Hindcast of short-term shoreline evolution

Estuarine, Coastaland She2fScience (1986) 23,499-512

Hindcast of Short-Term Shoreline Evolution

Luciana Bertotti and Luigi Cavaleri Istituto Studio Dinamica Grandi Masse-CNR, 1364 San Polo, 30125 Venezia, Italy

Received 30 May 1984 and in revisedform 8 August 1985

Keywords: littoral transport; beach erosion; shoreline; coastal currents; hind casting; wind waves; breaking waves; wind fields

A mathematical procedure is described with the aim of estimating short-term shoreline variations. The approach is fully deductive; the procedure being composed of different models that, starting from the atmospheric pressure distribution, evaluate in sequence the wind, current, wave fields, the breaking conditions, the coastal currents and finally the littoral transport. A comparison is made between visual observations of the shoreline evolution and, at one location, the variation of the bottom profile.

Introduction Our problem was to develop a general procedure to model and to hindcast sediment movement in a given coastal section during a defined period of time. This was one objec- tive of a much larger program aimed at understanding the biological and oceanographic dynamics of the northern Adriatic Sea, with major attention focused on the Italian coast, south of the PO River Delta (Figure 1).

There are two reasons why sediment movement in this area is important. The first one is its obvious influence on shaping the local beaches; the second is related to the capacity of the sediments to act as a trap for local pollutants. The large eutrophication often present in the area (Frignani & Ravaioli, 1982; Bregant et al., 1982) seems to be associated, among other reasons, with the stirring of the bottom sediments by wind waves and to their consequent release of large quantities of pollutants. In this paper we focus on the first problem.

There was an intensive period of observation from July till October 1978, a compromise between the warm season, favouring eutrophication, and the probability of a storm. During this period we have hindcasted the sediment movement and evaluated the consequent beach modifications. Usually this is done starting from some wave data taken more or less close to the area of interest. In our case no wave data were available, but we had at our disposal some mathematical models capable of estimating, starting from the general meteorological maps, first the wind field and then the wave field at any chosen location. The aim of this paper is a description of the modelling and a discussion of the results.

Modelling Our starting point had to be the meteorological maps of the Adriatic Sea during the two-month test period. From them we had to estimate the wind fields. These led to

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500 L. Bertotti B L. Cavaleri

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Hindcast of short-term shoreline evolution 501

the knowledge of the wave and current fields, from which the sediment transport at the different coastal locations could be obtained. Hence the sequence of models concerns wind, wave, general circulation, wave breaking, coastal current and sediment transport. Each one of these will be rapidly described. A more comprehensive description can be found in the quoted references.

Wind In principle the wind field in the open sea, particularly in a closed basin such as the Adriatic Sea, could be obtained by interpolation of the wind data collected at the various coastal meteorological stations. The complicated orography of most of the Adriatic coast, strongly influencing the local wind, excludes this method. A better starting point is the atmospheric pressure recorded at the same locations, a quantity much less sensitive to the local orography.

At each synoptic time (00 h, 03 h, 06 h, . . . GMT), the atmospheric pressure at the various meteorological stations is available from the meteorological maps or as digital data (Figure 1). The general pressure distribution is obtained by fitting a sixth order surface to the data by a least squares method (Bergamasco et al., 1986). This provides an estimate of the pressure field, in particular at the nodes of a grid covering the whole Adriatic Sea (not shown in the figure). The grid step size is 30 km.

Starting from the pressure distribution, the wind field is evaluated by numerically solving the nonlinear balance equation

where a, is the time derivative, c is the wind vector, V is the horizontal gradient operator,p is pressure, w is the wind vertical component, p is air density, z is vertical versor, g is the acceleration due to gravity, Y is surface friction. The assumption of a stationary and rotational solution, acceptable in a typical middle latitude storm, and focusing on .the geostrophic wind lead to the simplified equation

(2)

The limited north-south extension of the Adriatic Sea allows the use of a constant Coriolis parameter f. The introduction in (2) of the stream function w and of the divergence operator leads to the scalar equation

b2p-fv2y/+2J(Ux, u,>=o. P

(3)

The first two terms of (3) provide the linear solution corresponding to the geostrophic wind. The Jacobian of the two wind components U,, U, is the nonlinear part of the solution. Equation (3) is solved by iterative methods (Ames, 1965) providing the gradient wind at the grid nodes. The corresponding surface wind, the actual forcing function for waves and current, is evaluated on the basis of the wind speed and of the air stability conditions, using the transfer coefficients provided by Findlater et al. (1966). Solving equation (3) for the data from each synoptic time has provided a sequence of wind fields, at 3-hour intervals with a step grid size of 30 km, for the whole test period.

