CSHORE is a beach and dune morphological model developed by [16] to predict the transformation of irregular nonlinear waves using the time-averaged continuity equations. The model was previously used by many authors and researchers to determine temporal cross-shore beach profile evolution due to either wave action or an extreme storm event. It has been useful in determining sand dune response to storms and changes in tidal currents. Though CSHORE is limited only to cases of alongshore uniformity, it was found to be simple and robust to its purpose. According to [7] , the model uses linear wave theory and the Gaussian probability distribution to reduce the degree of empiricism. In his research, CSHORE was calibrated to predict the run-up of irregular waves, and the overtopping that can occur on coastal protection structures such as dykes and sand dunes as well as beaches. To that effect, 137 wave run-up tests were conducted and 97 wave overtopping tests.

Again, [16] came up with the continuity equations of bottom sediment which was written as follows.

$\left(1-n_p\right)\frac{\partial z_x}{\partial t}+\frac{\partial q_x}{\partial x}=0$ (1)

And

$\left(1-n_p\right)\frac{\partial z_y}{\partial t}+\frac{\partial q_y}{\partial y}=0$ (2)

where,

$n_p$ = porosity of the bottom sediment normally taken as n_p = 0.4.

$t$ = slow morphological time for change of the bottom elevation.

$q_y$ = longshore total sediment transport rate.

And again, the bottom elevation change $\Delta z_y$ during the interval $\Delta t$ is expressed as

$\Delta z_y=\frac{-\left|\Delta z_x\right|}{\left(1-n_p\right)A_x}\frac{\partial V_y}{\partial y}$(3)

where,

$A_x={\displaystyle {\int }_0^x\left|\Delta z_x\right|\text{d}x}$; (4)

and

$V_y={\displaystyle {\int }_0^{x_m}\text{d}x}{\displaystyle {\int }_t^{t+\Delta t}{q}_y\text{d}t}$ (5)

where,

$\Delta z_x$ = bottom elevation change based on Equation (2) during time t to (t + Δt).

$A_x$ = sum of the absolute value of $\Delta z_x$ across the cross-shore line.

$x_m$ = cross-shore distance of the line starting from the seaward boundary x = 0.

$V_y$ = longshore sediment volume across the entire cross-shore line during time t to (t + Δt).

These and all other mathematical derivation equations for the model were given in Kobayashi (2013).

CSHORE was modified to allow for simultaneous computation of more than one cross shorelines during calibration in experiments carried out at Dewey and Rehoboth in Delaware [7] . The bedload parameter b at a value of 0.002 was found not to cause sufficient erosion in the profile above MSL. This was attributed to the fact that the bedload parameter changes the profile thus affecting the computed hydrodynamics as well as the sediment transportation offshore of the beach. A value of b = 0.001 was found to give a better prediction of the erosion without accretion. The accretion volume was however predicted much better at b = 0.003 but the challenge was that at b = 0.003, no erosion was predicted. [17] indicated that the accurate prediction of both erosion and accretion is difficult due to the small differences between the bed-load transport and the transportation of sediments suspended in water. Figure 2 below shows the location of Dewey Beach in North Carolina.

And again, Figure 3 below shows model test results conducted during this investigation to qualify CSHORE for model tests on the study area.

One critical issue for most models was the assumption that during a storm event, the mean sea level remains constant while the profile change was only in the vertical so that the equilibrium of the dune profile was assumed to be based on the vertical profile change only [19] . However, [20] argued that numerical models were not very accurate since they tended to overestimate the actual volumes of sediment or rates of dune erosion [14] . [15] made a few suggestions to create large-scale physical data for dune erosion and some of the steps towards achieving better results were considered as follows.

Physical models have been used too over the years but are costly and not easy to make. However, many empirical formulas were derived from observations and analysis obtained from physical modelling. Physical modelling provides a basis for calibration and validation of numerical models.

An alternative solution to solving coastal engineering problems was the use of analytical models. Again, [15] pointed out that analytical models solved problems easily by utilizing marked simplifications in forcing conditions such as input data and boundary conditions. The advantage that analytical models will possess would be the simplicity in application which is assumed to be handy when initial conditions estimates are needed to drive the remainder of the model to give solutions. These models investigate the relationships between parameters and investigate causes and effects. Another basic assumption was proposed by [21] called the impact wave theory which assumed a linear relationship between the force F applied on the dune and the weight W of the sediment volume from the dune as follows:

ΔW = CEF (6)

where,

CE was assigned as an empirical coefficient.

In the relationship, the weight ΔW could also be written as:

ΔW = ΔVρs(1 − p)g (7)

where,

ρs = sediment density.

p = porosity.

g = acceleration due to gravity.

As suggested by [15] , analytical models could be calibrated based on the observed values from a physical model and application of the proper analytical method such as Least Square Method to derive optimum values of coefficients. Then, validation of the analytical model would include comparing between observed and calculated values for selected tests to determine if the values showed good agreements.

