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Continental Shelf Research 23 (2003) 1597–1613

*Correspondin

E-mail addre

ricamarg@mode

0278-4343/$ - see

doi:10.1016/S027

Numerical simulation of the tidal propagation in thecoastal region of Santos (Brazil, 24�S 46�W)

Joseph Hararia,*, Ricardo de Camargob

a Institute of Oceanography, S *ao Paulo University, S *ao Paulo SP 05508-900, Brazilb Institute of Astronomy, Geophysics and Atmospheric Sciences, S *ao Paulo University, Brazil

Received 19 February 2002; accepted 25 July 2003

Abstract

A sigma vertical co-ordinate model was implemented for the coastal region of Santos (Brazil, 24�S 46�W), aiming at

high-resolution barotropic tidal simulations. On a grid of 120� 80 cells and spacing of about 1 km, tidal elevations due

to the nine principal constituents were specified at the open boundaries, based on a model of the shelf. The coastal

model results were compared to observations and harmonic predictions in the Port of Santos. Tidal analyses of time

series relative to 5527 wet points, with 696 hourly results of elevations and surface currents each, generated maps of the

co-tidal lines and axes of the current ellipses, which numerical values were compared to harmonic constants of some

coastal stations. The obtained maps indicate the main characteristics of the tides in the coastal region of Santos, such as

the areas of amplification and main propagation directions. The model results generated maps of the potential, kinetic

and total energies associated to the tides. Residual depth-mean currents show tidal eddies related to coastal geometry

and bottom topography. Vertical profiles of the tidal currents have nearly constant intensities and directions along the

vertical, from the surface to the top of the bottom layer. The implemented modeling may be used for operational

predictions of tides and tidal currents in the study area, especially when considering grid nestings in the shallow inner

regions.

r 2003 Elsevier Ltd. All rights reserved.

Keywords: Hydrodynamical numerical modeling; Coastal region of Santos (SP,Brazil); Sigma vertical co-ordinate; Tidal currents;

Co-tidal maps; Current ellipses

1. Introduction

The region of this study is situated betweenlatitudes 23�400–24�300S and longitudes 46–47�W,in the State of S*ao Paulo, Brazil, including the cityof Santos and a complex estuary formed by the

g author.

sses: joharari@usp.br (J. Harari),

l.iag.usp.br (R. de Camargo).

front matter r 2003 Elsevier Ltd. All rights reserve

8-4343(03)00143-2

Channels of S*ao Vicente, Port of Santos andBertioga (Figs. 1a and b). This is an importantarea due to its large population and comprehen-sive economical activities, the main ones beingnavigation, fishing and leisure. Both sea- andinland-related exploitations have produced envir-onmental impacts, such as pollution and coastalerosion, which are becoming a major issue toscientists and technicians. This paper is a con-tribution to the knowledge of circulation in the

d.

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Fig. 1. Coastal area of Santos and the bathymetry considered

in the grid: (a) whole model domain, with positions of coastal

stations and (b) internal shallow areas.

J. Harari, R. de Camargo / Continental Shelf Research 23 (2003) 1597–16131598

coastal area of Santos, aiming to support theeconomical activities and other environmentalresearches, such as biological and geologicalstudies.

Several hydrodynamical numerical models in-cluding the State of S*ao Paulo shelf have beenrecently implemented by Harari (1985), CastroFilho (1985), Stech and Lorenzzetti (1992) andCirano and Campos (1996) among others. Thesemodels adopted grid spacings in the order of 10 kmand were used to different insights on the shelfhydrodynamics. Harari (1985) adopted the Galer-kin solution in the vertical, based on Heaps (1972)and Davies (1980), to simulate barotropic circula-tions due to tidal and meteorological effects;applications of this model were presented onHarari and Camargo (1994) and Camargo andHarari (1994). Castro Filho (1985) used a finitedifference solution on simulations of subtidalmeteorological effects. Stech and Lorenzzetti

(1992) studied the response of the region tocold fronts using a barotropic finite elementmodel. Finally, Cirano and Campos (1996)reproduced the water masses circulation, includ-ing the adjacent deep oceanic areas, through adiagnostic simulation based on a sigma co-ordinate model.

In the State of S*ao Paulo, several tide gaugeshave been permanently recording the sea levelvariations since the 1940s, the main ones being inUbatuba (23�300S 45�7.30W), Santos (23�570S46�18.50W) and Canan!eia (25�10S 47�55.50W).These data were analyzed in several reports, asthose of Mesquita and Harari (1983), Franco andHarari (1993) and Harari and Camargo (1995).The later processed yearly records collected at thePort of Santos, from 1944 to 1989, obtaining itstidal and mean sea level variabilities.

The main objectives of this paper are to presenta detailed study of the tidal characteristics in thecoastal area of Santos and to give support toengineering and environmental issues. The tidalsimulations may also be considered as a first steptowards the modeling of the whole coastalcirculation (including river outflows and meteor-ological forcing).

