Abstract. The Physical Problem

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1 Modelling and predicting beach evolution after groin construction in the north-east coast of Brazil F.C.B. Mascarenhas, E. Valentin!, A.L.T. da Costa Civil Engineering and Ocean Engineering Program (COPPE), Federal University ofrio de Janeiro, POBox , CEP: , Rio de Janeiro, RJ, Brazil Abstract The interference of man-made structures on natural phenomena as coastal processes are of particular interest when it is desired to detect and predict the evolution of the affected physical system and the environment. In this paper we propose the modelling of the coastline evolution of a beach in which it was built a groin in the past, and the sand transport was strongly modified after that. The modelling is made with respect to two approaches, analytical and numerical, starting from the diffusion equation for the sand transport due to waves in the breaking zone. The analytical solution is based on the theory developed by Pelnard- Considere (1956V and this approach is compared with a finite difference solution of the diffusion equation. In the numerical model we propose the use of an extended physical domain to prevent spurious behaviour of the solution at one of the boundaries, where a natural boundary condition is prescribed, related to an undisturbed coastline. Coastline and depth contours were given by field surveying data sets that are used for calibration of the numerical model at each one year of observed data. The calibrated model can be used to predict the future behavior of the coastline for times up to ten years from now and these predictions may allow an overview of the engineering works that would be made to avoid undesired disturbances in the coast and the associated environment. The Physical Problem The littoral of the Metropolitan Region of Fortaleza, Ceara State, in the northeastern coast of Brazil (Figure 1), has shown significant morphological changes during this century: about 6 million m* of coastal area have been eroded

2 84 Environmental Problems in Coastal Regions along 30 km of the littoral. The magnitude of these numbers thus becomes a matter of economic and social relevance, besides its intrinsic technical interest for coastal engineering. There is enough information about these changes which was compiled by Valentini & Rosman (1992, 1993)^ and Valentim (1994)\ such as field surveying and history of the coastal and port engineering works made there. There were analyzed 70 km of coastline around Mucuripe point and there were identified cause and effect links between these works and the physical processes in the region. Based on the cited studies, this paper analyzes the behaviour of a stretch of 10 km that corresponds to "Futuro" beach where morphological changes were observeded after the construction of a groin at its seaward end during the 70's. The main characteristics of the physical problem can be easily understood using the concept of the continuity equation for sand transport. The waves on breaking zone are responsible by the sand transport in the longshore and onshoreoffshore directions and in some cases the longshore transport is strongly bigger than the onshore-offshore one. It was proved, by previous studies, that it is the case of Fortaleza. It was also proved that in Fortaleza the wave climate is mainly governed by trade winds, resulting in an intense net longshore transport from right to left, for someone at the beach looking to the open sea. In this situation, the construction of a groin, which is a structure that interrupts the sand transport, without a sand bypass system, will obviously result in accumulation of sediment at the leeside and erosion at the seaside. This problem was studied by Pelnard-Considere in the 50's and an analytical solution was proposed based on the concept of continuity of sediment transport by means of a diffusion equation. The problem is that in the majority of the cases the nature does not match with the geometrical definitions used to solve the analytical problem. On the other hand, it is possible to solve the same problem using numerical techniques which are better for the geometrical description. This paper presents the solution for the problem described above by means of a numerical model developed in finite differences able to simulate the coastline evolution in a long term time scale. Analytical Solution Review The Pelnard-Considere (1956)' theory solves coastline evolution on time in oneline, where the coastline position y(x,t) depends on the longshore position x and time t. The basic hypothesis are: a) sand transport is governed by the longshore one due to waves at the breaking zone; b) longshore transport is considered proportional to the wave breaking angle (a**) and it is considered small; c) longshore transport takes place along beach profile from the top of the berm to a depth were the bottom is considered fixed; d) beach profile is considered in equilibrium, with low declivity and its evolution is parallel to the initial one; e)

