Buy eBook. Norsk English. Generating PDF file. In many cases, mathematical calculations and tests are necessary for predicting the outcomes of a given situation. We use numerical modelling for such tasks as calculating the stability of offshore wind turbines in extreme weather conditions, how a coastal area may be effected by tsunami or how a rock slide may be initiated and propagate over wide areas.
But, even these biogeochemical modules are completely dependent on the results obtained by the modules that represent the hydrodynamic patterns. The hydrodynamic conditions water levels, currents velocities, temperature, and salinity resulting, for example, from the complex interaction between tides, waves, storm surge, wave set-up, and river discharge, will define the main transportation patterns of sediments, larvae and pollutants with a direct effect on the ecosystems.
It has been demonstrated that this kind of models can accurately reproduce current velocity, flood extent, and water levels, being useful to complement risk assessment tools and early warning systems, because less computational resources are required and the numerical solutions are much faster obtained. Several authors used 2DH models with satisfactory results [ 14 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 ], though 3D models are required to properly represent several processes, like vertical stratification, vertical current profiles, turbulent mixing processes, sediment transport, turbidity, water quality, effects of salinity, and temperature gradients on river plumes or salt-wedge estuarine configurations [ 25 , 26 , 27 , 28 ].
Regarding the equations and their approximations, 3D models are normally based on the Navier-Stokes equations or its depth-integrated version, the shallow water equations, which are applicable when the horizontal scale is much greater than the vertical scale. Despite their wide applicability, shallow water equations are not able to properly represent small relative amplitude waves propagating in shallow water conditions, which is of upmost importance to simulate the superposition between waves and currents, the effects of the waves on the sediment transport, the waves interaction with the bottom, or other wave processes, as shoaling, reflection, refraction, diffraction and decomposition [ 16 , 30 ].
For this purpose, more sophisticated models are needed, like the ones of Boussinesq, Korteweg de Vries, Serre, or Green-Naghdi.
These models include additional terms that take into account the nonhydrostatic effects of free surface curvature. Boussinesq equations [ 31 , 32 ] are derived from the Navier-Stokes equations by depth-averaging them considering the pressure as nonhydrostatic [ 33 ]. Boussinesq-type equations can account nonlinear and dispersive effects considering various degrees of accuracy.
The Korteweg de Vries equation, which was first introduced by Boussinesq and rediscovered by Korteweg and de Vries [ 32 , 34 ] describes weakly nonlinear shallow water waves, allowing the representation of solitary wave solutions [ 35 ]. However, it must be considered that the wave dynamic becomes strongly nonlinear in the final stages of shoaling, particularly in the surf and swash zones.
To properly represent this phenomenon and provide a correct description of the waves up to the breaking point, fully nonlinear equations should be considered, as the Serre or Green-Naghdi equations. Serre equations [ 36 ], also known as the Su-Gardner equations, are deduced from the fundamental equations of fluid mechanics, but taking into account the incompressibility of the fluid, the vertical uniformity of the velocity field and the conservation laws [ 16 ]. Finally, Green-Naghdi nonlinear equations [ 37 ] considers the 3D water-wave problem with a free surface and a variable bottom, and taking into account that the fluid can be rotational [ 38 ].
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Nevertheless, there are already some works where the capabilities of different numerical models are compared. Walstra et al.
The same models but in a 2DH version were selected by Iglesias et al. For historical floods, the two models obtained very similar results, despite using different numerical approximations. They also demonstrate that the unstructured models present a higher computational efficiency. Through the implementation of numerical algorithms, the circulation in these systems can be reproduced and different hydrodynamic processes can be represented.
And knowledge about the hydrodynamic patterns influenced by bottom morphological changes may allow assessment and forecasting of the effects of hazardous and extreme events, anthropogenic intervention, or climate change. Hydrodynamic modeling has therefore been the focus of a large number of previous works in estuarine environments [ 11 , 14 , 17 , 21 , 22 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 ].
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Nowadays, the available powerful computational resources and complex numerical model suites allow the implementation of high-resolution studies with accurate results. Some examples are the works of Dias et al.