In the study, the Princeton Ocean Model (POM)(Blumberg and Mellor, 1987; Mellor, 1993) wasimplemented in the coastal region, for high-resolution tidal simulations, considering a gridspacing of about 1 km (Harari and Camargo,1998). This model used tidal oscillations at theboundaries imposed by the shelf model of Harariand Camargo (1994) and its results were comparedto observations and tidal predictions in the Port ofSantos, based on the analyses of Harari andCamargo (1995). The amplitudes and phasesobtained through the analyses of the time seriesgenerated by the model allowed the attainment ofco-tidal maps and ellipses of currents maps; theseresults were compared to the harmonic constantsof several coastal stations, which positions aregiven in Fig. 1a and Table 1 (FEMAR, 2000).Based on the tidal maps, characteristics of the tidalpropagation in the area will be presented anddiscussed. This study also included the computa-tion of the tidal energy, residual depth-meancurrents and their associated tidal eddies.

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Table 1

Coastal tidal stations in the modeled area: number of components derived by the tidal analysis, position, beginning and end of the tidal

records and general observations (FEMAR, 2000)

Tidal station (no. components) Latitude Beginning Observations

Longitude End

1 Pia@aguera (36 comp.) 23�52.50S 24 Jan 68 Tides with diurnal inequalities

46�22.60W 24 Dec 68 River and meteorological influences

2 Ilha Barnab!e (24 comp.) 23�55.70S 27 Sep 74 Tides with diurnal inequalities

46�19.90W 29 Nov 74

3 Torre Grande (32 comp.) 23�57.30S 01 Jan 56 Tides with diurnal inequalities

46�18.60W 23 Dec 56 (several other sampling periods)

4 Praticagem (26 comp.) 23�59.70S 05 Sep 95 Tides with diurnal inequalities

46�18.00W 07 Oct 95

5 Ilha Palmas (24 comp.) 24�00.50S 28 Mar 90 Tides with diurnal inequalities

46�19.60W 27 Apr 90 (several other sampling periods)

6 Ilha da Moela (35 comp.) 24�03.10S 12 Jul 62 Tides with diurnal inequalities

46�16.10W 12 Aug 62

7 Barra Peru!ıbe (82 comp.) 24�20.20S 11 Sep 81 Diurnal inequalities+river influences

47�00.50W 12 Oct 81 (several other sampling periods)

8 Guara !u(82 comp.) 24�22.80S 14 Jun 82 Tides with diurnal inequalities

46�59.20W 15 Jul 82

J. Harari, R. de Camargo / Continental Shelf Research 23 (2003) 1597–1613 1599

2. Methodology

2.1. The coastal model

The version of POM implemented has thefollowing characteristics (Blumberg and Mellor,1987): use of full 3D complete hydrodynamicalequations, written in flux form, under the hydro-static and Boussinesq approximations; verticalsolution based on sigma co-ordinates; second-order turbulent closure (2.0) for the verticaldiffusion; Smagorinsky parametrization for thehorizontal diffusion; mode splitting; explicit leap-frog method for the numerical integration in timeand the horizontal (centered in time and space);and implicit method for the vertical integration ofthe equations.

This model has been recently used for severaltidal simulations in coastal areas, such as the onesof Oey et al. (1985), Galperin and Mellor (1990),O’Connor (1991) and Stacey et al. (1995).

For the coastal region of Santos, a regular (EW–NS) cartesian grid of 120� 80 cells was used, withhorizontal spacing of approximately 1 km and5527 wet points. 11 sigma levels are placed at 0.0(surface), �0.03125, �0.0625, �0.125, �0.25,�0.5, �0.75, �0.875, �0.9375, �0.96875 and

�1.0 (bottom). The grid bathymetry was obtainedthrough digitalization of nautical maps of theBrazilian Navy, having depth-mean values thatrange from 2.0 to 53.5m (Figs. 1a and b). The timesteps are 15 s (for the external mode) and 300 s (forthe internal ones). The frictional (and filtering)parameters used in this model are similar to theones of Oey et al. (1985a), in their high-resolutionstudy of the Hudson–Raritan Estuary: constant inSmagorinsky horizontal diffusivity=0.01; bottomroughness parameter=0.002m; ratio of horizontalheat diffusivity to kinematic viscosity=1.0;advective terms of external mode updated at everyfive external time steps; finally, a temporalsmoother was applied to prevent solution splitting,in the following form (for variable V and timelevel N):

VFILTERED ¼ VN þ 0:05ðV N�1 � 2VN þ VNþ1Þ:

For simulations of barotropic tides, the associatedtidal energy and its propagation may be estimatedby using the depth-mean currents (U ; V ) andsurface elevations (Z) computed by the model. Foreach cell of total depth D; the sum of kinetic andpotential energies (per unit area) is given by

12r DðU2 þ V 2Þ þ gZ2� �

;

