Puesta a punto de las condiciones cromatográficas

Finally, spectral wave models are phase-average models, which rather than predicting the surface elevation of the waves through a suite of equations, resolve the evolution of the directional spectrum as waves propagate over varying water depths by means of the energy conservation equation (Hasselmann, 1971; Longuet-Higgins &

Stewart, 1961). Hence, these models compute the wave action, which consists of the spectral energy density divided by the intrinsic frequency. The new third-generation of wave models (which refers to those that account for all the physics relevant for the development of the sea state) predicts accurately the growth, decay and transformation of wind-generated waves and swells in the deep waters and shelf-seas. Thus, spectral models are capable of representing most of the wave transformation processes: shoaling, depth- and current-induced refraction, wind forcing, whitecapping, bottom friction dissipation, depth-induced breaking and non-linear quadruplet and triad wave-wave interactions. On the downside, as phase-averaged models, the diffraction cannot be calculated explicitly; nevertheless, in the case of SWAN (one of the most used wave propagation models) it can be modelled through a phase-decoupled refraction-diffraction approximation.

Regarding their applicability to study wave farm impacts, wave spectral models, and particularly SWAN, have been the most used numerical method (Carballo &

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Iglesias, 2013; Gonzalez, Zou & Pan, 2012; Iglesias & Carballo, 2014; Millar, Smith &

Reeve, 2007; Reeve et al., 2011b; Rusu & Guedes Soares, 2013; Smith, Millar &

Reeve, 2007; Smith, Pearce & Millar, 2010). As they solve the energy conservation equation, the implementation of the wave farms into the model is carried out by means of transmission coefficients that represent the absorption of wave energy brought about by the WECs. This approach does not account for the energy radiated, which will be of relevance in the case of wave-activated bodies. However, for overtopping devices, it can be assumed negligible, especially to study the effects of a wave farm on the coast.

Hence, in view of the aims and objectives of this thesis (Section 1.2) and considering the efficiency of wave spectral models to propagate long time series, the use of these models, in this case SWAN, is the most appropriate approach. A brief introduction of the model will be presented in the following section.

3.1.5.1 SWAN

Mathematical model

Simulating WAves Nearshore, SWAN (Booij, Ris & Holthuijsen, 1999), is a third-generation wave model that estimates the characteristics of the waves (significant wave height, peak period, mean direction, etc., or even more accurately, the directional wave spectrum) in coastal areas, lakes and estuaries from given wind, bottom and current conditions.

The model solves the spectral wave action balance equation without a priori assumptions on the shape of the wave spectrum. The wave field is described by the two-dimensional wave action density spectrum, N(,θ), where  is the angular wave frequency and θ is the wave direction. The wave action density spectrum is used in lieu of the energy density spectrum, for action density is conserved in the presence of currents whereas energy density is not; in any case, the wave energy spectrum may be

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computed from the wave action spectrum. The wave action balance equation is discretised by means of the finite difference method in time, geographic space (x,y), and spectral space (ω,θ). action over geographical space, with propagation velocities and in the and directions respectively; the fourth term quantifies the shifting of the relative frequency due to variations in depths and currents, with propagation velocity c in the  direction;

finally, the fifth term represents the effects of refraction induced either by depth variations or by currents, with propagation velocity in the direction. The expressions of the above propagation velocities are derived from linear wave theory. As for the right-hand side of the equation, S includes the source and sink terms of physical processes which generate, dissipate or redistribute wave energy:

4 3

nl nl in wc bot brk

S S   S   S S  S  S

where Snl4 refers to the redistribution of energy by nonlinear quadruplet wave–wave interactions, S_nl3 the non-linear triad redistribution of wave energy, S_in the transfer of energy from the wind to the waves and the dissipation of wave energy due to whitecapping, S_bot the sink term of energy dissipation by bottom friction and S_brk the energy dissipation in random waves due to depth-induced breaking.

cx c_y x y

c_ 

69 distributed over frequencies (σ) and directions (). The wave power magnitude is then calculated as

. (3.7)

The whole set of the governing equations regarding the spectral description of wind waves, the propagation of wave energy, the source and sinks, the influence of ambient current on waves, the modelling of obstacles and the wave-induced set-up can be found at SWAN’s manual (SWAN, 2007).

Numerical model application

SWAN has been developed to simulate coastal wave condition, and for this purpose the essential input data consists in a detailed bathymetry and the incident wave and wind field. The wave and wind data can be prescribed offshore coupling SWAN with larger scale models. In the case of the present document, the wave data is obtained with a three-hourly frequency from WaveWatch III, a third-generation offshore wave model consisting of global and regional nested grids with a resolution of approx. 50 km (Tolman, 2002b). In the same line, the wind data was provided from the Global Forecast System (GFS) weather model.

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The best practice to apply the model efficiently is through various grids, refining the grid towards the area of interest. In this line, the nesting concept is a very important implementation in order to reduce computational time and enhance accuracy, and it refers to computations conducted first on a coarse grid for a larger region and using the results as boundary conditions for a finer grid in the region of interest. The same types of coordinates (cartesian or spherical) have to be used in order to apply nesting. It is important to mention that curvilinear grids can be used for nested computations, but the boundaries should always be rectangular. Moreover, SWAN can also simulate unstructured grids, also called irregular grids and consists of triangles or tetrahedrons in an irregular pattern. This is relevant for complex bottom topographies in shallow areas and irregular shorelines.

Hence, as calculations are performed on a grid, SWAN is an Eulerian model that accounts for refractive propagation over varying bathymetries and current fields by solving the discrete balance equation. SWAN provides a representation of directional and non-directional spectrum at any point of the computational grids through spectral and time-dependant parameters of waves, e.g. wave height, peak or mean period, wave direction and energy transport.

On these grounds, the application of SWAN in this thesis will be conducted using two computational grids, a coarse grid from offshore to the coast and a high-resolution nested grid in the area of interest. The resolution of the nested grid allows the precise definition of the WEC position in the array and the simulation of their individual wakes with accuracy. This is a prerequisite to a detailed assessment of the wave farm effects (Carballo & Iglesias, 2013). The device that will be considered for the study is the WaveCat Overtopping WEC, as explained in Section 2.3.2 WEC technologies. The WEC-wave field interaction will be modelled by means of the wave transmission

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coefficients obtained in the laboratory tests (Figure 2.13) conducted at the laboratory of Porto by Fernandez et al. (2012). The transmission coefficients were calculated as the ratio between the wave heights measured in the lee and in front of the device under different wave conditions. The results showed that the wave transmission coefficient presented very small variability (K_t ~ 0.76), and therefore a constant value will be used in the medium- and long-term analysis. The limited range of wave conditions impeded the development of a frequency-dependant model; however, this is included in the future lines of research as part of the European WAVEIMPACT project lead by Prof Gregorio Iglesias and that is focussed on the interaction between a wave farm and the ocean through laboratory tests and numerical modelling (vid Section 10.2 Future Research lines)