964-1967 was
above-average, yet the Lake chlorinity increased.
To understand why changes in the hydrologic balance and
channel affect the Lake chlorinity one must examine the
physical processes governing the exchange of chlorides between
the Gulf of Venezuela and the Tablazo Bay-Straits of MaracaiboLake Iaracaibo system.
The chlorides enter the Bay from the Gulf of Venezuela
primarily through the channel between Zapara and San Carlos,
and diffuse southward through the Bay and Straits as a result
of tidal mixing. This diffusion of chlorides into the Straits
is balanced by the fresh water discharged from the Lake and
flowing northward. The velocities associated with the freshwater discharge are determined by the fresh-water budget of
the Lake and cross-sectional area of the channel through which
the fresh-water flows.
For a given cross-section the fresh-
water velocity will vary directly as the fresh-water budget
of the Lake, with high velocities occurring during wet periods
and low velocities during dry periods.
For a given fresh-
water budget of the Lake increases in cross-section will
reduce the fresh-water velocity, whereas decreases in the
-15-
cross-section increase the fresh-water velocity.
Increase in the fresh-water velocity will tend to "push"
the intruding chlorides northward toward the Gulf, and decreases
in the fresh-water velocity will allow the chlorides to move
south into the Straits and the Lake.
The chlorinity gradients resulting from these processes are,
in general, functions of both longitudinal and vertical
coordinates in a long narrow channel. This type of partially
mixed condition is typical of the Straits of Maracaibo and
Tablazo Bay, as can be seen from Figures
5-9.
These figures
are north-south sections from the Gulf of Venezuela to the
southern end of the Lake.
These figures, as well as Figures 10 and 11, indicate that
the chlorinity in the Lake has a different pattern and, therefore,
the physical processes governing the diffusion of chlorides in
the Lake are probably not the same as those in the Straits and
Tablazo Bay.
Figures
5-9
are also instructive in showing how the chlorinity
in the Straits has been affected by seasonal and long-term
variations in the fresh-water velocity.
In March 1954 (Figure 5),
when the fresh-water balance was negative, the 2.500 ppt
line of constant chlorinity (isochlor) intersected the bottom
of the Straits very near the northern entrance to the Lake.
Since the chlorinity at the bottom of the Lake near the center
was close to this value, it is reasonable to assume that the
South
North
0I Mt
nAj
_
Meters
9*00d
FIGURE 5.
9*3 0'
10'000
10* 30'
North-south cross-section from the Gulf of
Maracaibo showing chlorinity, C(ppt),
11000'
Venezuela to the south end
distribution in
March
1954,
of Lake
From Redfield(1955).
,
o'
South
North
0 r-.
0
10
Meters i
20
30
9*00'
FIGURE
9*30'
6.
IOod
10*3'
11000'
North- south cross-section from the Gulf of Venezuela to the south end of Lake
Maracaibo showing
chlorinity, C (ppt),
distribution in
May 1954.
From Redfield(1955).
South
North
0 rr-
-iO
-
10
10
Meters
20
- 20
SAA00-
30
I
9*30'
9*00
FIGURE
7.
30
I
10000
10030
II00'
North-south cross-section from the Gulf of Venezuela to the south end of Lake
Moracaibo showing chlorinity, C(ppt), distribution in March of 1963.
From
unpublished data of the Creole Oil Co.
South
North
0
---
n 0
-10
Meters
20
-20
-3
30
9030'
9000'
FIGURE
8.
10000'
11000'
10030'
North-south cross-section from the Gulf of Venezuela to the south end of Lake
Maracaibo showing chlorinity, C(ppt)
From unpublished data of the Creole
,
distribution in December
Oil Co.
1963.
North
South
10
10
-
'
- Meters
20-
-20
30-
-30
9*30'
9000'
FIGURE
9.
10*00'
10*30'
North-south cross-section from the Gulf of Venezuela to the south end of Lake
Maracaibo showing chlorinity, C(ppt), distribution in June 1966. From
unpublished data of the I.N.C.
1000
20
0
60 e a
Kilometers
40
80
a1,
t00
A
10
9
-o
Meters
20
20-
30-
30
Temperature - *C.
Meters
20
30
Chlorinity -
FIGURE
10.
ppt
Lake temperature and chlorinity along Section A A'
from Redfield et al (1955).
20
40
60
Kilometers
8,0
a
1QO
,
Ig0
,
0
1O
Meters
20
30
N)
Temperature -a'C
Meters
20
30
FIGURE 1I.
Lake temperature and chlorinity along Section BB
from Redfield et al (1955).
-23-
Lake was receiving water of this chlorinity from the Straits.
In May 1954 (Figure 6) the 2.500 ppt isochlor moved north,
almost to the Gulf of Venezuela, presumably as a result of
the positive fresh-water balance during this month.
A similar
shift in isochlors is seen by comparing March 1963 (Figure 7),
a dry month on the average, although no hydrological data is
available for this period, and December 1963 (Figure 8), a wet
month.
The data from 1963 also show how the chlorinity increased
both in the Lake and 'the Straits between 1954 and 1963.
The
most recent data is from a survey made in June 1966 by the
I.N.C. (Figure 9), which give an indication of the present
chlorinity distribution in the Straits and the Lake.
1.9 Flushing time of the Lake.-- The average time a molecule
of salt remains in the Lake determines how the Lake water will
react to factors altering the accession of water or salt into
the Lake.
This time is given'by the ratio of the Lake volume
to the fresh-water discharge.
The volume of Lake Maracaibo is
approximately 25 x 1010 cubic meters.
The total annual average
fresh-water discharge, as determined from Figure 2, is about'
50 x 109 cubic meters.
Dividing the Lake volume by the
fresh-water discharge gives a value of about 5 years as the
flushing, or residence, time for a salt molecule in the Lake.
1.10 Previous investigatons.-- Investigations of the salinity
intrusion problem in the Straits of Haracaibo and Tablazo Bay
were begun by Corona (1966) using the one-dimensional model for
a long narrow channel originated by Ippen and Harleman (1961).
In this method the salinity, S, and the longitudinal
velocity, u, are taken as instantaneous, average values over
each cross-section.
For Lake Maracaibo the salinity, S, is
related to the chlorinity, C, by the following formula:
S =
C
1.SOS
+0.0
0.10.1)
Because the channel is assumed to be long and narrow,
variations across the channel are neglected, and therefore,
variables are functions of time, t, and longitudinal distance
along the channel, x, only.
The one-dimensional conservation of salt equation for a
channel of constant cross-sectional area is given by:
LA CI/=.
c
(1.10.2)
where DI is the apparent diffusion coefficient. This coefficient
x
includes the mass transfer by turbulent diffusion and the
important nass transfer by internal currents caused by the
density difference between salt and fresh water.
The fluid
velocity, u, is equal to the sum of the tidal velocity, u(x,t),
and the fresh-water velocity, U, due to the outflow from the
Lake.
Equation (1.10.2) becomes:
~S
(~t~-U~
K#
(1.10.3)
The negative sign in the fresh-water velocity term appears
because the origin of x is taken at the ocean entrance and is
measured positive in the landward direction.
If the fresh-water discharge is relatively constant during
a period of time of the order of several days or a week, a
quasi-steady state salinity distribution should result in the
estuary. That is, the one-dimensional salinity distribution at
instants of time differing by one tidal cycle should be the
This should be expecially true at low water slack (L.W.S.)
same.
since the salinity distribution at high water slack may be more
influenced by daily variations in tidal amplitude.
At the moment of low water slack, the salinity at any
station should be close to its minimum value.
low water slack both
DS
Z>t
In addition, at
and u(x,t) are momentarily zero.
Equation (1.10.3) can then be written:
~U
0)SLWVS
C)
DS~
__
(1-10-4)
vhere SLS = salinity at low water slack and is a function of
x only.
The second assumption is that the salinity distribution at
-26-
high water slack is given by the low water slack distribution
curve displaced longitudinally by a distance equal to the tidal
excursion. This is equivalent to stating that within one-half
of a tidal period:
+ L
= aA
c
(1-10-5)
DX
C)t
Equation (1.10.4) can be integrated if the functional
dependence of D1 and boundary conditions on the salinity, S,
x
are given.