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J. Harari, R. de Camargo / Continental Shelf Research 23 (2003) 1597–16131600

where r is the density and g is the gravityacceleration. The vector of energy propagationhas components

rUD ðU2 þ V2Þ=2þ gZ� �

;�

rVD ðU2 þ V 2Þ=2þ gZ� ��

:

2.2. Observations and tidal forcing

Tidal elevations were specified at the openboundaries, based on the shelf model of Harariand Camargo (1994); the tidal oscillations werealso imposed on the first internal lines andcolumns of the grid, in order to avoid the needof current specifications, which first internalcomputations were extended to the open limits,as done by O’Connor (1991). The model resultswere initially compared to observations and tidalanalyses of the Port of Santos, computed byHarari and Camargo (1995), given in Table 2. Theobservations in the Port of Santos (Torre Grande)were taken for the model calibration due to thefact that it is a permanent tidal station and thecorrespondent observations have been carefullyedited and analyzed, giving highly reliable results(Harari and Camargo, 1995). In fact, the ampli-tudes and phases listed in Table 2 correspond tomean values of 46 yearly analyses of hourly tidalobservations. The ‘‘model calibration’’ consistedof small corrections on the amplitudes and phasesof the tidal components at the mesh boundaries

Table 2

Harmonic constants of the principal tidal components at Torre

Grande, Port of Santos (23�570S 46�18.50W), computed as

mean values of 46 yearly analyses of hourly sea level

observations (Harari and Camargo, 1995)

Tidal components Amplitudes (cm) Phases relative to

Greenwich (�)

Q1 3.0 99.4

O1 11.5 125.4

P1 2.3 182.7

K1 6.4 187.7

N2 5.1 235.2

M2 36.7 173.4

S2 23.1 179.0

K2 7.5 168.8

M3 5.6 358.7

after comparing the preliminary outputs of themodel with the correspondent values of Table 2(considering the adopted model parameters ofhorizontal spacing, time steps, number of sigmalevels, values of frictional coefficients, etc.). Asexamples of these corrections, the amplitudes ofM2 and S2 at the grid boundaries had to bemultiplied by 1.14 and 0.98 and their phasesdisplaced by –7.3� and –3.2�, respectively. Aftercalibrating the model by correcting its boundarytidal constants, a new processing generated thefinal results, and its analyses were compared to theharmonic constants of the tidal stations along thecoast, listed in Table 1.

The analyses of Harari and Camargo (1995) arerelative to 46 years of hourly tidal observations inthe Port of Santos, from 1944 to 1989; they wereperformed in a yearly basis, leading to time seriesof the tidal components amplitudes and phases. Inspite of being named as ‘‘harmonic constants’’they are not strictly constants and some variationsmay be observed, as shown in Fig. 2, whichpresents the amplitudes and phases of M2 tidalcomponent obtained in the harmonic analyses ofthe yearly records. Apart from the tidal analyses,Harari and Camargo (1995) also computed theevolution of monthly and seasonal mean sea levels,including their spectral analyses.

Fig. 2. M2 harmonic constants of Torre Grande, Port of

Santos, given by yearly analyses of sea level observations in the

period from 1944 to 1989 (Harari and Camargo, 1995).

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Variations of the amplitudes and phases of thetidal components in any coastal station may beattributed to some noise in the observations andthe spectral analysis, besides some long-term andlarge-scale global changes in the world ocean, suchas the decadal mean sea level rising in tropical andsub-tropical areas. As a matter of fact, these‘‘global changes’’ may be considered as theprimary cause of the observed linear trends inthe amplitudes and phases of M2, with values of–0.0162 cm/year and +0.0191�/year, respectively(Fig. 2). On the other hand, the corrections in theboundary values of the coastal model are due tothe fact that the spacing of the shelf grid is morethan 10 times the spacing of the coastal grid, thatway the shelf grid could not adequately representall the non-linear effects that modify the tides ontheir propagation towards the inner shallow areas.

2.3. The shelf model

The shelf model of Harari and Camargo (1994)is a linear, three-dimensional, barotropic spectralmodel, based on the Galerkin solution for thevertical dependence of the currents (Heaps, 1972;Davies, 1980); this model runs in a mesh with ahorizontal spacing of 13.89 km and 10 componentsin the currents expansions, which used cosine asbasis functions. In the shelf model, boundaryconditions were given by deep water tide gauges(Fig. 3a), which time series were analyzed throughthe response method (Munk and Cartwright,1966), giving harmonic constants published byIAPSO (1992) and Mesquita and Harari (2000);these constants were interpolated along the gridboundary (following the 100m isobath, approxi-mately). Fig. 3a also shows the coastal domainof this study embedded inside the larger shelfarea and an example of the co-tidal chartsobtained in the shelf, for the M2 tidal component(Figs. 3b and c).

