Ricardo de Camargo1, Joseph Harari2
& Amaury Caruzzo3

1, 3 Institute of Astronomy and Geophysics
2 Institute of Oceanography
University of São Paulo - Brazil

 In this study, the Princeton Ocean Model - POM was implemented for the coastal regions of Santos (Sao Paulo State) and Paranagua (Parana State). The models were processed for high resolution tidal simulations, considering a grid spacing of about 1 Km and tidal oscillations at the boundaries given by the shelf model of Harari & Camargo (1994). Characteristics of the tidal propagation in the modelled areas were obtained, leading to operational regional systems of tidal and tidal currents predictions (Harari & Camargo, 1997, 1998).
 The version of POM that was implemented has the following characteristics (Blumberg & Mellor, 1987; Mellor, 1993): use of the full 3D hydrodynamical equations, written in flux form, under the hydrostatic and Boussinesq approximations; vertical solution based on sigma coordinates; second order turbulent clousure for the vertical diffusion; Smagorinsky parametrization for the horizontal diffusion; mode splitting; explicit leapfrog method for the numerical integration in time and the horizontal; and implicit method for the vertical integration of the equations.
 Both models considered horizontal spacing of approximately 1 Km and 11 sigma levels, 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 time steps were 15 s (for the external mode) and 300 (for the internal modes); the frictional parameters used in the models were the same of Oey, Mellor & Hires (1985). For the coastal region of Santos, a regular cartesian grid of 120 x 80 cells was used, with 5527 wet points; for the coastal area of Paranagua, the regular grid has 90 x 70 cells, 3122 of them being wet.
 Tidal oscillations were specified both at the open boundaries and the first internal lines and columns of the grid, in order to avoid the need of current specifications. These boundary conditions had to be corrected, with supplementary tidal amplifications and delays, after comparing preliminary model results with tidal analyses of the Ports of Santos and Paranagua. The final results of the models matched the tidal predictions of these ports with errors in elevations of less than 5%.
 For Santos, the open boundary oscillations considered components Q1, O1, P1, K1, N2, M2, S2, K2 and M3, which account for more than 90% of the tidal energy in the area; on the other hand, the Paranagua simulations required additionally components MN4 and MS4 at the borders, due to their important contribution to the tidal range in the inner estuarine areas.
 The models run with uniform density and no meteorological effects, for a period of 31 days, starting from zero elevations and currents; the hydrodynamic equilibrium was reached at the end of approximately 2 days so that series of 696 hourly results (29 days) were obtained.
 The computed time series of surface elevations and currents, of all wet grid points, were analysed through the Harmonic Method of Tidal Analysis (Franco, 1988; Franco & Harari, 1987). The analyses results were used for mapping the elevation cotidal lines and the axes of the surface tidal current ellipses.
 Figures 1 to 4 present the cotidal maps for component M2, the most energetic among those considered in the models. The common features of all the principal tidal components are the significant amplitude gradients and phase delays towards the shallow internal regions; for instance, the M2 amplitudes ranged from 30 to 40 cm in Santos and from 30 to 50 cm in Paranagua. In fact, the models allowed very accurate computations of the tidal propagations in the estuaries, resolving variations of tenth of centimeters (in amplitude) and few minutes (in phase); as examples, Figures 5 and 6 present the M2 propagation into the internal channels in the region of Santos; strong tidal currents are associated to this shallow area circulation, with intensities reaching nearly 50 cm/s, as shown by the M2 tidal ellipses in Figure 7.

Fig. 1 - M2 amplitudes in Santos.

Fig. 2. M2 phases (relative to Greenwich) in Santos.

Fig. 3. M2 amplitudes in Paranaguá

Fig. 4. M2 phases (relative to Greenwich) in Paranaguá.

The  implemented  models  proved to  be useful for  both studies of characteristics of the tidal propagation and operational predictions of tides and tidal currents. Future improvements in the models should consider an increase of resolution, to a grid spacing of about 100 m, and the inclusion of meteorological effects and variations of density, in addition to the tides.

Fig. 5. M2  amplitudes in the shallow area of Santos.

Fig. 6. M2 phases in the shallow area of Santos.

Fig. 7. M2 current ellipses at the surface, in the shallow area of Santos

 The authors wish to thank Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Fundacão de Amparo a Pesquisa do Estado de São Paulo (FAPESP), for their finantial support.


Blumberg, A. F. & Mellor, G. L. 1987. A description of a three-dimensional coastal ocean circulation model. In: Three-Dimensional Coastal Ocean Models, vol 4, Ed: N. S. Heaps, American Geophysical Union, Washington, D. C., USA, p. 1-16.

Franco, A. S. 1988. Tides - fundamentals, analysis and prediction. Public. FCTH Fundação Centro Tecnológico de Hidráulica, Escola Politécnica da USP, 2a Edição - 249 p.

Franco, A. S. & Harari, J. 1987. Computer programs for tidal data checking, correction, analysis and prediction by the harmonic method. Relat. int. Inst. oceanogr. Univ. S Paulo, 16:1-65.

Harari, J. & Camargo, R. 1994. Simulação da propagação das nove principais componentes de maré na plataforma sudeste brasileira através de modelo numérico hidrodinâmico. Bolm Inst. oceanogr., S Paulo, 42 (1):35-54.

Harari, J. & Camargo, R. 1997. "Simulações da circulação de maré na região costeira de Santos (SP) com modelo numérico hidrodinâmico". Pesquisa Naval - Suplemento Especial da Revista Marítima Brasileira, n° 10, p. 173 - 188.

HarariI, J. & Camargo, R.  1998.  Modelagem numérica da região costeira de Santos (SP): circulação de maré. Rev. bras. oceanogr., S Paulo, 46(2):135-156.

Mellor, G. L. 1993. User's Guide for a three-dimensional, primitive equation, numerical ocean model. Princeton University, Internal Report, 35 p.

Oey, L. Y.; Mellor, G. L. & Hires, R. I. 1985. A three-dimensional simulation of the Hudson - Raritan Estuary. Part I: Description of the model and model simulations. Journal of Physical Oceanography, vol. 15, p. 1676-1692.