Medium-Term Effects of Dune Erosion and Longshore Sediment Transport on Beach–Dune Systems Evolution

Abstract

Beach–dune systems are highly dynamic features of the coastal system, the evolution of which is influenced by several processes that occur at very different spatial and temporal scales. To mitigate shoreline retreat that threatens extensive coastal areas worldwide, coastal erosion mitigation measures are implemented, aiming to make coastal areas resilient to the effects induced by coastal erosion and the anticipated climate change related to storms, flood events and sea level rise. Numerical modelling can support planned and sustainable coastal management from a medium-to-long-term perspective (decades). This research focuses on presenting contributions regarding the numerical modelling of subaerial beach dynamics (berm width and dune systems interactions) from a medium-term perspective. The method applied is based on a combination of the results of two simplified numerical models (the LTC and CS-Model). The results demonstrate the potential of the proposed combined model for medium-term projections, allowing for the interpretation of beach–dune dynamics and the evaluation of the importance of longshore and cross-shore sediment transport processes.

1. Introduction

Beach and dune systems represent a natural interface between the land and the sea. These coastal features play a crucial role in providing essential functions, working as a natural barrier against coastal hazards like floods and erosion, habitat and biodiversity preservation and tourism [1,2]. Projects of dune reinforcement and/or construction have been implemented, frequently combining artificial nourishments with other soft measures of dune preservation, such as planting vegetation, and implementing sand fences and walkways [3,4,5,6]. However, dune erosion caused by wave impact is a natural process and, as a result, dune reinforcement projects have a limited lifespan and typically require maintenance interventions. Understanding shoreline, beach and dune evolution is important for effectively managing coastal areas, supporting coastal management efforts, assisting in defining coastal erosion risk areas, identifying flood-prone areas, determining long-term interventions to preserve coastal systems and increasing coastal resilience [7,8,9].

The natural beach–dune system interactions involve various processes and time scales. These systems primarily respond to storm events that occur over hours to days, while their recovery can span years or decades, depending on factors such as sea level rise, the frequency and magnitude of storms, beach width, wind-blown sand transport, vegetation, wetting and drying cycles, and sediment supply [10,11,12]. Houser and Ellis [12] synthetize and link different time periods related to the main processes of dune evolution. According to the authors, in the mesoscale dimension, dune evolution is mainly related to aeolian transport potential and the intensity of storms. Over long periods, dune morphology is highly conditioned by the sediment supply that is dependent on alongshore and offshore sources. Based on the analysis of a dataset spanning a 400 km region, Beuzen et al. [13] attempted to identify the key parameters that control the spatial variability in storm-induced coastal erosion at a regional scale. They reported that erosion was primarily influenced by the exposure of the shoreline, runup height related to dune toe, pre-storm volume of sand on the beach, and pre-storm width of the beach. Sallenger [14] introduced a scale to classify the impact of storms in the beach–dune system and its response. It considers four regimes: swash, collision, overwash and breaching. In the swash regime, the wave runup is confined to the foreshore face of the beach. In the collision regime there is sediment transport from the dune to offshore, when the wave run-up erodes the base of the dune. In the overwash regime, wave overtops the dune and there is sediment deposition on the landward side of the dune. In the breaching regime, the dune crest is generally overtopped with important changes in the beach–dune topography.

In light of this, numerical modelling may serve as a valuable tool to support coastal management by facilitating discussions on intervention scenarios and related design parameters. In recent decades, the scientific community has devoted significant effort to studying and developing models describing the morphological evolution of coastal systems, including data-driven models, process-based models and physics-driven models [15,16,17]. Physics-driven models apply equations to describe hydrodynamics, waves, sediment transport and morphology [15,17,18]. The disadvantage of this class of models is that they require high computational effort and thus become inappropriate for long-term analysis (years to decades) on regional scales (kilometres) [18]. Data-driven models describe shoreline evolution based on observed behavior, but their applicability is limited by the availability of historical data [15].

Process-based models describe the dominant physical processes based on simplified equations. However, due to the complexity of describing all the processes and timescales, each model focuses on specific physical processes [17,19]. Generally, they are divided based on the temporal and spatial scales of the processes they cover, such as shoreline evolution due to longshore sediment transport, wave-driven, cross-shore change, beach profile evolution related to water levels and waves, and profile adjustment due to sea level rise [15,19]. In terms of spatial and temporal scales, longshore sediment gradients are associated with long-term changes in shoreline position. Thus, numerical models of shoreline evolution allow for long-term analysis (years to decades) in coastal domains spanning kilometres, calculating shoreline evolution based on longshore-sediment-transport gradients that occur alongshore [19]. Cross-shore processes are mainly related to the response to short-to-medium-term events (days to months), and related to storm events (days) and the seasonality of the wave climate (months) [19]. Therefore, cross-shore numerical models simulate beach profile evolution by computing cross-shore sediment transport within the beach profile.

Different types of numerical model for coastal zone evolution are presented in Hoagland et al. [16] and Antolínez et al. [15]. For instance, Hoagland et al. [16] identified 33 models and formulations focused on simulating event-scale processes of key phenomena or processes, ranging from dune erosion and breaching to overwash; they also identified 38 long-term models, among which 24 simulate some combination of the processes related to shoreline evolution, shoreline change, dune growth/erosion and overwash. Among these 24, only 2 combine longshore shoreline change with dune growth or erosion (COCOONED [15] and PAL20 [20]). In spite of that, coastal numerical modellers have devoted considerable effort to the development of beach-profile-evolution numerical models, including dune evolution and the effects of overtopping and overwash events. For instance, Figlus et al. [21] presented a study focused on wave overtopping and overwash of dunes, where results of laboratory tests were compared with numerical simulations carried out using the cross-shore process-based numerical model CSHORE [22,23]. Zhang and Larson [24] integrated the dune wave impact model used by Larson et al. [25,26] into a medium-term, physical-based model developed to reproduce the entire beach profile evolution due to waves and water levels. Berard et al. [27] applied the cross-shore process-based model XBeach to numerically simulate dune erosion. However, despite the good agreement highlighted in this research between field and laboratory tests and numerical results, the studies focused on short-to-medium-term periods (storms), and coastal engineering requires medium-to-long-term projections (years to decades).