The functional dependence of DI on x is assumed to be the
x
following form:
D
C
(1.10.6)
Therefore, at x = 0 (ocean entrance), D' = D', while at
0
x
x = -B (seaward of x = 0), D' -o oo , and for large x (in the
x
positive, or landward, direction), DI-->O.
X
That D. approaches an infinite value at x = -B is consistent
with the fact that if the estuary was imagined to extend to
x = -B, an infinite anount of mixing would be required to maintain
a constant salinity at low water slack.
The boundary conditions on the salinity are assumed to be
that SU- -:l0,and dS
tbLNS ->O0, as x ->OCC,
and S- I = So at
-27-
x = -B, where S is the ocean salinity.
The solution to equation (1.10.4) is then given by:
_U
3
WS --
B
(x,+)
(1-10.7)
so
If the salinity is knom
at low water slack for at least two
points (values of x) the parameters DI and B can be determined
from equation (1.10.7).
For actual field conditions, where the basic assumptions
used to derive the mathematical model are not exactly satisfied,
it has been found that two dimensionless salinity parameters:
and
Dt
0
UB
2'B
uOT
mhere u0 is the maxmum flood tide velocity at the ocean entrance,
and T is the tidal period, correlate with a dimensionless
combination of variables known as the "estuary number".
estuary number is defined as follows:
P
F
Qf T
The
-28-
where,
=
t
F
tidal prism, the volume of sea water entering
the estuary on flood tide.
= Froude number,
uo
, u being the maximum
flood tide velocity; g, the force per unity
mass due to gravity; and h, the mean channel
depth.
Qf = fresh-water discharge
T
= tidal period
Using salinity measurements obtained from the Straits of
Maracaibo and Tablazo Bay during surveys made by the I.N.C.,
Corona (1966) was able to calculate the values of D' and B
from equation (1.10.7) for various values of fresh-water
D
discharge and tidal velocity. The two salinity parameters, D
2'i\'B , were calculated and plotted against the estuary
and,
number
PT
2 , and the results are shown in Figures 12 and 13.
From these Tigures estimates of DI and B can be obtained for
given values of tidal velocity-and fresh-water discharge.
4aximum and minimum intrusion lengths for the Tablazo Bay-Straits
of Maracaibo estuary can then be predicted as shown in Table III
of Ippen (1966).
It should be kept in mind that the analysis described above
can be used with confidence only in the Straits and Tablazo Bay,
and does not apply to the diffusion of salt in the Lake proper.
It does however furnish information about the salinity at the
northern end of the Lake where the Straits join the Lake.
40
-I----I----I--1 I I
40
'
I
I
-I--i--i-i-li-
-
'
I
I
I
£I
1 -1s
0
0
1027rB
uT
5
I
I I1 A11
0.6
0.1
0.D
0.003
* (Estuary Number)
OfT
FIGURE 12. Basic salinity distribution parameters in the
Straits of Marocaibo. From Corona (1966).
50
-- r
r-rr-
a
h
-
I
r
I--r
-r
0
tidal amplitude
channel depth
10D'
UB
0. a003
--
QT /
0.0
Pt
0.1
0.9
PQF2 (Estuary Number)
FIGURE 13. Basic salinity distribution parameteirs in the
Straits of Maracaibo. From Corona (1966).
In this sense it gives a boundary condition for the diffusion of
salt into the Lake, and will therefore be important in an
analysis 'of the entire Tablazo Bay-Straits of Maracaibo-Lake
Maracaibo system.
An investigation which has bearing on the salinity problem
in the Lake proper is a model study by Stockhausen (1964).
In
this investigation a scale model of the Lake was constructed and
salt was added to simulate the salinity distribution in the Lake.
The model was then placed on a rotating platform and a wind stress
was applied.
The resulting salinity distribution measured in the
model agreed well with the prototype and indicated in a qualitative way .that the wind stress and rotation of the Earth play an
important role in the dynamics of the Lake.
These processes are
considerably different than those acting in the Straits and
Tablazo Bay, and any analysis of salt diffusion in the Lake must
take into account these same processes.
1.11 Scope of the present investigation.-- It has been emphasized
in previous sections that the physical processes affecting the
diffusion of salt in the Lake are different from those in the
Straits and Tablazo Bay.
The diffusion of salt in the Straits
and Tablazo Bay has been adequately described by Corona (1966) as
indicated in a previous paragraph.
To obtain a unified analysis
of the entire system it is necessary to determine the nature of
-31
the salt diffusion in the Lake.
-
The purpose of this work is to
develop a mathematical model of the dynamical processes affecting
the Lake circulation.
The diffusion of salt will play an
important part in this mathematical model, as will the effects
of wind stress and the rotation of the Earth.
The results of the theoretical investigation will be
compared with the field data obtained by Redfield et al (1955).
These data will be used because they provide the most complete
information on velocity and salinity profiles in the Lake.
That the dynamical picture has not changed appreciably since
Redfield t s study is suggested by the shapes of the isochlors
in Figures 5-9, however, more thorough field studies of
velocity and salinity profiles would be helpful for establishing
the validity of the theoretical model.
II.
DEVELOPIET OF THE IMATIEMATICAL
MODEL
2.1 Equations of motion.-- The equations required to describe
the diffusion of salinity and momentum in an incompressible
rotating fluid are the momentum equations, continuity equation,
equations of state, and salt diffusion equation.
For axially
symmetric flow, the momentum equations in the radial, azimuthal,
and vertical directions, respectively, are:
.. .
Sr
U LC) LA +4 .AA7 CL_
r
(2.1.1)
.
L C)L.
37.tA -- - V5-iL
C)r'
Lk
Ciif
,A-AY
C) Mr
C) t
ns
D-uX
3
3
ns LA
Ir
(2.1.2)
4 L-k
qL
r L)
+JL L)
+
2.1.3)
3 Aa
V.
C)e)
LA
y
+
C)-iIC')
C)
y
-33-
The equation of continuity:
)
(
(2.1.14)
O
The equation of state:
=-
(2.1
I .
P
.5)
The equation of salt diffusion:
-t.
..
C
LA
.
m3
t
Az
(2.1.6)
C)
D.[S
+
D_rS
\k
The dependent variables, r,
z, and t are the radial and
vertical coordinates and time, respectively, where r is positive
outwards and z is positive upward and parallel to the gravity
force.
The independent variables, u, v, w,
,
and S are the
radial, azimuthal and vertical velocities, pressure, density
and salinity, respectively.
-\
and
and vertical coefficients of eddy viscosity,
are the horizontal
and
\R
-34-
the horizontal and vertical coefficients of eddy diffusivity.
g is the force per unit mass due to gravity. Cx is a coefficient
relating the density to the salinity and f is the Coriolis
paramter, 2.0 sin
the Earth and
equator.
,
where .0
is the angular velocity of
P is the latitude measured northward from the
The 12KS system of units will be used throughout,
except where noted.
ant assumptions.-- The following assumptions will be
2.2
used in this work:
(1)
The motions are small enough so that the inertial terms
will be negligible.
(2)
Frictional effects are confined to small boundarylayers on the top and bottom.
(3)
The major portion of the Lake is in geostrophic
equilibrium, that is, the pressure is balanced by
the Coriolis force.
(h) The Coriolis parameter, f = 20 sin
4),
is a constant.
This implies that dynlamic effects due to the Earth's
sphericity can be ignored.
coefficients of viscosity and diffusion are
constant.
(5)Eddy
(6)
The variations in density are small enough to be
negligible, except where they are associated with
the gravity term. This is the Boussinesq approximation.
(7)
The density is a linear function of salinity.
inplies that CK is a constant.
(8)
The Lake is initially at rest and stratified in such
a way that the salinity is a linear function of the
depth only.
This
(9)
After the Lake has been set into motion the pressure
field can be separated into two portions. One portion
is a hydrostatic term resulting from the Lake's
initial stratification, and the other a small perturbation caused by the motion. Mathematically, this
can be stated:
f(r, :)
=f 0 (Z)
f4(,,)
(2.2.1)
(10) The salinity can be separated into three components.
The first two being a constant term and a tern which
is a linear function of the depth only. This describes
the condition of the Lake while it is at rest and is an
exact solution to equation (2.1.6). The third
component is a small perturbation resulting from the
motion. Mathematically, this can be stated:
S~
t,
=so+
sS
The first two terms, S. +
-)
v
(2.2.2)
, will be
referred to in later sections as the long-term
component.
