Calhoun: The NPS Institutional Archive

DSpace Repository

Theses and Dissertations 1. Thesis and Dissertation Collection, all items

1975-09

Tidal and current prediction for the Amazon's

North Channel using a

hydrodynamical-numerical model

Ferraz, Luiz Antonio de Carvalho

Monterey, California. Naval Postgraduate School

http://hdl.handle.net/10945/21094

Downloaded from NPS Archive: Calhoun

TIDAL AND CURRENT PREDICTIONFOR THE AMAZON'S NORTH CHANNEL

USING A HYDRODYNAMICAL-NUMERICAL MODEL

Luiz Antonio de Carvalho Ferraz

DUDLEY KNOX LIBRARY^^NAVAL P0ST6RADUATI SCHOOL

MONTEREY. CALIFORNIA M940

JAVAL POSTGRADU

Monterey, California

p.ui %j %0

HTIDAL AND CURRENT PREDICTIONFOR THE AMAZON'S NORTH CHANNEL

USING A HYDRODYNAMICAL-NUMERICAL MODEL

by

Luiz Antonio de Carvalho Ferraz

September 1975

Thesis Advisor Stevens P. Tucker

Approved for public release; distribution unlimited.

Un c i ass if i edSECURITY CLASSIFICATION OF THIS PAGE (When Date Entered)

REPORT DOCUMENTATION PAGE READ INSTRUCTIONSBEFORE COMPLETING FORM

1. REPORT NU M8ER 2. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER

4. "t ITLE (end Subtitle)

Tidal and Current Prediction for theAmazon's North Channel Using a Hydrodynamical-Numerical Model

5. TYPE OF REPORT ft PERIOD COVERED

Master's Thesis;September 19 7 5

6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR!"*;

Luiz Antonio de Carvalho Ferraz

8. CONTRACT OR GRANT NUMBERf.)

9. PERFORMING ORGANIZATION NAME AND ADDRESS

Naval Postgraduate SchoolMonterey, California 93940

10. PROGRAM ELEMENT. PROJECT TASKAREA ft WORK UNIT NUMBERS

II. CONTROLLING OFFICE NAME AND ADDRESS

Naval Postgraduate SchoolMonterey, California 93940

12. REPORT DATE

September 197513. NUMBER OF PAGES

14. MONITORING AGENCY NAME ft AODRESSf// different from Controlling Otllce)

Naval Postgraduate SchoolMonterey, California 93940

15. SECURITY CLASS, (of thle report)

Unclassified15«. DECLASSIFICATION/ DOWNGRADING

SCHEDULE

16. DISTRIBUTION ST ATEMEN T (ol thle Report)

Approved for public release; distribution unlimited

17. DISTRIBUTION STATEMENT (of the abetract entered In Block 20, If different from Report)

16. SUPPLEMENTARY NOTES

1). KEY WORDS (Continue on reveree elde It neceeeary end Identity by block number)

Hydrodynamical-numerical modelAmazon RiverNumerical modelTidal modelAmazon's tides

20. ABSTRACT (Continue on reveree elde It neceeeary and Identity by block number)

The hydrodynamical-numerical prediction model developed byW. Hansen is applied to the North Channel of the Amazon Riverfor computation of tides and currents; the results are com-pared with tidal prediction obtained by the harmonic methodand to actual current measurements. A medium size grid ofsquare mesh cells, 1800 m in length, represents the NorthChannel. The driving forces are the tides at the northern

DD I janM73 1473 EDITION OF I NOV 65 IS OBSOLETE

(Page 1) S/ N 0102-014-6601|

Unci assif iedSECURITY CLASSIFICATION OF THIS PAGE (When Date Entered)

UnclassifiedJuCUKtTY CLASSIFICATION OF THIS PAGEr»>>,n Dr(m Enlmrml

opening of the channel near the river's mouth and the riverdischarge into the channel at the southern end. The numer-ical results for tides were verified at three tidal stations,and it was observed that the tides predicted at the northernpart of the channel agreed, in the worse case, within 12% ofthe tidal range, but those predicted at the southern endwere unsatisfactorily reproduced. This fact is attributedto the size of the grid which is too coarse to describeadequately the variable and irregular cross -sections andbottom topography at the southern part of the channel. Thepredicted currents were in acceptable agreement with the fewavailable measurements.

DD Form 1473 „ . . _. ,

. l Jan 73 Unclns.si f ion —S/N 0102-014-G601 SECURITY CLASSIFICATION OF THIS PAGEfWh.n D.(. Enf.r.d,

Tidal and Current Prediction for the Amazon's North Channel

Using a Hydrodynamical-Numerical Model.

by

Luiz Antonio de Carvalho FerrazLieutenant-Commander, Brazilian Navy

B.S., United States Naval Postgraduate School, 1974,

Submitted in partial fulfillment of the

requirements for the degree of

MASTER OF SCIENCE IN OCEANOGRAPHY

C.I

DUDLEY KNOX LIBRARYNAVAL POSTGRADUATE SCHOOLMONTEREY, CALIFORNIA 93940

ABSTRACT

The hydrodynamical-numerical prediction model developed by

W. Hansen is applied to the North Channel of the Amazon River

for computation of tides and currents; the results are compared

with tidal prediction obtained by the harmonic method and to

actual current measurements. A medium size grid of square mesh

cells, 1800 m in length, represents the North Channel. The

driving forces are the tides at the northern opening of the

channel near the river's mouth and the river discharge into

the channel at the southern end. The numerical results for

tides were verified at three tidal stations, and it was ob-

served that the tides predicted at the northern part of the

channel agreed, in the worse case, within 12% of the tidal

range, but those predicted at the. southern end were unsatis-

factorily reproduced. This fact is attributed to the size of

the grid which is too coarse to describe adequately the var-

iable and irregular cross -sections and bottom topography at

the southern part of the channel. The predicted currents were

in acceptable agreement with the few available measurements.

TABLE OF CONTENTS

I. INTRODUCTION 13

A. REVIEW 13

B. OBJECTIVE 14

II. DESCRIPTION OF THE NORTH CHANNEL OF THEAMAZON RIVER 16

A. GENERAL -- 16

B. CHARACTERISTICS OF THE TIDES AND CURRENTSIN THE AMAZON RIVER 17

C. THE AREA OF STUDY 18

III. THE HANSEN HYDRODYNAMICAL MODEL 21

A. ASSUMPTIONS AND BASIC EQUATIONS 21

B. FINITE DIFFERENCE EQUATIONS 23

C. GRID SELECTION 28

D. INPUT OF TIDES AND RIVER DISCHARGE 29

E. TUNING THE MODEL 34

F. MODEL 34

IV. DISCUSSION 58

V. SUMMARY 63

APPENDIX A: NUMERICAL PROGRAMS 64

BIBLIOGRAPHY 81

INITIAL DISTRIBUTION LIST 83

LIST OF TABLES

I. Compilation of Some Discharge Measurements andInvestigations in the Amazon River and Estuary 32

II. Current Velocity and Direction Predicted at theThree Tide Stations by the HN-Model 37

III. Computer Output for Water Elevation and ResultantCurrent Velocity and Direction in 12 SelectedLocations 54

LIST OF FIGURES

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

North Channel of the Amazon 19

Grid Net Scheme 24

Grid Net Laid over the North Channel of the Amazon 30

Amazon River Cross-Section near Macapa,Looking Downstream 33

Tuning the Model 35

Current Velocity at the Three Tide Stations 38

(a

(b

(a

(b

(a

(b

(a

(b

(a

(b

(a

(b

(a

(b

(a

(b

(a

(b

Currents in the North Channel after 75 Hours 41

Currents in the North Channel after 76 Hours 41

Currents in the North Channel after 77 Hours 42

Currents in the North Channel after 78 Hours 42

Currents in the North Channel after 79 Hours 43

Currents in the North Channel after 80 Hours 43

Currents in the North Channel after 81 Hours 44

Currents in the North Channel after 82 Hours 44

Currents in the North Channel after 83 Hours 45

Currents in the North Channel after 84 Hours 45

Currents in the North Channel after 85 Hours 46

Currents in the North Channel after 86 Hours 46

Currents in the North Channel after 87 Hours 47

Currents in the North Channel after 88 Hours 47

Currents in the North Channel after 89 Hours 48

Currents in the North Channel after 90 Hours 48

Currents in the North Channel after 91 Hours 49

Currents in the North Channel after 92 Hours 49

16. (a) Currents in the North Channel after 93 Hours 50

(b) Currents in the North Channel after 94 Hours 50

17. (a) Currents in the North Channel after 95 Hours 51

(b) Currents in the North Channel after 96 Hours 51

18. (a) Currents in the North Channel after 97 Hours 52

(b) Currents in the North Channel after 98 Hours 52

19. Currents in the North Channel after 99 Hours 53

20. Prediction of Tides at Limao do Curua 55

21. Prediction of Tides at Acaituba 56

22. Prediction of Tides at Macapa 57

TABLE OF SYMBOLS AND ABBREVIATIONS

cm centimeter

cm 2 square centimeter

cm 3 cubic centimeter

cm/sec centimeter per second

cm 2 /sec cubic centimeter per second

d total difference

F friction force

f Coriolis parameter

g acceleration of gravity

H total depth (H = h + c)

h depth of water at mean water level

HM harmonic method of prediction of tides

HN hydrodynamical-numerical prediction model

hr hour

HTU,HTV depth at U- and V- grid points

HT high tide

HTZ symbolic depth at Z-grid points

K coefficient of horizontal kinematic eddy viscosity

km kilometer

km 2 square kilometer

L characteristic length

I length of the mesh

LT low tide

M,N orgrid coordinates

m meter

m 2 square meter

m 3• cubic meter

m 3 /sec cubic meter per second

n mile nautical mile

NE, ME grid delimeters

NEH, MEH near border grid coordinates(NEH = NE - 1, MEH = ME - 1)

p hydrostatic pressure

R bottom roughness coefficient

r coefficient of bottom friction

t time

u, v velocity components

U, V mean u and v components

V total velocity vector

Wf >. , W, >. wind speed components

X,Y or x,y space coordinates

Z, U, V grid points

a smoothing parameter

B 1 - a

6 partial difference

Y specific weight

X drag coefficient

c, water level anomaly

p density of the fluid

t wind stress

t° bottom stress

10

a) earth angular velocity

A difference, also, in figures, represents tide station

At half-time step

V 2 |^r + ^yr (Laplacian)

11

ACKNOWLEDGEMENT

I wish to express my sincere thanks to everyone who has

assisted in the preparation of this study, in particular to

the personnel of the Department of Oceanography of the Envi-

ronmental Prediction Research Facility, especially to Dr.

Taivo Laevastu, Mr. Kevin Rabe, and Mr. Arthur D. Stroud.

Finally to Dr. Stevens P. Tucker, my thesis advisor, my

gratitude for his patience, understanding, translation of

German references and bibliography and for his constant con-

structive criticism that greatly contributed to the completion

of this project.

12

I. INTRODUCTION

A. REVIEW

Pressure gradients are the dominant mechanisms in rivers,

narrow estuaries and some lakes which are opposed by friction

[21]. In large scale circulation the opposing forces are com-

binations of friction and the Coriolis force or the latter

alone.

Hydraulic models have been used for many years to reproduce

the physical characteristics of rivers and have proven to be

reliable, but space and operating problems make them a rela-

tively expensive tool of investigation. The appearance of high

speed computers made possible the numerical solution of the

differential equations used to describe flow. A physical model

can be transformed into a program which may be modified easily

to simulate different conditions and may be stored for future

utilization and alteration, thus allowing solutions to be ob-

tained rapidly and at a reasonable cost.

The hydrodynamical-numerical method developed by W. Hansen

makes it possible to determine the motion in oceanic areas, in

adjacent and marginal seas, in shelf areas as well as in estu-

aries and rivers [15, 18]. In many cases the method has been

applied successfully to the quantitative reproduction of tides

and storm surges. The numerical treatment of the hydrodynamical

equations requires the bathymetry, boundary geometry, and

meteorological and density structure to be considered.

13

This model has been applied to rivers such as the Eider,

the Ems and the Elbe, in Germany [9, 17, 18]. The Eider is a

tidal river in northern Germany with very irregular cross-

sections and a remarkably irregular depth distribution. Ap-

plication of the HN-model with a very fine mesh, to approximate

the cross-sections by a number of parallel canals of differing

width and depth, produced good results with a maximum deviation

of 30 cm between computed and observed tides, which is less

than 10% of the tidal range. Later computations with improve-

ments in the longitudinal direction from grid point to grid

point made it possible to reduce the deviations between the

computed and observed values to below 3.5 cm, which was about

II of the tidal range [17]. Similar applications of the HN-

model were made to the Ems and the Elbe, and the results were

in all cases of the same order of magnitude. For the Elbe

the differences were smaller than 2 cm [9, 17].

In this study the hydrodynamical-numerical prediction model

of Hansen is applied to the Amazon's North Channel and the

results are compared to field observations.

B. OBJECTIVE

The objective of this thesis is to predict by means of the

Hansen hydrodynamical-numerical model tides and currents which

exist in the North Channel of the Amazon River during an inter-

mediate river stage and to compare the numerical results with

available field measurements. The computed tides are compared

to prediction by harmonic method at three tide stations along

the channel and the currents arc compared to field observations

at selected locations.

14

A medium size square mesh grid is used to represent the

North Channel's configuration and bottom topography. The

numerical results will provide means of evaluating this rep

resentation in terms of grid size as well as of the merits

and limitations of the HN-model.

15

II. DESCRIPTION OF THE NORTH CHANNEL OF THE AMAZON RIVER

A. GENERAL

The Amazon River is located in South America, where it begins

its eastward flow from a chain of glacier-fed lakes in the Andes

Mountains in central Peru. It emerges from the eastern foothills

of the Andes, sweeps through the world's largest tropical rain

forest, and discharges into the Atlantic Ocean on the northern

coast of Brazil. En route to the Atlantic the river drains

about 7 x 10 6 km 2 of land, receiving the flow of over 200 tributaries,

The Amazon is the world's largest river in terms of drainage

area and discharge. Its discharge is subjected to seasonal

fluctuations. Two periods are generally considered: (1) that

from April to June, the rain season or the high water stage;

and (2) that from October to December, the season of low pre-

cipitation or low water stage.

Investigations and measurements made by the University of

Brazil, the Brazilian Navy, and the U.S. Geological Survey in

1963 and 1964 produced some astonishing numbers for the flow

of the river. The flow at Obidos (some 850 km from the mouth)

during the high stage measured about 216,000 m 3/sec, during low

stage about 72,500 m 3/sec, and during an intermediate stage,

164,520 m 3/sec. The average flow, estimated for the discharge

into the Atlantic Ocean, was slightly over 175,000 m 3/sec [2,19,20].

The river is generally deep, and in more than ten locations

depths greater than 100 m have been found. The width of the

16

channel ranges from about 2 km at its narrowest up to about

10 km in some places, with long sections having intermediate

values [2 , 19]

.

Gibbs [12] and Diegues [3] have shown that the Amazon waters

spread over the Atlantic Ocean in a variable layer, 10 to 20 m

thick, under which the oceanic water progresses towards the

shore. Gibbs [12] showed that during the high stage of the

river a vertical stratification ranges out over the continental

shelf, to distances of 80 to 230 km from the mouth, and during

the low stage it ranges out to 60 to 185 km. However, all

vertical isohalines indicative of turbulent mixing are limited

to the mouth of the river, and there is never penetration of

salt water into the river as occurs in some types of estuaries

[12, 13].

B. CHARACTERISTICS OF THE TIDES ANDCURRENTS IN THE AMAZON RIVER

The Amazon River seems to have the longest river tide- zone

in the world [1]. Tides are observed up to the vicinity of

Santarem harbor, some 800 km from the river's mouth, but during

extremely low flows tidal effects have reached upstream to

Obidos. The tides are of semidiurnal type with a maximum range

of about 5 m. Field measurements indicate that the tidal vari-

ation decreases upstream.

Tides and river flow combine to produce strong currents in

many sites upstream. The North Channel (Fig. 1) has a single

main entrance from the sea, but numerous small tributaries and

channels also offer secondary connections to the ocean.

17

In many of these channels a tidal bore is a peculiar character-

istic.

Franco [8] characterizes the current in the North Channel

in the vicinity of Macapa as "alternating-axial" (rectilinear

or reversing), that is, the alternating current component

superimposed on the main downstream river flow is maximum up-

stream during the high water, maximum downstream during the

low water and it is zero when the tide is at the mean water

level. There is a relatively strong permanent current down-

stream due to the enormous discharge of the river.