In fact, the most recent advances in numerical modelling focus on the development of medium-to-long-term models, integrating several physical processes without compromising computational efficiency. CoSMoS-Coast [28], LX-Shore [29] and GenCade [30] are shoreline evolution numerical models that attempt to integrate longshore and cross-shore processes in shoreline evolution. In these models, CERC [31] or Kamphuius [32] formulas are used to compute the longshore sediment transport. To obtain the sediment transport induced by wave action in the cross-shore direction, CoSMoS-Coast [28] and LX-Shore [29] apply the equilibrium shoreline model of Yates et al. [33], which simulates episodic beach erosion and recovery during periods of high and low energy, based on the difference between the instantaneous wave energy and the wave energy associated with the equilibrium shoreline position [28]. In GenCade [30], the cross-shore sediment transport is calculated based on semi-empirical algorithms that account for the effects of nonlinear waves (skewness and asymmetry), current and gravity in the nearshore zone. Larson et al. [25] introduced a medium-term, cross-shore numerical model (CS-Model) that integrates processes related to sandbar dynamics, dune erosion due to wave impact and dune recovery by windblown sand transport. In Larson et al.’s [25] model, the dune erosion process is simulated through the wave impact model presented in [26]. More recently, Palalane [20] combined Larson et al.’s [25] model with longshore-sediment-transport gradients, using an external loop that employs a shoreline evolution numerical model to compute longshore-sediment-transport gradients. Antolínez et al. [15] introduced COCOONED, a model that combines longshore processes related to longshore gradients, cross-shore processes due to waves and varying water levels, a dune erosion model based on equilibrium profile theory and profile adjustment by sediment supply.

The objective of this paper is to present contributions regarding the numerical modelling capacity to simulate dune evolution and beach width over a medium-to-long-term perspective, considering a combined effect of longshore-sediment-transport gradients and cross-shore sediment exchange between the beach berm and the dune. This involves the development of a method integrating the effects of both cross-shore and longshore-sediment-transport processes into the system evolution. Following the recommendation presented by Hanson et al. [17], the method integrates both components of sediment transport by combining the results provided by two existing simplified numerical models: LTC [34] that simulates longshore-sediment-transport processes, and CS-Model [25], to obtain the cross-shore sediment transport within the beach profiles. To achieve the main objective defined for the study, the research comprised two primary phases: (1) the development of a method to integrate cross-shore and longshore processes of sediment transport; (2) the application of the method to a predefined set of scenarios to discuss the beach–dune system’s evolution under the combined influence of longshore- and cross-shore-sediment-transport processes related to dune erosion due to wave impact. The study considers a regular and simple calculation domain, supporting results interpretation and assessment of the proposed methodology’s adequacy in describing the beach berm–dune interaction, considering both cross-shore and longshore-sediment-transport processes, without the influence of additional variables. Therefore, it was also considered a constant wave climate over a one-year simulation, while cross-shore effects showed their relevance in the overall morphological behaviours.

2. Methodology

In this study, the capabilities of numerical modelling were applied to evaluate the evolution of the dune system and beach berm position under the combined effects of longshore and cross-shore-sediment-transport processes associated with dune erosion caused by wave impact. To achieve the proposed goal, a model was developed that integrates the results of two existing numerical models, LTC [34] and CS-Model [25], which simulate medium-to-long-term longshore and cross-shore-sediment-transport processes, respectively. This section begins with the presentation of the main assumptions of the two models that are the basis for the study. Subsequently, the method to integrate the results of the two models is presented.

2.1. LTC and CS-Model

LTC (Long-Term Configuration) is a shoreline evolution numerical model designed for sandy beaches [34]. Figure 1 presents the main features of the model, which considers a numerical domain defining the bathymetry and topography of the coastal system through a grid of points (NX—number of points alongshore and NY—number of points across-shore). The points are spaced Δx in the longshore direction, and Δy in the cross-shore direction.

LTC assumes that each wave acts individually during a certain period of time (time step), and for each time step during the calculations it estimates the longshore-sediment-transport gradients along the coastal segments Δx, which are responsible for shoreline evolution [34,36]. For each segment, the wave parameters at breaking are estimated according to the linear wave theory of wave propagation, and the potential longshore sediment transport is computed based on CERC [31] or Kamphuis [32] formulas. Subsequently, the longshore-sediment-transport gradients are used to determine the shoreline position in each time step. The longshore-sediment-transport gradients are derived from the continuity equation (Equation (1) and Figure 1b), where V represents the volume of sediments in an infinitesimal section (∂x), Q is the longitudinal-sediment-transport rate, q_x represents any external supplies of sediments per unit of the cross-shore width and t represents the time. Finally, the model relates the volume variation over time intervals (∆t), with the variation of the longshore sediment transport over time (∆Q).

$$\frac{\partial V}{\partial t}=\left(\frac{\partial Q}{\partial x}+q_x\right)$$

(1)

