Hydrodynamical Numerical Modeling of Coastal Areas

Abstract

Coastal areas are important, due to the great population and activities that occur there. Oceanographic conditions in coastal waters differ in many aspects of those in the open ocean: In general, spatial and temporal variations are greater. Coastal management requires environmental monitoring, commonly done by direct measurements, remote sensing, and the processing of numerical models. This chapter deals with the hydrodynamical numerical modeling of coastal areas, presenting its basic concepts, implementation and use strategies, and examples of model outputs and practical applications. These models aim to reproduce the coastal circulation, by computing the time evolution of the spatial distribution of sea level, currents, temperature, salinity, and density—considering the respective forcing of tides, winds, ocean–atmosphere exchanges and river contributions. Several examples of model outputs and coupling are presented, with maps of the distribution of temperature and salinity, which are important seawater properties; waste discharges by submarine emissaries, giving support to public health service; fate of plastic debris in the ocean, among others. Despite the enormous feasibility and utility of modern high-resolution numerical models, some warning must be considered, such as checking all the inputs and comparing the results to reliable independent information before their dissemination.

1. Introduction

Coastal areas are especially important, due to the great population and the numerous activities that occur there, such as tourism, fishing, nautical sports, navigation, port activities, coastal engineering, waste discharge by submarine emissaries, among others [1]. Besides, these regions have been subject to intense pressure, associated to irregular buildings and sea level rising due to climate changes [2]. Nowadays, society does not admit polluted coastal waters, contaminated beaches, areas of marine erosion, inadequate marine buildings, or deformed landscapes. Thus, the proper management of the environmental conditions in coastal regions becomes an essential part of its economic, social, leisure, and sport activities [3, 4, 5].

Oceanographic conditions in coastal waters differ in many aspects of the conditions in the open ocean. In general, spatial and temporal variations are greater, and that is due to local factors such as river discharges, bottom topography, and coastline configuration. These effects produce greater seawater density gradients, stronger wind generated currents, higher tidal elevations, and more intense tidal currents. On the other hand, although most of the ocean dynamics is amplified in the shallow shelves when compared to the deep ocean, the energy dissipation at the shelves is much more effective, mostly due to bottom friction, i.e., friction of near bottom currents with the sea floor [6].

The direct effect of river discharge is to reduce the salinity of the surface layers and even the deep layers. In general, river discharge has significant seasonal variation, which causes seasonal fluctuations of salinity in coastal waters, much larger than in the open ocean. As rivers often carry suspended sediments, usually coastal waters are opaque; the deposition of these sediments decreases the depth, consequently resulting in navigation problems. Solar radiation penetrates to the bottom in the water columns of shallow places, causing temperature elevations higher than in the deep ocean. The wind stress acting at the ocean surface transfers more energy per volume in shallow waters. Also, the sun and moon tidal generation, being the same in the whole ocean, produce larger water transport in coastal regions. The effect of the coast as a boundary limit on ocean currents and waves is obvious, and here is one of the few situations in which man can exert significant influence on the ocean, through the construction of dikes and wave-breakers [7, 8].

Therefore, the coastal management requires qualified environmental monitoring, which is commonly done by direct measurements (including automatic equipment and gliders), remote sensing (from satellites, planes, drones, or towers), and by the processing of numerical models, both for the atmosphere and ocean. This chapter deals with the hydrodynamical numerical modeling of coastal areas, presenting its basic concepts, implementation and use strategies, and examples of model outputs and practical applications [9, 10, 11, 12].

2. Basic concepts

The numerical models aim to reproduce the coastal circulation, by computing the time evolution of the spatial distribution of sea level, currents, temperature, salinity, and density—considering the respective forcing of tides, winds, ocean–`atmosphere exchanges, and river contributions. The models solve the basic hydro-thermodynamic equations and may be used for hindcasts and forecasts. Once the ocean dynamics is modeled, its results may be used for several other ocean models, such as pollution dispersion, waves propagation, sediments transports, and the trophic chain.

An hydrodynamical numerical model consists therefore of the solution of the basic hydrodynamic equations through numerical methods, i.e., by using computational techniques [11]. Initially, the Navier–Stokes equations consider all the physical effects that induce ocean currents, namely seawater density variations, sea surface elevation gradients, sea level atmospheric pressure gradients, wind stress at the sea surface, tidal forcing, turbulence, and internal and bottom friction. At this point, the currents are computed, through the zonal, meridional, and vertical components. Afterwards, the continuity equation computes the sea surface level; equations of heat and salt conservation give the temperature and salinity of the seawater; and finally, the seawater state equation calculates the seawater density [13].