2.3 Scaling the eouations.-- Exact solutions to the system of
equations (2.1.1) - (2.1.6) are difficult to find. It has been
found, however, that for certain scales of motion the complexity
of these equations can be reduced while retaining the important
features of the flow.
A technique which has been successful is
to first reduce the equations to dimensionless form, identify
important terms in the scaled equations and then use the nethods
of singular perturbation theory to find an approximate solution.
-36-
Several investigators have used this method in the study
of diffusion processes in a rotating fluid.
Howard (1.963)
Greenspan and
investigated the time-dependent motions of a
homogeneous fluid in a rotating cylinder vhose vertical and
horizontal dimensions were approximately the same.
Barcilon
and Pedlosky (1967a, 1967b) were concerned with the steady
motion of a temerature-stratified fluid in a rotating
cylinder of similar dimensions as Greenspan and Howard.
Holton (1964) developed a model for the time-dependent motion
of a thermally stratified fluid in a rotating basin with a large
ratio of horizontal-to-vertical dimensions, but he rather
arbitrarily ignored the effects of temperature diffusion.
The following investigation will be concerned with the
time-dependent motions of a salt-stratified fluid in a rotating
basin of large horizontal dimensions and small vertical dimension.
In this respect it is similar to Holton's work, the major
difference being that the diffusion processes whnich Holton
neglected will be taken into account.
The appropriate scaling for a stratified fluid in neargeostrophic equilibrium, and subject to viscous forces, is
as follows:
-37-
gAAY
(f=V~f
__
L
V4P
VfL
%
S'
where the primed variables are dimensionless, L is a characteristic
horizontal dimension, H is a characteristic depth, and V is a
All other variables and
characteristic azimuthal velocity.
constants have been defined previously.
Certain dimensionless numbers will occur in the equations:
(1) Mechanical Rossby Number -
=
-
4L
This is a measure of how much the fluid departs from
a state of rigid rotation.
6
(2) Internal Rossby Number -
=
-
2
This is a measure of how much the fluid departs from
the initial linear vertical stratification.
(3)
E
Ekman Number -
This is a measure of the thickiess of the frictional
boundary layers at the top and bottom.
(h)
Aspect Ratio
-
/
L
(5)
Rotational Richardson Number
=
-
--
__V
This is a measure of the initial stratification.
(6)' Horizontal Prandtl Number (7)
Cr
.
Vertical Prandtl Number -
(8)
This is a measure of the anisotropic nature of the
turbulence.
The quantities &SH
and &Sv
occurring in (2) and
(5)
are estimates of the total variation of salinity from the center
of the Lake to the edge and the variation of salinity from the
top to the bottom, respectively.
In the perturbation expansion which follows it will be
assumed that the internal Rossby number, 6,
Rossby number,
,
the square root of the Elnan number,
and the rotational Richardson number,
of the same order.
the mechanical
There is,
for these assumptions.
(
,
,
are small and all
of course, no a priori justification
However, in Section III the theory will
be compared with actual field data and the values of important
constants estimated. The above dimensionless numbers can then
be calculated, and it will be showrn that assumptions made as
to their magnitude is, in fact, consistent with the observed
behaviour of the Lake.
The horizontal Prandtl number,
,
will be taken as
-39-
one (1), as suggested by Sverdrup (1942), and the vertical Prandtl
number, CUI , according to Hunk and Anderson is about 10.
The
physical dimensions of the Lake determine the aspect ratio.
Making use of the above, the dimensionless forms of the
equations are:
E
b'
QM~f
Lt"uc'U' + )r'
+
-4+E 3- -L
L 2.U -
2_u'
.'L
32U,
~j
(2.3.1)
I-Cs=
~
I..' ~Y'
r )I
DY11
FZ
3nS5
Lk'
+
8
+
fols
(2.3.2)
1+I
r
4-3-.a
yo 7
4.
E.c~U3
~t.
4+
~I~'s'8
Ax-,
+ E
4-'
I
XL
(U
'-
Iw
(2.3.3)
Jr
yo)__
+
xAY'~
(2-3-4)
-40-
E ;4A
(2.3.5)
1LAC)Z
-) t'
+
L[es"
E_-_X
-+-
The equation of state, equation (2.15), has been eliminated
by naking use of the Boussinesq approximation.
2.4 Perturbation expansion.-- Folloring the example of
Holton (1964) and Barcilon and Pedlosky (1967a, 1967b), the
independent variables, u, vp w,
, and S are expanded in terms
of the square root of the Ekman number, 0" =
, which is a
measure of the thickness of the frictional boundary layers at
the top and botton, and, as mentioned previously, is assumed to
be small:
tA(4
-~
E
E''
Fu{hI
Lk IE ( V8
)
Lk n 5 ( V: !: ) +
8:E
(2.h.1)
-41-
Z
cir(v1'') = O2 E
rir
AAY'
('7
"
l2.
r,'i
(nr
t')+
')
c r (
-,a't)
(
24.2)
7; ' t')
AY Yv- I
(2..3)
nh
E
(2.4-4)
'Ih81r
V
nhk
n 5o
(u
,
T
(Y.t)4
(2.6)
--- ) are the values of the variables in the
interior of the fluid, and (u' B
nB
,
---- ) and
V
(ut
nB
nB
are the correct'ions to the interior flow
,
in the bpundary layers at the top and bottom, respectively, and
=
E
,
=
t, and the stretched coordinates,
are functions of r,
needed to
. The boundary layer corrections are
satisfy the boundary conditions at the top and bottom, and must
vanish in the interior.
2.5 Interior expansion.-- The interior flow, to zero order, is
obtained by substituting the expanded form of the variables
(equations (2.h.1)
-
(2.h.5)) into equations (2.3.1)
- (2.3.5)
and collecting terms whose coefficients are of the order unity.
The resulting interior equations are:
-(1
'
OIL
_
_(2.5.1)
(2o.2
-L
0
(2.5.3)
-h3-
')'
9TO
Z~OL
(2.5.5)
=0
0
I2s
I
c3~
+
44 !4r
2.6 Boundary layer expansion.-- The zero-order expansion for
the corrections to the interior flow in the top boundary
layer (zt = 1) is found to be:
OT
: _*S
8
411
0
.4T.
(2.6.1)
r
t
Q05
=
)
(2.6.2)
it
(2.6.3)
I
L_
U*
775vo
r)
0
c)~tLr' t
(2.6.4)
(2.6.5)
Similarly, in the bottom boundary layer (zi = 0):
=--
+A
LAJL
L08
(2.6.7)
ii:
-
0
(2.6.6)
=
'e)M
(2.6.8)
.
,E8r\
D
/
A
'3X
+
Jro P
(2.6.9)
(2.6.10)
2.7 Boundary conditions.-- The boundary condtions to be
satisfied are:
at zt = 0
u=
0
v
at z'
Ut
=,
(rt), w
= 0, V'
= 0,
- 0
a)r0
at rt = 0
U1 ,
vt,
I
= 0=
*
Svt=
=0
/C (rt)is the dimensionless wind stress.
at rI = I
u'
0
1
0
where
= 0, wt = 0,
t,
31
are finite
.46-
at t'
u'
= 0
= vt
= W1
= St
= 0
2.8 Solutions.-- The boundary layer equations (2.6.1) - (2.6.10)
are the same as those first solved by Elonan (1905) with the
exception of equations (2.6.5) and 2.6.10) involving the
diffusion of salt, which were not used by Elnan because he was
concerned with a homogeneous fluid.
Since no solutions to
equations (2.6.5) and (2.6.10) can be found which allow So
S oB 0
to vanisa in the interior, except St
and S'
o13
oB
oB
the solutions to equations (2.6.1) - (2.6.10) for the boundary
conditions given in paragraph 2.7 must be the same as those
given by Ekman.
Q#t
These solutions are:
)t ) e
-47-
=-of V,O,-'
TlQ S
(2.8.4)
e
The boundary conditions for the vertical velocity are
obtained by integrating the continuity equations (2.5.4),
(2.6.4) and (2.6.9):
at z' = 0
(2.8.5)
0
Irl )
A
'w
I
at z, = 1
-05-1
O'
4..
JA
K>
(2.8.6)
(iXT'r
V )
a
7YL-
-148-
The results of the integration are:
at zt = 0
_
-
(2.8.7)
at zI = 1
,AAr
Ox
r
aI'L
(2.8.8)
V.dyL~~J
J
For the interior it is evident from equations (2.5.1) and
(2.5.3)
that the pressure, p0
function,
,
can be treated as a stream-
VI, which satisfies the equation:
(2.8.9)
+ #
L7-
+2t.