C. THE AREA OF STUDY

The Amazon discharges into the Atlantic in two branches.

The present study is limited to a region of the upper branch

which constitutes the North Channel, extending some 147 km from

the mouth. This channel is the navigable waterway from the

river's mouth to the ports of Macapa and Santana in the

Brazilian Federal Territory of Amapa (Fig. 1).

The Port of Santana is utilized mainly by the large vessels

that carry manganese ore extracted from the mines of Serra do

Navio, inland of Amapa. Navigation in the North Channel is

somewhat difficult due to natural conditions in the river, such

as enormous floating trees, sometimes attached to floating

grass "islands," pulled down from the adjacent margins by ero-

sion, strong currents, and a large tidal variation. The harbor

at Santana is of the floating type to compensate for the large

tidal range and for the seasonal fluctuations of the river

level. The port of Macapa consists mainly of a wharf where

18

19

small vessels and typical river boats load and unload passengers

and light cargoes.

The Channel is generally oriented in a SW-NE direction, has

an average depth of 20 m and is never narrower than 5.5 km. In

some places, as in the vicinity of Macapa, the configuration of

the river bed is quite variable, with irregular cross-sections

and bottom topography. The bottom is composed mainly of mud

and sand, and the banks and bars along the river are also of

these materials.

A tidal bore does not manifest itself in the North Channel,

but in the shallow sites near the mouth of the small tributar-

ies and channels discharging into the main river bores are of

very common occurrence.

20

III. THE HANSEN HYDRODYNAMICAL MODEL

A. ASSUMPTIONS AND BASIC EQUATIONS

The assumptions leading to a Hansen-type hydrodynamical

model are the following:

1. The fluid is homogeneous and incompressible;

2. Pressure is hydrostatic, and thus the changes in pressureare due solely to changes in water surface elevation;

3. Advection terms are neglectable;

4. The fluid is in hydrostatic equilibrium in the verticaldirection; and

5. The geographical and vertical variations of the Coriolisforce may be neglected.

The basic equations for the single layer hydrodynamical

model developed by Hansen follow from the application of these

assumptions to the equations of conservation of momentum.

Reference [10] presents the derivation of the equations.

Horizontal momentum equations are integrated over depth

to give:

«U . fV = - g II KV*U - I^El + l£l + X (1)

|V + a . . g |i + Kv2y . I^izl + lizl + Y (2)

fe |_CHU) + 6 (HV) . (3)6t 6x 6y

The wind stress components are represented by:

21

T, , = AW [W 2 + W 2]

! /2

(x) x L x y J

T r ,= XW [W 2 + W 2

]

1 / 2

(y) yL x y

J

(4)

(5)

The bottom stress components are given by

xb(x) = rU [U

2 + V 2]

1 /2

Tb(y) = rV [U

2 + V 2]

1/ 2

(6)

(7)

The various terms in the above equations being defined as:

K = coefficient of horizontal kinematic eddy viscosity

X = drag coefficient

W ,W = wind speed components

C = surface elevation

H = total depth (H = h + ?)

g = acceleration of gravity

f = Coriolis parameter

x,y = space coordinates

X,Y = components of external forces

U,V = water velocity components

( u = i/hudz >v irjvdz )

r = bottom friction coefficient

V =6

+6

Jx Zy

The equations for bottom friction and wind stress are

empirically developed. The wind stress terms as used in the

model are assumed to be quadratic expressions in wind velocity,

where A is the wind drag coefficient with a typical value of

22

3.2 x 10" [6, 7, 10]. The bottom stress is assumed to be

nonlinearly dependent on U and V, and like the wind stress

it was derived by empirical means. It was originally formu-

lated for a shallow water application and the use of this

value for deep water is questionable, since it probably should

be smaller in such an application. The value of r for this

particular case was recommended by Laevastu (personal commun-

ication) as 2.8 x 10 to 3.0 x 10

B. FINITE DIFFERENCE EQUATIONS

The differential equations are solved by a "leapfrog" or

central-difference scheme for time dependent solutions. At

the boundaries the values of the velocity components and sur-

face elevation are taken at actual points rather than from the

surrounding points as shown in Fig. 2.

The driving forces are the tides at the open boundaries

and the wind at the surface over the entire field. The tidal

values are computed at each time step and introduced as new

values of £ at each point on the boundary.

The finite-difference form of the equations is given below.

Using the equation of conservation of mass, the water elevation

is first calculated:

Ct + T (n,m) = c

t_T(n,m) - X[H*(n,m) U

t(n,m) - H*(n,m-1) U^n.m-l)

+ H^Cn-^m) Vt(n-l,m) - H*(n,m) V

t(n,m)] (8)

23

CN

O

E>.(A

o3

UD

D

>-Ctf

<

OCO

LU

zoi

M

00

zo

O —

0-

CM *-

I 9 1

2"h—I

—

k—I

—

A ^

©—I—e—|—©ro^-Z

24

The horizontal and vertical velocity components are then

determined from the respective horizontal and vertical momentum

equations

:

Ut+2x

(n,m) = [1 - (2xr/ h£+2t

(n,m) ) (Ut(n,m) 2 + V* (n ,m)2 )

1Aut(n ,m) ]

+ 2x^(^171) - I^[c

t+T(n,m + 1) -

Ct+T (n,m)]

+ 2TXt+2x

(n,m) (9)

Vt + 2T

(n,m) = [1 - (2xr/ H^+2

fn ,m) ) (V* (n ,m) 2 + Ul(n ,m) 2

) & V1(n,m) ]

- 2TfU t(n,m) - L% t+T

( n ,m) - Ct+T

(n+1 ,m)

]

+ 2TYt+2T

(n,m) (10)

where:

U^n.m) = aUt(n,m) + ^ [U

t(n-l,m) + U

t(n+l,m) + U

t(n,m+1)

+ Ut(n,m-1)] (11)

V (n,m) and t> (n,m) have expressions analogous to U (n,m),

Ut(n,m) = j [U

t(n,m-1) + U

t(n+l,m-l) + U

t(n,m) + U

t (n+l,m)],

*t *t (12)and V (n,m) is similarly analogous to U (n,m).

The time step is 2t . The total depth Hu

or Hy

at u- or

v-points is computed as:

H1 * 2T

,(n,m) = h (n,m) + I U t+T(n,m) + C

t+T(n,m + 1)] (13)

u (or v) K> u KI

The effects of wind are computed by:

25

t At t 2 t 2 1/ 1 (5px* - £**[(**] up ]A - I«| (14]

Yt

=H W

y [("x)2+

C*y>Y* " £|f (15)

where P is the barometric pressure. It is assumed that the

pressure gradient is zero for small areas and normal conditions

The stability of the finite-difference scheme is governed

by the Courant-Friedrich-Levy criterion which states that the

maximum length of the time step is determined by the grid size,

£, and maximum depth, Hmax

At =

*/2gH (16)& max * *

For this particular grid, a value of 30 seconds was used for

the one-half time step, At.

When the computational area contains small regions of great

depth, a "false-bottom" can be used in these sections [7], For

example, areas deeper than 500 m may be assumed to be 500 m

deep. This will often result in a considerable increase in

time step or a decrease in the grid length if total computa-

tion time is a critical factor. Experimental results have

shown that the errors introduced with this procedure are

acceptable in some cases for practical applications of the

model

.

The grid net is formed by three different sets of points

as shown in Fig. 2, the water elevation (Z-points) and the

two components of the velocity (U- and V-points) . Each of

these points has the same coordinate designation (N,M)

.

26

The coastline must pass through the U- and V-points and never

through the Z-points. The program reads the values of the

depth at U- and V-points as HTU and HTV, respectively.

The Z-points are for water, land and coast designation,

being read by the program as HTZ . They are treated symboli-

cally as 1, and -1 respectively. The HTU and HTV show the

chart depth in centimeters, while the coastline is symbolized

by -1 and the land by 0, depending on their location in the

grid. At the open boundaries the points HTZ have symbolic

values of -2 and -3 for the first and second boundary when

applicable. Outside these boundaries, values of for HTZ

are prescribed.

The coefficient of horizontal kinematic eddy viscosity is

related to the values of the U and V components. Hansen states

that the computations are always stable for values of $ =

(1 - ot) between and 0.5. This coefficient can also be used

as a tuning factor: the higher the value, the smoother the

current velocity distribution. In areas where the depth

distribution is irregular, as in this case, accelerations and

surface irregularities appear in the model. These abnormalities

can be corrected for by means of a proper smoothing of the

bottom or of the water surface elevation; in this model this

is accomplished by smoothing the water surface elevation, which

is a numerical artifact of the model.

In finite-difference form, the coefficient of horizontal

kinematic eddy viscosity is given by the relation:

K = A£ 2(1 - q)

4 At

27

For this model with a value of alpha equal to 0.983 recommended

by Laevastu (personal communication), K becomes 0.46 x 10"

cm 2 /sec .

•

High values of alpha must be used with caution, because

these can allow undesirable oscillations in the model. Also,

if too low a value is used to provide higher smoothing, the

current velocity may be overly damped.

As was pointed out in the assumptions of the Hansen model,

the mean advective terms, u.6u./<5x., have been neglected. This*

j i j&

is not necessarily a good assumption as these terms can be the

same order of magnitude as the local acceleration terms, 6u./<5t,

In the derivation of the time averaged equations of motion, the

turbulent contribution of the advective terms, as expressed by

the Reynold's stresses, have been included using the lineariz-

ing eddy diffusivity approximation. Hence, the non-linear

effects are not included but the amplitude effects of the

advective terms are included by the eddy diffusivity approx-

imation.

C. GRID SELECTION

The grid size is selected upon requirements of detail and

accuracy and the availability of computer core memory and time.

There is not a general rule for selection of the grid mesh,

but for open areas and round smooth bays, where the expected

values of current velocity and direction fields are smooth, a

fine mesh is not necessary, but for areas where the topography

is of primary importance, or in narrow channels, a fine mesh

may be required for reliable results.

28

For this particular model, a medium size grid was initially

considered satisfactory (see discussion on page 61 below).

Decreasing the size of the mesh will result in an increase of

time and computer memory. A 24 x 65 unit array of square mesh

cells having sides 1800 m in length was laid over the North

Channel of the Amazon River as it is represented in the Brazilian

Nautical Chart No. 220 (BRASIL - RIO AMAZONAS, carta 220 "Da

Barra Norte ao Porto de Santana," DHN - Rio de Janeiro)

.

Experiments performed at the Environmental Prediction

Research Facility (EPRF) have proven that there is not much

increase in accuracy for smaller grid size; for example, a

model for the San Diego Bay was run under three different

meshes, and except in computation of advection and dispersion

of pollutants, no significant differences were recorded between

fine and coarse grids.

D. INPUT OF TIDES AND RIVER DISCHARGE

The tides in this model were inputed at the northern opening

in the grid which was chosen and oriented such that the closest

tidal station in that location lay over or near a grid-point,

along the M = 2 grid line. This tidal entrance is parallel

to the y- coordinate.

The tides at the station LIMAO DO CURUA (Fig. 3) were

introduced with seven components, M2, S2, N2, K2 , L2, Kl and

MS4. Two other tide stations provide means of comparison for

the tides predicted by the HN-model: ACAITUBA and MACAPA,

58 km and 142 km upstream, respectively. Usually only four

components are introduced in the model, but the values of the

29

•J<cj

H<

2<i—

(

Jt—

i

PQ

WacH

QWE->

WcowPi

wPi

co

H

CJ

t—i

CO<

CJ2

CL,

o

oI—

I

HCJWPi

WHWH<CJi—

i

Q2

co j* * <

- ,o b Q>• "> #V 2° «? • —

1

<o w

2:~

cf z Z° -7 < ^

ffi# u Q'.

WPi PQo <2 -J

1 . CO CO- 122 o

tf> o cti •

' X tsi Pi CO

§j<

< m <ac 2

V m h t—

i

! ac QH Pi

'.'

Pi o oo

W oj CJ-o j > Cn] i

o P1 • »—

i

7 A H O pi

22 cj

lO /) i H SQ Pi

/ i—

(

< Q/ Pi DC z/ CJ CJ <

30

seven constituents introduced in this particular case were

considered quite significant.

The river discharge was inputed as permanent current,

averaged from the bottom to the surface, in a cross-section

near Macapa along the y-grid line at the southern end of the

grid. This input boundary was left open to insure continuity.

There is no adequate measurement of the river discharge

in the North Channel at the present time. There are, however,

some estimates and calculations of the river discharge at

Macapa and at the mouth and some current measurements made by

the Brazilian Navy for navigation purposes. Table I shows a

compilation of the more recent investigations and measurements

for the Amazon's discharge.

The flow of the Amazon is seasonal and can be correlated

with precipitation in the entire Amazon region. Diegues [3]

states that during the year the variation of the level of the

river is quite sufficient, the water level being closely re-

lated to precipitation in the Amazon Basin to a large degree

and the melting of snow in the Andes Mountains to a small

degree. Such variations are periodic and follow a yearly

cycle.

The value used for this model is an estimated averaged

value for Macapa, 175,000 m 3 /sec, and it is most generally

accepted. Such value which is expected to represent the mean

or intermediate stage for the river was distributed over the

cross-section in Macapa, as shown in Fig. 4, to produce the

mean velocities (permanent downstream flow) to be inputed in

grid points, in a manner similar to that described by Dronkers [4].

31

""vf «* -3-

vO <o vOcn cn cnrH rH rH

00 co CO #* 9t ^ COvo vO \£D rH rH CN] <N] i—

i

vO<D en cn CT) 03 03 r-- r-- 03 cnO i—

i

t—

1

rH cn cn rHP P +-» r-i rH P3 •v n 9k CD CD CD •»

o « PJ d m, A Cco 03 o3 03 03 03 if) Crt o3 oj

s S s M Cn] ,Q XI tsj eP p p 3 3 X) pQ 3 +->

rH rH rH o O •H •H O rHO O O CO CO CD o CO O

(D

to o O O "St O o CO CDP O o CD o o CD o r-- Oo3 <L) o lo o rH LO ** to OrC </) *l a •1 #* 1 9k #\ •t 0k

O \ \D CNl LO cn to r-~ CnJ LO(/) CO rH r^- VO vO CO o r-i t--

•H 6 Cn] rH rH (NJ CN] rHQ

X4-> u

C -H CD t». CTl LO <3" LO o o <*03 O C/) O) t^. LO O- to to cn <-i

<D O *v^ 1

S rH 6 r-t o rH I—

1

rH o CD rH<D

>

o CD CD "5f "5f o o COed o CD CD LO vO o o rHCD CM o •* CD o CO o o Cn)

P g 1

< o cni vO LO -5t o CD vOt—

1

en O CO rH r-- rH COr—

1

rH CN) to ^-i

rCP CD o O r-- to o o "tT3 6 CT> \o CO cn CO o o o•H rg CN] CM CD vO LO LO r-g

& #v #\ r* #* *. •s #»1

cni <N] CN] CNl Csl

rH rH

o\PP 0) rC ti X rC r* X cCD bO DO £ 03 bO bO z bO bO o3 CD> crj •H o <D •H •H o •H •H CD CNJ

•H p X -J 2 X EC -4 EC EC SOS CO Mh

o

CDc rg

o to c/) W C/) in rH rH 03 CNJ c•H o o O 2 3 (-* X a, X •HP 13 H3 T3 03 03 P P 03 +-> rHrt •H •H •H C r-1 3 3 U 3 03

u x> ,0 X o3 03 O O 03 o Xo o O o 2 s 2 2 s S O <D

-J•H

>H4J

to tO «* tO tO Q> o3

vO vO mo vO VO x: s<d 1 i 1 I 1 r- r-» to to +-> •H4-> *o CD en CD cni vO sO vO •o +-1

crj r-( Csl rsj cn) cn cn cn en 4-> (A>

Q 1 1

rHrH

i

COi rH i—

i

rH rH <i—

1

cu

CN]

0)

•p

wco•HP03

bO•HPV)

O>G•H

T3Ccrj

wPCd>

6CD

rH

3wcrj

<D

6

<D

bOrH

o3

XUW•Hh3

CD

SOt/i

*

<P XO rH

03

c 3o p•H to

p W03

rH XI•H cCU 03

6o PCJ o

>1 •H

OS—

I

(=;

w o—

1

Npq o3

<^

32

± <

Ec

- in

- *f

ro

- CM

cch-</>

z

oQ

Oz

oo

<Q_<<

cc

<LlJ

z

zoh-oUJ(/)

j

</>

(/)

oa:

o

zOM<

UJ

D

o o OCO

(^) Hidaa

33

E. TUNING THE MODEL

Tuning the model is a matter of patience. Several runs of

the program were made with different combinations of alpha and

the bottom friction parameter. After each run both the water

elevation and currents were compared to harmonic method and

current measurements available. The model is left to run for

some hours of real time with these combinations of parameters

until stability is attained.