3 Environmental Problems in Coastal Regions 85 the supply of sand is in the same amount of the flow transport capacity; f) there is no sinks or losses of sand in the domain of the problem. This situation is shown in Figure 2, and the mathematical description can be given using continuity equation that governs the problem, which for onedimension can be written as follows where: f = time (s) x,y = spatial dimensions (m) A = transversal section area (m^) Q = sand transport rate (mvs) q = sinks and losses of sand (in this case q = 0 mvms) As beach profile is always in equilibrium, the transversal section area can be written as Using (2) in the continuity equation M = AyD (2) The present case is schematically shown in Figure 3 for a straight coastline under the action of oblique breaking waves and with a groin at one side. The longshore sand transport rate Q can be given by the CERC formula (1984) *: g =& sin 2a*, (4) where go Ob. = longshore sand transport amplitude (mvs) = wave attack angle at the breaking point witch is given by -arctan (5) Ob = wave breaking angle related to jc-axis a, = coastline direction related to jc-axis Equation (4) can be written as Q = Q, sini 2 cu - arctan f (6) I L w%yjj The non-linearity in equation (6) can be eliminated using the hypothesis of small breaking angles, what in nature agrees with the wave refraction processes in gentle sloped beaches. So, the following approximation can be done:

4 86 Environmental Problems in Coastal Regions (7) As Qo and (%, are independent of x and /, equation (3) can be written as dy witch gives -^- = 6 y (8) ot ox 2O where e = -g^ (9) Expression (8) is an one-dimension diffusion equation with independent coefficient responsible by the intensity of the temporal evolution of coastline position)/. The domain of equation (8) is The initial condition is f>0 e 0<;c<oo (10) X*,0) = 0 orx*) (11) The boundary conditions are: a) At the position x = 0 there is an impervious groin, thus the sand transport at this point is zero that results. f F raal] (dy\ Q \ =QsSin\2 a,-arctan[f-\\\ =0.-. a,-arctan -^ =0 I L ^*'JJ^ v^^o W*lz=0 b) at the position jc = oo the coastline is considered fixed, so T\ =0 (13) &L. The analytical solution for equation (8) attending the conditions prescribed by equations (10) to (12) is given, using Laplace transforms on time, by

5 Environmental Problems in Coastal Regions 87 y(x, t) = 2 tan a& fi7 i erfc = \ (14) which is the same as exp,4e/ *,r - eric (IS) The Numerical Model and Virtual Extended Domain The methodology employed in this study is related to the numerical solution of the diffusion equation that describes the coastline evolution due to the interferences made on the sand transport, as described previously. The governing equation is given by expressions (8) and (9), which is a second order parabolic partial differential equation and can be solved, after the introduction of the initial and boundary conditions, by the Crank-Nicholson implicit finite difference scheme, which leads to a solution of second order accuracy. Introducing the approximations of that scheme into equation (8), the following discretized expression is obtained: At Ac" (16) where n, j = discrete integer notations for time and space At, Ax = rectangular grid sizes in the time and space dimensions After manipulation of equation (16), it is obtained the general discrete expression to be solved at each time step and for all the discrete locations/ where A/ (18) So, the left hand side of equation (17) is related to the unknown time level /i+l and the right hand side to the known preceding time level /?, starting from the initial conditions at the time step %=!, for which the discrete values of y are prescribed for all discrete^ values.

88 Environmental Problems in Coastal Regions The boundary conditions are applied to the extreme values on the %-axis, jc=0 and jt=l, where A is the length of the beach, and are approximated again by

6 88 Environmental Problems in Coastal Regions The boundary conditions are applied to the extreme values on the %-axis, jc=0 and jt=l, where A is the length of the beach, and are approximated again by first order finite differences expressions of the backward kind for x=l and forward kind for x=q: Ax These approximate equations, using eq. (12) into (20) and eq. (13) into (19), are introduced into the numerical model based on the general equation (17), as well the initial conditions. As will be pointed out in the next section, the boundary condition at x=l may introduce spurious behaviour in the solution at that boundary, due to the natural zero value prescribed condition which is usually troublesome in the sense of a numerical approach. To avoid this situation the spatial domain related to the.x-axis may be extended to two or three times the original beach length L, in order to assure that the undesired perturbed solutions are translated to a virtual and fictitious domain and restricted only to that region (Karmegam et al, 1991)*. Case Study and Results By an inspection of figures 4 and 5 we can see some differences between the behaviour of the numerical solution for the shoreline, with and without the use of an extended domain. While the shoreline remains practically unchanged, after one year of the groin construction, related both to the analytical as the numerical solution, for a time up to ten years ahead there is otherwise a difference of about 60 m at therightboundary between that solutions, when only the real domain had been considered. On the other hand, the two solutions for ten years after the groin construction present the same values when the extended domain is used. This numerical procedure does not disagree with the natural features. In fact the region described in this study has a very long shoreline, of about 30 km in the same alignment and under the same regime of littoral processes, but the monitoring area is 10 km length at the northern end, close to Mucuripe point - Fortaleza City, where the groin was built. To compare the simulated results with field data, it must be done in this monitored area, but the whole beach has to be considered in the model. Conclusions The behaviours in the analytical and numerical solutions, mentioned in the last section, suggest that the disturbance is propagated from the right boundary to the

Environmental Problems in Coastal Regions 89 left into the internal beach domain, which leads to a bad adjustment between the two solutions inside the last half of the beach line.