The longshore-sediment-transport volume gradient (m³) in Equation (2) is the difference between the sediment volumes Q_i and Q_i−1 in the Δx (m) segment, adding to eventual external supplies (q_ext). Based on the longshore-sediment-transport gradients, the model updates the bathymetry and topography of the domain at each time step (Δz), consequently adjusting the shoreline position through the distribution of the longshore-sediment-transport gradient (ΔV) between consecutive grid points in the modelled area (Δx), within the active width of the beach profile, W, extending from the closure depth to the wave runup limit (Equation (3)).

$$∆V=\left(Q_i-Q_{i-1}+q_{ext}\right)∆t=\left(∆Q+q_{ext}\right)∆t$$

(2)

$$∆z=\frac{∆V}{W\times ∆x}$$

(3)

For the boundary conditions definition, representing the exchange of sediments with the exterior of the domain, and consequently the balance between sediments going in and out the modelled area (QLN in the Northern border and QLS in the Southern border, according Figure 1a), the model allows for three options, namely [34,36]: no sedimentary exchanges with the exterior; extrapolation of the similar longshore sediments transport conditions in the neighbouring areas of the boundaries, giving continuity to the longshore-sediment-transport rates along the coast (typical of open coasts); and fixed volume entering or going out of the domain defined by the user.

The CS-Model [25] describes the cross-shore-sediment-transport processes and, consequently, the morphological evolution of cross-shore profiles, from a medium-term perspective (months to years), relying on a set of morphological parameters and equations (Figure 2). The cross-shore processes simulated by the model include the exchange of sediment material between the sandbar and the beach berm (q_B), the effect of wind-driven transport of sediments in the dune evolution (q_WS in the seaward face of the dune and q_WL in the landward face) and the sediments eroded from the dune due to wave impact (q_D), divided in the components q_S (from the dune to the berm, seaward) and q_L, if dune overwash occurs (landward). Overwash events are registered when the runup height exceeds the dune height [17].

The morphological parameters used by the CS-Model to describe the profile (Figure 2) are the dune height (S), the positions of the landward and seaward dune toe (Y_L and Y_S, respectively), the location of the beach berm crest (Y_B), the volume of the longshore sandbar (V_B), the height of the berm (D_B), the depth of closure (D_C), the landward and seaward dune face slopes (respectively β_L and β_S) and the foreshore slope (β_F). The beach profile in the CS-Model has a fixed shape from the berm crest to the depth of closure, resulting in the shoreline position (Y_G) of the relationship between the berm crest position, berm height and slope.

For each time step, the model computes the different components of cross-shore sediment transport, and these volumes of sediment are geometrically prescribed, updating the scheme of the cross-shore profile. In the CS-Model, the evolution of the berm position is given by Equation 4 and depends on the sediment exchanges between the submerged sandbar and the berm of the beach (q_B), the effect of wind-driven transport (q_WS), and the component of seaward dune erosion due to wave impact (q_S). According to Larson et al. [25], if overwash does not occur, the total volume eroded from the dune face (q_S) is totally transferred to the beach berm (q_S = q_D).

$$\frac{d{y}_B}{dt}=\frac{1}{D_B+D_C}\left({-q}_{WS}-q_B+q_S\right)$$

(4)

2.2. Wave Impact and Dune Erosion

The CS-Model applies the wave impact model developed by Larson et al. [26] to predict the erosion and retreat of coastal dunes, which is based on the wave impact theory presented by Fisher and Overton [25,37,38]. According to this model, if overwash does not occur, the cross-shore transport rate from the dune face to the berm due to wave impact is as given by Equation (5). This equation relates the transport from the dune with the wave runup height (R), the surge and tide level (Δh), the dune toe elevation ($z_D$), the swash period (T) and an empirical transport coefficient (C_S).

$$q_D=4C_S\frac{{(R+\Delta h-z_D)}^2}{T}, R>z_D-\Delta h$$

(5)

The model considers that dune erosion due to wave impact occurs when the runup height exceeds the dune foot elevation (R >${ z}_D-\Delta h$) [25]. In the model, runup is estimated according to Equation (6) and can be adjusted (R′) to include the velocity reduction due to friction resulting from the travel distance of the wave along the beach width (Equation (7)) [37].

$$R=a\sqrt{H_0{L}_0}$$

(6)

$$R^{\prime }=R \exp \left(-2k_f{Y}_{BD}\right)+z_D\left[1-\exp (-2k_f{Y}_{BD})\right]$$

(7)

In these equations:

• H₀ is the deepwater root-mean-square wave height [25] (m);
• L₀ is the deepwater wavelength (m);
• a is a coefficient representative of foreshore slope—about 0.15, according to Hanson et al. [37];
• R is the runup height (m);
• $k_{f }$ is a friction coefficient for waves travelling over the berm;
• Y_BD is the width of the beach, determined by the difference between the berm (Y_B) and the seaward foot dune (Y_S) positions (m).

2.3. Method to Combine Longshore and Cross-Shore Processes of Sediment Transport

The developed method for integrating longshore and cross-shore sediment transport processes due to dune erosion employs a cyclical structure which repeats a number of time steps, according to the user-defined number of waves to be simulated (NCAL). The sequence of this structure is presented in Figure 3. Initially, the user defines the numerical setup which includes the LTC mesh of the modelled domain, the morphological parameters that define each cross-shore profile, boundary conditions of the numerical domain, offshore wave parameters, wind action, water levels, runup effects and sediment characteristics. Regarding the mesh dimension, the user defines the number of alongshore and cross-shore grid points (NX and NY, respectively) and the distance between those points (Δx and Δy) (see Figure 1). Thus, NX corresponds to the number of cross-shore profiles to be modelled in the CS-Model, and for each one the user indicates morphological parameters that characterize its cross-shore configuration (parameters indicated in Figure 2).