)r,
~a2.
ft--ft-IPCM
DI )
0all
..-_.oarl
I.
with boundary conditions:
at z, = 0
E 1-
2.
12
q)-)
~~i2~
CrIr
)KA"~~TY
__
V.
(2.8.10)
DYL
dV.\c)21
)
I
7 Lv7J,
~
at z t = 1
~jat(
'I,
oI)
(2.8.11)
or
+
Dih.(
-I
ye
~
5~LA
~4=0
at rt = 0
'Va'
finite
(2.8.12)
at r' = 1
.
.
=0
(2.8.13)
C)r
at tt = 0
'T = 0
(2.8.11h)
To complete the solution-it is necessary to specify the
wind stress,
i
(rt,tl). For this investigation a particularly
useful form of the wind stress is:
r(P,-)
=U
Ia
)i-(
(2.8.15)
where u1 (t') is the unit stop-function, J (kr") is the Bessel
function of first kind and order one, and k is chosen such
that Jj(k) = 0.
Making use of Laplace transforms, the partial differential
equation (2.8.9) reduces to:
s
C-)
1
+
-f
ah1.
-
-os
(2.8.16)
while the boundary condition at zt = 0, equation (2.8.10),
becomes:
1/.
L
. 1'K
t~K1~
OT.
4
I ?tc'f
+]
-
-
O:.
(2.8.17)
ihere:
1
P7--
(2.8.18)
and:
cit)
00
e
0~
(2.8.19)
et
I4- (2(I, e
dt
(2.8.20)
0
Inverting equations (2.8.16) and (2.8.17) leads to the
solution:
(t)
-
(z~
IL)
(~t%)
tLA t)
]LA (t-)
(2.8.21)
(2.8.22)
From this the azimuthal velocity and the salinity perturbation
in the interior can be determined. The baroclinic and barotropic
modes of the azimuthal velocity are, respectively:
CLSQ. 3x
1
(1)(2.8.23)
(2.8.24)
The baroclinic mode, equation (2.8.23), responds to that
part of the pressure field for which the pressure is not a
function of the density only, whereas the barotropic mode,
equation (2.8.24), responds to that part of the pressure field
for which the pressure is a function of density only.
In a
barotropic fluid the lines of constant pressure and constant
density are everywhere parallel, which is not the case for a
baroclinic fluid.
In general, the pressure field of a fluid
can be considered to be composed of a baroclinic and a
barotropic component and the resulting velocity field uill have
a barotropic and baroclinic component as was found above.
For
a fluid which is in quasi-geostrophic equilibrium, as Lake
Maracaibo has been assumed to be, the barotropic velocity
component will be independent of the depth, whereas the
baroclinic mode irill have a dependence upon the depth.
This
is evident from equations (2.8.23) and (2.8.2).
The salinity perturbation is given by:
The zero order velocity and perturbation salinity fields
for the entire fluid are:
+ ell
_
Z e
C
= eI
I
LA
e
+
(2-8.27
(2.8.27)
4T
r
T
y-'t)
E
(2.8.23)
-
k#W4-iT
+
-Ar
-o U V: ) LA Q )
4e
4r(
(2.8.2>
-56-
III. APPLICATION OF THE THSORY TO
LAKE M1YAACAIBO
3.1 Estimating the size of important constants.-- The mathematical
model derived in Section II characterizes the time-dependent flow
of a rotating, stratified, and incompressible fluid for a rather
specific set of conditions.
These conditions are: (1) viscous
effects are confined to thin layers at the top and bottom of
the fluid; (2) the initial stratification of the fluid is small;
(3)
the fluid motion differs only slightly from a state of solid
rotation; (h) the stratification due to the fluid motion differs
only by a small amount from the initial stratification; and
(5)
the ratio of the horizontal to vertical dimensions is large.
In order to determine if such a model can be used to explain
the diffusion of salt and momentum in Lake H"aracaibo,
it
is
necessary to compare the theory with actual field data from the
Lake.
The most exhaustive field study of Lake
aracaibo was
made by Redfield et al (1955) in the Spring of 1954.
Figures
10 and 11 show Lake chlorinity and Figure 15 shows the azimuthal
velocity profiles obtained during this survey.
This data will
be used to determinc the validity of the theory for tho steadystate case only, since no suitable time series is
the investigation of tine-dependent motions.
available for
The theory can be compared
ith the data only if
numerical values of the constants, OC , f,
& SH' 4
'
).,
W.,
and,
, are known.
the
g, H, L, V, ASv,
These constants
have determined in the following manner:
C -
f -
the coefficient relating the density to the salinity
is given by Lafond (1951) as 1 .53 x 1o-3 ppt- 1 .
the Coriolis parameter, 2 C. sin 4 , is calculated for
a latitude of 90 501, which corresponds, approximately,
to the center of the Lake. -- , the angular velocity
of the Earth, is 0.73 x 10-4 seconds- 1 . Therefore:
f = 2 x 0.73 x 1oh~
x sin(9 0 501) = 2.48 x 10-5 sec.-1
g -
the force per unit mass due to gravity is 9.81 meters 2
sec.~1
H-
the characteristic depth of the Lake is approximately
30 meters. Actual depth and idealized depth are
compared in Figure 14.
L -
the characteristic radius of the Lake is approximately
60 x 1o3 meters as shown in Figure 14.
&SV- the variation in salinity from top to bottom is
obtained from the
Figures 10 and 11
the middle of the
bottom about 0.20
salinity by means
From
data of Redfield et al (1955).
the chlorinity, C, at the top in
Lake is about 0.80 ppt, and at the
ppt. Converting chlorinity to
of equation (1.10.1):
Sv = 1.805 x C(top) - 1.805 x C(bottom)
= 1.805 x (-1.20)
= -2.16 ppt
1
60
40
20
0
I
I
I
I
I
I
80
I
I
120
100
I
I
I
I
A
Kilometers
Meters
I
0
I
10
10
20
II
-
30
-
FIGURE
-
-
I
--------------------------------
Meters
20
-
30
14. Comparison of idealized Lake (dashed lines) with
actual Lkd (solid lines) along Sections AA' and BB.'
the variation in salinity from the middle of the Lake
to the edge of the Lake is also obtained from Redfield.
SH-
-
Figures 10 and 11 show that the chlorinity
varies from 0.80 ppt at the center to 0.70
edge, and, at the bottom, from 2.00 ppt at
to 1.00 ppt at the edge. Again making use
equation (1.10.1):
ASH (top)
=
at the top
ppt at the
the center
of
(0.70 - 0.80) x 1.805 = -0.18 ppt
ASH(bottom) = (1.00 - 2.00) x 1.805 = -1.81 ppt
The average of these two values:
ASH(top) + ASH(bottom) = -0.99 ppt
2
will be used.
-
the vertical eddy viscosity is determined by the size
of the frictional boundary layers. For a rotating
fluid the thickness of these boundaries is given by
.
the so-called Ekman depth,
In Lake Maracaibo this depth is approximately
as can be seen from Figure 15(c). Therefore:
Elman depth = 4.0
4 meters
=:
and
V
-
the characteristic velocity, and
-
the horizontal eddy diffusivity, are calculated in the
following way:
-60-
From Figure 15(c) the azimuthal velocity at the surface,
approximately midway between the center and edge of
the Lake, is 0.55 meters per second. The dimensionless
form of the theoretical steady-state (t = 00)
= 0,
= 10)
azimuthal velocity at the surface (z' 1,
and midway between the center and edge (r'
equation (2.8.26), is:
Recalling that:
Y
t~Ai
=
), from
Meters
20 .
0
FIGURE
0.2
(a)
0.4
15.
Velocity profiles at various po sitions in the
Lake as
The sense of
measured by Redfleld et al (1955).
the motion is counterclockwise.
0.4
0.2
(b)
Azimuthal Velocity,
0.
0
0.2
0.4
(c)
meters/second
0.6
-62-
then:
0
cr
bE.