In general, the model appears to be rather insensitive to

variations in the bottom friction coefficient, but it responds

quickly to small variations in alpha. The behavior of the tidal

curve at the same location for two different values of alpha

(a = 0.980, a = 0.975) and two different values of bottom fric-

tion parameter (r = 0.0028, r = 0.0030), is illustrated in Fig. 5

When changing alpha the bottom friction parameter was maintained

constant and vice-versa.

Due to a lack of adequate current measurements in the North

Channel the model was mainly adjusted for the tides, and the

velocities of the currents were considered satisfactory when

the tide agreement was reached and the currents approached the

values and directions of the few available observations.

F. MODEL

The calculated currents behave as described by Franco [8],

that is, they are what is termed "alternating-axial" or some-

times "rectilinear" or "reversing." The maximum downstream

velocities appear at low tide (LT) and they are greater at the

first LT than in the second. Similarly, maximum upstream

34

o inoo r^CD <J)

o ooCO C\J

o oo o

in

UJDC

Do

II 1

1

JZ

'xO-o

II

if

7

/f o"0^

f

"00

11 f^ 1

1 1 1 I

8CvJ

o oo o1

<>J

o oo oCM T—

HID

O

UU3

OO

OO

wH-

1

CJI—

I

PL,

Phwocj

oHcjI—

I

«:Ph

wac

Q<QWCD

<accj

co

<aca,

<

>o<

JWQO

wan

CJ

i—

i

anH

QWCJz<uCO

Q<H<HCO

OCJ

CO

<a.

<

oweq

2:<E-«

CO

OCJ

QWl-H

<H2

CO

35

velocities occur at the high tide (HT) . Slack water corresponds

to the crossings of the mean level.

At the grid points representing the tidal stations, the

respective maximum and minimum predicted values of current were

as follows:

(1) At Limao do Curua, grid coordinates (17,2), 196 cm/sec

and 183 cm/sec for the first and second LT and 162 cm/sec and

157 cm/sec for the first and second HT. The direction changed

from 062° (downstream) to 242° (upstream)

.

(2) At Acaituba, grid coordinates (11,26), 115 cm/sec and

107 cm/sec for the first and second LT and 90 cm/sec and 87

cm/sec for the first and second HT. The direction changed from

032° (downstream) to 210° (upstream)

.

(3) At Macapa, grid coordinates (7,63), 62 cm/sec and 60

cm/sec for the first and second LT and 42 cm/sec and 40 cm/sec

for the first and second HT. The direction changed from 024°

(upstream) to 201° (downstream)

.

At these locations a phase lag of about 40 minutes between

the instant of the high tide and low tide and the peak of the

current is observed, but it is possible that this lag may be

smaller, since the output of one hour interval of the program

did not allow a precise determination of the exact instant of

the low and high tide (Table II).

Figure 6 shows a plot of the current velocities at the

three mentioned tide stations for the last 24 hours for a

simulated 100 hours (real time) run of the program.

36

TIME LIMAO DO CURUA ACAITUBA MACAPA

VEL DIR VEL DIR VEL DIR

cm/ sec cm/sec cm/ sec

76 73 242 80 210 40 196< :ht77 11 062 43 211 42 220

78 100 061 11 027 33 206

79 165 062 60 031 13 202

80 196 062<;lt91 031 21 029

81 192 063 109 032 43 026

82 151 061 115 032< LT54 025

83 75 062 103 032 60 024<:lt

84 29 242 65 033 62 024

85 121 242 12 203 58 024

86 162 242<:ht71 209,^HT

41 024

87 150 240 90 209 5 174

88 98 242 84 210 32 194

89 25 242 58 210 40 201<:ht

90 60 061 13 213 36 206

91 132 062 41 030 21 200

92 174 06 0<. LT

76 031 9 049

93 183 062 97 031 35 029

94 158 062 107 032 <It 49 027

95 98 062 103 032 56 026<LT96 6 060 76 032 60 026

97 93 240 16 037 58 026

98 151 24 2<HT 56 209 47 027

99 157 242 86 209 12 043

100 118A .

242 87 210 25 194

TABLE II - Current velocity and direction predicted by the

HN-model at the three tidal stations. The left-hand columnindicates the last 24 hrs of a 100-hr (real time) computerrun. The right-hand columns indicate the approximate time

of the low tides (LT) and high tides (HT)

.

37

200^_LIMAO DO CURUA

N

-200U TIME (hr)

ACAITUBA

- 150 TIME (hr)

MACAPA

TIME (hr)

CURRENT AT TIDAL STATIONS

FIGURE 6

38

Plots of the currents in the North Channel are shown in

Figs. 7 (a,b) through 19 in which the currents are scaled so

that 1 cm corresponds to 157 cm/sec and the arrows indicate

the directions. The full lines on the graphs represent the

symbolic boundaries, the triangles represent the location of

the tidal stations, and the ellipses the location of selected

points. (It should be noted that the plotting subroutines used

for these plots are not included in the program of Appendix A.)

The water elevation and the resultant current velocity at the

selected points are shown in Table III, which is a reproduction

of the computer output for the special points.

The tides predicted by the model are shown in Figs. 20

through 22 for the last 24 hours of a simulated 100 hours (real

time) period. In these figures is included the difference

between the prediction by the numerical model (HN) and the

prediction by harmonic method (HM) . For Limao do Curua (Fig.

20) the maximum difference was 19 cm which represents 4% of

the tidal range and a phase lag of less than 30 minutes is

observed only at the first LT. For Acaituba (Fig. 21) the

maximum difference was 41 cm which represents 11.7% of the

tidal range but a phase lag of about 20 minutes is observed

between the HN and HM prediction. This can be partially

explained by the fact that the actual tide station lies in

much shallower water than the closest grid point chosen to

represent it and also because the distance from this grid

point to the station is about 2 km.

39

For Macapa (Fig. 20) the results were not satisfactory.

An attempt to improve the results, by smoothing the bottom,

resulted in the curve labeled HN1 in the figure, but further

attempts were abandoned because they would not be realistic

with the actual mesh size.

40

-I o• • • 6 '

«

• <M>

Eo

o

©

-a3

r^VSbH 3d y.-Ujd iNiibdno jjs N&?skd SbH 9t! aiJi.-id no;

41

O O -U f \ T T

?o?tv/ r / pg Sjy 1

/1/'

iNdbd '0 • m -'..•• SbH 8^ d31Jb IN^bdHO 3JS ND2U

42

o o1

U .1 A. f r

? O ?rH .

oe

Eooo

;:

- -. • .•.--•

43

rt

, Ll ig y_q |tj•

..... -•. LM • Mi

44

- -' - ]J LN3bdn3 3dS KlQZdWd:

45

D II I

I Il -

-J U| ! / /. / ".-rJ

. iG Si -v..-

46

" - -•

.:.

47

SbH 65 H^ldU lNldtiDO 3JS .

.-.. . •

' N u

48

SbH 26 dJldb iSJ3dan3 OJS NOZbWU

49

;

. 3SHP3 Od hi!

50

, . u o 3 I

«iil

'

O O 3, 1 ,

: S'ifyy.1 V

*

o 3E-o_ • < . * *

t i i i" 'J s

; ) : : *m -.

<• t r . E ' i / I . . . , .

8 , 'I

f r /u

'

« / 4 k i u•

• ok i i I «

-. -.-.:-..•

1

T '•

' ''

• _TCN W : ...:..-. ,0 -

1

r

A r<

—

.J

oi^

,..:.-.,...-.:,

(

'

r r

ro. 4 . ro ..... . .

T

T

1—f

1 1

r i

f T

...• -.

> * « t

-.-.-.;-. ;

f

•

r•

r

r

1 1

1 \

1 1

'

* c *

. ::..•-. 1 .-. . -. :

i 1 \ '

< ^ « ~T" .-•...-.:-.-.-. ,

i . r 1 ^ \, r ( . ....-.-.•.:-. 0O - - - ;

! .

r 1 ^ h o

«. V \ \ S v 1 3 I, J) UJ

, s \ \ S S \—1 TL >

^—i, \ \ \ v\ h-\+ a =

L-1 , ^ \ \ \ ..

I-—, «;

'

. ^ V \ \ V V ^~|^ u-c « ' < « ' |v O .--..-.

f

-. -. -. - = r c « * . « <-)<;

t

1 •'

T"|. < v \ \ \ -v -irr-

-. -. -. -~~\-

-. y \ ^ <. ^ ^10

,T],\\\\S\ i| .:-.::. - 1 c \\*«.\\1

. . r \ w ^. \ \ \ r~ • ~ Tt^»vVT S

t- f b- ^ > b)

o -.:-.: t ** f(J

* » «. \ g) T

r ^ \

f l

7L1\ \ \

r 1 i

r 1 [

:-.-.-.,-.-.-.: "U1 1

T 1 1

f i r

,-.-..

*r \

, \ —

r 1 !

r r \

f T 1 ^^ :

' T 1 f

r f 1

I 1 !

• ' "-

r \

V

r f <

^ .6: CO

F f 1

00

'

'^\ o- • b ' ' '^ '

. . -. -. . r^v^ <»« T & \ 1 \ ,

:

, v \ \ .i -v-- -. v \ \ ', T Lt^

"l. ;','/' ''-.

'-.'-.'-.

. . . , ^\ W1 , >

o 1: : -. -.:-.-. , C> r r

(5 , ,«- ^

.

••

'

: -J;

: i / ; :^

TION

POINT

-f-

f

3'-

f *

[ ;

o.

/ /

/

T

'1

1 1

]

' (—^ii

TION

POINT

5!

/ /

/ /

,

:

'rn

1 v -

So f ]

; 1' J 1

< f

\ 1

r[J:

.

V)UI,

' • _1 ' 00 LU ' >

1

• ' K i^ ,-JU ^

,

1 |

—

<Ui -f \ > • ir,

!, < uj -|T\ \ \ V *

i-co ac

f \ \ V

<

*1 .

• 1 , 1 r 1 * 1 -I.N

3'-A 31

- 31JU 1 n;3bi

-

]

.

- ii

r^I t!

- -- :" "

.

51

o o Lc

'J u / /

I // /

1/ /

t I I

1

/ '

,

)•

• CN

OV*EVoo

25i- a.

<ui— U4H-00<

"» \ N Nl N

L_\

o-Q3

i-i°i-i

:- f"

•-.

.••-•: .

• SbH ge y^iju . ;. c^s nqzi

52

ShH 66 ajidd iN3bdno ojs no?uwu

53

I Iff

I St'. I

(75 Hr)

(76 Hr)

( \ : l . .u -

1

. 7 . , _ 7 -. .

7TTT

.251

-jjLli II lAJMUJl;

J . 2

. . I <7.

-IHL^I Ijf. |n 3J H SPF1.1.U I'JI'.IS

l~4. 5

. l . .>

; i

""i

—

~i l . . i- : l

.j..u

12 i

1 --,. 3

i 7. ,7 {2

). >0

,117

131.

i . . 2 1 -

1 Jl .OJ

?. .257.

1 14. <l

51 ., '7 J.

7r. i

2 1 . . ' :

3

(77 Hr)

717-hTTT

(78 Hr)

ol . I

1.'

. . / 7

.

I -i. 7n

. C 26

13S.21

2 > . .'

I-. I ,

. 1 1 4 J .

lii. M>. ,21 J.

KO .1 i

7..261.

I 40 . )

j 7 . , '•- i.

I ' 7.4

!. , 23)

. i' j 1 .

I J I . -.I

II . ,2L^

1 I . . I 1?. J

i. ,2j->

2g44j 0.

(79 Hr)

-92

1. 5 . . ' 1 .

J. I

7

;

l >.l 3

• 35

4. 6

24 . , 44 .

24 . 7 3

. 34 .

it. 70

1-. . ?

38 .2

d . . 3 5'

73. 3

I 3 . , ? '

(80 Br)

(81 Hr)

. . "34 .

-22. u43. . ,-7

- I > . pi I

II. i 71

I .. '

M. . 2-1

73.03

i 43

-oO. 7

. 7. , 43 57

-34 . -»' -4 / ,«J4

44. , 77

52..

(82 Br)

(63 3r)

(84 Hr)

»CO0J

j

(85 Hr)

. 3 .'1

- 1 II. ..

I .' 7 . , « :

-in)1. i

J '. . . 2 2

32.

M

. 4 3.

-1 33.5-.

51., )).

-I )4.4l

. J . 3 •> .

- 30. I. 3

. . 5

-t)3.=4

-74.7

6 1., 2 7

''-Silt-

Jl . , 2 7

-67773

3 7., dt .

-7*77

1 . , =13

- 7 2.4.

7., Jl,

-3V. I

. J . , 2 '.

•..06

1 d'».

-1 J. <; J

;. , 34.

-2 0.0

7., 26

-3J.1)

74 . , 54 .

' 5 . . 6 2.

- 77. 5)

5-.. , 2 7.

"-42. 7

33., 2d.

•72 .-.

-44.15

., 3

-72.15

3 5., 91.

70., ;i.

-36. 7

5 3., 24

(86 Hr)

1 7.1.6.

>L .,2 7

2. 7s -..7

:i4.

'3. 3

J). ,225

-4.20

1 2. ,21 J

43 . .rj

I 7 . , 2 I

36.2 2

14. , 1 )).

16.47

24., 3.3.

-1 2.C

, 1. , 2 .

3133^.

(67 Hr)

(88 Hr)

•>. ?

li.J

1- = . 1•»

.. ,20 1

IjJ.J 1 --.. 7

..,20 1

.,21'.

10 7.

30. .743.

92.43

I0..271.

47.1

5. , 174

1-3. ^5'. ,210

1 J<J.H

. .'54

1 i I ) . 8 3

. 3 . , 2 7 .

74.2

' 2. , 194

7u»0-i

(89 Hr)

. I 3. 7(_

• . , 2 7

5. l7

2 00.

|4fc.4»

,.,201 ,224

147. 2",

34. 221

.

145.1

7

.5. ,23

I -,3. 52

4 7 . , 2 I

1 » 1 .66

>0. ,25-).

I4H. 15

67. ,270.

125. J

43. , 2J1

(90 Hr)

4. 7

1 II

'3.24

,227

112.65

4 4. ,242.

1 14. 33

I . . »1

115.43

>7. ,263

1 16. 32

54. ,26<).

1 IB.

2

36. ,236

'? 7t,00

(91 Hr)

27.2 7

29., 2 2

5 3.60

4. ,27

,7. 77

262.

6 3.7

21 . ,203

3 3120

(92 Hr)

1 15.

1.2., 'I

60. 13

, 72. >•.. 22 .

5.i:

. , o2

J34B0O.

(93 Hr)

>.42

2i3.

-32.4

4 0., 74

33t,*^,

(94 Hr)

-•(6.-7

,5 . ,

-U7.3

>7. , 40

(95 Hr)

(96 Hr)

-4b. 6-4

- 7. . 39.

3q . , 3 3

-9i).*2

79., 5V.

74. 14

, 2 7.

75. , 6 56. , 2 7

-^6-77T?

. 7. . 32

-6d. JJ

- 3l.-

37. , 2o

TTTT

34 4 2-0 <:

(97 Hr)

-4 6-, 5-4

47.

rt-»i4

1 •! . , 3 6.

—3 2.3 3

57. , 61

-52. O?

4). , 2

-50 .6

. . 85.

-35. 7

53., 26

(98 Hr) 1 - 4 ., 2 7

1

3 3.5-7

,142.

-,3.32

20. , 2i 7.

2^.2 3

13., 40.

16. Jt

. , 140,

(99 Hr-)

3-7 4.ft0

26 5 . 7 i, . , 2 ) 1

--tl 9 . ?o

> 3 . , 2 2 3

- 4*,'3

37. ,2 )S.

34.4

M.,237.

74. 7-

, ,276,

2o.3

12. . 43

IV.. il TTTTTtT

4. , 253

"cTTT

(100 Hr) • 2^2.

rrrv-—rrnr. 1 ;

.