Sediment transport at a point in the nearshore zone is a vector with both longshore and cross-shore components [39]. In accordance with USACE—United States Army Corps of Engineers [39], at each time step, both LTC and the CS-Model (for each of the NX profiles) are run, and the sediment transport volumes computed by each model and for each cross-shore profile are integrated according to the following sequence (Figure 3):

(1)

LTC is run to obtain the longshore sediment transport gradients at each cross-shore profile (ΔQ_longshore);

(2)

The bathymetry and topography of the LTC numerical domain are updated through Δz, resulting from the distribution of the volume of the longshore-sediment-transport gradient computed for each beach profile across its active width. The active width of the profiles is defined by the distance between the depth of closure and the runup limit;

(3)

The new shoreline position in LTC is determined and compared with the previous position, allowing for the estimation of shoreline position variation due to the effects of longitudinal-sediment-transport gradients at each coastal segment, directly related to each cross-shore profile (ΔY_B);

(4)

The morphological parameters used by the CS-Model to define each cross-shore profile are updated to incorporate the effects of the longshore-sediment-transport gradients, defining a new berm position by adding ΔY_B;

(5)

The CS-Model is run to obtain the effects of cross-shore sediment transport processes (eventual wave attack to the dune and consequent dune erosion) at each cross-shore profile (ΔQ_cross-shore);

(6)

The volume of sediments resulting from dune erosion is used to update the bathymetry and topography of the LTC through the distribution of the cross-shore volumes in the active profile width (Δz);

(7)

The parameters that define the cross-shore morphology of the profiles are updated based on the impacts induced by the cross-shore effects (new berm position, new dune toe positions and new sandbar volume).

3. Numerical Setup and Assessed Scenarios

The developed model was applied to a set of conceptual scenarios, enabling the evaluation of beach morphology evolution induced by dune erosion due to cross-shore processes of sediment transport and the resulting longshore-sediment-transport gradients. This section begins with the presentation of a sensitivity analysis developed for the parameters involved in simulating dune erosion, which served as the basis for defining the variables of the numerical model. Subsequently, supported by this analysis, the modelling setup and the evaluated scenarios are presented, being defined in accordance with the goals of the study (giving relevance to dune erosion processes).

3.1. Sensibility of the CS-Model Results to Dune Erosion

According to Equations (5)–(7), the CS-Model estimates dune erosion in relation to the geometric characteristics of the profile, namely the berm height and the beach width. It is observed that the adjusted runup height (R′) diminishes as the beach width increases, converging towards the berm height of the beach profile. Under the assumption of a constant initial berm height, this convergence is more rapid for narrower beaches when the friction coefficient ($k_f$) is higher. As an illustrative example, considering a constant wave climate characterized by H₀ = 3 m and T = 10.55 s, and a beach profile with the berm height and the dune toe at 2.8 m, Figure 4 presents the evolution of the adjusted runup height as a function of the beach width, for different values of friction coefficient ($k_f$ ranging from 0.01 to 0.1).

Based on the analysis of Figure 4, when $k_f$ is 0.05 and the beach width is equal to 100 m, the runup is equal to the berm height, but for $k_f$ = 0.01 the runup is higher than the berm height. This means that, for a specific beach width, considering the condition that wave impact on dune face results from the relationship between the runup and berm height, dune erosion due to wave impact is strongly dependent on the value adopted by $k_f$. For instance, considering a beach width of 100 m, when $k_f$ = 0.05 there is no sediment transport from the dune face, but for $k_f$ = 0.01, dune erosion occurs due to wave impact.

Figure 5 focuses on analysing the importance of the empirical transport coefficient (C_S) in the model to determine the erosion volume of the dune. Considering the adjusted runup height obtained for $k_f$ = 0.05, the figure compares the volume eroded from the dune for different values of transport coefficient. The influence of the empirical coefficient is relevant for lower beach widths and tends to zero for beach widths of 20 m or 30 m.

Assuming a beach profile with the initial shape given by h = 0.127x^2/3 [40] in the submerged zone, and an inclination of 1.5% in the emerged zone, with an active beach width of 1500 m (see Section 3.2), Figure 6 compares the bathymetric and topography variation (Δz) of the beach profiles and the consequent impact on the shoreline position (ΔY_B) at the end of a one-year period, considering the constant wave climate previously referred to. The effect is presented and compared for two situations of beach width (Y_BD), combined with two different values of the friction coefficient ($k_f$ = 0.01 and $k_f$ = 0.05) and Cs = 0.1.

Additionally, Larson et al. [26] proposed C_S values based on the analysis of various data sets (

Table 1. Empirical transport coefficients obtained by Larson et al. [26], based on the analysis of data sets.

Data Sets	Cs
Large wave tank data [41,42,43]	1.80 × 10−3
Hughes and Chiu small-scale laboratory data [44]	0.82 × 10−3
Kubota et al. field data HA94 [45]	1.60 × 10−3
Kubota et al. field data HA97 [45]	0.92 × 10−3
Birkemeier et al. field data [46]	0.13 × 10−3

), and developed an empirical relationship to obtain C_S (Equation (8)).

$$C_S=A{e}^{-b\frac{H_0}{D_{50}}}$$

(8)

where:

• A = 1.34 × 10⁻³ and b = 3.19 × 10⁻⁴;
• H₀ is the deepwater root-mean-square wave height (m);
• D₅₀ is the medium grain size (m).