OA
TT~~
A'
Applying the numerical values of C(
,
g, f,
, and
L SV calculated previously:
S
2L(z x 155 x to _c(,131
A v. 2.&VS %i o*"S
E'aX
'2.A't Vo-
513.(0 x io
The dimensional azimuthal velocity is related to the
dimensionless velocity, as follows:
r\'
= V rro
The dimensional azimuthal velocity at the surface, midway
between the center and edge of the Lake is then:
L
2
0,.
-63-
The constant, k, is the first root of the equation
J (k) = 0. From Jahnke and Emde (1945), k = 3.83,
,and J1 (3.83) = 0.58.
2
IMatching the observed Lake velocity of 0.55 meters
per second with the theoretical velocity:
0.55
j
--.
____
Y. I
2-
This equation contains two unknoims, *Ayand V.
Therefore, a second equation is needed. The second
equation is provided by the salinity perturbation term.
From equation (2.3.29) the dimensionless form of the
salinity perturbation is:
8.O x \O To
Y
Recalling the scaling procedure of paragraph 2.3,
the dimensional form of the salinity perturbation is
VF L S,
<I H
V
-
-
.XO
X-s
.S
xOS
V
85-I
The horizontal variation of the theoretical salinity
perturbation, 6 (S Szi), from the center of the Lake
to the adge is given by:
2,%8 11
eo~o3(32)
3
S-q4.~*x o
V
where the values of JO(3.83) and Jo(o.00) have been
determined from Jahnke and Erde (1945).
Equating the theoretical salinity variation to the
actual average variation in the Lake, ASH = -0.99,
as calculated previously:
0.9 4X3 .. 0
V
IP\V
2.Lc0 x \
Solving the two simultaneous equations:
o.90 = V
Vas
4Kj
3\
0
0-10-L
4
F -iT Xo-3
determines the values of %yand V:
V = 0.03 meters per second
2
,r= 3.17 x 102 meters seconds-1
In paragraph 2.3 it was pointed out that the horizontal
is a out one (1), and
-, = -Q. /v,
Prandtl number,
6 /96, is about
the vertical Prandtl number, C-A =
10, for most cases of geophysical interest.
these ratios:
Applying
1
y = 3.17 x 10 2 meters 2 seconds-
reters 2 seconds
= 1.98 x 10~
1
The values of these constants as they have been computed
above are given in Table 1.
They can also be used to calculate
ihe magnitude of the important dimensionless parameters:
Rotational Richardson H4umber,
30.0
=
oLLk
(
-66--
Mechanical Rossby Nlunber,_C
cE-v
-0L
12.48k25P.16~5
= 0.L8I
O.514 x 1c,
x
(o0o.ItoI
Internal Rossby Numbe,±-r
ET
%
1- 53x.
x.\x30.0
( 2.&A8?%1\5~ fx(JOo
=0.20O
Ekman Number
E
(30.l
=
.89j
2.48
x
%o-2-
x \ O-5
(-0.C
Symbol
Parameter Name
Value
Units
O<
Coefficient relating salinity and density
1.53 x 10-3
ppt-1
f
Coriolis paramter
2.48 x 10-5
seconds-1
g
Force per unit mass due to gravity
9.81
meters-seconds-2
H
Characteristic depth
30.0
L
Characteristic radius
60.0
ASH
Magnitude of top-to-bottom salinity
variation
-0.99
ppt
as
Magnitude of middle-to-edge salinity
variation
-2.16
ppt
Y
V
meters
x 103
meters
Horizontal eddy viscosity
3.17 x 102
meters2-secondsl
Vertical eddy viscosity
1.98 x 10~4
meters2seconds 1
Horizontal eddy diffusivity
3.17 x 102
meters2seconds1
Vertical eddy diffusivity
1.98 x 10-
meters 2-seconds1
Characteristic azimuthal velocity
0.08
meters-seconds~1
TABLE 1. Numerical values of important constants.
-68-
Aspect Ratio,
-
L
30.0
(00. 0 xoK10:
5.00 x 16~
In paragraph 2.3 it was assumed that g
,
were small and of approximately the same size.
C-
E
,
and E.
The above calcu-
lations indicate that this is fairly consistent with actual Lake
conditions.
Applying the numbers in Table 1 to tho steady-state
dimensional form of the equations (2.8.26) - (2.8.28) the
theoretical velocity profiles for the Lake can be determined:
LA
0.08[e l
+ 1)
-
(3.1.1)
S(3.1.2)
e WFT vA
e
77L
AK1>
-69-
VA
-se
Av = i.LA 3 Y,Ii
(3.1.3)
T r7"T4
,&M a -VAI
goL
Using equations (3.1.1) - (3.1.3) the radial, azimuthal
and vertical velocities at r = 30 kilometers have been computed
for various depths.
The results are given in Table 2 and
Figures 16 (a), (b), and (c).
Depth below
surface
(meters)
z
(meters)
u
(meters per
second)
v
(meters per
second)
w
(meters per
second)
0,00
0
30
0,03
0,55
3
6
27
21
0.02
0.01
0.48
0.3
9
21
0.001
0.38
0.88
"
12
18
0.3
0.30
"
15
-0.001
-0.001
0.88
15
0.88
"
12
-0.002
0.27
0.87
"
21
9
-0.006
0.21
it
24
6
-0.02
0.16
27
3
-0.02
0.10
0.86
0.60
0.26
30
0
0.00
0.00
0.00
16
-
0.52 x 10~9
"
0.80
t
"
TABLE 2. Values of theoretical radial, azimuthal and vertical
velocities at r = 30 kilometers.
0.5
0.0
0
,
,
,
,
,
-0.05
0.0
0.05
,2
10
10 -
Meters
20
20-
300.0
0.5
(a) Azimuthal Velocity, v,
meters/second
FIGURE
16.
-0.05
0.0
0.05
(b)Radial Velocit y, U,
meters/second
Theoretical velocity profiles at r = 30 kilometers.
Sense of the azimuthal velocity Is counterclockwise.
0.0
30
(c) Vertical Velocityw,
meters /second
(x10 5 )
-71-
The total salinity field is given by the sum of the longterm components:
so
.2.Z-a
+
E
and the dimensional form of the perturbation component,
equation (2.8.29):
0.95 .0(A Y)
'L'
The sum of the long-term and perturbation components
gives:
S
2.7
- 2.1S Z-O.9 5
N4
(3.1.h)
The salinity field has been calculated from equation (3.1.4)
for: (a) r = 0 kilometers, (b) r = 30 kilometers, (c) r = 60 kilo-
meters at various depths. The salinity has been converted to
chlorinity, C, by equation (1.10.1):
S -0-.03
I.805
and the results are shown in Table 3.
(1.10.1)
-72-
Chlorinity, C, ppt
Depth belor
surface
(meters)
r = 0km.
6
9
12
30
27
2h
21
18
15
18
21
0
3
21t
27
30
TABLE, 3.
r = 30 km. r = 60 lm.
(n,-ters)
0.80
1.0
1.16
1.27
0.h3
0.5h
0.66
0.78
0.90
15
1.ho
1.02
0.6
12
1.52
1.15
0.60
9
1.65
1.27
6
1.77
0.93
1.38
1.oh
3
0
1.88
2.00
1.'1
1.6T
1.16
1.30
0.92
0.03
0.22
0.31
0.b6
0.53
Values of theoretical chlorinity.
Figure 17 is a cross-section through the idealized Lake
shoving the theoretical chlorinity distribution. Chlorinity
has been used as the indicator of salt in all figures in order
to facilitate comparison with the field data, which are given
in terns of chlorinity.
3.2 Transient respQnse.-- Since there are no time series of
salinity and velocity measurements in the Lake it is impossible
to verify the validity of the time-dependent velocity and
salinity fields predicted by the theory.
However,
it
is of
interest to determine the scales associated irith the timedependence,
as given by the theory, so compirisons can be made
0
I
I
60
40
20
a
*
a
.
a-
80
a
a
100
I
120
Kilometers
0
10
Meters
20
20
30
30
FIGURE
17. Cross section through Lake showing theoretical
chlorinity(solinity) prof ile.
-74-
when sufficient field data is available.
In accordance with equations (2.8.23) and (2.8.24), and the
discussion following (for definitions of baroclinic and barotropic modes), the transient mode of the baroclinic mode has
the following tire--dependence:
(2.8.23)
The dimensional form of equation (2.8.23) is:
V
(if~VI
t~cyE
(3.2.1)
From equation (3.2.1) it can be seen that the time-dependent
term decays to 1/e of its initial value in the time:
tQ'/e)
H7-
(3.2.2)
Making use of the numbors from Table 1, this time is found
to be:
t (1/e) =.