502 L. Bertotti & L. Cavaleri

Waves

Figure 1 (b) shows the area south of the PO River Delta where the hindcast had to be carried out. As it will be mentioned later, the local variation of beaches depends on the spatial divergence of the sediment flux, that is highly sensitive to the wave energy input. Had we used a grid model to hindcast the wave conditions, the spatial resolution of the grid would not have been small enough to take into account the small, but important, variations of the coastal bottom topography from place to place. Besides the area of interest is very shallow (see Figure l), hence, for a proper estimate of the wave conditions, all the shallow water processes had to be taken into account. This has led to the choice of a ray spectral model capable of providing, given the wind field, the wave directional spectrum at specified times and positions.

The model is briefly described as follows [a comprehensive description is given by Cavaleri & Malanotte Rizzoli (1981) and Cavaleri & Bertotti (1982)]. The wave directional spectrum is discretized into a number of components, specified as frequencyfand direction 13. Usually 20 frequencies and 36 directions are considered, the latter ones reduced to 19 when the model is used close to the coast. Given the bottom topography and the coordinates of the location of interest, the approaching path of each component of given f and 0 is tracked back by numerical integration of the refraction laws. The path is uniquely determined by f,O and the location of the target point, and it is stored on permanent file.

The actual energy approaching at a specified time the target point along this path is evaluated by integration of the energy equation

aF

at= S, (4)

where F = F(f$) is the energy density of the wave component, t is time and S summarizes the energy input-output along the path. The original energy balance equation from which (4) is derived includes also an advection term (e.g. see the SWAMP Group, 1985). This is not considered in this case because, using the ray technique, equation (4) describes the energy balance of a wave group while proceeding along the ray.

The output of energy in (4) corresponds to the dissipative processes acting in deep (whitecapping) and shallow water (bottom friction and percolation). The input is due to wind action, that is estimated on the basis of Phillips’ and Miles’ processes. The references for these processes are respectively Hasselmann (1974), Shemdin et al. (1978) for friction and percolation, Phillips (1957) and Miles (1957).

While approaching the target point along the path, the wind is evaluated point by point, instant by instant, using the predetermined wind fields. The integration of (4) is carried out in wave number space so that the shallow water processes (refraction and shoaling) are automatically taken into account.

Integration of (4) for each wave component provides the wave directional spectrum at the specified time and location. This requires the procedure to be repeated for each estimate and for each point. The computer time is anyhow negligible, the procedure being mostly based on pre-calculations. Once the directional spectrum is known, by straight- forward integration the standard quantities, such as the energy spectrum, significant and r.m.s. wave height, peak and mean period, and mean direction are obtained. Examples of results will be given in the next section.

For our aim the waves and the related sediment transport had to be estimated all along the coastline shown in Figure l(b). The question was at how many points we had to carry


out the calculations. Still with an eye on the obvious requirement of keeping the amount of calculations within reasonable limits, the mandatory condition was to have the target points close enough to make any intermediate coastal section sufficiently uniform, or at least with a constant trend. Consideration of the detailed bottom topography and some tests with the wave model led us to the choice of the 17 points shown in Figure 1 (b). Their intermediate average distance is about 5 km. Their average depth is 5 m. Using the wind fields previously evaluated we have then obtained the wave history for the same period at each one of the 17 chosen locations.

General circulation Figure l(b) clearly shows the shallow northern part of the Adriatic Sea, and the deeper conditions of the southern portion. In addition, the presence of water masses with largely different characteristics, particularly in the warm season, leads to strong stratification. It follows that any sound modelling of the general circulation must be based on a multilayer description of the basin. Such a model has been proposed by Malanotte Rizzoli & Bergamasco (1983), and applied to describe the general circulation of the Adriatic Sea during the July-October 1978 test period.