When applying the CS-Model to conduct numerical studies in three real study cases, Palalane et al. [47] utilized C_S values of 1 × 10⁻³ for Barra-Vagueira beaches (Portugal), 1 × 10⁻⁴ for Macaneta (Mozambique) and 8 × 10⁻⁴ for Angelholm (Sweden). Kato and Udo [48] considered 1.7 × 10⁻⁴ for Hasaki (Japan) and explained that the lower value, compared to those employed by Palalane et al. [47], is due to the high level at which the dune toe is situated in Hasaki beach, preventing dune erosion. Regarding the friction coefficient, Hanson et al. [37] mentioned that they employed a value of $k_{f }$ = 0.02 for developing numerical simulations using the CS-Model. Palalane et al. [47] considered values between 0.1 and 0.2.

The presented analysis helps with the interpretation of the adequate values to represent wave attacks on dunes and their impact on transferring sediment from the dune to the beach berm. Based on the developed analysis, the values of $k_{f }$ = 0.05 and C_S = 0.1 were adopted in this study. These values were chosen to ensure that various wave dune-attack behaviours, including scenarios where no dune attack occurs, were attained across a range of beach widths from 15 to 85 m. Therefore, this study aims to show a generic evaluation and, thus, the adopted values intend to represent illustrative scenarios of beach–dune interaction. For specific real case studies, these coefficients should be calibrated for each specific site.

3.2. Numerical Setup

The numerical simulations spanned a one-year period (2920 time steps, with each time step representing a 3-h interval). To address the most negative impacts of dune erosion related to wave impact, a high energetic wave climate was adopted in the numerical simulations. The wave climate was defined to be regular throughout the numerical simulations, with an offshore wave height of 3 m, a wave direction of 80° (anticlockwise angle with the initial shoreline) and a wave period of 10.55 s. The values chosen to define the wave climate represent an energetic environment which characterizes the wave climate of the Portuguese West coast, exposed to the energetic wave climate of the Atlantic Ocean [34,49,50].

The numerical study was conducted using a mesh composed of 15 profiles spaced 100 m in the longshore direction and 251 points in the cross-shore direction, spaced 20 m (5000 × 1400 m²) apart. The initial shoreline position was defined with a rectilinear configuration (Figure 7).

Regarding the parameters required by the LTC numerical model, the topography was assumed to have a constant slope equal to 1.5%, and the bathymetry was defined following the Dean [40] profile (h = Ax^m, where the sediment-dependent scale parameter (A) was set to 0.127, and the m parameter, related to beach exposure to wave energy, was defined as 2/3). The active beach profile was established with a closure depth of 15 m and a maximum runup limit of 5 m.

The CERC [31] formula was used to calculate the potential longshore sediment transport, with the calibration coefficient set to 0.1. The resulting potential longshore sediment transport with the prescribed wave conditions and model parameters was close to 2.5 × 10⁶ m³/year. Longshore sediment transport at the boundaries of the domain were defined as extrapolation of the neighbouring conditions, which means that if no perturbations are imposed on longshore sediment transport in the domain (e.g., sediment input from the dune), the shoreline position will be constant across time.

Table 2. Cross-shore morphology of the CS-Model beach profiles (see also Figure 2).

βL	βS	YB	YG	S	DB	Dc	Vb
(rad)	(rad)	(m)	(m)	(m)	(m)	(m)	(m3/m)
0.32	0.24	3000	2957.21	6	2.8	15	129.46

presents the values adopted for the main variables involved in the CS-Model simulation of the cross-shore processes within the beach profiles that are common to all profiles of the study area. In the context of the problem addressed in this article, and as mentioned previously, the parameter related to wind-blown sand transport was set to zero, and the initial sandbar volume of the beach profiles was calculated and defined to be in equilibrium with the wave climate. Thus, there were no sediment exchanges within the beach profiles related to wind transport, or between the berm and the submerged sandbar. The configuration of beach profiles was defined to ensure that the dune maintained its trapezoidal shape during the numerical simulation. The dune height (S) was defined as sufficiently high to prevent overwash events, and the profile’s berm height (D_B) was set at 2.8 m. Based on the previous analysis, and as already mentioned, the ${\mathrm{k}}_{\mathrm{f}}$parameter was defined as 0.05, and C_S was set to 0.1. The water level was kept constant throughout the numerical simulations (Δh = 0 m).

3.3. Assessed Scenarios

The exposed method of integrating LTC and CS-Model results was applied to a set of scenarios defined to study and discuss the combined effects of longshore and cross-shore sediment transport processes in the shoreline evolution resulting from the wave impact on dunes. The assessed scenarios differ from each one in the initial beach width alongshore, which results from the different seaward-dune-foot-position value considered for each cross-shore profile alongshore (Y_S). Figure 8 schematises the scenarios, illustrating for each one the initial positions of the berm and the seaward dune foot.

In scenarios S1, S2 and S3, the initial berm is consistent alongshore, and the beach width is 15 m in width in S1, 85 m in S2 and 50 m in S3. For scenarios S4 to S6, the initial beach width varies alongshore. In scenario S4, the northern border of the numerical domain was defined with a 15 m beach width, increasing alongshore and reaching an 85 m beach width at the southern border. Scenario S5 is the reverse of scenario S4, with higher beach width at North and smaller at South. In scenario S6, an initial beach width of 15 m was considered in the central area of the numerical domain (profiles P6 to P10), with a width of 50 m in the remaining area of the domain.

4. Results

Given the potential insights provided by the model developed to integrate the effects induced by cross-shore- and longshore-sediment-transport processes on shoreline evolution, this section begins by presenting the results of the sediment balance for the different modelled scenarios in the numerical domain. This involves discussing and analysing the volumes of sediments transported alongshore and cross-shore in the beach profiles. Following this, the evolution of the morphological parameters that define the beach profiles is presented. This allows for an examination of the influence of both longshore and cross-shore sediment transport processes on shoreline position, seaward dune foot position, and berm width.