, 2 J

1

oTu2 2 3.

14 3.44

. . ,'36.

125.6;

.,271,

TABLE II

CurrentI. Computer Output for WaterVelocity and Direction in 12 S

Elevation and Resultantelected Locations

54

o

Z XX X

Z XX X

<Don

DuOoO<X

I I L . I L

/

J I I

o o a

(™) 1H9I3H HWH- NH

55

z 21

1

Z XI I

CMUJ

=>

CO

D

<<

oco -U

I L J I

i"1) 1H9I3H (iamo) WH-NH

5 6

I I

z 2 zx x x

CNCM

D

<<<2

UJ

o"eo

("0 1H9I3H

57

IV. DISCUSSION

Before discussing the results obtained upon applying the

hydrodynamical-numerical prediction model developed by Hansen

to the Amazon's North Channel, it is worthwhile to recall some

of its properties or characteristics.

The model represents topographic features in terms of grid

points. To obtain a good bathymetric representation a small

mesh is required, which may sometimes lead to an extremely large

array. It is therefore necessary to compromise between spatial

resolution and the amount of computer core memory and time

available

.

The time resolution is also dependent upon availability of

computer time; in general, the smaller the time step the greater

the computer time required.

The flow is described by finite difference methods of solu-

tion of the equations of motion, which include terms of differ-

ent orders and degrees, each one describing specific characteristics

of the flow. These depend on the external forces applied, loca-

tion, shape, bathymetry and properties inherent to the fluid.

In describing the flow by these equations, it happens that some

of the terms become dominant, so that others can be neglected

or included in parameters such as K. In addition, Hansen's

model neglects the advective non-linear terms, which appears

not to cause problems if medium or coarse grids are used but

seems to be significant for small grids, as mentioned before.

58

The finite difference method of solution of the equations

of motion makes continuous computation impossible. The calcu-

lations are performed by time and spatial increments which can

lead to enormous errors if the proper precautions are not taken.

One example is that of the eddy diffusivity coefficient, which

is affected directly by the grid size and the time step since

it is defined in terms of spatial and temporal steps.

The action of viscosity in fluids is of a complicated nature

It may be assumed for the water motion in almost all tidal

channels that a complete state of turbulence occurs, so that

the flow is practically independent of the exact- value of the

Reynold's number. Because of the mathematical difficulties

involved, the general equations of fluid motion, which include

the viscous forces, are not generally solved in applied hydro-

dynamics. However, the result obtained by completely neglecting

the bottom friction is, only under the most favorable circum-

stances, a satisfactory approximation to the actual fluid

motion. Therefore, to determine the flow conditions in a

channel, the influence of friction cannot be neglected [4].

To include friction, because the velocity varies in the

horizontal as well as in the vertical directions in the cross-

sections of the channel, there remains only the possibility

of experimentally determining it by tuning the model so that

predicted values match as closely as possible the observations.

In tidal channels and inland rivers, the river beds are

usually irregular and form hydraulically rough surfaces, tur-

bulence being generated by the irregularities of the bottom.

59

At the bottom the velocity is zero and increases in the

vertical direction until it reaches a maximum value at some

distance from the free surface. From this point to the sur-

face the velocities decrease slightly. Observations in very

wide channels have shown that the sides of the channel have

practically no influence on the velocity distribution in the

central region [4]

.

In engineering and practical problems the determination

of the mean velocity in a cross-section is usually sufficient,

thus making it necessary to have a relationship between the

mean velocity and the frictional forces.

Hansen's model assumes that the bottom stress is directly

dependent on U and V but only indirectly on X and t, and the

same constant friction coefficient is used in all computations

(for this case r = 0.0029).

Finally, the setting of the boundaries and the representa-

tion of the coast line and margins is also a matter of personal

judgement, and because of this, often the representation of the

real boundaries along the grid-points is somewhat difficult and

may be questionable.

For our particular case, some facts have contributed to

make adequate conclusions impossible. First, the river dis-

charge is inputed as a permanent flow using an estimated value,

because at the present time there are no available measurements

of the discharge in the Amazon's North Channel. The distribu-

tion of the mean velocities in the cross -sect ion was made at

only five grid-points which received relatively large values,

clearly affecting the resolution.

60

Second, there is insufficient adequate current measure-

ments to provide a better adjustment of the predicted current

velocities, either for the improvement of the estimated river

discharge, neither for posterior verifications.

Third, the only tidal station in the middle portion of the

Channel, ACAITUBA, had its constituents determined in a very-

short period of time, and the harmonic prediction for that

station, using only few constituents, certainly is not the

best. Furthermore, as mentioned before, this station is lo-

cated about 2 km from the closest grid-point, which in turn

is located in much deeper water.

In spite of these facts the model produced numerical results

that, if not optimal, are at least acceptable.

The tides predicted at the northern station, LI MAO DO CURUA,

agreed with the harmonic prediction within 4% of the tidal range

and no phase lag is observed. The maximum difference for the

station at ACAITUBA is about 12% of the tidal range, and a

phase lag of about 20 minutes is observed, but both may be

attributed to the fact that the actual station is over shallower

water than it is represented in the grid-point and to the dis-

tance from the grid-point to it.

At MACAPA, however, the results are unsatisfactory. This

can be, at least partially, attributed to the fact that the

grid size is too coarse to describe correctly the bottom to-

pography and the cross-sections in the southern 18 km of the

Channel, both very irregular. In fact, smoothing the large

gradients between the depths at U- and V- grid-points of same

61

coordinate designation by making them shallower resulted in a

better curve, as is illustrated in Fig. 22. No further im-

provement was tried because, due to the actual mesh size, the

new "false-bottom" would not be realistic, and the procedure

was abandoned. A similar situation is reported by Ramming [17]

when modelling the Eider River in Germany: the first attempts

to reproduce the tides in areas of very irregular cross-sections

and depths were not satisfactory, in particular with regard to

the steep flood part of the tidal curve. Differences of about

60 cm were recorded in a 3.8 m tidal range. Representing the

cross-section by a fine structure of depth distribution, i.e.,

using a finer grid, led to results in which the maximum devia-

tion was less than 10% of the tidal range and later improvements

reduced this deviation to less than \% of the tidal range [17]

.

The prediction of currents seems reasonable. At least at

the four points where our current predictions were compared

to actual observations made by the Brazilian Navy, in identical

conditions of river stage, the results are quite satisfactory.

62

V. SUMMARY

The hydrodynamical-numerical prediction model developed by

W. Hansen was applied to the North Channel of the Amazon River

for prediction of tides and currents.

The tides predicted by the model agreed, in the worse case,

to within 121 of the tidal variations at two stations, but the

results at a third location weren't satisfactory. This is

attributed to the fact that the irregular shape of the channel

in the vicinity of this station was not adequately represented

by the grid. Improvement in the results was obtained by smooth-

ing the bottom at the U- and V- grid-points in order to decrease

the gradients existing between these points, but no further

adjustments were tried because they would be not realistic with

the mesh size used. The currents predicted by the model appear

to be satisfactorily reproduced, but more measurements are

required to allow further conclusions.

It is the intention of the author to continue the investi-

gations in this field to achieve improved results, both by

decreasing the mesh dimensions and by using new measurements

to be obtained in the near future. Continuation of the study

will also be in the form of a two-layer HN-model which the

author is preparing for the Amazon Estuary, for which more

data and information are available.

63

APPENDIX A

NUMERICAL PROGRAMS

oLUM-J 5!CL >-< LULU LUQ- ><X CO OQ O< <(J< h- J— **-« kh lOzoa id 2: h-t- o h- k-_j cc< a. lu h- zLU 2TQ.O 2: CO 3 LLLL 2 LL LUO <T < >-< LU I— OO h* O 3Oc^'-'uos: q »-i o ^-2Ilu_j»-h _i —* h- 2:2 a. 2 <-*>«IO < h- CO OO O h- CO2*-trvJh- y~ O 2 •-<•-< I— >-< CO t—oa:< lu a u. a »-i- 3 •»- 2 2•"-1 c£2XO -' O O <<X •Q.t—< o •-<

>-2;rQt-Ht— 2: > _J_Ji— 2co~J cj O00 3< -. . j OQCO-.20 CL-•i^ 10000 a < aaz iQQ. _jQ< LUST 1-* ~Z.cC O C£«:a>-'OC£ < fvi

IJJSMQ LL —* LU 1-1 LULUO> Itl OQi<s<r-5 •*<— — (— \— f-i— >-i^i— 1-1 i—01 c£_j«. 21- 22 22-jo<2 I- <UJQ^UQt'J < CO«-« LU LL *-4 •-<< 2: *-<

_IXLU2UJ>-«CO *— 3 O Q—«>-t LL CO<h-LL»-»>cOO • 22 H- 22'-' t~2 Qcoco XO >-4 >-*l— LU>—1 »-* h- _»>-«|— J— -< ^_|_0OCO|—-'ixO^Qc:a:>—co O I— colu co <lu H-22i-f-a.0:010 >—co2 h-c*: co a:co o£ acc^u. ,oac cc lu.-hh-.22luLU JZZIUmO *-"LL) ZK LU LU LULUOW- 111 LU COOO'-o-.O COs:_j<r oxa:o \~ h- oz^-a h- h-t— coi— m- h- aiaoZ)lU>COr-s|h-LU COlUQC OLULU2 LU LULUI— 2LULU UJ h- 0.G.O JZO<<0-JKKh-22iXlU< H--J ZSLU t-iSO 51 SZUJO^S 2 COZ>> 2 LUhhH->— t-«|-.LU <<t/>z<z^:qj< o<3^2_jo<o <r <<2coo < < <x. ox •—3: <>-»-<•-< lu

—i<o>-.<2<:o o^ujq< ,-'o:iij nci— c^gc ooc a: •—• i— »— _^ u_ u-«'-<ou.^^a:<x *— <tct*~, <3L".-ta iL<co <T2<r<i— _i2< < ll<<>-i— » ll^zz- i-^-oo'-uo i <h- oa.<i--«u-a. a.Luo.a.co<oa. occlluql <<a: oqomhdqzzci:i-m OO OX < OTOH-LU LL >-« CSiOO O NJujLLCOiO LU 2 2 ZZmwGSi C^JUJOU. QLU<rLU OLUO UJOLULUi—h-.a-:'_U h-LU»-iH-OXXtOcOh- '-•^-'cOcO.-.'-'OOLL<l— OtDiuco o >H(.Q.tu.OK t— »-< >— L- LL H- >y> L- <I(— coco I—^-XX< 3 :u .2:2: ^O.Q_2c£>-'0200 t~< ^O <^) <LL<< < 0< cOGLak-l— 3 -CZOCl- V)

>-OZWZ -JUJ hOQ X—«LU<I'-<LL^-"-*LL'J_LL>-'a: — »~<QLULULULUa.a. ^LL2IO'-<—i OLLX'-LqCQ2Q <cO'-UO J.QZ I—OOOXOluQQOOOOluCO-<SQQOuULUO:uOO>-<I— t—aOOOOO V-OX0C«-O 3UJ>-il— 2cOUJ30-LUOlUlu lUKZL'JQiMD Q3m h- zh-<i<z h-

a:iijzzu;iJu.kL/) ^: — z'jjozt-j^oz ^^^^^z—srot-^-i-i _j2ri: <u-i-—ioiQCo^<ox<—< 1 h- •-•t—o^f-Lu'^^ici'-'Oi «:^ c^2) ci -<<i<;n^a:0 |jjuj>-'00 lulu'uo>,(— HX'.UQ.JZUJLLiJiiDI- UJ-J -JjOLUuLJQic^ari-UOOLULLLL-JOOLUUJcnOcQOt— Q.Q.LU3D -Q'^>-'

x rvji— (~»5:z:ci.'~"ui_aaiO'^)i~-3- i K^hOQOi-uZhJjiiH.t-t-i-iOSHHiSQzss^l-Da:<^2 3a.i:2;Dai:D;/^ ZiZZ 2 -<2< <2TOO<IO 32ZOQhDddZ

co djjci; 2:<a.co»-«cO'o<< ,--'<'-,>_•0 ,-<'-'0'J30—«<?!:«-<XX<<<3:i:^<2»—^oul Z! ZZ--JCL2K -•

2; - a. 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11-----?

11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11

LU cCt. -.~ J-\ m -? *-» -% ^.^»^.^«> ^5

^ ^ujZli- ** -* ^"^ < co »—<— 5:5:^:^:5:—l ^i2 of-<o ™» m » o>—^^— •.•.•,»•.•. •.

< I— Oct - -^w<2« O O OO-acQ zzzzz^: -«X JZOI-^h hOI«- h--»<l— 22<<t<0 —.w— *->-.—

- o-* -JOl/)i-.v. —2Q-OOC0 —«t— —+-Q.Q.0.LJ »-<0>0>nj-3 LU II II i.U

q;h-<wX J>HJZt3'-'w '-LI'-« <<t<<<Z3:-irO_lhs.O'Jt-k-l- «»00<t<<—«--iiO<lLUCLJ« «<««CD''Q!flUOUUOOOOOOQOIIIIIIw«HTni;^^J_

I

CCcoLU a.O to_J 2< <> >a:

CO >-0il-OcO ct:<r 1

-ID <02LUO CO020«-<•—< 020-«LL> -"OOco

LU 1-OcQDaa COCD LLona. •-< 2LLLU a:2LU—i^js: LULUClO

1— Q.O *CO \-x 00 1-

<LL < 02_J CCh- 2LULU OO <cooa:1—

1

LULU IiOiu_ M>sJ 0»-<LUO

»—t ft LLCOOOor-j-JciQ

64

> >-a: ao oCD CQ

h- t- 0000 00 h- H-CL j- en o 00 Z(-4 O t-H a. X z z <-*

t: U_ a UL z h- —I >-4 OO O<t •• t— *• •—4 c£ zz aaC 1— o O o o <t <t >—»—

1

u3 3 UJ O UJ -J z a: a: 10 _iO LUQ-h- 00 00 < UJ UJ h-< <3a: azz 00 a UJ > > Oct -<a. <H-||JJ h- f- K ^.-» ^— —

«

13 < < Ouj1- 0£ <00 Z <IoOoO zrzt- a: oo 5T> UJ

o XUJ-J ^~ »•<-< 1— 1— UJUJ f-h- O Z) oo<ooQ-O Z -JI-U-< oOZ O oozz a:r£U_ Z3 Z Z I— 00UJ >~i <r^u_3 t— UJ a H-UJ»-i oc-jtjo t- oo <T < &AX.2Lh- h- ZiilwO Zq: zcto OO z 00 H-t— DO«hZ> h- OQOLU -ho: _i i-ioca oo • uj l- z O 'J50u_u_a<ICl _J 1*4 OO < OZ? _J .-.M* a: h-HH z zz az z o h-QoOoO Q-O i—

i

aui- h-f-O 1—

O

•cf oa: >-i >—«>—toOOO H»o •—• 1— 0- —<UJ|— h- UJ o O zza. z<-< i— uia X XOUJUJJ-OmZ Z5 DmZZ fvl h-k Ul h-i—

a

UJ'JJq: <u_ 00 a f-^h-i— <H-t—oahOK Q.-J Oa.w-< -H OZoO a OZI- ZZUJl-hii- •» z iflzzozoa:«ah<:>—oz< <mGQ 00 auia: <n Q.UO <<l-Z_jUJ 1/) a UJUJU, OUJOUJi—<•—it— Z5z i-i-z.-«—• _iaa ZZUJ zzo SZZUJ.JO a —

<

cxujQ:5;ci::z>Q(Z)00O Z>< o CZ>a o H-(<(_ u, -«< CCcC—'>-«ooo o t— "2Q£ooo:oo<ujujo•" Q.hhh UJ JZS>> —

1

STUI o 2__J UJUJ OUJ h- < xv-o z> zs a:i-zzDZza t—4 >-^ a: luocs: u.ck:< aa^MQiQ O0 f— i- ujo cco at: q: a' cc _jo<GOaUJLUoO q:ioqq CD QUJm oo OLUm OU_wZ Z3 •—•CO O OOujijj<u.-•-'o'Siac c£ — zz a—i UJ Q-O IJL U- U- U. •-• Q a 3: u. a. u_ a. u_ k t—