The model is based on the numerical integration of the shallow-water momentum equations for a Boussinesq, hydrostatic, incompressible fluid. The momentum equations are fully time-dependent. Vertical integration of the hydrostatic relationship allows the splitting of the pressure field into a barotropic component, depending upon the sea-level distribution, and a baroclinic one, depending on the density field in the interior water mass. The momentum equations are coupled with thermodynamic equations for the temperature and salinity. The two latter equations are considered in their fully nonlinear form while the horizontal momentum equations are linearized. The system is closed by an equation of state for sea water.

The input to the model is given by the bottom topography, the wind stress field, the air-sea interface heat and evaporation fluxes, the sea level, temperature and salinity distribution at open boundaries and at the river mouths.

The model is run on a 7.5 km grid with a smaller nested grid close to the coast. The description of the edge circulation is obtained by a boundary layer analysis in terms of a dimensionless diffusivity parameter (Malanotte Rizzoli & Dell’Orto, 1981).

It was not the intent of the project to analyse the small details of circulation and sediment transport. Any discontinuity of the coast was ignored, and similarly, the existence of any artificial structure such as the two piers bordering a river mouth, etc., were also ignored. This attitude was backed by the limited number of structures actually present in the area, and by the smoothness of the coast. So the implicit hypothesis was that any external influence would have only local consequences, without any effect on the general movement of sediments, in particular on the budget of the single sections.

Breaking We have previously mentioned that at any target point, in correspondence with each coastal section, we have obtained the time series of the two-dimensional wave energy spectrum, then reduced it to the r.m.s. wave height H,,,,,., the peak period Tr,, and the mean direction 8. Knowing the local bottom profile and the orientation of the coast, this wave has been shoaled by numerical procedure, taking refraction and shoaling into account while continuously checking the breaking condition.

Breaking has been checked following the breaking criterion given by the ‘ Shore

504 L. Berrotti & L. Cavaleri

Protection Manual ’ (1977), relating wave height, period and slope of the beach. This point will be further dealt with in the discussion section.

Coastal current To evaluate the coastal current due to breaking two different approaches have been followed in the literature. Both the approaches rely on the assumption of a steady and constant current, and they are based on the equilibrium between the onshore flux of longshore momentum due to waves, the frictional drag due to the longshore current and the horizontal mixing across the surf zone (e.g., see Komar, 1983a). Longuet-Higgins (1970a) and Komar & Inman (1970) aimed at determining an average value of the current within the breaker zone, and their results were later shown by Komar (1975) to be equivalent to

?ji = 1~17(gH,)&in ab cos ab, (5)

Hb being the height of the breaking wave, ab its angle of approach with respect to the shore, g the acceleration due to gravity. Equation (5) was shown to fit well both the laboratory and field data.

On a more complicated basis Longuet-Higgins (19706), Kraus & Sasaki (1979) and Symonds & Huntley (1980), among others, have worked to provide the transversal profile of the current across the breaker zone. The results, particularly those of Kraus & Sasaki (1979), are very good, but computationally heavy.

From the practical point of view equation (5) seems highly sufficient for all the applications (see Komar, 1983a), and we chose it for the procedure. The current in the nearshore has been so obtained by the superposition of the longshore current due to waves and the current associated with the general circulation previously described.

Transport Many formulas relate the littoral transport to the incoming wave energy. Such an approach is, however, not physically sound when the local current is not due only to waves, but, as in our case, also to other simultaneous reasons, such as tide, general circulation or river discharge. In this case, it is convenient to use.

v 1, = 0.28(E&-, (6)

%I

where E is the wave energy, c is the phase velocity, tl is the ratio between group and phase velocity, 6 is the littoral current and u, is the maximum bottom horizontal orbital velocity of the waves evaluated at the breaker zone. Equation (6), proposed by Bagnold (1963), was applied to the littoral drift evaluation by Inman & Bagnold (1963). The coefficient 0.28 was determined by Komar & Inman (1970) as a best fit to the available littoral drift measurements from different locations. For engineering purposes, equation (6), giving the immersed weight transport, is better transformed to the apparent volumetric transport, taking into account the characteristics of the material moved. For quartz sand, the main constituent of the beaches in the area under study, (6) is transformed to

v Q, = 2.5(Ecn), -%

%I (7)


Here the wave and current parameters are in mks units (W m-i) and Q, has units of m3 day-- ‘. Komar (1983~) provides an extensive and clear discussion of the use and accuracy of these formulae.