The open boundary oscillations of both shelfand coastal models considered components Q1,O1, P1, K1, N2, M2, S2, K2 and M3, whichaccount for more than 90% of the tidal energy inthe shelf area (Mesquita and Harari, 1983, 2000;Harari and Camargo, 1995). For instance, in thePort of Santos, at Torre Grande, 23�570S

46�18.50W (Fig. 1a), whose amplitudes (and phasesrelative to Greenwich) are given in Table 2,no other component reaches 3 cm in amplitude.On the other hand, M3 component must beincluded in the simulations since it has significantamplitude due to resonant effects in the shelf, asstudied by Huthnance (1980).

2.4. The coastal model setup

In order to avoid any baroclinic forcing inthe simulations, all the wet points of thecoastal grid have the same salinity (35 psu) andminimum exponential decay of temperature (T ;in �C) as depth (z; in m) increases: T ¼ 5þ15 expð�z=1000Þ:

The coastal model ran with the above specifiedtidal forcing, uniform density distribution and nometeorological effects, for a period of 31 days,starting from zero elevations and currents; thehydrodynamic equilibrium was reached at the endof approximately 2 days so that series of 696hourly results (29 days) were obtained. Thisrelatively quick hydrodynamic equilibrium is atypical feature of limited-area models whenbarotropic tidal simulations use consistent valuesof sea level oscillations at the boundaries. In orderto assure this statement, two additional periods of29 days of calculation were taken into considera-tion. Fig. 4 shows the evolution of the total kineticand potential energy in the model area, confirmingthat the equilibrium was effectively reached afterthe two first days of processing. Besides, the co-tidal maps obtained by the analyses of the resultsfrom days 32 to 61 and 62 to 91 reproduce exactlythe ones obtained in the period from days 2 to 31.On the other hand, if baroclinic conditions wereconsidered, the hydrodynamic equilibrium of themodel could only be reached at the end of severalmonths of processing. Fig. 4 also indicates that theopen boundary conditions used in the tidalsimulations allow the propagation of energy bothinside and outside the grid area.

Another interesting feature presented in Fig. 4 isthe contrast between spring and neap tides,yielding to a decay of energy after days 12 and26, at the beginning of neap tide periods.

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Fig. 3. Shelf model of Harari and Camargo (1994): (a) location of deep sea tide gauges and location of the coastal domain of this study

embedded inside the larger shelf area; (b) M2 harmonic constants of amplitude and (c) phase relative to Greenwich.

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3. Model results

Fig. 5 compares the record of sea surfaceelevations at Torre Grande, in the Port of Santos,with model outputs and harmonic predictions(using the same nine tidal components). For theparticular period of processing, with resultsbeginning on 00 GMT 30 December 1979, whichcorresponds to 48 h of simulation time and is thetime origin for valid results (t ¼ 00 h), the rootmean square (RMS) of the differences of 696hourly model results to observations is 25.6 cmand to harmonic predictions is only 1.3 cm. The

former is due to the absence of any meteorologicaleffect in the model run; these effects are particu-larly important when cold front incursions causesignificant variations on the mean sea level.That can be observed in the simulation period(Fig. 5) between times 24 and 84 h (mean levelincrease), 84 and 144 h (decrease) and finally 624and 660 h (increase). On the other hand, a tidalprediction at Torre Grande considering 44 tidalcomponents gave a RMS of the differencesto the correspondent model results of 5.4 cm(mainly due to the absence of some minor tidalcomponents and shallow water components, not

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J. Harari, R. de Camargo / Continental Shelf Research 23 (2003) 1597–1613 1603

included in the boundary oscillations, such asMU2, 2N2, MK3, M4 and MS4, which haveamplitudes of 2.1, 2.0, 2.5, 2.6 and 2.2 cm in thePort of Santos).

The computed time series of surface elevationsand currents, from all wet grid points, were

Fig. 4. Time evolution of the spatial means of potential, kinetic

and total tidal energy in the model area.

Fig. 5. Tidal elevations in Torre Grande, Port of Santos: model results

predictions, observations and differences of model results to observa

analyzed through the Harmonic Method of TidalAnalysis (Franco, 1988; Franco and Harari, 1987).This method is based on the Fourier decomposi-tion of the observed time series, interpolation ofamplitudes and phases for the exact tidal frequen-cies and inclusion of nodal corrections. Thisprocedure is widely used and agrees with otherones, based on minimum square differences (in thetime domain) and the response method (whichconsiders local responses to the tidal potential andtransfer functions). The analyses results were usedfor mapping the elevation co-tidal lines (co-rangeand co-phase lines) and the axes of the surfacetidal current ellipses. Figs. 6–9 present some ofthese maps, for components K1, M2 and M3,which have the most significant currents in eachtidal band.

Since the model outputs selection for presenta-tion was made depending on the features of thetidal propagation, some maps are relative to thewhole domain of the model while some arerestricted to the shallow region, which includesS*ao Vicente, Bertioga and the Port of SantosChannels (and adjacent coastal area). Maps of thedeeper region (Fig. 1a) present one every six sets of

, harmonic predictions, differences of model results to harmonic

tions; time origin t ¼ 0 h at 00 GMT 30 December 1979.