—

Z5I— OQCcvQi -3(*<r< u_ 00 LU f— 00 UJ OO LUC2 Z 2 JO O ZZO00Q.<ZODDQ c?o too UJULO < uju,a UJOUJ iu>-hcs:q o< UJ UJ UJ—1 *-* Ul LU5!f-Q Oua.O'X'u.ODii i-a 2; \— OoO l-kl-OUJO>l-2lJJU •-•> \— V— f— 1— H- CL>~0_>>-t z^i xoo <t Q »—* < zz< a — >—< i- c' cc Z <r Z << oo <oa >-n~z.v- 1— oi'-t oiv-H-uja:-^z:K-_) Q zi-u, uj uj «-« oo x^k-ZZU. oaiu. —t.'J—<— 2 1 >-H

?:D3zzaK^o<<c UJ » ~<Z>UJ a: *-oa ZZQ2 aooQjjzJO I-ZOZ3QD3U.Ql-OU.'-' ,.UUJI-l-ZK ZI-LUQQ.M ^o^Qd woctiiai-azujic a:u_za ujOujluOQujwoxi^z ^<ia—<<'/)oo>-<ijjr-: ~£;zu.ujOiuci:Zci:ujcl a.;?-—<za£ujS3soi-a)0—<a s: ^>z'H> - i.UJ •h3<J<oj|-UTI ?r -O—t 1—OX Z)—"UX Z" 5"cci— LUh- Oai x;h 2: ,^s:^jc ».;^'r)c:

ad!— OllZ:Z>-h:X^'—<!— 1—G HrtO 2 UjO -l j .jj —)O uj _) ij--. ^o IjjUJUL oz'H-OJCUJOUJ jj— —lUJaJ00 u_a: c£Q 000.0- — _J 1 ooz DZOuQZOJO^>-"y-3a:t-S:i; ZJJSUHUI- f-CX>".ZH-

Oujo—'Uj uuZO—»Zujuj:£ujuj Z—'O -2 z oz z a:o uj>-h 2'-''- ujC)>-*>— z zz< ^<zZo:zQaa«2:a.MQQOQ::s<:io-'«;MZ<ZM3>Miio'J<«hh--JOi-hD —4>—4i—l>OZ«—

<

II II II II II II II II II II II II II II II II

-JIIUl

o—10-to-• 11 11 11 11 z -s:11 11 11 11 11 11 aaa.zzzz »x -UJujMMDDoa'JjDD «.UUi<LD-ijjjjjjjj-jjjzzjis

n ti 11 11 11 11 11 11 11 ii 11 11 11 n11 11 11 11

_j —z z ^-—•» zti

Z -ZSZ Z5ZZ)>0 -'I- H-l5- ^ ac -azacai dlujo

3z3^—<o o—<uj'ua: <aoo>

11 11 n 11 11 11 o> 11 n-* —

. _j •» -j-

t: xa<_i >oj -» „^. -Z)>-JI- ZZS.Z>«"> o•. »-w •— UJ I—0_l ZcQZcQmuhj

1 um —« r-i >* r^ *-• a, on< —1 —< rnjsxnz zzaaaa:a:a:cx;a:^Hhi-hh-zz>»»»

65

oooooooooooooooooooooooooooooooooooooooooooo

2SS5;s:jzz5:5:s?:zz5:z3:3:2:£25:zrz>:ss2Szs2ss:tz5:5:v5:t5:z<<<<<r<<<<<<i<<<<<<i<i<<<<<<<<t<<<<<<<t<<<i<<<<<<<<

<x<:do<?: c\inQ_ *• •-2T'—«o •^•Of*- -4- •-

OI2 »+- * > •— ».r\j—

.

OOkUJ t-X «»vj-C5—«wm»-X ».^:uj(_> ^r\j>jriLn>>-.x ••uu^:a:2 in—<v0J5o<z>i— *—(-! vo^c: •- iso ••UJ •«»— ».>^.<t- •• #-

«-*3» . •+— i— <-»tn »-ro w m*:>r\j< »>LU XJ^or-a. ••o ^o

2 k~—>a • « •"-'Z^stz •»

•ro •• »0 X — n^-O »-<\l •> -t

^>^IJh inc\!JJ~»—~ f\J

5: ~\— lu »-oo vo—aroio'^- —CO *-SI'-« •" K'tdMCSx' H-r>>o ^^> vtx -•—xo oz •»— X *-'-J ^ ••—•—• -~Z. —J

<OUJDZ —-.0>—i>-i Q.

>-(/>a: «« *•(— ••O *—«r\lO ••

•> ».<iujuj »• —- «-r*-o— » •"•

n<;cDh-su> tnvt— <M--t— ^xo_ •> »• •-t— *o f\i q. »-»> r»- vo

«.>.<JOLLJ>—'UJ «.w<r^J «-* •>

CD «~r5Z_J * -tOQ. T\I~»0 -J"

i— h- zcrx *• ••> i\i2< «oz c\i

UJIU ••ID*— S_J^£l/> > •• •o—< *—

a^QG Q •••—'•>•"• t— —.-^.cjro •» Q

acoC <—<w-'o>'-« xmo—>-» -J

UU •» *-c_DO "O —• •* -Z—^ Z5

i/)ha:>: »0 •> m<ff- «>OCQ •qq >sl --ice*: so<ni-— -»(mq- —zz * oq »-oo ..wxm>-*< co

t» »-o ».•••. »-» cC -— — *"—• *"* -J

JLLL. >-JZU-JZ r) -O —NUhOO S'-h «-Oi/i h--»r-i-*rOw l— -<

Z> -<r<oiO xin—in—<Oh-^t-_j *joi «-^, »->o^— <°- - -

luuj> • »-o_J^-t— m>i- ^-•o-^-^n

30j f—c^ 1—• »•>—iUJ •>^-O --* ^0" cs^ •*

Q.Q- xuj<<joo-J sj-oOr-cn i*-<<t-

OoS O X)LU~.> — i^>CNi-J ^— (MsOOOOOOOOOOOOOOOOOO o

5 x2< .. --J ^-^n -lu^ —*< ^-ooooooooooooooooo *

iCx ioi- -j *o -—1^«^ 2!>3sn ii ii it ii ii >r I H II • ii »' ^ ^M " " t r< r'

u^ ^^".X^^^^Fw^ .«« ,— , .— .ajojSajtnuj

o o o—

<

o o oo

66

ooooooooooocoooooooooooooooooooooooooooooooooooo*d-4-v$- ^vi-tnininLntnintnininir^NG'-QvO-o-O sOvOvOO^-^r-i^r^-r^r^r^-f^-r-cocoaoooooaaaoajcocoo^o^C^

<<<<i<<t<r<;<<< <<<<<<r<<<<i<K<ci<<<i<i<<:<K< <<<<<<<<<<<<<<». »o o -o m

Qu —nOi »-l_) .^-^-^i O<KD'JZ fMLH •• »o. •> »z>-iu —'vor~- »* *

UOHUJ »x »*>to«-»—

m

X ••^ujo —»r<jrj:ir\>—

.

<cot— -—• o^ •• »iz•o »-lu »-K ••>-'-» -o- »^X^-tOcO •• •* Of\)-»—*r .4-.—ia:i_) c\i«—.-*^ir.in —••vj- ». *-h- >— — IT, *":0<>—» ini«£>o«j<: «-t±j Ovor-o. *-0 -O

Z t— "3Q- •- • *•— O-J-2••CO • »OI — «tO "^M •> "4"

0>~<X_J.— iTifvjaj •—>•»<•» ^vl

s; «-v- lu -lo >o«a:ODN —

»

•-cm ».2r>— •• •i^a.'M O-— t—0>O -*i> <fX »-IO Q2 -t— X *0 'NJ *.—"^ »2 _)

<ioiuo2: w—.o>-»»-i Q-

•• »-<iuJixj » •»>• »-^~o~"" *• "~*

^qdh-2I_i> m >*•<-..cNjNt—. mxa. » »• •+- -ocnjo.— >r~- -o*•> OO LU .—i LU ».*-»<f^J •-— •O »0 2:_J • +c5a>:-0 -4-

2!c0< -> c\IZL< *OZ c\)

<(X f^Ot— ~-<XO-— Psl«-< —••Ot—O^oO > •> -o-*<t —O *•*—* •>•»•' t— -^•~»C\JrO » 3<^t—0> 1-' x inO "-»>•«- -J

• »oo *o —• •• *<z-**~ Oi/>-n!S! »o •• m^*r~ -oca ••

>S «-_Ja:2: .o<NJ>—— fMCi. —

.

»- »Oq *-oo ••~»Xin—-< oo

wnZ mo ^q<-»c\jo—»—

'

OZ- -o£<;CL <\l< »•_!> nMJJZo * •* •• —a:—Z ——-jo>jZu.J2 O »-0 «—.r~OH-i-i•OS;.-* •Oi/} I

. —i -^ rO— I— »—

< 1

—

Z3 *<r<tc£0 xin—mw<iaH<jOx *Q- *-v0i£— <Q. - ••'-*

rvj2 •» (X<I *• —• »>21—I••<-»»—»t—

•• »-0-Ji;H- in^f ».z:-*o^rm>-<>>2<OLU orxJ- in *-^Z

eg

—I

o vo m>j- •• ^-i • o

ONO OCsJ

MvOO ^^-i ^_i so in Tai4<wJ -i-oo—en r-<^-N<1 _ ^. ^ -a. -I- f\j>—w o-wt-cxia jh

oo -h i rH o <\i coo -coiu-o — ^-^rvJ-i -r*2^ _"r<_

b-*UJUJC\JC0-HUJUJ MlOCL^CDH ON3^>0!a<Oj(M-l|JJNC.^<OHJNe£ ro ^^ ro ^o mo zh < -uj-i z+—«.-323*im ^-LLiJ-Luina.

-^^(Ln^.t) —— -« cn^i-1-iUjr^'Xl.-lUJlU —• -l_'JJU. vrc_ir— v^^^u.^^--.^^-^ .-_.-„„ „ ._

^o^r^ooo^^^uvo^in^ina^ 2S"5S^SSSSOfM<M OO OOO oc\JO 7;^

67

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOCOOOCOOOCOOO

<t<<i«<<<<<<«<«<<<«<<<<i<<«<<<<<<<«<<«<<<<'t<i

0.

IU

LU LUs: —. oo•. LUUJ CC

r~< ZZ UJII * •• h-

S *-«.-4 LU•* II II 2!

~» 5:51— <LU * *0 CC

Z ———I <•> LULU »» Q.

LU -^ II •- • II Z3ZD —

J

-^ —. ». .. 11 11 •••••• z:

Zf*- •• II 5: * •"'-, II II «—

'

4-^^u ..2: ^s:_j • » <<t-••II II

—"—»«— <l- •> »•" —»—» i—^-. <rs:iu<i— » —• ••••••• co—•«-• —— njn inzz *—>-* r3-»

^o^c^'ZZLl- lTZ-^-^^w 1^«m-d.w«i . oltm-i—-a:oc on o

m2 • .X *rtl -</>*r>J=>~< Z -*•«-*—I<~-hiM •" XX-JLULU O^ Onu.Oh<M.(-*+J32:zz> o <ko»iiio

£za-Q. < ^ o

^r-^Z3-»-IC5Q l-0< -«Z — -OOZTH ll-OW11-'.^ < *_._<

«~ w /07Ht\luN^u-\000>t t-.UJ l JJL)in>0NI,--J0O'M-( «-^rfl 00.0000-hQ* -Q>t- I I

HHK«Si«SSS5SSS°S -+**1*™**;« - *2*™™™"rV•£?«£

22.J,, ^onjQnQQQQO II II O II II UQOOOQO— JQOQQOI-U.<(M CZOO'J" M

S?r7a<<i<<d<S<<<<<2: 11 ziw««< ^rvj^-<<<<<t20H-n 11 11 it 11 11 XX

0<0<\\cr\-J-t~-cO&0~*<^i -O rgro^-fMrnvrtn--O—« '"O <f LT> -O Is- 3)

rO.'0''OrorOro<J-<MrM^Lri O i^jniANNNOkvO 3 OvOvO-O'OOoooooo o O Ol—

(J\ ~t —ILU

t—

1

LU^ OC*H H-

X> »•

on fQ

5 a: *=^«**«

K ox>—•• _J •• ^o:lu

LU —»t— rsz)*r H- - o ^Z 1-*

*: •»-* •» •> •»» .-•<•> _i r-l-IX

LU o o II II II z>

x> r\i a: t—i r~* >~i CC

l-H <H «* •. «• »-3)

». h- •- at . —.-^—''Za LU—'OO •* »—^ Mt-4 *>

iZ CEI-O < WW <"«-<

•• cc -o 2: ^ivjZ5 II

UJl— -ZZ~4 (.0 5TX2'-'

vOr»Ln—< v0°0rOroromo OO^f

< OOLU ^-i^a:

68

ooooooooooooooooouoooooooooooooooooooooooooooooo^<f-^-<Cvfxf>fvf^}- 1r>LrMAir, ^L^tnu^u^u^^oo<)>Ov£>vOO^of^r^^r^r^f^^^r^f^cccocoaDoooocococo

t •

ooooUJ II II

_l sz> zz_J r» '

< XX

— --. LU

< 00 </>

• • aa: i-v- a.

• • z>

o o —-«• CL

cc m o ss:in CO 0>-« • *• ^-5

]jj ™ 0-2:2 z

X CO ON Z» :*£

t__ OO I- 1- OL, cm (M XX uj^ t. •* • • X

00 i-o* zz<}- o o << oc

CO <M CO Of\l • • OCO O • • U-

CM ~» — OO — —CO — >© •—— 00 *- <1 -•

„ «r ,co OS" • >-<-* —» I— —

*

ro »o •« • -O ujuj • • < <~Jooco 2:0.0 1x12:0 zsh-i- a oar>

~*\~u_ — oco _i—o •> -_j -1 o <r *<

^n^^ . ,Z . .£ ~ . . OO-M- II II-- ---II 33iiujo^^s: c^jO!r o moths o ™o • • SI II 2:2:-^ ^" £*•-5 II II - . II » II fc • II . I] I

»• • II » II OO t II II I • - II M -L.-5 --—, ^'

-2oo^ " ^^ii.^l^^>^^~^r^^>3'ti§go>li^^^liuj^

^§Qm^§n>mDDO3DhIm»0»X>^3>hmI0QQmmO<J0Q<UOO2Q.Q.

KP5SK £§ SS S5 SSS ££ S2 5S 22222S5S5222^ S CM * J

—

to> O< CO0£ *•

oC com< r-

•H- ijj c0

T r-

69

So2m«^9S00200000000ooooo0o000°oooooooaooooooooo

ii.iiii«.i5:i.i5.j;s:^j:j:i5:rg Z;j;2iS 2z:j;z275:jsr5:s5'r5:s:yj:j5:rT<£<<<<<«3<<<<<<<<<<<<<<<<<<<<<i<<<<<<<<<<1<<<t< ^ <:r<f<3. < 5

<Q.

<

ini o ^ v. ,. ^» xx—• r\i #•> «- »—i —

.

II — .x

r-( U- •>

fM *. .

<~* Xm

-H »X

<r—

—I h- X N*I/) <r m *

•* o t—

<

o ll i—

<

>I—

4

• Q X a:QC u. 2 m <o UJ »-H »• a

o o xLL o Q • x-J X 00 o< X —

<

u_— :0X o IS VX 00 X "v»aj

o x cr . -XC\J < •~4 ><S)h-•h h- 9* alZX 00 X i-UOI-0> x o h-0<Tr\i <i CM ^UJ•» X * *~<l/)00o «o rn XI-

IU r-H • t-oz00 •» o ID »•>—

•

X a. aooin »» K «Q.

X *•ii XCO

m Nt ou_v~*- • < X -Xo <\l X in a• -H a. —t'-TlZ.

in LU j 9-d.f—t

u_ »• < xo*-

II int— a.tl a: -ooQ

C* UJ oi— UJ h- • Csoo

Ln o> x o t-o<t v, _m x — xmIU-

—Q- o *o rf> xi— o^ » m — in--»v0< iu —i • i-n0Z —i •Js-r\j i— tn »xr«->-o oo •• o x *-< u-t^- • • » » x—iw -J •* x >J- clou r- «oco ii x m—

.