Going back to Figure l(b), having available, for each one of the 17 locations there indicated, the time series of the wave characteristics and of the nearshore current, by using (7), we have evaluated the littoral transport at the same locations for the whole test period. By continuity the budget of each coastal section between two sequential points is the divergence of the transport, i.e., the difference between the input-output at the two bordering locations (e.g., see Komar, 19836). This implicitly assumes that the on-offshore movement of sand is negligible compared to the littoral one.

The positive and negative budget of each section is reflected in the variation of the shoreline. However, this is not a definitive parameter, being very sensitive to the shift of material in the on-offshore direction associated with fair or stormy periods. The typical movement of material from the berm to the submerged bar consequent to a storm can imply a variation of the apparent coastline larger than that due to the actual budget of the area. This implies that any check of the ‘ wet edge ’ of a beach, if related to the sediment budget, should be carried out under the same conditions, with possibly the same bottom profile, typically one year later in the same season. This was not our case; our test period lasted two months, during which some early storms were likely to initiate the construction of the winter bar. On the other hand we had only the variation of the ‘ wet edge ’ at some locations, and only at one of these were we capable of obtaining the bottom profile normal to the beach, before and after a storm.

In this sense the verification of the overall procedure is not definitive, being associated only with a single point. Nevertheless reasoned indications can be obtained also from the points where only external evidence was available.

Results

As expected for that season, the test period was mainly characterized by fair weather with very little wind. One stormy period was present in the latter 30 days and we will limit our report to it.

Figure 2 shows the wave history at the 17 locations [see Figure l(b)] from 15 September till 16 October 1978 (only daily values are shown). The ordered numbers in the ordinates are the reference zeros for the corresponding stations. The scale for significant wave height is such that interval between horizontal lines is equivalent to one metre.

Basically three storms were present during this month representing the two typical situations acting on the area. Around 21 and 28 September an active cold wind ‘ bora ‘, was blowing from the northeast. From Figure l(b) some characteristics of the wave field can be anticipated. The wave height grows moving towards the south, because of the increased fetch length and the shadowing effect of the PO Delta. The former point is associated also with an increase in the wave period [e.g., see the SWAMP Group (1985) for wave energy and period dependence on fetch]. So, looking at Figure 2 in correspondence with the above mentioned dates, we see that the wave height grows from 0.5 m at point 17 to 1.5 m at point 1. At the same time, moving southward the coast bends facing more directly the northeast waves, with a consequent decrease in the coastal current [formula (5)]. Figure 3 gives the time series of the coastal current at the 17 locations. The superposition of the single diagrams makes quite evident the different values of the simultaneous coastal currents.


16

2

0

1; 2-O 3-o 10 15

SEPTEMBER OCTOBER

Figure 2. Time history of the significant wave height at the 17 locations shown in Figure l(b) during September-October 1978. Daily values are shown. For each point the zero reference is at the corresponding figure on the vertical axis. One point step equals one metre.

The second type of storm was present around 3 October. A strong southeast wind sirocco blew over most of the Adriatic for more than a day, leading to a stormy situation followed by swell. In contrast to the other case there was no practical variation of the offshore wave height in correspondence with the different locations, the fetch difference being negligible compared to the actual one covered by the storm. Nevertheless, because of the orientation of the coast and of the refraction effect, there was a large increase in wave height from south to north, going from 0.5 m at point 1 to almost 3 m at point 17 (Figure 2). Besides, the wave angle of approach favoured the transport, causing the large increase in sediment movement moving towards the northern points. This is clearly shown in Figure 4(a) where apart from the variations within the short distance, the spatial trend is unmistakable (a positive transport is meant to move to the north). The sediment budget for each section, corresponding to the same situation (12 GMT, 3 October 1978), is shown in Figure 4(b). The general tendency is clearly that of erosion, as it had to be expected from Figure 4(a), because the area is practically acting as a source of sediments without any closure at the northern end.

The final budget for the whole coast at the end of the test period is shown in Figure 5. Coastal erosion is clearly the general tendency. A comparison with Figure 4(b) reveals the strong similarity between the two diagrams. This suggests that the overall transport during the two months was dominated by the storm of early October, because of its intensity and the low peak frequency consequent to the larger fetch from the southeast. This is confirmed by the time series (not shown) of the coastal transport at the single locations.

