Hydrodynamic wave analysis as basis for sustainable beach design

The regulation and stabilization of movable parts of shoreline is a complex task, as the sea and shore interaction is quite complex is such coastal areas. If the analysis concerns beaches, either natural or newly built ones, the task becomes even more complex as they are used for rest and recreation, and so an appropriate maintenance must also be provided. Numerical modelling is conducted to enable proper shaping and maintenance of beaches. In this paper, the CGWAVE model is used to optimize a beach made of granular material, in order to ensure its sustainability.


Authors:
Željan Pernat, Mijo Vranješ solution.The procedure is repeated until the results of the solution under study are satisfactory.These results then become the basis for a detailed elaboration of the final design solution.

Basic wave agitation process equation of the applied numerical CGWAVE model
The approximation of the coastal wave motion can be made using the basic shape of the mild slope equation [1,2]: where C is the wave velocity, C g the wave group velocity, f (x, y) the complex two-dimensional potential, and ω is the angular frequency.The equation ( 1) is a two-dimensional elliptic partial differential equation that derives solutions of unknown potentials f (x, y) in the horizontal plane.The equation ( 1) is valid for the mild ∇ D kD /  1 (D-depth, k-wavenumber) [3].It includes all forms of wave transformations in coastal areas such as diffraction, refraction, shallow water effects, and reflection.In order to include nonlinear processes in the model, like wave refraction and friction at the bottom, the basic equation ( 1) has to be extended [4,5]: where W d is the coefficient combining the effects of wave breaking and bottom friction [5].
Once the boundary conditions are set, the mild slope equation in the inner area of a model discretized by a triangular finite element grid [5] can be solved using the CGWAVE model [7].
The elliptic mild slope equation can be written as a matrix after the discretization.After the term modification, it can be written in the following formulation due to conditions that guarantee its convergence [5,6]: Where [A] is the coefficient matrix; {f} is an unknown vector of potential values in network nodes; {f} the boundary conditions vector, and [A*] is the complex conjugated transposed matrix of the matrix [A].The equation is solved using an iterative method of conjugated gradients, and the solving algorithm implemented in the CGWAVE is given in literature [5,7].
Once the solution is found for the potential f, various cinematic parameters for each node of the finite element grid can be calculated [5].

Introduction
Previous practice in the design of beaches made of granular materials at the Croatian Adriatic coast has pointed to the need of conducting the hydrodynamic wave analysis by numerical modelling of wave agitation.Beach shaping and achieving beach stability is a very complex and highly sensitive task.The beach sustainability is primarily reflected in its ability to retain the existing (or artificially defined) configuration in terms of numerous effects of either natural or artificial nature.The wave activity exerts the biggest natural impact on sea beaches, and so a detailed knowledge of the wave field in the studied water zone is needed.This can only be obtained by detailed numerical wave agitation modelling, or by long-term field measurements.The possibility of using such an analytic procedure as the basis for designing sustainable beaches made of granular materials is presented in this paper on an example of the Hotel Split beach.

Basic approach
A basic approach or sequence of procedures that lead to a good-quality beach design solution using the wave agitation modelling is shown in Figure 1.
First, the wave agitation modelling is conducted for the existing coastline configuration and the corresponding results are analysed.Based on these results, the decision on the need to intervene in the beach water area is made.If results show that the intervention is needed, then a new solution is created.The wave agitation modelling is conducted once again for the new