But before solving this set of equations, the modeler should define the region of interest, which may be the whole global ocean, or a huge ocean basin (such as the South Atlantic Ocean), any shelf region (extending from the coastline to the shelf break) or even an inner estuary. Apart from the global models, the remaining ones are called limited-area models and require lateral boundary conditions. On the other hand, both global and limited-area models require the specification of surface conditions, which consist of meteorological data. In fact, coupled atmospheric–ocean models do not require specifying sea surface conditions, but these models are not the subject of this chapter.

Here, the main objective is related to limited-area models for coastal areas, from the coastline until the shelf break, excluding then deep ocean regions. And for this kind of model, the specification of lateral and surface boundary conditions is essential.

The lateral boundary conditions inform the oceanic oscillations that propagate across the open borders into the model region, consisting of the mean sea level, tidal oscillations, river discharges, and vertical profile of currents, temperature, and salinity. The sea surface conditions inform about the wind stress, the atmospheric pressure gradients, and the mass and heat exchanges at the air–sea interface [14].

Once delimited the region of interest and the sources of boundary condition data, the next step is the gridding of the region. Now the model resolution is defined, specifically the horizontal and vertical spacings for the points where the ocean parameters will be computed, once the model advances in time, with a given time step, starting from a given initial hydrodynamic condition.

Basically, two approaches are considered for gridding: Regular grids perform the computations at equally spaced points (the standard of the finite difference method), while irregular grids perform at irregularly distributed grid points (with the finite difference or the finite element methods).

The grid extension and spacing, the time step, and the period of model integration are chosen as a compromise between the available computational resources and the objectives of the numerical modeling. For example, modeling a region of 200 km x 100 km, with maximum depth of 100 m: if the user chooses an horizontal spacing of 1 km and a vertical one of 10 m, that will require much less computational effort than a choice of spacings of 100 m and 2 m, respectively. Moreover, the smaller the grid spacings, the smaller is the time step, to keep stability in the model computations. Among the objectives, they can range from very short phenomena simulations (such as a cold front intrusion during a few days) to very long ones (such as inter-annual variability of the ocean circulation).

After gridding the model area, bathymetric data are interpolated into the grid points, to define the wet points and levels, opposed to the dry ones (continental, island, or below the sea floor points).

Figure 1 presents two grids and respective bathymetry, used for modeling the circulation in the Shelf of Sao Paulo State (Brazil) and its inner Estuary of Santos—Sao Vicente. The former has a horizontal spacing of 1 km and the latter of 250 m. The bathymetric data were extracted from nautical charts and local hydrographic surveys. Figure 2 presents currents computed for the large grid (1 km horizontal spacing) and for a nested grid (250 m spacing), both for the time 21 h GMT of 24/July/2021. Note that for the larger grid, one every nine computed vectors were plotted, while for the smaller grid, one every three. Evidently, high-resolution grids give much more details of the hydrodynamics, as can be easily detected by the currents of São Vicente Channel and the Port of Santos (Figure 2).

Figure 1.

Grids and respective bathymetry of Sao Paulo State Shelf (above) and Santos—Sao Vicente Estuary (below).

Figure 2.

Currents computed for the large grid (1 km horizontal spacing, above) and for the small grid (250 m spacing, below).

At this point, it is worth to present a very useful technique used in hydrodynamic numerical modeling, named as “grid nesting”. In this case, a smaller grid, with much higher resolution, is nested within the bigger grid, so that the lateral boundary conditions of the smaller grid are defined by the results of the bigger one. This technique is often used when high resolution and detailed results are required only in a specific sub-region of the bigger grid.

3. Implementation and use strategies

Hydrodynamical numerical models of coastal areas may be implemented for a variety of purposes, such as:

• Forecast of sea level and coastal currents, for ocean users and navigation security [15, 16].

• Hindcast of long-term coastal circulation, to estimate coastal sediment erosion or deposition.

• Simulating of coastal currents to subsidize coastal engineering work, such as building of marinas, ports, dikes, and breakers [17].

But, most of all, the greatest utility of the hydrodynamic models is obtained when coupling them to other numerical models, such as:

• Models of surface gravity wave propagation, allowing the estimation of current–wave interactions [18].