We have mentioned in the introduction that basically our experimental data consisted of a visual observation of the shoreline variations. So, as a first-order comparison, assuming a


tY W m 0 I- O


100

1

x103 (m3)

50'-

so--

SECTION bl -lOO-

Figure4. (a) Fluxofsedimentson3 October 1978atthe 17 1ocationsshowninFigure l(b). A positive sign means transport is to the north. (b) The budget for the 16 coastal sections during the same day. ith section is included between iand i+ 1 points.

loo-=103

(m31

50.-

SECTION -1OOL

Figure 5. Final budget for the 16 coastal sections in Figure l(b) after two months of modelling.

20

(cm) 1 1 lo-

0

-lo.- . .

5

(ml

0

. SECTION 1 -5

Figure 6. Depth variation in each section estimated on the basis of the results shown in Figure 5. The corresponding horizontal variation of the shoreline is given by the scale on the right. Dots show the visual estimates of the shoreline position.

permanent bottom profile, we have transformed the final budget into a layer of sand and then into the consequent shift of the apparent shoreline. For this the coastal extension and the bottom profile of each section have been taken into account. Note however that the bottom profile is very similar along the coast, allowing an almost constant ratio between the moved layer of sand and the associated horizontal variation of the coastline. This is shown by the double scale in Figure 6, where, as compared to the hindcast (solid line), the dots show also the experimental evidence at the end of the period.


2 - 2

(ml

imi .

1. .

I K-4

1

. . -

0 .yq . -r 7. 0 - .I . . 15 20 25 3: 5 10 15

t OCTOBER

Figure 7. Time history of the estimated horizontal shoreline variation at section 14 during the simulation period. Daily values are shown. Dots represent visual observations. Arrows indicate timing of the bottom profiles in Figure 8.

-LL---- * 1 I 1-L 0 LO 00 120 160 cm1 200

Figure 8. Bottom profiles at section 14. Their timing is indicated by arrows in Figure 7.

Still within the expected uncertainty of the measurement (the variations are not very large) we have estimated an average 30% error, with a tendency to underestimate the erosion at several locations. A better insight can be gained using the sequential observations of the coastline and the bottom profiles available at section 14. Figure 7 shows the hindcast evolution of the shoreline (solid line) as compared to the observations (dots). After early October there is great discrepancy between the models results and the measurements. The difference then disappears in time, the actual coastline gradually approaching the hindcast. The expected explanation comes from the bottom profiles shown in Figure 8. Their timing is indicated by the arrows in Figure 7. Till the end of September the beach had still its typical summer profile, while, after the first heavy storm, a submerged bar was built, with a consequent erosion of the berm and an apparent regression of the shoreline. During the subsequent days the persistence of fair weather (see Figure 2 for the wave history) favoured the erosion of the bar and the gradual backfeeding of the berm, with a tendency to reshape the beach in the standard summer profile. The consequence was a better fit between the hindcast and the experimental results.

Discussion

The description of the modelling given earlier was used to reach the results reported in the last section. It is worthwhile to point out that the description represents only the basic scheme of the procedure followed for the actual calculations, but neglects all the ‘ variations

510 L. Bertotti & L. Cavaieri

on the theme ’ attempted along the way. Komar (1983~) provides a clear discussion of the approximations used in the evaluation of the coastal transport. A very apparent one is the collapse of the two-dimensional wave spectrum into a monochromatic, long-crested wave. So a random sea is reduced to a sinusoid, neatly repeating its cycle every Tseconds. This is quite an approximation. The actual distribution of energy over the whole frequency range of a wave spectrum leads to a wide breaking zone, where breaking is not a repetitive phenomenon, but it randomly varies in space and time. While studying the coastal set-up, we have done an extensive study of breaking, and we have put forward evidence of its strong variations (Bertotti & Cavaleri, 1985). Even more, the breaking criterion of a random sea is not well-established, whereas on the coastal current and transport are very sensitive to the characteristics of the breaking wave.

If we turn our attention to the directionality of the sea it is immediately evident that, as the coastal current Z, [formula 51 is not linearly dependent on the angle of approach, the overall effect of the different directional components, if estimated through the same formula, cannot coincide with that associated with a wave running in the mean direction. Battjes (1972) has shown that an overestimate by as much as 1OOo/0 is possible in these cases.