0)L~o
.5\xkto '
=
x
- x\o
(5.00
xY\Idl
q
Seconds
(
'
SiLX\.98
x
to
.% 8oacs
The theory thus predicts that in approximately 19 days the
salinity and baroclinic mode of the velocity field will have
adjusted themselves to any change in the wind stress.
The response of the barotropic mode for the azimuthal
velocity (equation (2.8.24)), the radial and azimuthal velocities
in the boundary layers, and the vertical velocity in the interior
and the boundary layer is instantaneous in the theory presented
here. This is because the time scaling used in this analysis
has resulted in the time-dependence of the barotropic mode
being filtered out.
However, Holton (1964) has shown that the
response of these velocities to changes in the wind stress is
of the order of'E
f- .
Applying the values of Table 1:
E -P=
0.9 '1 x 1O'
4.2ct x 10
x 2.48 x I O
seconds
49
This means that the barotro-pic modes of the internal velocity
-76-
and the boundary layer velocities reach equilibrium much faster
than the baroclinic mode.
Other time-dependent motions which have been filtered out
by the scaling procedure are the internal and external gravity
wave modes wfich give rise to internal and external seiches.
Little is known about the internal motions in Lake Maracaibo
because there are no time series of temperature or salinity.
However, the tidal gauge at La Ensenada, at the north end of
the Lake, has provided sufficient data for determining the nature
of some of the external motions due to seiching.
Power spectrum
analysis of tidal records from La Ensenada was performed on Lake
surface records for the periods January 7 - February 3, 1966
and March h - March 31,
1966.
The numerical method of spectral
analysis described by Blackman and Tukey (1958) was coded in
FORTRAN IV and the IBMI System 360 in the
.I.T.
Laboratory was used to compute the spectra.
Civil Engineering
These spectra are
shown in Figures 18 and 19.
The semi-diurnal and diurnal tides are present in the Lake
as indicated by spectral peaks at 12 and 24 hours.
The peak at
6 hours in both records corresponds to the fundamental seiche
mode of the longest horizontal Lake dimension, the 5-hour peak
to the seiche mode of the shortest Lake dimension.
9 hours has not been explained, although it
The poak at
has been suggestbed
FIGURE 18.
0.5 Frequency, cycles/hour
0.4
0.3
0.2
0.1
Spectrum of Lake level during the period January 7. - February 3, 1966,
calculated from unpublished dato of the I.N.C.
-1
10
l0~ -
-10
-3
-3
1-
o10
CD
C
-
-4
-4
- 1o
1o-0-
10
o
-5
z
-a -5
-10
-
10
0
0.1
0.2
0.3
0.4
0.5
Frequency, cycles/hour
FIGURE 19.
Spectrum of Lake level during the period March 4 -March
calculated from unpublished dato of the I.N.C.
31,
1966.,
-79-
by Professor A. T. Ippen that it may be a seiche mode of the
combined Lake Ml-aracaibo-Tablazo Bay system.
-80-
IV. DISCIBSIUON
4.1
Comoarison of theory and data.-- Before beginning the
discussion of the theoretical results it
should be pointed out
that the chlorinity has been assumed to be linearly related
to the salinity by equation (1.10.1).
As a result, lines of
constant salinity will also be lines of constant chlorinity.
Furthermore, the density has been assumed to be a linear function
of salinity through equation (2.1.5) and lines of constant
density will also be lines of constant salinity and chlorinity.
In the following discussion the terms salinity, chlorinity and
density will be used interchangeably.
The theoretical results for the salinity and velocity profiles
are compared with the field measurements made by Iedfield et
al (1955) in Figures 20, 21, 22, and 23.
The major discrepancy
between theory hand field data is found in the vertical distribution
of chlorinity.
From Figures 20 and 21 it
is
evident that while
the slopes of the isochlors as predicted by the theoretical
analysis show fair agreont with the field data, the vertical
distributions in Figuro 22 do not.
This lack of agreement in
the vertical chlorinity'profiles has a noticeable effect on the
corresponding theorctical and actul aziruthal velocity profil3
0O
40
20
I
I
I
I
60
80
100
120
.
Kilometers
0
10
Meters
20
20
30
30
FIGURE
20. Theoretical
chlorinity (solid lines)
chlorinity ( dashed lines)
along
compared
Section AA.
with actual
20
40
60
80
100
120
Ki Iometers
Meters
20
30
Chlorinity - ppt
FIGURE 21.
-O
0
Theoretical chlorinity (solid lines) compared with actual
chlorinity (dashed lines) along Sec tion BB.
Chlorinity, ppt
.00
2.00
-,
Chlorinity, ppt
1.00
Chlorinity,
2.00
1.00
ppt
2.00
10
Meters
20
20
30
1.00
(a)
FIGURE
2 2.
2.00
1.00
(b)
2.00
1.00
2.00
Vertical profiles of theoretical (solid lines) and actual (dashed lines) chlorinity
at:(a) 60 kilometers, (b) 30 kilometers , (c) 0 kilometers, from the center of
the Lake. Actual chlorinity from Redfield et at (1955).
rrrm Position of actual bottom
30
0
0.2
0
0.4
0.2
0.4
0
.
0.2
0.4
0.6
. 9. .I
.
,
0
Meters
20
le
0
0
02 0.4
(a)
0.4
02
(b)
0
I
I
I
0.2
I
0.4
I-
'130
0.6
(c)
Azimuthal Velocity, meters/second
FIGURE 23.
Theoretical azimuthal velocity profiles(solid lines) compared with
velocity profiles(dashed lines) measured by Redfield et al (1955).
Station locations are shown in Figure I.
rrm,
Position
of actual bottom
given in Figure 23.
Figure 23(c) shows that the actual azi-
muthal velocity has a rapid increase in shear at about 17 meters
which is not evident in the theoretical profile.
This can be
related to differences in the vertical chlorinity profiles in
the following way.
Differentiating equation (26.51)
partially with respect
to z, and equation (2.5.1) with respect to r,
and eliminating
the pressure, p, gives:
Since the long-term salinity is
equation (4.1.1)
a function of z only,
can be written in terms of the total salinity
field, S:
(.1.
2)
For the steady-stato case, the following functional
relationshio holds:
s1 =
S'('(
.1.3)
-86-
Applying the chain-rule for partial derivatives:
where the subscripts refer to the variable being held constant.
Substituting equation (4.1.4) into equation
(4.1.2):
Equation (4.1.5) demonstrates the manner in which the shear
of the azimuthal velocity depends upon both the vertical chlorinity
distribution and the slope .of the isochlors.
From equation (2.8.29)
and Figure 22 it can be seen that in the theoretical analysis
the vertical chlorinity,
(,':-)
is a constant.
that the shear of the azimuthal velocity,
This iplies
, ill
be a
only.
The
function of the slope of the isochlors,
,
field measurements by Redfield shou that this is
also true, in
some areas, for the actua1 Lake.
The regions in which the actual
Lake chlorinity has an approximwtely constant gradient are near
the edge of the Lake as illustrated by Figures 22(a) and 22(b).
It
is
also near the outer parts of the Lake whoro the s
e of
-87-
the theoretical and actual azimuthal velocity profiles show best
agreement, as in Figures 23(a) and 23(b).
must be treated with caution since it
These comparisons
is in this area that the
geometry of the actual Lake differs considerably from the idealized
right-circular cylinder of constant thickness used in the
theoretical analysis.
In the central part of the Lake where the theoretical and
actual chlorinity profiles of Figure 22(c) do not agree well
one would also expect a corresponding difference in velocity
profiles.
The sharp change in the chlorinity gradient which
occurs at about 17-20 meters in the actual Lake should give
rise to a correspondingly sharp increase in the shear of the
azimuthal velocity, as can be seen from equation
(4.1.5).
The
actual velocity profile from the central part of the Lake does,
in fact, show this to be the case in Figure 23(c).
The theoreti-
cal profiles of both chlorinity and azimuthal velocity in this
area, as seen in Figures 22(c) and 23(c), do not show this sharp
change.
One must conclude from this that the theoretical model
represents the basic dynarics of the Lake correctly, but oversimplifies the vertical diffusion of salt by assuring a constant
gradient for the long-term component.