X<~- in •. j— iq. .O'-tu. >—• in v.m* ••<J

V X •• I! XCO X~*ULCT> -J >^ •» -^ -•—»—>-* m -4- ou_v~ inu_N. »• x v. »x —. uj iuvcm< •» • < X »-x »•>- ro i» c^m uj 2: 2> w^a o <m x m o. - »x —. •> \- c£^ z ••

•-« X-J<i —» • -h a. —«cr>z »xm «vn X x—

•

- ~4 —

*

-J •><<_) X in uj j a. 1-' i— in Div."-* a. ox —< 11 11

» —• •• * -JU_*-<XO oo» •-< • •• 2T «f\J 11 zz<t in——^ » 11 lohu. z 'hqoii —1 s: 2—i >* >trn ^11 oc *-ooo ot— 00 •• «x t a:—» •• —. —

.

•• x^»— -» II oi uj o U^iZQ^J « lux — O in2! — •_J< r> 1-1 I— UJ h- «ooo zqujllX ac <x—

<

in r<1 -fX Q-^<Q- _l •• Q H- UJ COZUJ^LUOOLU ffO X ii h •>

—J ><i-4- < *•• UJ s: U,»-<l—>sQO Q-X •> O U,-^ •» s0 ^" »-^-»0 ^<»-< Q- 21 < *2<I»n.'—' UJoomrO • O • -* —< r<*i

Q. ~4Xrn » 11 — uj < oi , IZ-.h-ujo -< 5: «m 11 11 11O 11 •—»-« -«> h- a: <i X—i—<x 0>-h «o •- c£. Q.U, s S S-J »-»-»_jr\| -» f^ 00 < D. I— -JQ!MU_>-<I— >>-i 11 ijj X. •> • •• •»> »-r^<—. x —.uj —.a. o -cc -cjy- anxm a. q •. — -»

«-" —»w •< x •—a ujv. o 2xO- u_ujo_j-* ox x—s-'^:—-H _i •-jx-^ - s: w s^voo 2: ujltiOxoou-cdi— i\ u_ ^4 — ^o »m t.otr> » w* »(\j< •• 0_J'-» hJh K-. _j »-o—<:JJCJ ZlLros O > •> iAZ^Zv+2>C•> x ^—•^'O —< c^s;t-H h-o-J x o *-o 00—0 •- •• • —4 "4— »w *.-«. •>

"-«J2 _j SMJ »—» II —< »•— Oh- • •••.'NiX^X *• (^>r0 Q Q • «<~s) ^O'^l-^^l OOh- » »--^<—. •—

1-H~^ U. >-' Q Q-C0X>-'l— OCLOI— -••-i-* QC Zm ^H~—it— <-^i— >}-

•- r. UJ —X •—*cr\ - —.a _lXcu X XU- * KHUJaZJ » ••co—.X <ij_ II X II X II X 11

<.'-* -J >-< •—.^ -ft—

• t)«^'.ij <t^O X Q *-Z "JO. <»!>-< II II mOO X—X—'~^»2TIl-J •» «^-—. ^-< -tt —1

•—• C\J_|CL X »0 <-0—O —'Ia^^3!h->-'>h. »0 X~- «»•>»••.•.••

O- *••-*»-< \i—ivixu^ —«2— Xvi-X I\IhI-*Al";iIy-i (£! Ii ZZZ2?.^Za:_jLiji— ^: x~"_j<i~-< —i"- o^ •—< r-v. n<^ fjo^^iMijxu.^ o^-* ^-—w^-^*— -_•.<— ». •. «I<o ^^ » •. ^-4—«r\j (\jrO—4 <j-(NJ» (\|f\Jf\J—(K)I C0l^-C\J ro^. •>•.<»•>•.••••

00 •• •(s.j.n •- «» •. »»x »• 'DJ^r*- »-~4 • •« • »-oo ••—< •»•••« » » •>< "^v »• •> •o"^. o-^r\jr-4*\i—<oj

vom^vtvt mMjr-»co~<o> c\JnJ--j- xujx xoox 1- x «-xxxxxx ».\'i)x x •• o-ooooo^of-icoco—(—4 uj 00 io ao co u. co jj—<—i—Kum »-in—• jmj. m a: u^ -* in 'naMn in inro"v-nn mx ^h^4-^^<—4—<~4OH^HHDH-iwwHH3HH-t3w ^->v^- ».— jwX—"-'^-'—ww.— ^-..^. ^u-\ vQ O >0 s0 v0 ^0 -OX X-» XV- II I— >vr— II h-t— i— -^1— !— I— I— h-h- »-»—

I— I— h~--vh~ V-\— h- f— >-*Y—\— h-h- || h->-<X!— H-'-H< <f\|< <0O<w<<l<<<I< || <<< <—

.

I—— |_ t—I—

I— K—

zzzzzh-zzzz zi-_jzzks:_is •X"-<x zinz^sszr:<i:rr sx zzzzzzzctoiGicx-tatxaioiaixxcju-^fxaa o—«x QZOxoQoooaiQQo-.a ».^ci:a:i:a:tv:a:3:a. a. q. a. a.o a. o. o. a. 2c ex.o »-• a. o-o u. lu u_ uj u. cc u_ >-4 u,m u_ u_ u_ u. iu u. ro ul a. ll^u.—.^0.0.0.0.0.0.0.

—< ^-4 '-I ^-< —I --^ .—I ^-4(\J

ro^in-oOLn^o^cooc^-j-'nvDf^co—< <\j !T» «j- in vo r"-cocMNiro f\j<^o r>-

rO'-O—4—4C\jrn-n-<>rn vf mtMrvjcNJfMJNJCO CO CO co JO CO co CO CO ^ -f sO-O^ ^3"

70

«<«<<««<t<<<<<t<<<«<<<<< <l<i <1^ <<<<1<<<1^<<5 <<<<<<

o_ «. -^*-«^)<> 3 -i—oo»— uj *x

r* »x ••^.uij

°0 <Z>»- -—*-" »'J> *-UJ «-(—^ ^X^ZJeQ -'"", 5£ •--»•—<cr:o* »-^- »• .-H-l

—

x :*i><\j< »-uj

^ *- •><<*> •> »OI^ -«.«-.%» Z>>-*X_II—• cocov. 2: t>j— uj •</>to M _ir\ ^(vj „5;_ »i— mm • r>>o -*:>z »-t—«o> z »-i—x <_>

O XX - $£ - -Z_j-iCL cocq. >-ooct: • ••—

X.^ II •» •.<UJ|_U ••

^ v^dc *:coi-2;_j>""* *"» —» «"» -^ '«» ^ .^ —* *^ m m Xfl. »• *-V—LU IXI -» 'U UJ IJJ iJJ —. UJ UJ OJ LULU OOOOl— ».-i> i/IUJ —ii 1

1

ZZUjZZZZTUJZZ^ZX f-l-Z OOZJ.•"» Z »»•••. Z •.»•• •.»- Z2iu Zco< •><—t ^ •. .-« «h ^ _< .. ^ ^ ^ _j hmm <q, •.rrvOf—•I II ~« II H If II -• II II II III- OOU •Ol-O^ooZ Z II ZZZZIIZZZ Z< CLO.-. ' Q _ »••••Z»«»'»»-Z»'»> «-»- LL <I'-HKO> >-*.-. -, •»«,«, ^ -» •. *» -» -^ ^.x Z5>u- a:> - aia:om—•OLnom—OLOo mf- lu *. »old »o00inm>i-00tnm-<l-0 >oa luujOx oo-nIS -Z2 **- — f~« »>»*»».,_i».».«. »-uj XXOf— >SI -Jet 21o —» • o -* o ~« »• o —< o —«o f— i— a. •» oq •</)

•4" O ~* —*POvj- No^^-irO->t'0 ZuJ </>r>«j?' 'in3ii ii ii ii ii it ii ii it ii ii it c£ ki-oq zjz »-^:<aS:S:S:5:5:2:S:S:S:2:2: SLIJ <<m •.•.(_)-••<•.•.•••.•.*»••.•.•. «-H- K-LLI >_IZU._JZ—. —. _ _. .^ _ — ««».-» ^ _» «< IIOJ + Z1<-* »Q^2:—2: 2:-»5:-.s:~x~2: ?:— 2:— s:—2:— 2:3 »-i->-.co z> -<<^q•»m •— » »o ••m »>o ^if\ •-^. »o »Lf\ »o *.m * o.a_c£<t »jui »-a.ZOZiftZ^Z-tZ-OZOZ'-OZ^Z^ZvOZvOZo UJUJU,*-. ^j2T *Cl< •»— »-— —«— »»rf •««- »-w «—!—««* »-w »-w *.»•• •.-_.—< —^ QQ q£ •• (J-JZhr«0-^>j .O -0 Z5 ~j3 sO Z>—< Z) >s0>—<> 0>>-<>_J-^Xi — s;< >>?"<rOLiJf— nOI— —II— —«t— ("OH- vJ-K-O^—4^—<>— rO^ -tl- -0^OO>-"»303i,i:Q> -»—

.

^q/.-. >_<lL)

X 11 X II X II X II X 11 X 11 X II X 11 X il X 11 X 11 Xrr. »LrvO—-iiJ'-Uv- SS Ini<f<wj^-»2! >»^3--^ 2»— ^^— r^~-> 1Z —• 2T —* r5! --*2 --« 2!»— .^-^i. ^J—( »mh— I— H-u. •> • C\j »-o_ *•(— •> »•

*•.•.•>•.•.»•-•.•.«.•.•.••.•.»•.•.•.•.•. •-5-U. «--0 »-<KOO 2T^ CQZ5 *':2'JJ.—<>ZZ2.222Z2ZZ2Z2?7 2Z^2<JZZ-2^^Xa.X"i:.<:a luuj>».w Ol-Da.OU-*»,»..»,w«t.s.»--( '*.— «—~.—>-»»*«— «——--.-^w^.^^,. _.<oomco- - - ->ZS3> -lla;< fs .-._>

^^.^••^•^••••••••••••-•••**»^#*»-»»»-^»»*» **^ •« >sfc •» •• •* «J fc *"H"» H* »»LJLj^^<I •*

—<^.r^LA^-<r\|—»r\j—^rNj^v^^ ,<5^ir%i^4tM>^tN;»-<-4- -nX cr ^ 3SKXX<H-tII lUr-OQ. »<Dt— _jnOnO-.OsOO ^OOvO 0^0 /WO-Q-O O O-O O-OO 0-o-^^.»-<^—<—«rg—« II n II n 2TV- »< ^lu»->^<F-) ^^_)^-r^^^^_^-^^_|_(_4^4^^^-(^^^,_<_< r-4—1^-— >^^-~-^»~*^-(— ^iI-»-"UJLU «IOO.^d3l-

i_i—^(— 1— 1— h-i-zmLn * -^:^:z o^<o •• »-<oi-m-i-i-f-i-i~i-hi-hhKi-(-hi~i-Ki-i-i-'i<«««wsNZZ'-|i-<a: 00 •• Kn*-jzZZZZZZZZZZZZ/'ZZZZZZZZZZ^SZZSSSSwH^w-hl-O aCSO—1< »•>-<fc_ t_—<t_^,«-, >_^H^^, f_—.^, f-.^_i—,—1 ^,i-.H-i >-1 «^ioi;t:cx jvcc-Xcxa: D>zzi-Qrasijj< —«s<i

aaaaaa.Q.Q.aQ.Q.aaaao.aa.ciaaaa.u.a.ij.u.u.Lta.iiOQQiiuoiXiJJ^uaco< »--j

~Z --<(N),-r»^moO ^h (X) -O sT LA O CO 1-4O—* 1<"* ^J" LTi IAOOOOOOOOK- r^r^r*-f,*-r»rs-.

OOOOOOOOUJ ~tCO

71

OOOOOOOOOOOOOOOOOOOOOOOClOOOOOOOOOOOOOOOOOOOOOOOoir»sO^-co'>o—<<^iro^-u\ •or-coa>o~Hf\jfriNj-'-n-i'i~-'JO'JN 0'^<Nifn-J'iri ^or— coo> o—'cm -o -.j-moi>-ooo-o--<c\iCOCOCOCC^OC^ Jv0\0>^g,0'Q^^OQOOOOOOOOHH-t^-.H^H-.«—(r\Jf\jr\jr\IO\jr\j!\)rvc\Jr\jrOrOrOr\J c\J r\j cm r\j r\i "M (M r\i f\J (M r\] rxi (\J r\i ro ro ro ro ro ro ro ro ro ro *o ro < o rom i- ro ro orOrOiomrOro.O'OirorOrorO.-OrorvlT^tvjrvJr>J^fvji>JrNgf>OrvjrNJ^jrvlr>JtvJ^OrvjrvJr»jfvjrvs(rNj>v)'>JivJ ^f^r^r^'N(^^r>ONJr>0^v|NJr'0^lr--jrvjr-sJr«orvjr^l'vjrvj

<<t<<3,<<<<<<!<<< <<<<<<<<<!<<<<<*<<<< <<I<<<t<<<<I<<<<I<<<<I•- »o in •K-

s^-».—4 ^0 «-»

CSJin •» •> 1-4

*-"^0 I*" J" •" •>

> ».w rg^ «4-

•»-ttO—— '-^ •—

•

-Mo:in>v. _l X^w«<ooo < <_lvO^ - "IS LUZ "• •>

„>,— N$" •» »- QiO — N.»* ON-»- <(-H t—«>^

(M—.—t~.LfMO -~ V~ Owin »-co o>- in oo <r rOa:Z>ot°~o. «-o >o H-H- «-»<•> .wo-4'Z •» Z3 O^-^^O »c\J •> * UJQ. a:—in<\iiu~~.-». (NJ -Z.SL <-vO-OiODN «— oo **^a.M;J« h- O-O —.r--

>tX vio O 5: t—4>—

r\j .—.—< »-^ —I OUJ X^-*.0>-» 1-< Cl OX i-o<rsjin —4 »o< > r~ ^» +•.vO ».*^rvjO •» z I—

«•» ••C-O'—' • —

»

UJLU <-.m>t — f\Jvt-—

.

LH >o •». •»

vOfNICL—'>r>- o LU -*^0—«• <i rv) •••* •> 002 >—< "^

>tc5 0-2:-»0 4- S oc\J2:<i «oz <\J az5 t\IQi

>—<0—.C\]>—

'

-^ Z-J — <> .. »-o— <* \- <o (J>*>_— -»C\J^ •» O o o *^ •^ Qi—imo->^ -J m vO 0^(\ir»- — ro —» <t *-"<

••nO •—"vj »-r>- Qu <o —1 —ir^fNj -> 0^ -* «• •«

3 z>z m rO »-rO •» m -5 -..o

m^tr-- -oca » a:o <n »-r0 * >-< —i fh t—fatf

vO(\l>—"»(\|Q_ •** DO CO -^(MiA *•

*

>t t—

•

•>x

•wim-< CO OUJ ^^ -<-^m romro o —4 "-* OJ<^r»<— f\jo-,M 00 -3 m-^co ix^ ».r<*i UJ • o w +r\j<t •-_)> •> 4-UJ o •» •• >» ^ ••ro •• a: fO U.I X——«:—»s: •«-»•»»_J OUJ t-4 C0i-<rO -^rsjin Q. « a: <!>-«

3 »-0 «"—'<NOh X —• r\—iwco (NJrOfN-* 1 -^ CL + f

h-—.r—I^»r<->^=»1... '— Ol- q.—(MorvJ rO fO-» ro —>1

-•in ••o

lin^m--<Ot— < <m— uj • -^^.-^~? o> •a in —t-^ O••sQ^— <CL * •* Tl- O- *-LAa^vO ^roa^ «- in •—

«

CO •o m*-. si-j •»< —» *^% _ =i < -«coa. i ooco—

•

r—i *«• •—i --a: »•

m>t ••2:-»Ovi-m -J 0fNJr-4 +— i^inrow <t o ro •~»<I ro

vorvi-^ «*in «*•— O - UJ « Or^iA-3 O—irMO —( UJ CO o-* r\l

»-^»O "-» <X~~ ^ - Z t- •-coco »• r^vt-HJJ • q; f\J a'— in

^- UO .—4 LT\ »f-» ^(-t U-O •» >tf\J—

<

v-< in—<f-ai: rO a • <!•-< •mr\i

ujin

C\J> -s*--~ t-i\] QC~ —

»

—1 •ff'l— • • » r>_ ^^ w o -!( •

— Q._2

UJOOI—-a-o^o

1 1 1vO OO

—

-

o— IT.