4.1. Sediment Transport Volumes

In scenarios S1, S2 and S3, there are no sediment transport gradients alongshore as all the profiles along the numerical domain have the same initial morphology. Regarding the cross-shore processes, in scenario S1 sediment transport occurs from the dune face to the berm as the beach width is just 15 m and there is wave collision with the dune (Figure 9). In scenarios S2 and S3, the total cross-shore sediment transport is almost nil (in scenario S3, the 50 m back width allows some residual volumes to be transferred from the dune to the berm). Therefore, the differences in the effects induced by the cross-shore processes in scenarios S1, S2 and S3 result from the initial beach width. The larger initial beach width prevents wave impact on the dune and consequent dune erosion (scenarios S2 and S3), while in scenario S1 dune erosion occurs, and the sediments from the dune feed the numerical domain, with a resulting impact on the shoreline-advance evolution.

In scenarios S4, S5 and S6, it was observed that both longshore and cross-shore sediment-transport-processes occur in the beach profiles. Figure 10 and Figure 11 present the cumulative sediment transport in each beach profile of the numerical domain, due to cross-shore and longshore effects, respectively.

In scenarios S4 and S5, both longshore and cross-shore sediment transport patterns lead to an increase in the sediment volume in the majority of the beach profiles. Regarding the cross-shore effects, a higher sediment transport from the dune face is observed in the profiles with a small initial beach width (Figure 10a,b). This effect decreases with the increase in beach width, and tends to be nil in the profiles with an initial beach width higher than 50 m. Therefore, dune erosion promotes different impacts between adjacent beach profiles, resulting in the emergence of longshore-sediment-transport gradients.

In scenario S6, the effects of cross-shore and longshore processes can be grouped based on the initial beach width. The central profiles of the numerical domain (P6 to P10) have volume gains due to cross-shore impact related to dune erosion (Figure 10c), but the longshore effects lead to volume loss in these beach profiles (Figure 11c). The remaining profiles (P1 to P5 and P11 to P15) show the irrelevance of cross-shore processes, as the beach width is large enough to avoid the wave attack of the dune. However, it was observed that these profiles receive sediment volumes from the longshore-sediment-transport gradients that resulted from the dune erosion in the profiles P6 to P10.

The relationship between the effects induced by cross-shore- and longshore-sediment-transport components impacts the sediment balance in the numerical domain.

Table 3. Sediment balances within the numerical domain for every assessed scenario (m³).

	S1	S2	S3	S4	S5	S6
Qin (North)	2,248,400	2,248,400	2,248,400	2,301,389	2,230,283	2,219,080
Qout (South)	2,248,400	2,248,400	2,248,400	2,264,110	2,192,673	2,275,124
ΔVlong = Qin − Qout	0	0	0	37,279	37,610	−56,044
ΔVcross = ΔVdune	181,073	3	3669	34,111	34,110	69,838
ΔVtotwl	181,073	3	3669	72,530	72,576	14,036

and Figure 12 present a summary of the sediment balance in the numerical domain (ΔV_total). For each scenario, the volumes of sediments going into the domain by the northern boundary (Q_in) and going out at the southern boundary (Q_out) are assessed, allowing us to quantify the longshore impacts (ΔV_long) in each scenario. Additionally, variations in the cross-shore volumes (ΔV_cross) due to sediment transport processes, resulting from the dune erosion as result of wave impact on the dune system (ΔV_dune), are also quantified. Finally, the total variation in volumes (ΔV_total) in the modelled domain are estimated, considering both longshore and cross-shore processes.

In scenario S1, a gain of sediment volume in the numerical domain arising from the cross-shore sediment transport processes (ΔV_long = 0) is observed. In scenario S4, the dune erosion at North increases the southward-longshore-sediment transport (Q_in), with less intensity at South (Q_out), representing a total gain in the domain. In scenario S5, the dune erosion at South promotes shoreline advance, decreasing the longshore sediment transport going out at the southern boundary, also decreasing the volumes going into the domain at North, but with a lower expression, representing again a total gain in the domain. In scenario S6, it is observed that the dune erosion in the central area of the domain promotes a shoreline orientation at the northern border that leads to lower sediment volumes going in, and the shoreline orientation at the southern border leads to a higher sediment volume going out. This results in a loss of sand volume in the study area due to the longshore effects, and that is compensated by the dune erosion in the cross-shore processes (Table 3).

4.2. Evolution of Berm and Dune Foot Positions

The relationship between the effects of cross-shore and longshore sediment transport has impacts on the morphological evolution of the beach, including shoreline configuration, berm position (Y_B), seaward dune foot position (Y_S) and consequent beach width (Y_BD, resulting from the distance from the berm to the dune foot). The numerical results indicate that, in scenarios S2 and S3, the beach parameters remain in their initial positions throughout the numerical simulation because there are no longshore-sediment-transport gradients, and the beach width prevents dune erosion. However, in the other scenarios, impacts on beach morphology evolution are observed. Figure 13 illustrates the evolution of the berm and seaward dune foot positions for scenarios S1, S4, S5 and S6. The upper figures display the coastal domain area, including the berm and dune foot positions. The lower figures provide a zoom-in on the numerical domain area where the berm position is located.

The evolution of Y_S and Y_B in scenario S1, as shown in Figure 13a, is solely influenced by cross-shore effects. The dune erosion due to the wave impact leads to landward dune foot retreat, and the berm position moves seaward, benefiting from the sediments coming from the dune face. As all profiles alongshore experience the same impact, the shoreline position maintains its linear configuration throughout the simulation.