CSHORE was applied to model other beaches with known and measured field data on one of the barrier islands adjacent to the study area. The idea was to carry out as mant tests as possible and compare the results to known measured data and assess the validity of using CSHORE on the study area. Results proved assuring with an increased confidence to using CSHORE as the model for the investigation. Findings pointed out to the effect of longshore sediment transportation as a major force influencing erosion of the barrier islands. [4] noted that the prevailing direction of approach of the waves acting on the barrier islands was from the southeast which caused the south to north direction of longshore sediment. However the overall predominant direction of longshore sediment transport was from north to south but there were several cases where the predominant direction of sediment transport was from south to north [22] . Clearwater Beach Island and Caladesi Island into Hurricane Pass and as well as Sand Key and portions of Honeymoon Island up to Estero Island all experienced direction reversal of sediment transportation. Sea level rise has become a major reason the frequency of storm events has increased in the past years and many coastal communities will experience increased occurrences of storms and with high intensity. Many sandy beaches will suffer continued erosion and, in most cases, the cross-shore profile changes by retreating landward. Eroded sediments may be washed offshore but remain in the erosive condition or in the case of overtopping, may be deposited landward. [19] noted that the changes of the shoreline in the long shore were directly proportional to the sea level rise. Understanding dune evolution on a sand beach is therefore an important issue to accurately predict the flow circulation and the changes to the bathymetry. However, their evolution is still poorly understood due to their complex behavior [23] . Figure 4 below shows the details of transects at Madeira Beach as per measured field data.

Model simulations for Madeira beach using different parameters were conducted and the results are as shown in Figure 5 below as a comparative analysis of the CSHORE performance and justification for use in this investigation.

Water elevations, water circulation and tidal currents data sets were obtained from the Coastal Hazard Systems (CHS) and these inputs were used in running of the CSHORE model. On the CHS data website, data sets were obtained from points in the ocean at offsets less than 5 miles from the Sand Key beach. These points were appropriate because they would provide the exact wave climate data for waves that affected the study area by means of a set of input parameters derived by [17] as per the governing equations of the model. Some of the important parameters to be considered in the model which were detailed in [16] include:

Gamma (Υ)—an empirical factor used to adjust the variation of the model wave height against the actual measured wave height and ranges between 0.5 to 1.0. It is suggested to use a conservative value of 0.7 if, in the study area, there is no measured wave height data.

Mean grain size (D50)—the median sediment diameter of the beach sand soil in mm.

Wf—sediment fall velocity measured in m/s.

SG—sediment specific gravity of the beach sand.

EFFB—suspension efficiency due to wave breaking and the suggested range between 0.002 and 0.01.

EFFF—suspension efficiency due to bottom friction with a typical value of 0.01.

SLP—suspended load parameter with a range of 0.1 - 0.4.

SLPOT—suspended load parameter associated with the wave overtopping rate which ranges between 0.1 and 3.6.

Tan phi—the sediment limiting slope value measured as tan Ø and calibrated in the range of tan Ø.

BLP—bedload parameter determined to range between 0.001 to 0.004 and typically 0.002.

RWH—runup wire height (m).

JCREST—initial crest location for L = 1 which is the crest node of the maximum bottom elevation for the input bottom profile.

RCREST—initial crest height (m) for L = 1 which is measured from the input bottom elevation and usually corresponds to an input of the elevation data into the model.

AWD—swash velocity parameter which expresses the horizontal velocity U as a function of the water depth h in the wet and dry zone.

EWD—output exceedance probability which compares measured values corresponding to 2% of incident irregular waves and ranges between 0.01 and 0.015.

Table 1 below shows values used in the setting up of the numerical model for the Sand Key beach.

The depth at seaward boundary (m) was 3.2 m and the friction factor for the bottom interaction between water and seabed was 0.015. The value of the Still water level (SWL) above the elevation Z = 0 m was 0.573 m. Tidal currents were incorporated in the data for water elevations hence the cross-shore volume flux associated with the temporary variation of the still water level was included.

Some of the assumptions used in this study according to [16] include the following:

Beach profile data used in the validation of the model performance at Sand Key Beach was the data measured and recorded by the USGS from 17 October 2018 until 11 September 2023. A series of profiles were surveyed as shown in Figure 6 below. The model simulation was run with beach profiles extracted at approximately the same profile lines and results were analyzed.

And again, Figure 7 below shows the water circulation conditions, and wave climate of the period selected in the simulation. From the data provided, the maximum wave height recorded was approximately 3.4 m whilst the maximum tidal elevation reached up to about 1 m.

Wave analysis was conducted using the Surface Water Modelling System (SMS) software with a grid angle set to 112.0 degrees. The reference time matching that of the simulation period was set to model the wave energy at Sandy Key Beach. The model showed that the frequency of waves ranged from a minimum of 0.04 to 0.4 Hz which translated to 3.0 to 25 second waves. The method used to generate waves was the shallow water method for generating a spectrum from parameters from the Coastal Hazard Systems to generate wave energy frequency and direction. Figure 8 below shows the wave spectra of the wave conditions.

Subsequently, Figure 9 below shows the distribution of wave energy vs frequency and direction for the parameters modeled in the SMS interface during analysis of wave conditions.