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However, every modeling system has its own advantages and limitations, and model solutions will display uncertainties related with errors, calibration parameters, or model assumptions and forcing functions. Given the need for accurate forecasts, finding and implementing new solutions to avoid such errors is crucial.ersopebe.tk
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A single model can have biases, high variability, or inaccuracies related with the specification of initial conditions or the representation of physical processes in the models, causing large uncertainties in numerical prediction systems that can affect the reliability of the obtained results [ 52 ]. So, why not use several models to reduce uncertainties? The e nsembles technique is based on the combination of several numerical models solutions and can improve the forecast results when compared to a single model-based approach, even if the number of members in the models ensemble is as few as two [ 53 ].
Ensemble modeling is a method that consists of running two or more related numerical models with the same conditions and then synthesizing the results into a single solution. This single solution will improve the accuracy of the final forecast, compared to the solutions provided by each model. Two types of ensembles can be defined: single-model and multi-model ensembles.
The multi-model ensemble , as its name suggests, considers different numerical models that present different structural complexities. Each model is run using the same initial and boundary conditions. The multi-model ensemble clearly outperforms both single models and the single-model ensemble [ 60 , 61 , 62 ].
To combine the outputs of different models, several statistical techniques can be applied, considering the mean, the median, a linear regression, weighted average, or linear programming techniques, among others [ 52 , 57 , 63 , 64 , 65 ]. The bias is used to measure the extent to which the ensemble result, averaged over the ensemble members, differs from the target function. The variance can be defined as the extent to which the considered ensemble models disagree [ 66 ]. The ensembles technique can thus be used to improve single models forecasts, reduce their uncertainty, and provide the most accurate results for a variety of sectors, as previously mentioned.
In this project, the ensembles technique will be applied to two of the main estuaries of the Portuguese coast: the Douro and Minho estuaries. The need for an accurate forecasting based on numerical models becomes clear when the Douro and Minho estuarine regions are analyzed. Despite presenting similar seasonal river flow regimes Figure 2 , with minimum values in summer and maximum values in the rainy winter season, flow values and patterns differ. The difference between these two systems is also evident when river flood peak discharges associated with different return periods are estimated Table 2.
The different river discharge patterns suffice to support the need for a focused and local hydrodynamic characterization of the two estuaries when trying to avoid future risks related with human interventions dredging, ports, alterations in the estuarine banks, designing of breakwaters, etc. However, a comparison of the main estuaries characteristics and bathymetric conditions Table 1 and Figures 1 and 3 , reinforces this necessity. For the river flood peak discharges analysis, we assumed that the probability distribution for extreme flow events follows a Gumbel law [ 67 ].
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It should be noted that the presented extreme values refer to average daily extremes, since annual instantaneous peak values were not available. Thus, higher flood peak extremes are expected for both estuaries. River flood average daily extreme discharges Q associated with different return periods T. The Douro bathymetry Figure 1 was extracted from a bathymetric survey performed by the IH. The sand spit topography was taken from a topographical survey [ 68 ], and the adjacent coastal bathymetry was obtained from the Bathymetric Model of Douro IH.
For both estuaries, additional ocean bathymetric data was extracted from the GEBCO database [ 69 ], and the different data sets considered were interpolated using a Kriging algorithm [ 70 , 71 ]. The Douro River is one of the major rivers of the Iberian Peninsula. The Douro is a highly dynamic narrow estuary with torrential regimes that produce strong currents and recurrent severe floods that cause serious damage to the riverine populations and navigation problems [ 68 , 72 ].
Its dynamics is mainly forced by freshwater flows, being very dependent on highly variable natural conditions and on the hydropower production schedule of the Crestuma-Lever dam and of the other 50 national and international river basin dams. This sand spit is made up of fluvial and maritime sediments, and its morphodynamics is conditioned by natural wind, rainfall, river flow, waves, tides and storm surges and human breakwaters and dams construction, sand extraction, and dredging processes [ 68 , 75 , 76 ].
To prevent erosion of the sand spit or its excessive migration into the navigation channel, maintaining both navigability and bank protection, a new detached breakwater was built parallel to the head of the sandbar, and the existing northern breakwater was extended. Historical records reveal ruptures or partial destruction of the sandbar during river flood events, allowing for a rapid discharge of excess water and reducing the risk of urban flooding [ 68 ].
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