To obtain closer agree-
ment botween theory and field data it would be necessary to
assune a more conMlicated forn- for the long-term chlorinity
distribution, or reexamine the assumption of a constant eddy
-83-
diffusion coefficient.
Since the scale of motions at the bottom
of the Lake is considerably smaller than at the top, it might be
expected that the eddy diffusivity would decrease from top to
bottom.
4.2 Inadequacies of the theory.-- While the theory described in
the present study shows fair agreement with actual conditions in
the Lake there are certain aspects of the problem for which the
theory is either incomplete or unsatisfactory.
The most important
of these have to do with the wind stress, temperature effects,
tidal effects and salinity diffusion advection.
(1) Wind stress - In this analysis the wind stress was
chosen to be of a simple axisyTmetric form. More complex
forms of the wind stress could have been analyzed as long as
they were axisymmetric.
In view of the fact that there is
very little wind data from th6 south end of the Lake it seemed
pointless to make the problem more complex than available data
warranted.
It should be pointed out that the limited data
available indicate that the wind over the south end of the Lake
is from the west during most of the year.
Since the wind over the
north end of the Lake is predominantly from the northeast, the
resulting wind stress over the actual Lake is not markedly
different from the theoretical wind stress.
Further refinements
in the analysis could be nade by changing to a rectangular
coordinate system, in which case, the deviations from axial
syirmatry could be investigated.
(2) Temperature effects - The effects of temperature upon
the density field have not been included in this problem.
The
temperature-dependence of the density can be rritten approximateji:
where
(4.2.1)
OCT
I+o(
) =-..
eO is the density of water at 00 C., T is the temperature
in OC. and O(-
is the coefficient of thermal expansion.
The
Handbook of Physics and Chemistry (1964) gives 3.22 x 10~
OC1
for the value of M, at 290 C., which is a representative
temperature of the Lake.
Since the maximui variation of
temperature in the Lake is of the order of 20 C., from Figures
10 and 11, the maximum density variation due to temperature
differences is:
LAz
(-T)bffl
(3?L-
V.I
-90-
The maximum density variation due to changes in salinity
can be determined using equation (2.1 .5) and the values of
and
ASy given in Table 1:
V-
= 3
x0
YOA
S
per cc.
This result shows that the density variation due to temperature differences is nearly an order of magnitude less than that
due to salinity differences.
Therefore, it seems reasonable
to assume that the Lake is essentially isothermal.
However, it is noted that the shapes of the isotherms in
the Lake, from Figures 10 and 11, are very similar to the shapes
of the isochlors, and, therefore, the processes governing the
diffusion of heat are probably similar to those governing the
diffusion of the chlorides.
(3)
Tidal effects - The Lake communicates with the Gulf of
Venezuela by means of the Straits of IMaracaibo and Tablazo Bay,
permitting the tidal wave to enter the Lake.
As a result of
friction and the large size of the Lake compared to the width
of tho Straits, the tidal effects are much smaller in the Lake
than in the Straits of
Haracaibo or Tablazo Bay.
As pointed
-91-
out in paragraph 1.7, measurements by Redfield showed that
tidal velocities in the Lake are of the order of 0.05-0.1 meters
per second. This is almost an order of magnitude less than the
wind-driven currents, which are of the order of 0.5 meters per
second. The fact that the isochlors have a domed shape, which
is very different from the shape of isochlors resulting from
tidal mixing in estuaries or channels, also indicates that the
tidal mixing does not play an important in the diffusion of
salt in the Lake.
(4)Salinity
advection and diffusion - The theory, as
developed in this study, has provided a rationalization for the
domed shape of the isochlors observed in the Lake.
It does not
explain, however, the basic long-term stratification of the Lake.
The long-term salinity stratification of the Lake is determined
by the amount of salt intrusion from the Gulf into the Straits
of Maracaibo.
For a partially mixed estuary such as Tablazo
Bay and the Straits of Maracaibo, the isochlors slope dowmward
away from the source of high salinity, as can be seen in
Figures
5-9.
The water of highest chlorinity at the bottom of
the Straits, at the north end of the Lake where the Straits
enter the Lake, flows into the Lake along the bottom and provides
the highest chlorinity :mter observcd there.
Because the estuary
is only partially mixed the chlorinity at the northern entrance
-92-
to the Lake varies with depth.
If the Lake were motionless,
that is there were no wind-driven currents, it would be expected
that the Lake isochlors would be essentially horizontal and would
reflect the conditions found at the northern entrance to the
Lake.
Water of a particular chlorinity would intrude into the
Lake at a level appropriate to its density.
When the Lake is
set into motion these isochlors would then acquire the domed
shape in order to balance the induced velocity field.
In the
theory, the mechanism by which this occurs is that of vertical
advection and turbulent diffusion, and the theory predicts that
balance will be achieved in approximately 19 days.
As long
as the stratification in the partially mixed estuary remains
constant, sufficient chlorides will enter the Lake and maintain
the conditions described above.
However, the stratification
does not remain constant, but varies considerably from wet
season to dry season as the fresh-water velocity in the channel
varies.
It has also been suggested that the change in the
depth of the navigation channel affected the amount of salt
intrusion into the estuary and, therefore, into the Lake.
If the salinity in the estuary decrease, as actually occurs
during the wet season (see Figures 6 and 8), it would be
expected that the mixing processes in the Lake would eventually
result in a uniform chloririty throughout t'he Lake.
The length
of time required for this to occur can be estimated by considering
-93-
the characteristic time for the vertical diffusion of chloride.
This characteristic time is given by the term:
t(v e Ac
Using the values in Table 1:
-.
4.52 x
0-
= 518
ag
=
LA
\O
Secoitis
jears
The cycle associated with wet-dry periods is predominantly
annual, writh the wet season lasting 6 to 8 months, as given
in Figure 3. Because the time required for complete mixing
of the Lake is relatively long compared to the period during
which the chlorinity in the Straits would be expected to be
lower (the wet season),
the long-term chlorinity profile in
the Lake would remain fairly constant.
Another reason for a decrease in chlorinity in the Lake
would be dilution by fresh-water run-off and prdcipitation
over the Lake.
In paragraph 1.9 the time required to dilute
the Lake in this .rmOr was calculated to be about 5 years.
Hence, to produce an effect on the Lake it
would be necessary
-94-
to have a long-term change in the hydrologic balance, or to
construct a device which would permanently inhibit the intrusion
of chlorides from the Gulf of Venezuela into the Straits.
The converse problem, associated with a large increase in
the chlorinity of the Straits as was observed between 1958 and
1966, cannot be adequately treated by the methods described in
this study.
However, this study does provide estimates of the
coefficients of eddy viscosity and eddy diffusivity which would
be of value in any investigation relating to the rapid increase
in Lake chlorinity.
V. CONCLUSION
5.1 Sumay. --
Using perturbation techniques, an analytical
model of the diffusion of salt and momentum in Lake
has been established.
aracaibo
This model gives estimates of diffusion
time scales and predicts chlorinity and velocity in the Lake
as a function of position and time.
The time required for the
velocity field to reach equilibrium after a change in the
applied wind stress has been found to be about 19 days.
The
steady-state solutions for the velocity and chlorinity show
fair agreement ith
et al (1955).
data obtained in a survey made by Redfield
No field data are available to determine the
validity of the theoretical time-dependent motions.
5.2 Recommendations for further work.-- One of the most important
areas in which work is needed is that of deterining the actual
conditions in the Lake.
An annual survey, in which temperature,
salinity and velocity are determined, would be extremely useful.
Such a survey would be desirable for determining the changes
in the salt content of the Lake and the effects which these
changes might have on the dynuamics of the Lake.
information from various parts of the Lake,
Ieteorological
particularly fro
the south end, iTould also to esential for' det err.ining the wind
stress.
-96-
Another important aspect of the problem is that of matching
the salinity intrusion in the Straits with the mixing of the
salt in the Lake.
This analysis might begin with the one-
dimensional method of Ippen and Harleman (1961)
Section I.
described in
The work of Corona (1966) has indicated that this
method can be used with confidence in the Straits of Maracaibo
and Tablazo Bay.
Using the results of this work, the salinity
distribution along the channel can be estimated for a given
fresh-watcr discharge and channel geometry.
Knowledge of the
salinity distribution along the channel provides the necessary
boundary conditions for the diffusion of salt into the Lake.