Concerning the scenarios with an initially variable beach width alongshore (S4, S5 and S6), it is observed that the sand transferred from the dune face to the berm at the profiles where the initial beach widths were lower than 50 m tends to move the berm position seaward, and the dune foot position retreats. However, in these scenarios, a seaward advance of the berm position is also observed in the profiles where cross-shore processes do not occur. This reveals that the longshore sediment transport contributes to the seaward movement of the berm position.

Generally, it is also observed that the transfer of sediment from the dune to the beach berm results in gains in the beach width. Figure 14 illustrates the evolution of the beach width in seven profiles within the calculation domain for the six scenarios evaluated. All profiles in scenario S1 exhibit gains in beach width, increasing the width by approximately 23.26 m after 1 year of simulation. Scenarios S2 and S3 keep the beach width along all the simulation, 85 m and 50 m, respectively.

The results of scenarios S4 and S5 show that the beach width gain is also observed in profiles with an initial berm position higher than 50 m, indicating a slight positive impact of the longshore process because cross-shore sediment transport does not occur in these profiles. The exception is the profile located at the limit of the numerical domains with an initial beach width of 85 m (P1 in scenario S4 and P15 in scenario S5), due to the boundary conditions defined (extrapolation of nearby three cells average longshore sediment transport). In these situations, it is observed that the shoreline orientation results in negative gradients of longshore sediment transport, leading to a shoreline retreat. Consequently, it is observed that the longshore sediment transport has a negative impact in the beach width (decrease in the initial beach width by approximately 0.19 m).

In scenario S6, the longshore sediment transport leads to a beach width gain in the profiles with an initial beach width of 15 m (P6 to P10). In this scenario, these profiles exhibit a beach berm gain of approximately 22.7 m at the end of the simulation (final width of about 37.7 m), falling below the gain observed in scenario S1, where only cross-shore sediment transport processes impacted the shoreline evolution. In fact, when the values of both components of sediment transport are compared, it is observed that longshore processes have a negative impact (loss of volume), in spite of being lower than the gain of sediment volume in the profiles, due to the cross-shore component of sediment transport (Figure 10c and Figure 11c).

5. Discussion

The evolution of coastal morphology is highly dynamic and complex to describe due to the extensive set of processes influencing its evolution, as well as the different temporal and spatial scales at which these processes occur [1,51]. Globally, it is observed that vast stretches of sandy beaches face coastal erosion issues, reducing the available beach area and exposing the dune system to the energetic actions of the sea [52]. Dune systems represent the last barrier separating the land from the sea, playing a crucial role in several ecosystem services (for instance, functions related to storm and flood protection, tourism and culture) [53,54,55]. Given their importance, restructuring and stabilization are currently a priority for coastal management, aiming to make these systems resilient to the negative effects induced by coastal erosion and the anticipated impacts of climate change, including the projected future sea level rise and the increased frequency and intensity of storm and flood events [8,56,57].

The methodology applied in the present study to evaluate the combined effect of cross-shore and longshore-sediment-transport processes in shoreline and dune evolution from a medium-term perspective is based on integrating the results of simplified numerical models, specifically LTC to simulate longshore effects and shoreline evolution, and the CS-Model to simulate cross-shore sediment transport processes and their consequent effects. This approach was adopted based on the assumption that this numerical framework can be valuable for supporting coastal management. As highlighted by Larson et al. [11], although storms are short-duration events, the sediment transport occurring during annual storms can be similar to the total transport that occurs during the rest of the year under no-storm conditions, and this can impact regional coastal processes (large scale). In Larson et al. [25], the authors state that some coastal interventions, such as artificial nourishment, cause short- and long-term changes to beach morphology. Additionally, French and Burningham [58] emphasized the importance of improving the capacity of medium-term projections for strategic coastal management.

The method of combining two numerical models to develop the study was based on the proposal outlined by Hanson et al. [17], which suggested that, given the high number of existing numerical models, future efforts in numerical modelling of coastal zones evolution should focus on integrating existing models and incorporating processes that occur at different temporal scales. Palalane et al. [20] followed a similar approach to integrate cross-shore and longshore processes. According to those authors, longshore processes are obtained based on an external loop, while in the presented approach, both cross-shore and longshore processes are computed for each time step, allowing a total integration between processes. Furthermore, in Palalane et al. [20] the shoreline position results from the sand distribution in the active profile and is not dependent on bathymetry or topography. In contrast, in the presented model, based on the numerical assumptions of the shoreline evolution model used to compute the longshore effects (LTC), at each time step, after the distribution of the longshore sediment transport in the profile, the limits of the active beach profile are adjusted in the adjacent areas. Therefore, the shoreline position also depends on the bathymetry and topography, which are updated at each time step [34,36]. Although not shown in this work, the developed model allows us to consider the variation in sandbar volume associated with seasonal wave changes, and also accounts for dune recovery induced by aeolian processes, as both of these processes are simulated by the CS-Model. These two aspects are neglect in the COCOONED numerical model [15]. For dune erosion due to wave impact, the numerical CS-Model approach not only considers the runup level but also accounts for wave energy dissipation along the berm width. In spite of the outlined potentialities of the integrated model, it has not yet developed a complete integration of longshore and cross-shore processes in the presence of coastal structures, such as groins, breakwaters or longitudinal revetments, which are common in coastal environments. Future developments must investigate this integration, as the LTC numerical model has the ability to simulate the effects of coastal structures on longshore sediment transport and shoreline evolution [34,36,59].