Wind-wave data
As it is positioned within the Adriatic Seal (Fig. 4), the area of the hotel Split beach is a relatively well protected water area exposed to wind waves generated in the Brač and Split channels (III.and IV.quadrant waves) Waves relevant for shaping the beach and shore profile are the waves generated by the following winds: SE-Jugo, SSW-Lebić and WSW-Maestral.The wind called Jugo, dominant in the winter period, generates powerful waves that carry most of the energy to the coastline throughout the year.In other words, Jugo exerts the most powerful effect and influence on the shape and sustainability of the beach.On the other hand, Lebić is fleeting and rare.It generates waves that destroy the beach by scattering the beach material.Maestral often blows in summer, which also contributes to the formation of sandy beaches, and affects their sustainability.Thus, the wave analysis must be conducted for this particular beach in two directions of incident waves (Jugo and Maestral), because these waves are the biggest contributors to its sustainability, as the beach can not be protected from waves generated by the Lebić wind.Since the wave measurement data are not available for the area under study, deep-water wave parameters must be defined for water agitation simulations.These parameters imply deep-water wave spectra for incident wave directions based on wind measurements [10].Many methods can be used to obtain these parameters (e.g. the Gröen-Dorrenstein method), and a number of wave forecasting methods are also available.Wave parameters that will be used in the analysis of the beach under study are given in the literature [11] for the five-year recurrence interval (Table 1).When modelling wave agitation for these types of water areas that are often exposed to wind waves with main direction variations at the local level, and when wave terms are described by a power spectrum, it is necessary to take into account a certain directional dispersion around the basic direction of incident waves [2,12].The directional Hydrodynamic wave analysis as basis for sustainable beach design bathymetry and coastal boundary, are required for creation of the model field.A realistic graphical environment, e.g.ortophoto surface (Figure 2), can also be entered in order to maintain visibility and regulate the area setting.According to the bottom bathymetry (Figure 2), the depths present an average slope of approximately 1:15, which is significantly less than the maximum slope for the mild slope equation [4,5,8].Also, it is very important to have an appropriate knowledge of the sea level fluctuation statistics for the area under study.As no sea level measurements are available for the Hotel Split beach locality, the survey data collected in the period from 1995 to 2005 [2] at the Split tide gauge station, located at the City Port (Figure 3), will be used.The mean sea level (MSL) will be used for all numerical wave agitation simulations, taking into account the fact that in case of stormy wind (Jugo) the sea level can rise by more than 0.50 m above the mean level.
Where x n and x s are coordinates of this local system, and i δ is the angle of an incident wave to the coastal edge obtained by iteration [5,13] (Figure 6).The existing condition of the coastal line, and new solutions for the Hotel Split beach coastal line, consist of just a few typical coastal profiles, which contribute to the complexity of wave field deformations in the direct water area with their reflective characteristics.Related reflection coefficients (Figure 7) were obtained from literature [11,12,13,14].

Development of calculation model
As already mentioned, the modal area is discretised by the triangular finite element grid.The size of the grid depends on the local wavelength λ [11].The minimum finite element grid resolution requires 8-10 points per wavelength [3,5,13].As the discussed port area occupies an approximate area of 146.000 m 2 , it is necessary to select the most appropriate minimum finite element so that the time of analysis can be considered acceptable.The triangular element with the side length of about 2.0 m was selected, and the grid with approximately 60,000 finite elements and 36,000 nodes was generated (Figure 5).

Coastal boundary conditions
The boundary condition for coastal line is written in general form as [3]: Where K r is the coastal reflection coefficient (between 0 and 1), and n is the normal to the coastal edge.The boundary condition (4) for a finite element (Figure 6) on the coastal edge is written as follows [5,13]: H s -significant wave height T 0 -mean wave period T p -spectral peak period (5)

Figure 8. Wave breaking line for the existing beach state, SE direction of incident waves and selected control lines
Wave height H sb at the wave breaking point is directly related to the wave power [10,15]: Where C gb is the local group wave velocity and α b is the angle between the wave direction, and the bottom slope direction at the wave breaking point.Based on the wave strength P l data, it is possible to determine the amount of its potential volumetric transport along the coast Q l [11].Since we have no data on the particle size distribution of beach materials for the considered beach, only the data on the wave power P l are presented, in order to detect beach parts with possible beach material transport problems.Numerical wave agitation modelling results are presented as fields of significant wave heights H s with incident wave vector directions marked in the corresponding figures (Figures 9 to 12), and with vectors of total orbital velocities at the bottom u d and their vectors (Figures 13  to 16).Lastly, variations of H s and P l in control lines are shown in Figures 17 and 18.The results show that waves from the SE direction have a much greater impact on the beach than waves from the WSW direction.Results concerning the existing state present parts of coast with possible beach material transport problems (bringing and removal), and are reflected in increased values of orbital velocities at the bottom u d for the SE incident direction (Figure 13) and wave power (Figure 18).Orbital velocities at the bottom are u d highest at the central part of the beach where they reach values of 1,30 m/s (Figure 13), and the wave power P l , is most prominent at the westernmost part of the beach within the first 20 meters of the control line, with the maximum value of approximately 5x103 N/s (Figure 18).These areas, where higher u d and P l values were registered, are also the most exposed parts of the actual field state.Certain interventions are needed in these areas in order to border.The total two-dimensional complex potential f (x, y) along an open border can be expressed as: f I ncident wave potential, f r reflected wave potential, and f s scattered wave potential.Further parsing and simplifications [1,3,13] give the total boundary condition for an open border: Where x n is the local system coordinate along the open boundary, placed perpendicular to the tangent of the circle segment, and R is the radius of a circle.Since an open border represents the line of incident wave generation, incident wave conditions have to be defined by the deep water wave power spectrum at an open border.The JONSWAP [13] was selected as the frequency range for numerical water area simulations, with different parameters for Jugo (SE) and Maestral (WSW) directions (Table 1), and directional dispersion around the main direction of incident waves.