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Fig. 6. (a) Elevation amplitudes, (b) tidal phases relative to Greenwich and (c) axes of the current ellipses at the surface for tidal

component K1, in the internal shallow areas.

J. Harari, R. de Camargo / Continental Shelf Research 23 (2003) 1597–16131604

axes, while maps of the shallow area (Fig. 1b)show the computed ellipses axes of all wet gridpoints.

Table 3 presents the comparison of the analysesof model results with harmonic constants ofcoastal stations, published by FEMAR (2000).

Results of the tidal energy are given in Fig. 10,with the distributions of the (temporal) meanpotential, kinetic and total energy per area for aperiod of 29 days of numerical simulation.

The time series of depth-mean currents com-puted by the model were used to estimate theresidual depth-mean currents, simply by averagingin time the 696 hourly results of each grid point(covering 58 complete cycles of the semi-diurnal

tides); the corresponding map is presented inFig. 11.

Finally, an instantaneous vertical section of thehorizontal currents magnitude along column 75 ofthe grid (46.39�W), between lines 1 and 55 (24.42and 23.99�S) is shown in Fig. 12, which corre-sponds to typical maximum spring tide currentscomputed by the model (in the deeper region ofthe grid).

4. Discussion

One critical point of the hydrodynamicalnumerical modeling is the formulation of the open

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Fig. 7. (a) Elevation amplitudes, (b) tidal phases relative to Greenwich and (c) axes of the current ellipses at the surface for tidal

component M2, in the whole model domain.

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boundary conditions (Blumberg and Kantha,1985). Several solutions may be taken intoconsideration, and the ones adopted here arerelated to ‘‘no gradient values’’ and the‘‘clamped/radiational condition’’. The former sim-ply extends the first internal calculations to theopen boundary points, and was used for thecurrent fields (depth-mean currents and values atevery sigma level). The latter may be expressedthrough the following expression, for any variableV dependent on space x and time t:

qV

qtþ c

qV

qx¼ �

V � Vobs

Tf

� �;

where c is the phase speed of the propagatingdisturbances (given as the square root of the local

depth multiplied by the gravity acceleration), Vobs

correspond to observed (or expected) values of V

at the boundary, and Tf is the ‘‘relaxation period’’.When Tf tends to zero, the boundary valuesbecome totally clamped to the observed values(in the present case, to the harmonic elevationsgiven by the coarser grid model); when Tf tends toinfinite, the boundary conditions are purely radia-tional; and for a limited value of Tf ; the conditionsare named ‘‘partially clamped’’.

The combination of the elevation clampedconditions (to harmonic oscillations) and nogradient values for the currents proved to beefficient in the simulations here performed, since itallowed the income and outcome of energy in themodel area, as shown in Fig. 4. Tests with purely

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Fig. 8. (a) Elevation amplitudes, (b) tidal phases relative to Greenwich and (c) axes of the current ellipses at the surface for tidal

component M2, in the internal shallow areas.

J. Harari, R. de Camargo / Continental Shelf Research 23 (2003) 1597–16131606

radiational boundary conditions to the currentsled to results extremely similar to the ones herepresented. This is due to the fact that thesimulations were restricted to barotropic tides.When wind or baroclinic forcings are included inthe simulations, the totally clamped conditionsmust be replaced by partially clamped ones forelevations and radiational for currents, in order toavoid an energy accumulation in the model area;in this case, the option of no gradient conditionsfor currents also produces reliable results. Thetime series of the total kinetic and potential energyin the model area, presented in Fig. 4, clearly showthat the open boundary conditions used in thetidal simulations allow the propagation of energyinto and outside the grid.

Using the values listed in Table 2, the tidalcharacteristics in the region can be inferred: theratio between the main diurnal and semi-diurnalcomponents is 0.30 so the tides in the study areaare semi-diurnal with significant diurnal inequal-ities; the range of spring tides (defined as twice theamplitude) is 119.60 cm and the range of neap tidesis 27.20 cm. Considering the whole area, somelimited regions have the influence of river dis-charges on the tides (Table 1). Strong meteorolo-gical effects in the study area may causedisplacements of about 1m in the mean sea level(Camargo and Harari, 1994). As a matter of fact,meteorological effects in the model region aremuch variable but, in the absence of any extremeevent, they cause much weaker currents and

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Fig. 9. (a) Elevation amplitudes, (b) tidal phases relative to Greenwich and (c) axes of the current ellipses at the surface for tidal

component M3, in the internal shallow areas.

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elevations than those due to tidal forcing; on theother hand, the occurrence of strong fronts maycause meteorologically induced circulations stron-ger than the tidally generated ones.