<IUUJ UJ

t— in »• ll. oO «* -*<! l/)4 •^•^-J -^-~ • t ~»~»-~^ a o 00 ^ »-X CQZ.2 j?' 51'™'

—» 21 Q2^ ». ^<-5~5^)Q. -> -s •}a o o o •» —*< 1 •• *k • • * --^

^0 X _J -^—« tl i t» h II •- •> ••^j i 1 o •-< >— 4- fNjr-lr-^OO ^-( —<S

>—4 >-^ ^~ *-* •» •» fc •—

4

H-t—

<

3ZDLUUJO.C73 OO •» rfiOOO'Ji-n'O^"'^"'^

II X»^ I' » II I'

<^huju.5Ts;-»—.

-»il -DOII

'jjz3

II «•

512:

oooO •0<h— -*OOoO CC 2 UJ 2! co 00 >-» U1JJ'—'—

<

LULULU • •»

-». .r—1. *

ZrOror-j«—<rvirMV—

irMro^ino \- -~2 ^o^^^^^^^oor^O^niro^inNOr^coo^fvj.-o <f30-irvjro.ro~^r'^ r

~r, n0r_,^^,-,^,—,-.—.-i^rviiXirvjrvjrMrofMrNjrNjOTXJCOco coa.O>o>rjNrj>^000

^f^ujuj^r^rororot^roro^rorororo^rorot^ro^rororororoi-o rOSro^roo-i.-ominm

O-hXoO^ ^Ot— >~n)0

72

OOOOOQOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

<<<<<<<t<i<<<<<t<<<<r<<<<<<<<<<<<<<<<i<<<<-<«a:<<<<<<<<<

I

>oX+

CC<>+UJ_1

<>+O-J<>+CLr><>w*

-* *--< <

in O in CT> t-

O -A —

(

-4 UJ

in in » in CO

•» «J> vO •> +r- oo T-4 —i •^

o t—t m (NJ 2:

in om •• oo in *

>* - o »• ^->t i—i » 2oin mo —«-^ <S\0 >»

mo •>!—

1

mm »~-4 r^l

»-LA com •t—

•

r-*'S\ ->

f^--» .—i —

»

o^ r\j— <o—

—

'fi f-- —4—»—» | H'»',» 3Q

tA22 •"21 ^Sl i_n.—<r—

I

».—< —

<

3-

•»»»»» ~.a~ »" -« - 1 1— JO + + —»_)

<J-~<.—

<

S O —J "-i ^vtSIT r^vs -^<

1

r> v. (ni

o — ^x — -»

— + 2:2: 2:

3

m —cOmZ—(fliA-Zin »o *rgmoz:..mx—inm-—

>

iniOvt—m<s:inm 1 -

x m •;r^»ujuj r<"> •>

.-»—.-.xrvJ rur^"-! 5T CD

|

mz2oz-zzHZinzzHZ-zzNZ«>Ju.zszz>swj^z ::z-^Q

a);o2z-zz^

"" o o o

rvj —

.

00 in 03+ rvl i- vO n0-» + < m in

OS (M— _i ~t —4>*• *<}-3: LL m - vO •*

inzui vO-t sOf^~—1^*1^ :z _J invO mvo•>NJ •—

«

< -4 10 -Hinf-«>-»mivj Q ••—I — r-4

<f +-^w •—

<

in •• co ••

-y> lt\ in + t— 0^ ^or~X -4-.-H— mo inoH- -s -S LU r-tin -4 in

fi.XX—» -ro O --^ •..-»

lUi-UUi-t 2T<f^ in—

.

CO—OZS^-i^^ • vO •

* »-»-4 ~D^-<5> m UjUJlflO mo_)—<—< *— H- O 2 5" ^-4r—

4

—4—4

< — X—X Z - —. | .— 1 •

UJ t| II — ~» Wi-I ^-1—»—.ro-*-»rOcc n 11 s 11 > II 11 >r^" 1 s~ 1

^s: »• «* CZZLX. - - 11 --it

73

oooooooooooooooooooooooooooooooooooooooooooooooo«—ir\im &m vO r- -oc^o—< f\i<^>4- in or— oo c> o—<<\i in n3-m\o r— .n j\ o —« cvjco <t-m o p— <x> cr>o—

<

<xjm ^-m .o f"- -x>

CQ:ociD;o^cocoa3co^ax9>a>Os av>c^C^^oooooOoOOO'^^^^^^^^^rOrn(n^rnrO'T>rn(^r^roin^rO'n<^o>^(^vt<fvf<r<rvt-^xfvfN^

ssxi5:5:5;5:s5:2isj:ss:5:s5:5:5:2:t>:s:2ssztss:s:tssss:5:s:j:5:s:szss£

-J-

>+

>+O-J>+

O> •

# h- "N, CO

< o *» in

in o in oh-miA <t- cc o-vj- m m voujso*- r*» ~i ^—

'

n m in in tneamo <\im > mr-4- * r- n. - + ~*m

fs» (nj r*. M « o> ir\ mi— cc •>•-<

.* m in vosr-o- h- 3 in*in m CMn m c^m » in r» o m > omoo<j- .. ^ ^-^ m * z * -o r- * + mcr>

m^-in mo in in ino «-> in m in ^ -j ~min,* --n m in *o 3 -o — * ^- ^J £

in m f\im -in com *acin — oo m > ^.^^^t/TT.tiN-. m— r-— v0— <<-~ -<— — r*" — + x xn .

^wm^st- m- m- m- n +- -* ^ - J ww ?

5 „-zinzz-zmZ zmzx-zmzzo-z-zmzj-z-ZinZjSDO --s

^in^in^mminin.nm^inmmmmu^mmmin^3 o o

74

*in o< H-

1

<<^ -jn x>»- oZ i—

«

%^ o^ CO z«"% Q. LUSi > Q.» «• Ofz CO LU>»» < CLCOCL 4- ZM- _,

O•—4

s: o•• z lu -*

^* CO *^—<0D CL LU ZcM>a z> ox —

•

mm> -K- Zl- ^>•. «. 4- «— —

«

•—

«

•*» *~

om~» ^S I— f- ^mm •. »-z lu

^ «o v. a:

mm«- «^:»£ lu cq mm o •-«-J

mmct-o or5 > * *'-• 0«-0 i-h • * U) ™

,~# ^v LL.O?>-l inU z* ^i co^ ^ m ch > <* >t^^ *_ ^ ^ - *v<>* ^ - 2 £ j:

+,

D- £

zz- oj: z ^ -<* o,* <* c* *> -* *^.^^ o_j-* m — oocn) -o — •* •-« r^ P Trt>OQ~ t z z< - x> oo or ujoc* - ~< <* > - *

r>>£ .~ v-o< oj lu zro-n- i-ui *<*~ i^ - ^ r ^ ~ ^*Z.Z.^.h-~. i ujq. o-« -• a- i - — i -« -•- z

XT "^ >_>->•; --

^

a?2^2z^-Li!^= 5Ssri5S22z5z22J2s25oS?2525o5?zSu.oSS2--o^-z-lu* -

azafg|Qrs:;gKOoI^^rJ03jl3u.5oS2£oQCOoou.

-^ a. I X <-*-h—'—<—<-^ro roromO

75

OOOOOOOOOOOQOOOOOOOOOOOOOOOOOOOOOOOQOOOOOOOOOOOOh-r^l^-COCOCO X) CO X) 00 CQ CO OO^O^OJ>0> !>CM>'>0> OOOOOOOOOO'^h-h-<hhh»'-ww(M.M(\J(MN«i-^^^4-<*--t<J"v}-vi-^-vtvf-4-vt-.fvj-v^ Nj-vt-t^-vt LnininininninininininnininininininintninintnininM^J^r>jr»jr>Jr^^r>jrvjr^r>Or^fvjrv^ir^i>o^^>l^r>JrvjfvorvJrv|rot^zzisrs5:isss:5:2S5:sri5:ssszszs2Ss<:z5:sj'S£>:5:5;2sx?:ss!:25:<<<<<<<<<<<<!<<<<<<<<<<<<<<<<r<<<<<<<:<<i<<i<i<<<i<<<<i<

««•

«•

in<

in o

— co— Ou

~» #hh z Ma: ~ <> co4- a. |

UJ Z>_) it —

O Z-I Z —> —CO+ inmco a.

a. vOOQ.- > o3 OCMD « Z> . »•-<. —

»

»—

i

# vo nO^os s: i—

m o i— o •• •• »-z z 00

^ ^- in u) o«-< oo-*rO On c> o CT- CD OO OOCO > 3 -i-

Of- (XI f-ro ^ 4. m s: o - -.—& ^ »• u.

1U

vj- o>cr> s o>^ 00 six— ^iz z >

££m So ^m a^o - -o zz - o*

- <ui..u£^~ S~ £~ <^~ J *£ ? >=£ J_ at >

5 + ^-^ ff>- ^5->w — ( >— — *Z OCh>>-. cDCOoOrriSrZZ 0~~—2

^^Si5iS|S|g|^.3.«i.HJ

. 55So.»ii^«Sl!l5~£g«J 3!*

r^^^O^rxi^^in^r-co^O^CxJ^^insOr-co^O^rvjro^^O^rxiO^^m^n^^ os O

8 S >

76

ooooOO—ICNJ

* »»o O •» »-o ina. ••—*oc -u 4-^.-4 vQ<IOOZ r\)ia » *CL. •• —Z^O >--vO^ *$" *•

' <> _"5 »— •« > ««w n\J—OOh'JJ »-X k^O—»«-*tr\

• «-X NiUJO — r\ia£'r>>«*»x »-uj*:a:z mw<vOOco

»-(.D -UJ »K •->-»>+• »• »*;x^oco •• -t *<D(%j^»—

.

S" »-l— —<C£0 (NJ—^^-LfMn —...4- . ^^—h- —m »>cQvO—' in^:>(\j< »-aj Ovor-n. *-o >o

Z ^-50. •> •• ""-O^Z •» <^„w«l «. >qx —«4"0 »"C\j •» -4" H-

0>"-<X_jh. tr»rMLU~—>-* c\i Z2: ••»— uj •.(/) ^-iCDN — LLI

•»r\l ••T't-H • »-^Q_rj'j3>_ h- Zo>o -i^> --tx —xo o o —z ».j-x «>o <\j •—»-* »*z —i a. m<OLUDZ w-»0>—•*"• Q- 2C -ft

^ .. •>2T_j»-i i^in—t »~0< > O *>-i/)a; •• «•(— «»-0 *•—»c\jt_) •> O ^*

•» •-<! UJ UJ •• ~» •Is- 0>* » *» ""*

i«£ co h- 2: _i> in>t— c\jvt'- in > 2:

•"^couj"—tuj •»—»<r-vj •—« »• o r£

O »oz_j •> ^oa.^"U -* *-2: «-

ZcQ< -> rvl2!< .OZ OJ "->< ^ —"~"<Q. •-roOt— «-'<o— fNJi-^ ~-> —. ~» —. Z w ">-<•

•Oi~Oi>i<-o > •• -o>^< I— •* ai iu uj »o -"'s-

o —, . •1 c\jro •• o iu z z z —• 1/) 2:0

<~4h-0>— IinO^>-» -J Z - - •• II SI CD rr*

C£> • "QCCtL *--£>~4t^i ••{*- Cl •••-»•-« »"H ZO <K ZlPv

-00 -a — - -Z—' o -• 11 11 "I +& + — °°

t/5*>J2: »-0 •« m^r* -oca - 11 z Z Z -~u. fNJ O---

>s: »-_ict:s: vocm-——.ooa. ~ 2: •• » m •« <ro»oq «-oo —--x--n^< CO »«.*»« vOZ •« occo

</>rvJZ -^T\0 <tO<>-MO — O m O *0 — v.f\j

r>z ~*<a- fM< »-J> <fiu m en vt o -•-< ^ 5 •

+ +*+** «-.af"S —«•— -J -• •" •" * ^^ 2- ^^>_JZLL_)2 ZJ -O -NOH- - vO -- >0 MO • ^•.^v-i »OoO h-«^— rO— >— >-i r-4 ^ fO vt i-UJ ^ <£ •

Z> -<<r^O xm—ia-<0>- M " « H -* <* ^^^ X^^>M2^<* -.°SJ»< m ^*°23 «^~ ^^^;o.jih m-4- z—o-*m -.—o-^n—o— ^o-o ^T"

>->^<(j>uj OCX)— -m »wOi/> — -Zi^Z^2oS-*2Q * -^ *t*JT

o2o?:0O- ^~Z^J<~au. lna>~ Z-Z-Z-Z-ZZ^UJUiOOH wC>

zt,:^ ;^^ S^3^ocQ<0^^^^uJAi^in^in>tinsi->D^ii.zs:ss;-fs»----ro--.s:

hk2 S|oj< ^3m ^2SSiKxfeS55z55555555552,0,0^H5 ,

"QoKSS^

^E5on£l^£?3Smo^£^~ oooaaa: a:a*^a:PC^i-oou.u.<o<zozzu.SSS^di^<^3 -SSz^>^h-QO^diro^a.acxaaaa.aa.<oo™aioa:<o<<-

^fxjm>tmo -<rsjf0>tm-3 2m<rrM^a-irsi^ o<rr>-coc7>o-.rNjrOvrm^r«.OfM 00 ,^ ^^,^^ a -^ -t^ -•^ fM r\j ro ^1 cm rvjrM

^^ *-._, o3C «->

UJ'XJ

00Z.Z.Z-

11

ooooooooooooooooocoooooooooooooooooooooooocooooo

mm irnTi ialt> inmmmmmmmmmmmmmmmmmmmm vo ^oo o oO --0 ^o o o oo -o -o -o -o voo »oo o

2;s:Z5:5:5:r3:5:5:j:r:t5:ssss:!:mj:2:i:5:Z5:si:s:s5:ss5:s^sj:5:r:s22:J:si

Oo

oo•

«-»ro ••

com lurom _jmvt h-

»-h -^O •• -«.-»—.—»—. LUa: o» —

*

-. lu lu lu lu lu zo io m lu z z z z z •»

o# • Z •> • •• • * -*2* & *» • * ,~« ,-* i—t ^ -^ ||

H-l -» <f _t || II II II II Z• CO •• nzzzzz*-

Q COO O Z •.•.•>•••• —

.

lu o<m • . „»» «^, *«, «. -» mt— <M + —< -.OOOOOnGV— + «— »• O M f'l -^ ^ <o •q »^2I O "' •.•.•.•«•» •—I «—.«•».«,

I 21 •» • •• ^ t-« i-« .H -* >0 —.LULUUJa. »-z -* h -^ (\i m >f ir\ ii lu z z z

z~- II II II II II II ~sl z - ••

lu ^»o o. 2: z si s z z •• .. _« _i _i

CO oz <t •• •.•.•.•.* .—. —< II II II

_, z<r z ~~~~-~ZHZZZ-. o <— - z z z z z z z •• -. ^_ ww (— „•.*».•.•. z .. ^ -» .^

—oo O ZZZZZZ— —omoI oo ZO -» — — >-.—. — wO mro<t-<5

ou >-o a o o o o o o z _, * •— OZ 00 D ZZZZZZ< -O-4-O—. olu * •» <<<<<!<•• *^ ~4 ro >d-

za> coz # - ~ " " .H"^ « ^. ^- o — c <^~ ~ — — — —— s: -z z ^ zvoo z z sz^s oa ~z _j os-z~z-z—z-,zin « m —» -. ~~ -

£^ * Z -^ ^ i-z z a. - *o -o -o -o »o ..oz--^-o-^otvO-O Z Z ZOOZ O ••Z > 00 -ZMZolZ+Z^^-OZ —~,^0 -Xro,i_^-2:v0>,

„ ,^ w v- 2cO-«— OO Z—• •• Q^w ••—• ••** »>—• •"-» »— •—<0-0—••»•.•»»»»••

vO^r, O CD CXOO O — =»UJ m -J M Q^QhOhO^QhQ ^<i -M Z 'OZ-ZO Z-fs^ -x -^ ^, ^^ -.»_ q<ij o uJZ<rH^M»iio<?tsj-<f'r(< ii aro^^^wYi-.f^.nm ^ ^ rf .co5 »Z <:*!- — ni M & n a" II c< ll c£ M a:z— -^ -oo n (yi n >.o n oo»* ? + ?0*+ oSunucC I - J- iz-z-z-z-z-z- :

*z:~z->z-z~

^^ a O Ow^O 5tf"ll h H -zzzzzzzzzzz '^uT22

^^m^^'n'o i?":t

,

<?i- 'rU°Z^??0^aj-. *-C^cTco«^co^co^coo>'000^)0.j-m^m vfm<t-

s?^ii.S-.^--^~^.jj .a. "^ ^-i .ujDoooooaooooooo-o^ooooooo^r-r „ *. *. »-<0 •> z —I CO^-"—mh-ooOZ "*

55°^°«2~<5Si-ko3Sooos-j HOZZZZZZZZZZZZZZJZZZZZZZZ

MMO<o<3<<«<ou-<QQD>^^o^ozaaaQ.CLiiac.a.Q.aaaaMaaaaaaaa.