We have done some sensitivity studies in this direction, by propagating the discrete components of the wave spectrum toward the shore, and checking the random breaking along the way. As expected in some cases the results turned out to be quite different from the monochromatic, unidirectional approach depending basically on the frequency range and the directional spreading of the wave spectrum. In these conditions, why should one cling to the monochromatic, unidirectional approach? In our opinion there are basically two reasons. First it is the way through which most of the practical formulae have been obtained, and the only way to use them is to use the same conditions. Second our knowledge of the field, even if greatly improved in recent years, is still so crude as to make worthless any approach which is very relined in only one section of the procedure.

For the same reason we have neglected the influence of the current on the refraction path of the single waves. It should be pointed out that the general circulation in the Adriatic Sea is weak, and the circulation close to the shore is totally dominated by breaking, when present. So there is practically no consequence on the wave path almost up to the breaking point. We ran some numerical experiments within the breaker zone refracting the single components of the spectrum, but the implicit approximations are such that we look to these more as numerical exercises, rather than a representation of the physical situation.

We have then to consider the errors of the other models, say for wind, wave and shoaling. By direct experience the first two have been established at about 150,d (Bergmasco et al., 1985; Cavaleri & Malanotte Rizzoli, 1981). The difficulty with shoaling is associated with the nonlinearity. While most of the calculations are usually based on the linear theory, there is large evidence that the importance of nonlinearity grows while approaching very shallow water conditions. The attempts made up to now to overcome the problem, even if very attractive from the scientific point of view, are not practical, either because of the different limitations introduced in the solution, as is the case of cnoidal theory, or due to the heavy computations required (Freilich & Guza, 1984).

After this pessimistic account, it might be surprising that the final results of the procedure are acceptable. The explanation lies in the fact that the models described above have, for different reasons, very little bias. It follows that, if the initial hypotheses are satisfied, we should expect for the results an error of the same order as that of the models. Besides, and this is particularly true on a straight uniform beach, if an error is present, we

Hindcast sf short-term shoreline evolution 511

can expect the consequences to be similar at all the points. Hence our hindcast could be quantitatively wrong, but qualitatively correct.

Clearly a weak point of the procedure is the lack of any consideration for the modifi- cation of the bottom profile. On the other hand, there were very little data available for it. Therefore neither was it possible to operate with the empirical method used by Fox & Davis (1973) on Lake Michigan, nor had it any meaning, from our point of view, to work out some predictions without any possibility of verifying them.

The formal verification of the overall procedure has been limited by the scarceness of the data, particularly of the bottom profile. On the other hand, the data, where available, were consistent with the results. Hence, within these limits, we think the results are positive. Also, as the procedure is made of blocks singularly verified on previous occasions, there is no evident reason why their joint use should not be positive.

While this can be accepted in the short term, problems arise in the eventual long-term use. A purely deterministic approach, such as the one we have followed, loses validity in time, as the hindcast and reality will sooner or later diverge. The question is how far one can go with this approach or how it can be improved. The reply depends on the more or less complicated orography of the area, on the climatology of the basin, on the knowledge of the eventual sediment sources, as rivers, and on the amount of human intervention on the coast. It also seems likely that, if we want to extrapolate for a long term, some kind of randomness of the occurring events has to be included in the model, as discussed in Davis (1978). In connection with other programs, we are now considering the possibility of such a long-term experiment in the Adriatic Sea.

Acknowledgements

Luciana Bertotti carried out this work as an oceanographic expert for the Regione Emilia- Romagna, which sponsored the whole project. All the calculations of the circulation model have been executed by Paola Malanotte Rizzoli and Andrea Bergamasco. Dr Malanotte Rizzoli was also responsible for the oceanographic group. The data of the shore line evolution were part of the overall project.

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Ephraums, J., Golding, B., Greenwood, A., Guddal, J., Gunther, H., Hassehnann, K., Hasselmann, S., Joseph, P., Kawai, S., Komen, G. J., Lawson, L., Linne, H., Long, R. B., Lybanon, M., Maeland, E., Rosenthal, W., Toba, Y., Uji, T. & de Voogt, W. J. P. 1985 Sea Wave Modelling Project (SWAMP), an intercomparison study of wind wave prediction models. In Ocean Wave Modeling. Plenum Press, New York. 256 pp.

Symonds, G. & Huntley, D. A. 1980 Waves and currents over nearshore bar systems, Canadian Coastal Conference 1980. Proceedings of the National Research Council, Canada. pp. 6478.

Documents

Hindcast of short-term shoreline evolution