An analysis concerned with salt flux into the Lake could
expand the theory described in Section II of this study and,
in this manner, a unified analysis of the Lake MaracaiboStraits of Maracaibo-Tzablazo Bay system would be achieved.
Such an analysis would provide a means for predicting
changes in Lake salinity as a result of changes in the hydrologic
balance and channel geometry.
It would be extremely useful,
for example, in evaluating techniques for reducing the chloride
content of the Like, such as those mentionod by Partheniades (1966).
-97-
VI. BIBLIOGRAPHY
Barcilon, V., and Pedlosky, J.
1967a. Linear theory of rotating stratified fluid motions.
J. Fluid Mech., 29, 1-19.
1967b.
A unified theory of homogeneous and stratified
rotating fluids. J. Fluid Mech., 29, 609-621.
Blacka1an, R. B., and Tukey, J. W.
1958.
The measurement of power spectra.
Dover Publi-
cations, New York.
Corona-Chuecos, L. F.
1964.
Balance hidrologico del Lago de 'Maracaibo.
Instituto
Nacional de Canalizaciones, Caracas, Venezuela.
1966.
Salinity intrusion and sedimentation in the Straits
of Maracaibo. M.S. Thesis, Massachusetts Institute
of Technology, Department of Civil Engineering.
Eknan, V. W.
On the influence of the Earth's rotation on ocean
1905.
currents. Arkiv Math. Astr. Fysik. Stockholm,
2 (11).
Greenspan, H. P., and Howard, L. N.
On a time dependent motion of a rotating fluid.
1963.
J. Fluid ech., 17, 385-44.
Holton, J.
1964.
t.
The role of viscosity in stratified rotating fluids.
Ph.D. Thesis, Massachusetts Institute of Technology,
Department of Meteorology.
Ippen, A. T.
1966.
Salinity intrusion in tidal estuaries. In Estuary
and Coastline Hydrodynamics (A. T. Ippen~ed.),
McGraw Hill, 1Hw York.
-98-
Ippen, A. T., and Harleman, D. R. F.
1961.
One dimensional analysis of salinity intrusion in
estuaries. Corrittee on Tidal Hydraulics, Corps of
Eigineers Tech. Bull. No. 5, Vicksburg, Mississippi.
Jahnke, E., and Emde, F.
Tables of functions.
1945.
Dover Publications, New York.
LaFond, E. C.
Processing oceanographic data. U. S. Navy Hydrographic
1951.
Office Pub. No. 614., Washington, D. C.
Munk, W., and Anderson, E. R.
Notes on a theory of the thermocline.
1948.
Res., 7, 276-295.
J. Marine
Partheniades, E.
1966.
Field investigations to determine sediment sources
and salinity intrusion in the Maracaibo Estuary,
Venezuela. M.I.T. Hydrodynamics Laboratory Tech.
Report No. 94, Cambridge, Massachusetts.
Redfield, A. C., Ketchum, B., and Bumpus, D.
The hydrography of Lake Maracaibo, Venezuela. Woods
1955.
Hole Oceanographic Institution, Woods Hole,
Massachusetts. Unpublished manuscript.
Stockhausen, P. J.
1964. A study of the salinity stratification of a distorted
rotating lake model. S.B. Thesis, Massachusetts
Institute of Technology, Department of Civil
Engineering.
Sverdrup, H. U., Johnson, M. W., and Fleming, R. H.
The oceans; their physics, chemistry, and general
1942.
biology. Prentice-Hall, New York.
1964.
The handbook of physics and chemistry (R. C. Weast, ed.).
The Chemical Rubber Co., Cleveland, Ohio.
-99-
VII. LIST OF SYMBOLS*
Subscrits
B
-
indicates a boundary layer variable
I
-
indicates an interior variable
n -
an integer, 0,1,2,----; and indicates the order of the
perturbation expansion
Superscripts
-
indicates a dimensionless quantity
(1) -
indicates the baroclinic mode
(2) -
indicates the barotropic mode
I II
-
indicates the boundary layer at the top of the Lake
indicates the boundary layer at the bottom of the Lake
Prefixes
S
-
indicates a perturbation quantity
-
indicates the maximum variation of a quantity
* Symbols used in Section I (Introduction) are not listed.
-100-
Symbols
S - 0.03
C
-
chlorinity =
E
-
Ekman number = )%
f
-
Coriolis parameter = 2.0. sin 4
g
-
force per unit mass due to gravity
H
-
a characteristic depth of the Lake
k -
1.805
the first root of the Bessel function of first kind and
order one (1), J, (k) = 0.
Jo -
Bessel function of the first kind and order zero (0)
Ji -
Bessel function of the first kind and order one (1)
L
-
a characteristic radius of the Lake
p
-
pressure
r
-
radial coordinate
S
-
salinity = 1.805 C + 0.03
t
-
time
u
-
radial velocity
0, for t (0
U_1 (t) -
the unit step function
1, for t > 0
v
-
azimuthal velocity
w
-
vertical velocity
z
-
vertical coordinate
CK
-
coefficient relating the density to the salinity
-
rotational Richardson nunber =L
IAk
-101-
-
ratio of horizontal to vertical eddy viscosity
-
horizontal Prandtl number =
-
vertical Prandtl number =
E
-
mechanical Rossby number =
0
-
internal Rossby number =
-
horizontal eddy diffusivity
-
vertical eddy diffusivity
-
aspect ratio
0
4-
=
-
=
4
.
horizontal eddy viscosity
-
vertical eddy viscosity
-
stretched vertical coordinate in the top boundary
layer =
stretched vertical coordinate in the bottom boundary
layer =
*/t
-
density
-
wind stress applied at the surface of the Lake
-
latitude as measured from the Equator
-
stream function
- Laplace transform of the stream function
- -
angular velocity of the Ear th
=
J
e
-102-
VIII. LIST OF FIGUIRES
Title
fgu ae're
a
1
Lake Maracaibo.
2
2
Average net monthly hydrologic balance.
7
3
Net monthly hydrologic balance for the period
1944-1960, 1964-1967.
8
h
Chlorinity in the epilimnion of Lake Maracaibo
13
during 1914-1960, 1964-1967.
5
16
North-south cross-section from the Gulf of
Venezuela to the south end of Lake Maracaibo showing chlorinity distribution in March 1954.
6
North-south cross-section from the Gulf of
17
Venezuela to the south end of Lake Maracaibo showing chlorinity distribution in May 1954.
7
18
North-south cross-section from the Gulf of
Venezuela to the south end of Lake Maracaibo showing chlorinity distribution in March 1963.
8
19
North-south cross-section from the Gulf of
Venezuela to the south end of Lake Maracaibo showing chlorinity distribution in December 1963.
9
20
North-south cross-s'ection from the Gulf of
Venezuela to the south end of Lake Maracaibo showing chlorinity distribution in June 1966.
10
Lake temperature and chlorinity along Section
AA' from Redfield et al (1955).
21
11
Lake temperature and chlorinity along Section
BB' from Redfield et al (1955).
22
12
Basic salinity distribution parameter's in the
Straits of aracaibo.
29
13
Basic sali nity distribution paraecters in the
Straits of ,aracaibo.
29
-103-
Figure
Title
Page
14
Comparison of idealized Lake with actual
Lake along Sections AA and BB'.
58
15
Velocity profiles at various positions in the
Lake as measured by Redfield et al (1955).
61
16
Theoretical velocity profiles at r = 30 ki.
70
17
Cross-section through the Lake showing theoretical chlorinity (salinity) profile.
73
18
Spectrum of Lake level during the period
77
January 7 - February 3, 1966.
19
Spectrum of Lake level during the period
March 4 - March 31, 1966.
78
20
Theoretical chlorinity compared with actual
chlorinity along Section AA'.
81
21
Theoretical chlorinity compared with actual
chlorinity along Section BB.
82
22
Vertical profiles of theoretical and actual
chlorinity at: (a) 60 km., (b) 30 km., (c) 0 km.,
from the center of the Lake.
83
23
Theoretical azimuthal velocity profiles
compared with velocity profiles measured by
84
Redfield et al (1955).
-104-
IX. LIST OF TABLES
Table
Title
Page
1
Numerical values of important constants.
67
2'
Values of theoretical radial, azimuthal
and vertical velocities at r = 30 km.
69
3
Values of theoretical chlorinity.
72