Table 3 shows a reasonable agreement betweenthe amplitudes and phases of the tidal componentsin several coastal stations with correspondentvalues derived through the analyses of the timeseries computed by the model. Some discrepanciesmay be attributed to local effects on the tides notrepresented in the model, especially topographicand river influences. On the other hand, some tidalrecords that generated the tidal constants in thecoast are limited to a few months and thecorrespondent harmonic constants may have sig-

nificant deviations. For example, the amplitudesand phases at Torre Grande, computed as meanvalues of 46 years of data, given in Table 2 (Harariand Camargo, 1995), differ from those presentedin Table 3, published by FEMAR (2000) andbased on approximately 1 year of observations.

The coastal stations listed in Tables 1 and 3 arelocated mainly along the Channel of the Port ofSantos (the first six) and in the extreme south ofthe model area (the last two). An importantinformation given by the model in the region ofthe Port is about the propagation of the tidalwaves. For example, considering the principal tidalcomponent M2, Table 3 and Fig. 8 show itsprogression from I. da Moela to I. das Palmas,

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Table 3

Amplitudes and phases of the nine principal tidal components as given by the analyses of sea level records (FEMAR, 2000) and time

series generated by the model run (at the specified latitudes and longitudes)

Tidal comp. Pia@aguera (1) �23.875�S, �46.377�W I. Barnab!e (2) �23.928�S, �46.332�W

Amp.(cm) Phase (�) Amp. (cm) Phase (�) Amp. (cm) Phase (�) Amp. (cm) Phase (�)

Q1 3.7 95 3.10 99.53 2.5 091 3.05 99.04

O1 13.1 130 11.63 126.59 11.4 125 11.56 126.19

P1 1.9 208 2.35 184.11 2.1 197 2.33 183.68

K1 5.7 208 6.43 189.07 6.2 205 6.39 188.62

N2 6.5 244 5.38 237.40 5.1 231 5.26 236.38

M2 37.7 178 38.77 174.92 38.4 173 37.81 174.22

S2 23.7 184 24.38 181.02 23.3 178 23.72 180.22

K2 6.4 184 7.93 170.83 6.3 178 7.72 170.03

M3 6.8 020 6.14 2.84 5.5 015 5.82 1.28

Tidal comp. Torre Grande (3) �23.955�S, �46.310�W Praticagem (4) �23.995�S, �46.300�W

Amp. (cm) Phase (�) Amp. (cm) Phase (�) Amp. (cm) Phase (�) Amp. (cm) Phase (�)

Q1 2.5 098 3.03 98.06 4.8 083 2.99 96.76

O1 11.5 123 11.43 125.34 12.9 129 11.27 123.81

P1 2.3 181 2.30 182.81 2.2 193 2.26 180.84

K1 6.3 188 6.31 187.76 6.5 198 6.19 185.74

N2 5.4 234 5.02 234.52 4.7 232 4.72 231.30

M2 36.4 175 36.21 172.85 33.9 166 34.04 170.46

S2 22.5 181 22.63 178.51 24.9 176 21.17 175.47

K2 7.4 172 7.36 168.29 6.8 177 6.89 165.19

M3 4.9 004 5.32 357.87 5.2 343 4.64 351.71

Tidal comp. I. das Palmas (5) �24.008�S, �46.327�W I. da Moela (6) �24.052�S, �46.268�W

Amp. (cm) Phase (�) Amp. (cm) Phase (�) Amp. (cm) Phase (�) Amp. (cm) Phase (�)

Q1 3.2 073 2.99 96.27 4.7 098 2.95 96.37

O1 9.6 126 11.23 123.25 11.5 114 11.19 123.10

P1 2.2 149 2.25 180.22 2.8 177 2.18 179.36

K1 6.7 151 6.16 185.13 8.6 177 5.97 184.20

N2 4.6 234 4.63 230.17 3.0 203 4.47 229.22

M2 35.3 168 33.36 169.63 32.6 160 32.25 168.85

S2 22.9 165 20.70 174.38 23.1 168 20.11 173.70

K2 6.2 165 6.74 164.08 6.3 168 6.54 163.41

M3 4.5 328 4.45 349.38 4.5 313 4.06 348.34

Tidal comp. Barra Peru!ıbe (7) �24.337�S, �47.008�W Guara !u (8) �24.380�S, �46.987�W

Amp. (cm) Phase (�) Amp. (cm) Phase (�) Amp. (cm) Phase (�) Amp. (cm) Phase (�)

Q1 2.1 096 3.06 93.44 3.4 101 3.06 93.43

O1 9.6 123 11.42 122.38 12.8 120 11.42 122.37

P1 1.7 171 2.59 174.64 2.2 174 2.59 174.47

K1 5.0 175 7.10 179.16 6.5 178 7.10 178.98

N2 4.8 226 5.00 230.30 5.1 229 4.99 230.28

M2 31.6 175 36.39 169.06 34.8 165 36.40 169.01

S2 21.2 179 21.79 173.62 23.6 172 21.78 173.60

K2 5.8 179 7.09 163.31 6.4 173 7.09 163.29

M3 4.9 348 5.61 341.93 5.1 325 5.61 341.79

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Fig. 10. Mean tidal energy per unit area over a period of 29 days of simulation: (a) potential, (b) kinetic and (c) total.