(M^f^i.^m^.xv-nm.'O a. O o ozmmmm vO^^

CM CC O OJQC

oqooooooooooooooooooooooooooooooooooooqooooooooo(Mc\j(Mro(MC\jp^i(X)f\i /nff>.-nromi'Ororr»('r) r^vf %t- st -j- vt* <j- ^^t^t- tir\mmir\uMr\ini.rwur\ soo -oo -o o -o •€> ovOOOOvOO^OvOO OvOvOOO>O iO>0'0'0>0>0-0^000^>Om:\OOvO>0 -0-0 0-0*0-0 O O O O O O -O

S2S5:5;ix5:sj:>:2:5:j:iSZS5:rs:ssEsss5:s2TX5:5:2:j:?:i:r2s:5:5:tsz25:<<<<<<<<<<<<<<<<<<<<<<<<i<.<t<<<<i<<r<<<<<<r<<i<t<i<<i<t<:<<

>* a >v •> -id r> - -o moo z *>» a. —-«cr: -o vj-—.-^ oin cm a •• -z: 1—«o -—01s- »f -,~4 «•« • <>3C ••>— • > »-^-» »>CM«-»

m* - O sO 00*~UJ -X »->frO— w,f»>». x uj ul -I -^wo •^f\io:.n>-—n. oo io - 3: -tu^aiz: in—.<oo'jjoo >k s ii <3i- -t-~i so^ •- -x-s:—. *v s s: -o *>'jj -h- ->-— j- * •«

in —OX S^X^OCO v}- -Or\J——«. (_> uj - »"4- •» »-i— i— —»m "TjvO—

00 »-» Q <\| bii><M< -UJ Oor~a -'J)

nh ^ • Z I— ->0l »• »• -w3v}-2•. o i—— >0 -m - -OX — vi-O -fNi -

X hi —<rj\ u_ x>>—<X_ji— incvjuj— «»—•

00 «/) 2:>-h •• 2- »+-UJ -oo O^-tXOOf*->», OO ii -<\i -Tr—* - •^aojOw^ x <*-« z o>o -^> tx —-io•» o s: - z -h-x -o <m — —« •>;<:

o - x - »<oujdz >—-—o>—•—•

• O >C0 h- *1 - -2T_J 1-' fv»m—i »-o<0% t J-V. o >.00C£ * -i— -o -MOm ^> >—«>v uj - »-=ii-iJuj • —» *r^o—' -,. U_ O - ""3 *£C0H-5--J> m<4-—'<\J^—

.

II». Oo oo xa - - -t— o<no_—>^

i_ ii _juj o ««:>i/>ujh-iuj »—'<n ••»—

^ UJ |_ LUi/) O -OZ!_l - 4-O0.^C-»iJUj(V (x > O ZcO< - -> <MZ< »02-tqj ^ ££ X 2T <0- -oaCDl— <—<io— r\J>-H~K ,__ uj h-vd* h-. 3i~o^oo > - -o—

<

i_4 g_ I— 2 - h- Q -'", - - - i— —»—»r\r<> -

IK z li» UJO U, <-^»-0>i-i XiAO->-»2 o < cc • -< cc> •• ••o: or: »sO'-<rsJ -^»

^^ u. x o sns >2: -ooc:5: .or\j .^io.

^^ z __».-.t_5 —.->—• u_ - -Oo -oo »-»-XLn^<^<j _ OOO"-* I— II OOO O O0NJ2 -u-\0 >d"0<«-''NiO

^,v o oiaTc^uj 2Tf- • • • 3Z -ac<a. f\i< -_|>^_ _ djdi < stvj-^j- -z. - -o - - - «-a:~2: --

.

Z22 l~cx:u.U_LL O >_j2ii._IZ O -O -—.r>-

•• ».q: OUJ - - - h-< »2;i-< ••Qoo h- —. —i—. >-n —

»

ii z:s:uj o

— »-o uj ^j_ . ^ o -<<roiO iin>-^^<im^^ it! 'li'iitli- wu.ni -i -oox -a. -so^— <a.^x-^; , , _,«,«!< uj<—<—i-^ ^> ^2: -o_< — -.s;_j ~<»^:' »— 1T1 •.•••• 3: a: - - - a - -o-jxk m<r -z:—

o

^-^xm-J -*™x X—OOO— a. »Z<Ouu vOrvj— *l-»

V.SvO^ • • »^>X \~c£*-* «->^LU -—.0— O—o 00 f*- uj -* •J* -t o^ ^^ X'u<<ioo_j >4*i/>t-«iri -r*i^oofMso^

_^ rvjm^ - •— xa.n^fl m -a. -i- -

O— fM>-4U-«

^, >>>x c3^v:I/^3—-o - coo -aiij-i> >- »:<:>NJzsT^^S o -s-^- --—-J- u ^-*'D^os\ -x oi-saoaoH ^j— z: cc^-x.

Z.HL J^ ^ — - - - —4 ir> ^•jj-h-i n.^ ~5Xoi<r - -_J t-ua -ujco -

k ,.vlt • 5:^2 1— 1— l— - hX -O - - — - -UJOI~< - XO— Q. —

??E-^""^^M^7^-a £lx£S£££2 ZZWS^< -7^ Q.O0X

£5££o3gS§ii>55SS5su.>ii.u.u.o>u.icii.u.^oooc«u^oaS<^^2z->-

2S£££°^35S«-*st- O OOO OOOON .^CM

t-^ Q^ t—« t—» r-4

3:

79

oooooooooooooooooooooooosOr^r^-r~-r-r^-h-r-r^r^r^-oocnco'X)coooaDcocoococ7>C7^sOvOvO OvDJ?0.0>0-0-0-O^OvOsOvOvOnOvDX!OnO-0-0fsJfV r>sjrvjrvjrxjr-Jrvjr>vlr^J ,vjr>j-vjf>Orvlr>o-v|-vjrvJ->J'vjrvj-vji^

<i<r<<<<<<<<i<< <i< < <:<<<<:<<< <<too

tooUJQ.

-J ^>-<! ^*H-.-I —

.

«— OLU •-"

in oa •«o -JO •>

.. uj xst >H- ~< «»CNJ <t •• V.•— f— rf\ «-»

H- Z«— •-* OO UJoO t

_J QCUJ X «*

Q. ctoj r\J U.

> =5C£ — *•• OO in ••

«-* uj »• Xin oq x —

•

\0 ^ —»tn *.. CfZS-i o^ t-t"»\ •

c\j -» m—. -4-

<— 5!2<I^ u.

H UQ -—'

*-

O -mX * in

_j zj--i- •» — ••

(X Oo— 3: !\i X — -"•

•- i— cc *• 3^ h a!— <MX.fi u. ^ r> ^00 x^m^ in—»-» "Z.-^—»—»—»2I

—»*- UJ »-< • "SO *OZ>Z3^> ••

<fuj jqsx x<t •—^o:ciia;a:-<^-_i ujz—>m ma^Nt-r} 11 Z3p3Z) 11

Oh- <mw hou-^mZ^z^hf->~. c£ <r\) "^O *0 •»•••.•.*•Ot— ujo •—« c\ia: »»Z-"" —<-*—»-*-"»

h-UJX " «a.X —— 'I M «i " —

•

-*-» <oo<f-x O Hr^w>-< «-l >-<—<_o^'r\ I2W—<j- 'J_U_ •« II M ~ + ~ -Nt-'M

»- -Ooo xsifNi ••oocMOO'-ts:^—«—"— >3Dor ».t— vi-u-*"'!-'-1 • * •**"—"-"-< •• •*

<j-z —i «oz •o^t——..—ww—

o

—»£X.fX, X •*>-•&. fXrvj—*i*J —»—»-«-«-» '"» •"

>j-<r intuiux ONt»-«>Z •* • * ••>o_;-!_J •• Cl ^1' ITN _l • —* •»•*— h— — H— >——"—

<

co <r r-i^-i ;.<«* »-x » • •• * :0

zdz^-^aj x i— y-x *-* —*x '-nm -J" inr-*-»>o

<0'J^UJhhIj/JJ.IAhOO-»'»-»OOI-0•..-HlJJ 'Dw^-t—UJww— COCO<MfNJ<MCO00 ( 30

-*oO'iOSt>—O H— Q. I— t—H —<-^—

<

w LU -«i-:r2:r~s: ssizzooozz-zdh-^LL^sra:— ^cci— dc a: q; i-i—<< <r< —« »-i ^-.^p

»-Q txoa.zsx u.u.u-ao.a:a:Qia.Q--< a.a:uJ

Z5 rom >t 3m v0 r» co cmd m^mvOQ. OO OO-OOOOO-H —i~-,-t^H- coco CO H- CO CO <v 00 ao co cocococo

o o

80

BIBLIOGRAPHY

1. Defant, A., Physical Oceanography , v. 2, Pergamon Press,New York, N.Y. , 1961.

2. Departamento Nacional da Producao Mineral, DivulgacaoTecnica No. 1, As Mais Recentes Medicoes do Rio Amazonas

,

by S. de Souza and others, Belem, PA, 1964.

3. Diegues , F.M.F., "Introducao a Oceanografia do EstuarioAmazonico, n Anais Hidrograf icos , v. XXIX, P. 129-157,Rio de Janeiro, RJ , 1972.

4. Dronkers , J.J., Tidal Computations in Rivers and CoastalWaters , Wiley, New York, N.Y., 1964.

5. Department of Energy, Mines and Resources of Canada, Manu-script Report Series No. 17, Simulation of Tidal Motion inComplex River Systems and Inlets by a Method of OverlappingSegments , by R.F. Henry, 1971.

6. Environmental Prediction Research Facility Technical PaperNo. 3-72, A Description of the EPRF Hydrodynamical-NumericalModel , by T. Laevastu and K. Rabe, March 1972.

7. Environmental Prediction Research Facility Technical PaperNo. 1-74, A Vertically Integrated Hydrodynamical-NumericalModel (W. Hansen Type) , by Oceanography Department of EPRF,January 1974.

8. Franco, A.S., Livro Texto de Mares para o Curso de Hidro-grafia para Oficiais , Directoria de Hidrografia do Ministerioda Marinha do Brasil, Rio de Janeiro, RJ , 1964.

.9. Hansen, W. , "Recent Development in Tidal Theory," Proceedingsof the Symposium on Mathemat ical-Hydrodynamica 1 Investiga-tions of Physical Processes in the Sea -Moscow, May 1966 ,

Hamburg, February 1968.

10. Hansen, W. , "Theorie zur Errechnung des Wasserstandes und

der Strommungen in Randmeeren nebst Anwendungen ," Tellus ,

v. 8, p. 287-300, 1956.

11. Henderson, F.M., Open Channel Flow , MacMillan, New York,

N.Y. , 1967.

12. Gibbs, R.J. , "Circulation in the Amazon River Estuary and

Adjacent Atlantic Ocean," Journal of Marine Research,

v.* 28, p. 113-123, 1970.

81

13. Gibbs, R.J., "Amazon River Estuarine System," GeologicalSociety of America Memoir No. 133

, p. 85-88, 1972.

14. Ippen, A.T., Estuary and Coastline Hydrodynamics , McGraw-Hill, New York, N.Y., 1966.

15. Institute fur Meereskunde der Universitat Hamburg, ReportNo. 18-71, Hydrodynamical Numerical Investigations andHorizontal Dispersion of Seston in the River Elbe , byH.G. Ramming, February 1971.

16. Neuman, G., and Pierson, W.J., Principles of PhysicalOceanography , Prentice Hall, Englewood Cliffs, N.J., 1966.

17. Ramming, H.G., "Shallow Water Tides," Proceedings of theSymposium on Mathematical -Hydrodynamical Investigationsof Physical Processes in the Sea - Moscow, May 1966 ,

Hamburg, February 1968.

18. Ramming, H.G., "Investigations of Motion Processes inShallow Water Areas and Estuaries," Proceedings of theSymposium on Coastal Geodesy in Munich , p. 439-452, July1970.

19. U.S. Geological Survey Circular 484, Amazon River Investi-gations, Reconnaissance Measurements of July 1963 , byR.E. Oltman, H.O.R. Sternberg, F.C. Ames and L.C. Davis Jr1964.

20. U.S. Geological Survey Circular 552, ReconnaissanceInvestigations of the Discharge and Water Quality of the

Amazon River , by R. Oltman, 1968.

21. Von Arx, W.S., An Introduction to Physical Oceanography,

Addison Wesley, Reading, Ms, 1962.

82

INITIAL DISTRIBUTION LIST

No. Copies

1. Defense Documentation Center 2

Cameron StationAlexandria, Virginia 22314

2. Library, Code 0212 2

Naval Postgraduate SchoolMonterey, California 93940

3. Department Chairman, Code 58 3

Department of OceanographyNaval Postgraduate SchoolMonterey, California 93940

4. Dr. Stevens P. Tucker, Code 58 Tx (Thesis Advisor) 4

Department of OceanographyNaval Postgraduate SchoolMonterey, California 93940

5. CC. Luiz Antonio de Carvalho Ferraz 6

Diretoria de Hidrografia e NavegacaoPrimeiro Distrito NavalRio de Janeiro, RJ 20.000 Brazil

6. Diretor 3

Diretoria de Hidrografia e NavegacaoPrimeiro Distrito NavalRio de Janeiro, RJ 20.000 Brazil

7. Dr. Edward B. Thornton, Code 58 Tm 1

Department of OceanographyNaval Postgraduate SchoolMonterey, California 93940

8. Dr. Warren Thompson, Code 58 Th 1

Department of OceanographyNaval Postgraduate SchoolMonterey, California 93940

Dr. Taivo LaevastuDepartment of OceanographyEnvironmental Prediction Research Facility

Naval Postgraduate SchoolMonterey, California 939-0

83

10. Mr. Kevin RabeDepartment of OceanographyEnvironmental Prediction Research FacilityNaval Postgraduate SchoolMonterey, California 93940

11. Office of Naval ResearchCode 480Arlington, Va 22217

12. Dr. Robert E. StevensonScientific Liaison Office, ONRScripps Institution of OceanographyLa Jolla, Ca 92037

13. Petroleo Brasileiro SACentro de Pesquisas e DesenvolvimentoRio de Janeiro, RJ 20.000 Brazil

14. DiretorInstituto Oceanograf icoUniversidade de Sao PauloAlameda Eduardo Prado 698Sao Paulo, SP 01000, Brazil

15. DiretorSegundo Distrito da Divisao de AguasBelem do Para, PA, 66.000 Brazil

16. DiretorDivisao do Porto de SantanaIndustria e Comercio de Minerios (ICOMI)Macapa, Amapa - Brazil

17. Dr. Walther HansenAttn. Institut fur MeereskundeUniversitat HamburgHamburg, Germany

18. Oceanographer of the NavyHoffman Building No. 2

200 Stovall StreetAlexandria, Va 22332

19. U.S. Geological SurveyAttn. LibraryWashington, DC 20390

20. Library, Code 3330Naval Oceanographic OfficeWashington, DC 20373

84

21. SIO LibraryUniversity of California, San DiegoP.O. Box 2367La Jolla, Ca 92037

22. Department of Oceanography LibraryUniversity of WashingtonSeattle, Wa 98105

23. Department of Oceanography LibraryOregon State UniversityCorvallis, Oregon 97331

24. Commanding OfficerFleet Numerical Weather CentralMonterey, Ca 93940

25. Commanding OfficerEnvironmental Prediction Research FacilityMonterey, Ca 93940

26. Department of the NavyCommander, Oceanographic System PacificBox 1390FPO San Francisco 96610

ThesisF2645c.l

1S28Q6Ferraz

Tidal and currentprediction for the

Amazon's North Chan-nel using a hydrody-namical -numerical model,

Thesis

F2645c.l

lGr.806

Ferraz

Tidal and current

prediction for the

Amazon's North Chan-

nel using a hydrody-

namical -numerical model.

thesF2645

Tidal and current prediction for the Ama

||ili!i|l||iil|iii||H

III I II II II

3 2768 002 06557 5DUDLEY KNOX LIBRARY