Analysis of the wave modelling results
A series of numerical simulations of wave agitation was conducted in the water area with one sea level (MSL) and with two incident wave directions (SE and WSW).However, results presented in this paper are only related to the existing state, and to the final design solution for the Hotel Split beach [4].Different parameters important for the beach morphodynamics and evaluation of its sustainability can be obtained through wave agitation modelling [2,3,10].An important parameter is the wave height H s , which indicates the potential quantity of energy that is brought to the coast by wave action.Furthermore, significant parameters also include the horizontal orbital velocity of wave particles at the bottom u d (orbital velocity at the bottom), and the wave power (coastal wave energy flow) P l : Both parameters are used to estimate the size and direction of potential transport of beach material caused by wave-induced flow [5,15].Dominant coastal currents due to waves point to the biggest sediment transport in the area of wave refraction [15], which is a very interesting phenomenon [10,11].The following condition is applied for determination of the wave breaking line, [10]: where H sb i d b represent significant wave height and depth values at the wave breaking point.Two linear control lines have been determined for the case of the beach under study.These lines are valid for all cases of incident waves for the existing state and the design state (Figure 8).
Željan Pernat, Mijo Vranješ ).However, significant wave height differences were registered within the area of wave refraction near interventions, which is especially evident at the control lines (Figure 17).Wave heights for WSW direction in the final project design are up to 3.5 times smaller than the existing wave heights, while they are up to 2 times smaller for the SE direction (Figure 17).This is because new parts of the coast are planned with relatively low reflection coefficients (Figure 7), which affects all wave fields in the area of wave refraction.As for orbital velocities at the bottom u d for the final design solution, it is evident that values for the dominant SE direction in the entire beach area have been significantly reduced (Figure 14).Orbital velocity decrease at the bottom of the western and central parts of the beach can be noticed for incidental WSW direction, while the values at the eastern part have slightly increased, but with better vector direction.Wave power P l , with highest values at the western part of the existing beach for incidental SE direction, has decreased significantly due to interventions (Figure 18).Wave power for the WSW direction has remained at approximately the same level for the final design solution as for the existing situation (Figure 18).

Conclusion
The shape of beach with granular materials can be optimized in order to ensure its sustainability by means of a detailed numerical wave agitation modelling.At the same time, it is essential to have good knowledge of specific wind-wave conditions in the area under study.The numerical wave agitation modelling produces a set of parameters that are highly significant for the morphodynamics and sustainability Schlüsselwörter: Strände, körniges Material, Nachhaltigkeit der Strände, numerische Modellierung Hydrodynamic wave analysis as basis for sustainable beach design Primljen / Received: 29.6.2009.Ispravljen / Corrected: 29.12.2011.Prihvaćen / Accepted: 8.6.2012.Dostupno online / Available online: 16.7.2012.

Figure 1 .
Figure 1.Sequence of hydrodynamic wave analysis procedures for the design of sustainable beaches

Figure 4 .
Figure 4. Wider water area of Hotel Split beach

Figure 2 .
Figure 2. Existing state and bathymetry of local waters at the Hotel Split beach, ortophotographic view

Figure 3 .
Figure 3.Typical relations compared to the tide gauge datum as registered at the Split tide gauge station, adapted from [9]

Figure 6 .
Figure 6.Boundary element and an incident wave with respect to the edge

Figure 7 .
Figure 7. Coastal reflection coefficients per sections, for the existing condition and the final design solution Open boundary conditions Before defining this boundary condition, it should be noted that in the CGWAVE numerical model the open boundary is a semicircle because of the Sommerfeld radiation condition [11].In simpler terms, all the waves that enter the model area through an open border, and then scatter due to obstacles in the waters of the model, have to leak back through the open

Figure 9 .Figure 10 .Figure 11 .Figure 12 .Figure 13 .
Figure 9. H s field and velocity vectors for the existing situation (SE) Very good and stable gravel beach has been obtained on the basis of many years of observation and study of wave action at the coastline of the hotel Split beach, and based on results obtained by this numerical model.The shore coarseness was defined according to literature[8].The crushed stone from nearby limestone quarries was used.Sharp edges of grains were ground in just one winter season, and rounded pebbles were produced.As the beach was recharged with stone material in 2010, just before the bathing season, the Sava gravels in a layer of about 10 cm in thickness were spread on the surface and at the beach face.During the winter season of 2010/11 the shoreline was fully mixed and fully rounded.Currently an average grain size measures d 50 =32 mm in diameter.Shore conditions in the winter and summer are shown in Figures19 to 22 .

Figure 21 .
Figure 21.Beach condition in August 2011 Figure 22.Beach condition in August 2011 (detailed view)