• Water-quality models, for simulating the concentration and dispersion of pollutants in the seawater.

• Models of sediment transports, including the deposition and re-suspension of particles at the sea floor, and dredging operations [19].

• Morphodynamic models, to estimate the coastline evolution along time [20].

• Particle displacements models, for estimating the positions of missing boats, supporting boat racing, or estimating plastic trajectories.

• Trophic chain models, for studying the influence of physical parameters (currents, temperature, and salinity) in the oceanic tropic levels [21] .

Any model implementation and use strategy should consider the purpose of the modeling, the available computational resources, and the access to data. Each coupled model requires correspondent data for processing, such as pollutant spill data, or quantity of dredged materials. Data are important not only for initial and boundary conditions specification, but for calibrating and validating model results [22]. The calibration consists in comparing model results with a dataset, in order to obtain the best fitted values of the coefficients of forcing and dissipation. Once calibrated a model, validation is the comparison of its results with another dataset, to infer the quality of the model outputs.

In fact, any model, whether hydrodynamic or coupled, must have its results compared to independent information, either by direct measurements, remote sensing data, or even other validated models. Several statistic methods are used to compare model results to independent information, such as correlation coefficient (CC) or root mean square error (RMSE) [23, 24, 25].

Figure 3 shows time series of model results of sea-level elevation and surface current component as compared to measurements, including statistical comparison parameters, at the position 24° 0,757′ S 46° 19,579′ W (coastal region of Santos). For the sea level, CC is 0.9707 ± 0.0038 and RMSE 0.0690 m; meanwhile, for the North current component the correspondent values are CC = 0.5815 ± 0.0430 and RMSE = 0.0610 m/s. This is a common feature when assessing hydrodynamic numerical model quality results: Sea-level parameters are much better than the correspondent current component ones; this is because currents exhibit much larger temporal and spatial variability than sea level.

Figure 3.

Comparing model outputs (blue) to measurements (red) of sea level (above) and surface currents (below), for a simulation on January/February 2009.

4. Model outputs and practical applications

This section presents several examples of model outputs, with emphasis on the results of models coupled to the hydrodynamic ones (Figures 4–6).

Figure 4.

Hydrodynamical numerical model computation of temperature for Sao Paulo State Shelf (above) and salinity for Santos—Sao Vicente Estuary (below), at 21 h GMT of 24/July/2021.

Figure 5.

Enterococus concentration as computed by a water quality model, at 10 h GMT 07/August/2016, due to the operation of submarine emissaries located at the black triangles [26].

Figure 6.

Monthly mean currents at Sao Paulo State shelf in March 2017 computed by hydrodynamical model (above), and particle trajectories during this month, computed by a particle tracking model, starting at the beginning of the month on the positions marked by yellow triangles (middle and below).

5. Conclusions

Numerical modeling, and particularly hydrodynamical modeling of coastal areas, became a very powerful methodology for environmental analysis and monitoring. Several examples of model outputs and coupling were presented in this chapter.

Temperature and salinity are important seawater properties; in general, temperature has latitudinal variability, while salinity may be strongly affected by rivers, hence presenting significant variations in small distances (Figure 4). Monitoring of waste discharges by submarine emissaries may be performed through enterococcus concentration mapping, with real-time model processing, giving support to public health service (Figure 5). One of the most difficult issues of modern society is determining the fate of plastic debris in the ocean, and again numerical modeling may give an important contribution, through particle tracking models, as shown in Figure 6.

Despite the enormous feasibility and utility of modern high-resolution numerical models, some warning must be considered. Great care should be taken when processing a model, checking all the inputs, especially in terms of units, times, geographic positions, precision, and uncertainties. Prior to the model processing, investigating the region behavior is important, to determine the most important processes involved [27]. The model itself must be reliable, already tested by the scientific community in several different conditions. And, finally, the model results should be compared to reliable independent information before disseminating its results.

Acknowledgments

The authors acknowledge the Coordination of Improvement of Higher Education Personnel—CAPES—for the support to Project “Morphodynamic Response of Brazilian Southeast Beaches to the Effects of Sea Level Rise and Extreme Oceanographic and Meteorological Events up to 2100”, process 88881.146051 / 2017-2101, and the Foundation of Support to Research of Sao Paulo State—FAPESP—for the support to Project in Public Policies “System for the Alert of Floods for the Coast of São Paulo, with a focus on Impacts of Climate Change”, process 2017 / 50336-6.