Fig. 11. Residual depth-mean currents at the end of 29 days of

tidal simulation, in the internal shallow areas.

Fig. 12. Magnitude of the tidal currents in the vertical section

along 46.39�W, from 24.42 to 23.99�S (column 75, lines 01–55

of the model grid) on 16 January 1980 20:00 GMT.

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Praticagem, Torre Grande, I. Barnab!e and Pia@a-guera with a variation of phase from 168.85� to169.63�, 170.46�, 172.85�, 174.22� and 174.92�; thecorrespondent amplitudes are increased from 32.25 to33.36, 34.04, 36.21, 37.81 and 38.77 cm. These phasedelay and amplitude increase nearly reproduce theanalyses of observations, as shown in Table 3, exceptfor stations Praticagem, which exhibits an unexpecteddecrease in amplitude and phase, I. Barnab!e, whichphase should be greater than that of Torre Grande,and Pia@aguera, that presents a small amplitudedecrease when compared to I. Barnab!e.

The model results indicate that the principaltidal components have significant gradients to thewestern part of the modeled area and towards theshallow internal regions of Santos–S*ao Vicente–Bertioga estuary (Figs. 6a–9a). The most energeticconstituent in the area is M2, which meanamplitude is 32.96 cm (Figs. 7a and 8a), followedby S2, which mean is 20.28 cm. O1 is theconstituent that presents the lowest amplificationrate (maximum/minimum amplitude results�100%), reaching at most 5.8%, while the mostamplified component is M3, with rate up to100.6% (Fig. 9a, including in the rate computationvalues of the deeper region).

The diurnal components O1 and K1 propagatefrom South to North, the semi-diurnal M2 and S2from Southeast to Northwest (Fig. 7b) and thethird-diurnal M3 travels from South to North. Thephase differences of the components in themodeled area correspond to delays of 29.08minfor M2 (Fig. 7b) and 32.16min for S2, while themaximum delay is 68.53min for K1.

The co-phase lines configurations suggest tidalwaves predominantly progressive in the deeperregion (except for O1) until the entrance of theChannels; far inside the estuarine area, stationaryconditions are predominant, as the phase varia-tions decrease (Figs. 6a–9a). As a matter of fact,the Port of Santos has progressive tidal waves, butin the Channels of S*ao Vicente and Bertioga occurthe encounters of waves that enter them from bothentrances (see K1, M2 and M3 phases in Figs. 6b,8b and 9b); such features are also described inseveral local reports.

M2 and S2 axes of the ellipses correspond tomaximum currents much stronger than those of

the remaining components, reaching 47.8 and29.1 cm/s, respectively (Fig. 8c); on the other hand,the maximum currents of O1, K1 and M3 are only7.2, 6.2 and 10.0 cm/s (Figs. 6c and 9c). Aninteresting feature of the tidal current componentsis the nearly unidirectional M3 currents in thedeeper region of the study area.

Next, mean values for a period of 29 days of thepotential, kinetic and total energy (per unit area)are given (Figs. 10a–c). The potential tidal energyin the model area has (spatial) minimum, meanand maximum values of 332, 408 and 549 J/m2,with distribution that closely follows those of M2and S2 amplitudes, with strong gradients to thewest and also towards the region of the channels;the kinetic energy reaches (spatial) minimum,mean and maximum values of 0, 44 and 258 J/m2, following the bathymetric contours, beingstronger in the deeper region and much weakerwhen approaching the coastline, but it is enhancedby the intense currents at the channels. Finally, thetotal energy has (spatial) minimum, mean andmaximum values of 371, 453 and 718 J/m2, withincreasing values towards SW and the innerchannels. The energy propagation vectors aredirected towards the regions of higher levels oftidal energy.

The residual depth-mean currents (Fig. 11) showthree areas with closed cells that may be related toeddy formations: along the beaches of Santos untilthe entrance of the Port, in the vicinity of I. daMoela and at the entrance of the Channel ofBertioga. Two other minor eddy-like features arepresent far inside the Channel of the Port and atthe eastern extreme of the beach of Guaruj!a. Theseeddies are generated by coastal geometry andbottom topography, due to non-linear effects(Zimmerman, 1981); they are permanent barotro-pic features and have a signature in the localsediment transports.

Although only barotropic conditions wereconsidered in the simulations, the three-dimen-sional model processing allowed to obtain thevertical structure of the tidal currents; as a matterof fact, the vertical profiles of the tidal currentspresent almost constant intensities and directionsalong the vertical, but in the bottom boundarylayer the friction causes a decay of the current

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intensities (Fig. 12). The current profiles computedby the model are comparable to some sparsemeasurements in the model area.