Figure 10 below shows variation of model results to the actual measured data. The model output (green line) over-predicted erosion and deviated from the actual measurements. The model representation of the bed perturbations showed a steeper landward shift of the sand berm. The shoreline retreats further towards the land and there is significant sand loss when compared to the nourished beach profile. The overall changes are however not caused by a one-time storm event but a compounded effect of wave condition variations and seasonal changes. As depth increased below the 1m depth level, the model reproduced the bed slope movement a meter deeper than the actual measurements.

The changes across both the measured data and the model results which agree at the sand dune slopes give evidence of the cross-shore processes such as over wash of the sand dune. This was illustrated by both onshore and offshore changes of the bed profile. Deviation of the model results to the actual measured data in the swash zone was observed though the magnitude was not as great as compared to Line 01-08 results. Model results showed a good agreement in the bed profile variations along the cross-shore profiles especially on the sand dune zone as well as on the slope direction and angle of the sand berm as it retreated landward. Figure 11 below shows the results.

Localized variations of the model predictions against the actual measured data were identified. Compared to the cases in line 01-08 results as well as line 09-10 results the model results did not show a wider margin of deviation from the measured data but a balance between eroded profile and accretion along the cross shoreline. As depth increased, the measured data showed a uniform sediment flux with a constant gradient whilst the model indicated the potential for more sediment transportation at lower depths before merging into a final bed form with a constant or no slope. Figure 12 below shows the results as plotted.

Model results showed a good agreement with measured data. The bigger size of the sand dune provides more protection to the landward sand. More erosion resistance was generated and there was less sediment migration back into the ocean. The results in the model supported the conclusion by [3] that bed slope temporal evolution was dependent and influenced by dune slope and height and position. Figure 13 below shows the plot of the results.

The overall trend and magnitude of contour change of both the model and the measured profile followed closely with one another and showed a similar pattern as shown in Figure 14 below. The slope of the sand berm at elevation 0m was comparatively steeper for the measured data than the model prediction. It is worth noting that the cross-shore profile changes show a landward retreat of the shoreline therefore beach nourishment helps the recovery process of the beach and minimizes the shift of the shoreline as caused by the influence of the longshore transport gradient.

The root mean square error (RMSE) and the mean absolute error (MAE) were computed and Table 2 below shows values for the model error.

However, other methods could be employed such as the Bayesian method, where parameter distributions are estimated, though the method will not allow statistically independent events to merge [26] . The low values of the root mean square error indicate that the model fits the actual measured data. Observations showed that beach nourishment in 2018 had a huge positive impact in fortifying the beach berm hence a wider and higher beach berm as shown on the initial beach profiles measured in 2018. Model predictions generally show that the cross-shore model over-predicts the erosion compared to the actual data. This over prediction is more common where the beach profile is submerged. CSHORE generates forcing in the swash zone by hydrostatic oscillations of short waves of different period therefore synthesizing wave groupings. Because CSHORE is a time averaged numerical model, the time averaging of the swash generates errors that cannot be accounted for only when the correct parameters are used in the model input data [27] . As a result, a monochromatic wave could simulate the wave energy and sediment transportation to some good extent but will give a lower prediction of the wave run up under random wave conditions [28] . Figure 15 below shows model evaluation using 5% and 10% error bands.

Error bands showed model deviations in the positive spectrum thus raising a validation question on the results. It is important to note that, further investigations would need to be carried out using different models that predict cross shore sediment transportation. However, the better results on the sand dune do not justify the complexities at the profiles below the MS and since the model does not do better in modelling accretion which introduces beach recovery in later periods as the water depth is forced to change rapidly in the swash zone, the model limitations are acknowledged in the investigations. However, in a bid to give a more accurate idea on what can be expected on a beach after a severe storm, the model gives a better understanding of what to expect from impacts of a storm on a temporal basis for better decision-making during design and maintenance of sand beaches.

According to [12] , beaches have a resilient feature that they are partially self-healing. During storms, some berm sand moves offshore thereby creating large bars that cause waves to break further offshore reducing damage to dunes and infrastructure. Within months following a storm, the sand is returned to the beach by calm weather waves. Increasing sea level rise due to global warming adds significant risk to beach recovery. In the barrier islands, there is high development such that retreat is impossible. A more likely approach would be to maintain the groin systems to aid the beach in trapping sand to counteract the effects of the rising sea level. Due to the increased frequency of severe storms, the barrier islands no longer recover at a fast pace than before human influence hence beach nourishment after some period would help the beaches recover from the changing climate. Beach nourishment would also raise nearshore profiles [3] . The refurbishment of the groin system helps in that the new groins are designed with the rising sea level in mind and hence will be able to help in the recovery of the beach in the long term before the next phase of refurbishing the groins several years in the future. It is important to note that when modelling beach response and recovery, two or more storm events affect the beach erosion and recovery dynamics because it is likely that two or more storm events occurring within a few days of each other may yield more extensive beach erosion than a single event [29] . The compounding effect of many storms has the effect of causing the particles to transition from a very low flow state to an extremely high flow state thus resulting in more erosion [30] .