TIDAL AND CURRENT PREDICTION FOR 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.u i %j %0 H TIDAL AND CURRENT PREDICTION FOR 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. Unci ass if i ed SECURITY CLASSIFICATION OF THIS PAGE (When Date Entered) REPORT DOCUMENTATION PAGE READ INSTRUCTIONS BEFORE 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 the Amazon's North Channel Using a Hydro dynamical-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 School Monterey, California 93940 10. PROGRAM ELEMENT. PROJECT TASK AREA ft WORK UNIT NUMBERS II. CONTROLLING OFFICE NAME AND ADDRESS Naval Postgraduate School Monterey, California 93940 12. REPORT DATE September 1975 13. NUMBER OF PAGES 14. MONITORING AGENCY NAME ft AODRESSf// different from Controlling Otllce) Naval Postgraduate School Monterey, California 93940 15. SECURITY CLASS, (of thle report) Unclassified 15«. 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 model Amazon River Numerical model Tidal model Amazon's tides 20. ABSTRACT (Continue on reveree elde It neceeeary and Identity by block number) 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 com- pared 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 The driving forces are the tides at the northern DD I janM73 1473 EDITION OF I NOV 65 IS OBSOLETE (Page 1) S/N 0102-014-6601 | Unci assif ied SECURITY CLASSIFICATION OF THIS PAGE (When Date Entered) Unclassified JuCUKtTY CLASSIFICATION OF THIS PAGEr»>>,n Dr(m Enlmrml opening of the channel near the river's mouth and the river discharge 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 northern part of the channel agreed, in the worse case, within 12% of the tidal range, but those predicted at the southern end were unsatisfactorily reproduced. This fact is attributed to the size of the grid which is too coarse to describe adequately the variable 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. 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 Ferraz Lieutenant-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 LIBRARY NAVAL POSTGRADUATE SCHOOL MONTEREY, 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 THE AMAZON RIVER 16 A. GENERAL -- 16 B. CHARACTERISTICS OF THE TIDES AND CURRENTS IN THE AMAZON RIVER 17 C. THE AREA OF STUDY 18 III. THE HANSEN HYDRO DYNAMICAL 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 and Investigations in the Amazon River and Estuary 32 II. Current Velocity and Direction Predicted at the Three Tide Stations by the HN-Model 37 III. Computer Output for Water Elevation and Resultant Current Velocity and Direction in 12 Selected Locations 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 cm2 square centimeter cm3 cubic centimeter cm/sec centimeter per second cm2/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 km2 square kilometer L characteristic length I length of the mesh LT low tide M,N or grid coordinates m meter m2 square meter m3 • cubic meter m3/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 V2 |^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 106 km2 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 m3/sec, during low stage about 72,500 m3/sec, and during an intermediate stage, 164,520 m3/sec. The average flow, estimated for the discharge into the Atlantic Ocean, was slightly over 175,000 m3/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 AND CURRENTS 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 pressure are due solely to changes in water surface elevation; 3. Advection terms are neglectable; 4. The fluid is in hydrostatic equilibrium in the vertical direction; and 5. The geographical and vertical variations of the Coriolis force 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 [W2 + W2] !/2 (x) xL x yJ Tr , = XW [W2 + W2]1/2 (y) yL x yJ (4) (5) The bottom stress components are given by xb(x) = rU [U2 + V2]1/2 Tb(y) = rV [U2 + V2]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) = ct_T(n,m) - X[H*(n,m) Ut(n,m) - H*(n,m-1) U^n.m-l) + H^Cn-^m) Vt(n-l,m) - H*(n,m) Vt(n,m)] (8) 23 CN O E >. (A o 3 U D D >- Ctf < O CO LU z o i M 00 z o 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^[ct+T(n,m+1) - Ct+T(n,m)] + 2TXt+2x(n,m) (9) Vt + 2T(n,m) = [1 - (2xr/ H^+2fn ,m) ) (V* (n ,m) 2 + Ul (n ,m) 2) & V1 (n,m) ] - 2TfUt(n,m) - L%t+T(n,m) - C t+T (n+1 ,m) ] + 2TYt+2T(n,m) (10) where: U^n.m) = aUt(n,m) + ^ [Ut(n-l,m) + Ut(n+l,m) + Ut(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 [Ut(n,m-1) + Ut(n+l,m-l) + Ut(n,m) + Ut(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 Ut+T(n,m) + Ct+T(n,m+1)] (13) u (or v)K> uK I The effects of wind are computed by: 25 t At t2 t 2 1/ 1 (5p x* - £**[(**] ♦ up ]A - I«|0 (14] Yt = HWy [("x)2+ C*y>Y* " £|f (15) where P0 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, H max 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, 0 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 0 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 0 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" cm2/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— ( ►J t— i PQ W ac H Q W E-> W co w Pi w Pi co H CJ t—i CO < CJ 2 CL, o o I— I H CJ W Pi W H W H < CJ i— i Q 2 co j * * < - ,o b Q >• "> #V 2 ° «? • — 1 < o w 2: ~ cf z Z ° -7 < ^ ffi # u Q '. W Pi PQ o < 2 -J 1 . CO CO - 12 2 o tf> o cti • ' X tsi Pi CO §j < < m < ac 2 V m h t— i ■ ! ac Q H Pi '.' Pi o o o W oj CJ -o j > Cn] i o P 1 • »— ■ i 7 A H O pi 2 2 cj lO /) i H S Q 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 m3/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 rH rH rH o O •H •H O rH O O O CO CO CD o CO O (D to o O O "St O o CO CD P O o CD o o CD o r-- O o3 u C -H CD t». CTl LO <3" LO o o <* 03 O C/) O) t^. LO O- to to cn <-i o CD CD "5f "5f o o CO ed o CD CD LO vO o o rH CD CM o •* CD o CO o o Cn) P g 1 < o cni vO LO -5t o CD vO t— 1 en O CO rH r-- rH CO r— 1 rH CN) to ^-i rC P CD o O r-- to o o "t T3 6 CT> \o CO cn CO o o o •H rg CN] CM CD vO LO LO r-g & #v #\ r* #* *. •s #» 1 cni crj •H o rH rt •H •H •H C r-1 3 3 U 3 03 u x> ,0 X o3 03 O O 03 o X o o O o 2 s 2 2 s S O ■H 4J to tO «* tO tO Q> o3 vO vO mo vO VO x: s •H 4-> *o CD en CD cni vO sO vO •o +-1 crj r-( Csl rsj cn) cn cn cn en 4-> (A> Q 1 1 rH rH i CO i rH i— i rH rH < i— 1 cu CN] 0) •p w c o •H P 03 bO •H P V) O > G •H T3 C crj w P C d> 6 CD rH 3 w crj 1 •H OS ►— I (=; w o — 1 N pq o3 < ^ 32 ± < E c - in - *f ro - CM cc h- z o Q O z o o < Q_ < < cc < LlJ z z o h- o UJ (/) j (/) o a: o z O M < UJ D o o O CO (^) 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 in oo r^ CD J o o o o CM T— HID O UU3 O O O O w H- 1 CJ I— I PL, Ph w o cj o H cj I— I «: Ph w ac Q < Q W CD < ac cj co < ac a, < > o < ►J W Q O w an CJ i— i an H Q W CJ z < u CO Q < H < H CO O CJ CO < a. < o w eq 2: < E-« CO O CJ Q W l-H < H 2 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 LI MAO DO CURUA ACAITUBA MACAPA VEL DIR VEL DIR VEL DIR cm/ sec 0 cm/sec 0 cm/ sec 0 76 73 242 80 210 40 196< :ht 77 11 062 43 211 42 220 78 100 061 11 027 33 206 79 165 062 60 031 13 202 80 196 062< ;lt 91 031 21 029 81 192 063 109 032 43 026 82 151 061 115 032< LT 54 025 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< :ht 71 209,^ 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 92 174 060< . 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 E o o © 0 -a 3 r^V SbH 3d y.-Ujd iNiibdno jjs N&?skd SbH 9t! aiJi.-id no; 41 O O -U f \ T T ?o?tv/ r / p g Sjy1/1/' iNdbd '0 • m -'..•• SbH 8^ d31Jb IN^bdHO 3JS ND2U 42 o o 1 U .1 A. f r ? 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V)UI , ' • _1 ' 00 LU ' > 1 • ' K i^ , -JU ^ , 1 | — • ir, !, < uj -|T\ \ \ V * i-co 0 a c f \ \ V < 0 * 1 . • 1 , 1 r 1 * 1 -I.N 3 '- A 31 - 31JU 1 n; 3b i - ] . - ii r^ I t! - -- :" " . 51 o o ■ Lc 'J u / / I // / 1/ / t I I 1 / ' , ) • • CN O V *■ E V o o 25 i- a. ; 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. . . ' I-. I , . 1 1 4 J . lii. M >. ,21 J. KO .1 i 7..261. I 40 . 0 ) 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 . , ? 0 ' (80 Br) (81 Hr) . . "34 . -22. u 43. . ,-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 0 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 0 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 0 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 J0. ,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 0 -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 265. 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 oTu 2 2 3. 14 3.44 . . ,'36. 125.6; .,271, TABLE II Current I. Computer Output for Water Velocity and Direction in 12 S Elevation and Resultant elected Locations 54 o Z X X X Z X X X < D on D u O o O < X I I L . I L / J I I o o a (™) 1H9I3H HWH- NH 55 z 2 1 1 Z X I I CM UJ => CO D < < o co -U I L J I i"1) 1H9I3H (iamo) WH-NH 5 6 I I z 2 z x x x CN CM 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 o LU M -J 5! 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