It is important to point out that the modelresults are especially suitable in the deeper areas;on the other hand, modeling of the inner channelsshould be considered as preliminary: although thecomputed elevations may be regarded as correct,and the estimated depth-mean currents reproduceknown transport features, the exact reproductionof the influence of topography and the coastline onthe currents requires a finer grid. For instance, theChannel of the Port of Santos has complexbathymetry, being shallow in the margins andquite deep at the middle, due to periodic dredges,made in order to guarantee the navigation of largevessels. Aiming an improvement of the currentssimulations in the channels, the implementation ofa nested grid in the shallow part of the presentmodel, with a horizontal spacing of about 100m, isactually in progress.

Apart from nesting grids, the model (andcomputational grid) here presented will now runwith meteorological and baroclinic forcings inaddition to the tides, which will allow to simulatethe complete coastal circulation and the interac-tions between tides and the other circulationcomponents, as done by Cummins and Oey(1997), Masson and Cummins (1999) and Cum-mins et al., 2000.

5. Conclusions

The primary objective of this research is todescribe the tidal circulation of the coastal regionof Santos in detail. Next steps should considermeteorological and density gradient effects inaddition to the tides and nesting grids in theshallow interior regions.

The model results indicate that the principal tidalcomponents have significant amplitude gradients tothe western part of the model area and towards theshallow internal regions of Santos–S*ao Vicente–Bertioga estuary. The most energetic constituent inthe area is M2, followed by S2; O1 is the constituentthat presents the lowest amplification rate, while themost amplified component is M3.

The diurnal components O1 and K1 propa-gate from South to North, the semi-diurnal M2and S2 from Southeast to Northwest and thethird-diurnal M3 travels from South to North.The co-phase lines configurations suggest tidalwaves predominantly progressive in the deeperregion (except for O1) until the entrance of theChannels; far inside the estuarine area, stationaryconditions are predominant, as the phase varia-tions decrease. As a matter of fact, the Port ofSantos has progressive tidal waves, but in theChannels of S*ao Vicente and Bertioga occur theencounters of waves that enter them from bothentrances.

M2 and S2 axes of the ellipses correspond tomaximum currents much stronger than those ofthe remaining components.

The potential tidal energy in the model area hasdistribution that closely follows those of M2 andS2 amplitudes, with strong gradients to the westand also towards the region of the channels; thekinetic energy follows the bathymetric contours,being stronger in the deeper region and muchweaker when approaching the coastline, but it isenhanced by the intense currents at the channels;finally, the total energy has increasing valuestowards SW and the inner channels.

The residual depth-mean currents show threeareas with closed cells that may be related to eddyformations: along the beaches of Santos until theentrance of the Port, in the vicinity of I. da Moelaand at the entrance of the Channel of Bertioga.

Vertical profiles of the tidal currents presentalmost constant intensities and directions alongthe vertical, but in the bottom boundary layer thefriction causes a decay of the current intensities.

The implemented model proved to be useful forboth studies of characteristics of the tidal propaga-tion and operational predictions of tides and tidalcurrents. The deviations of the harmonic analyses ofmodel results to sea level observations (Table 3)correspond to global errors in tidal predictions lessthan 15min and 5 cms, which are insignificant inmost of the practical applications. On the otherhand, modeling is the only possible way of extendingthe elevation predictions to other points rather thanthose of the observed ones, besides providing alsocurrents predictions. Only the currents of the inner

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shallow areas must be regarded as preliminary,because the model grid does not resolve themadequately, but the elevations and transportscomputed in these areas are reliable.

Future improvements in the model shouldconsider: increase of resolution, through nestedgrids with spacings of about 100m, a crucial pointfor a better representation of currents in theshallow internal areas; the inclusion of meteor-ological effects and variations of density, inaddition to the tides, in order to model the generalcirculation and the interaction between tides andthe other contributors to the circulation; andfinally the adoption of a variable grid, allowingareas of flooding and bottom exposure.

The implemented modeling may be used inseveral engineering and environmental studies. Forexample, initial results of a high-resolution nestedgrid for the shallow inner areas have beenobtained, considering the circulation generatedby tides and typical winds; and the correspondentcurrents were used to model the dispersion ofpassive substances (and properties) in the Port andthe Bay of Santos (Harari et al., 2000).

Acknowledgements

The authors wish to thank Conselho Nacional deDesenvolvimento Cient!ıfico e Tecnol!ogico (CNPq)and Funda@*ao de Amparo "a Pesquisa do Estado deS*ao Paulo (FAPESP), for their finantial support,and Instituto Oceanogr!afico da Universidade deS*ao Paulo (IOUSP) and Instituto de Astronomia,Geof!ısica e Ci#encias Atmosf!ericas da Universidadede S*ao Paulo (IAGUSP), for providing all thenecessary means to acomplish the research.

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