7 Environmental Problems in Coastal Regions 89 left into the internal beach domain, which leads to a bad adjustment between the two solutions inside the last half of the beach line. As we expected, the difference is a maximum at Jc=Z,, where a strong boundary condition of exactly zero prescribed dy/dx is used in the modelling. The differences however disappear for the whole beach length when the extended domain is introduced in the numerical solution, so justifying the adoption of the virtual length artifice. As the numerical technique described here works well in this real case, the next steps in this study are: calibrate the model using field data in the same time intervals of field surveying, and simulate several future situations. Actually the groin is almost full and sand bypass takes place at the groin toe. This situation is being incorporated to the presented model. In previous studies (Valentini, 1994)* it was identified that there is an important contribution of sand transport by wind. It will be useful to continue this study including the sand transport by wind contribution in the numerical model and perform numerical simulations for these regions. It was also identified that the general characteristics of the study area are common to many others in the coastline of the Cear6 State, Brazil. So the model established for the sediment budget and shoreline evolution can be applied to other regions, including as a preventive measure, which makes this study as a contribution to the regional coastal management program. References 1. Pelnard-Considere. Essai de Theorie de ('Evolution des Formes de Rivage en Paiges de sables et de Galets - 4- Journees de 1'Hydraulique, Les Energies de la Mer, 1956, Question II, Rapport n- 1, pp Valentini, E. & Rosman, P.C.C. Coastal Erosion in Fortaleza/CE - Revista Brasileira de Engenharia, RBE, Caderno Recursos Hidricos, 1992, vol. 10, n^ 1, Rio de Janeiro, Brazil (in Portuguese). 3. Valentini, E. & Rosman, P.C.C. Managing Beach Erosion in Fortaleza, Brazil, in Coastal Zone'93. Proceedings of the 7th Symposium on Coastal and Ocean Management, ASCE, New Orleans, USA, Valentini, E. Assessment of Littoral Processes and Consequences to Coastal Management Program in Ceard, D.Sc. Thesis, Ocean Engineering Program, COPPE, Federal University of Rio de Janeiro, 80p, Rio de Janeiro, Brazil, 1994, (in Portuguese). 5. CERC, Coastal Engineering Research Center, Shore Protection Manual, Department of the Army, Waterways Experiment Station, U. S. Army Corps of Engineers, Washington, DC, USA, Karmegam, M., Rangapathy, V. & Haribabu, S. Development of Conveyance Routing, in Moving Boundaries'91, Fluid Flow, v.l, pp , Proceedings of the 1st International Conference on Computational Modeling offree and Moving Boundaries, Southampton, UK, 1991.

8 90 Environmental Problems in Coastal Regions Fortateza - CE,- --> Brasilia-FD f Atlantic Ocean Fortalezacity Mucuripe point Harbour / ^ Futuro\ beach Rio de Janeiro - RJ Figure 1 - Geographycal localization of the study area. Coastline Figure 2 - Representation of coastline evolution in one-line model. groin wave attack direction x = 0 x (coastline alignment) Figure 3 - Plant view of the physical problem.

9 Environmental Problems in Coastal Regions 91 e Distance along shore (m) a Analyt. lyear Numer.lyear A Analyt. loyears Numei. loyears Figure 4 - Analytical and numerical solutions for coastline evolution, without extended domain Distance along shore (m) D Analyt lyear Numer.lyear A Analyt loyears Numer. loyears Figure 5 - Analytical and numerical solutions for coastline evolution, with extended domain.

SHORELINE SOLUTION FOR TAPERED BEACH FILL By Todd L. Walton Jr., ~ Member, ASCE

SHORELINE SOLUTION FOR TAPERED BEACH FILL By Todd L. Walton Jr., ~ Member, ASCE ABSTRACT: An analytical solution for the evolution of the planform for a beach fill with tapered ends is presented, and solution