Generally, the results suggest that the cross-shore processes result in sediment transport from the dune face to the berm (this is also reported by D’Alessandro and Tomasicchio [60], based on physical model tests), but the effects on shoreline evolution are also dependent on the longshore-sediment-transport gradients. Therefore, the coupling of the two models yields promising results and is of interest to coastal management.

The updating of beach morphology at each time step based on longshore- and cross-shore-sediment-transport processes provides the model with the capability to reproduce the evolution of interactions between processes acting at different temporal scales. According to Larson et al. [25], the potential for integration is crucial to analysing the performance of coastal-erosion-mitigation interventions, particularly in the case of artificial sediment-nourishment, the effectiveness of which is influenced by sediment transport in both cross-shore and alongshore directions, occurring at various temporal scales. Based on monitoring work performed in Portugal (Mira beach), Fontán-Bouzas et al. [61] indicated that wave impacts on the foot of the dune occur not only during storm events but also during spring tides and mild wave conditions. Furthermore, the authors indicated that more stable dune systems with higher dunes are less impacted by waves. Also, Sabatier et al. [62], based on the analysis of monitoring works performed in France, concluded that at medium-to-long-term timescales, shoreline and coastal dunes are controlled by both cross-shore and longshore processes.

The assessed scenarios considered dune erosion under specific values of friction coefficient and empirical coefficient of sand transport. However, the discussion of runup height and dune erosion allows for variable magnitude and trends of sand erosion based on the parameters $k_f$, Cs and beach width (Section 3.1). Thus, under the assumption of developing the study, the numerical results enabled the interpretation of dune erosion by considering different initial beach widths, as well as their importance to shoreline evolution. Considering the adopted wave climate and sediment grain size (H₀ = 3 m and D₅₀ = 0.5 mm), the application of the empirical equation to obtain C_S proposed by Larson et al. [26] results in a Cs value of 0.035 × 10⁻³. Considering this value and the scenario of alongshore beach width of 15 m, the cross-shore effect after one year of simulation represents a shoreline seaward advance of approximately 0.11 m, a final beach width of 16.55 m and a total sediment transferred from the dune system to the berm of close to 12 × 10³ m³ (8.63 m³/m). The chosen conditions for the developed study represented much higher impacts, to better demonstrate the sediment transport mechanisms. As recommended by Larson et al. [26], practical applications of the dune erosion model require the analysis of the variability of dune evolution for a range of values of Cs.

The results of the proposed model also have the potential to assess the evolution of various morphological indicators, such as dune position and its volume robustness, shoreline position and effective beach width. The control of these indicators is important from a coastal management perspective, helping to determine the need for and nature of intervention. According to Coelho et al. [63] and País-Barbosa et al. [59], the physical performance of coastal-erosion-mitigation interventions depends on the objectives defined for the project. These objectives may include increasing the volume of the dune system to enhance its robustness, reducing the frequency of floods and overtopping, expanding the usable beach area, decreasing the frequency of maintenance work on existing structures, or a combination of these objectives. Whitehead et al. [64] and Parson et al. [65] indicated that the economic value attributed to beaches by their users is strongly influenced by beach width, and higher values are not necessarily linked with larger beach berms. De Paula et al. [66] highlighted that there must exist a minimum width of the emerged beach for recreational functions to be developed.

The numerical simulations presented in this research were developed considering a medium-term perspective (1-year period), but coastal management requires long-term projections. To discuss the importance of beach–dune interactions from a long-term perspective, the scenario of alongshore beach width equal to 15 m was assessed over a 10-year period. The results suggest that the sediment transport from the dune face is more intense in the first year of the numerical simulation (two thirds of the total transported volumes from the dune to the domain), and the beach–dune system tends towards equilibrium, with a decrease in the volume eroded from the dune over time (Figure 15), which justifies the time span adopted in the presented simulations. At the end of the numerical simulation, the beach width of the beach profiles was close to 49.50 m. Thus, increases in beach width due to gains induced by cross-shore transport from the dune lead to a decrease in runup height over time, consequently making the dune less vulnerable to wave impact.

Future numerical modelling research work must investigate other parameters that intervene in the dynamics of sediment transport in the dune, such as wind [67] and vegetation [2,51], both related to aeolian sediment transport, sea water levels related to tides, storm surges and water tables [12]; the work must also study the ability of the model to simulate dune-recovery processes.

6. Conclusions

The objective of this study was to contribute to the improvement of the numerical modelling capacity to represent the evolution of the interactions of beach–dune systems from a medium-term perspective. To achieve the research objective, the first part of the study focused on developing a method that relies on the results of two medium-term numerical models, the LTC and CS-Model, to integrate longshore- and cross-shore-sediment-transport processes. In the second part of the study, the developed method was applied to a set of scenarios defined to investigate the dynamics of the beach–dune system and the influence of longshore- and cross-shore-sediment-transport processes due to dune erosion caused by wave impact on the dune system. It is considered that the proposed approach, which consisted in applying two already validated numerical models, has the ability and is appropriate to represent both sediment transport processes, allowing us to understand and analyse the influence of each component of sediment transport on morphology evolution, as its interaction was demonstrated. The numerical results highlight the importance of integrating both cross-shore- and longshore-sediment-transport processes.

The evolution of shoreline position, as well as the sediment budget within the calculation domain, results from the interaction between the effects induced by both cross-shore- and longshore-sediment-transport processes. Regarding the cross-shore processes, it is observed that wave impact on the dune system leads to sediment exchanges with the beach berm, causing the dune to lose volume and the foot dune position to shift landward. The sediments transported from the dune face cause the shoreline position to move seaward, resulting in gains in terms of beach width. The gradients of longshore sediment transport, which occur due to changes in shoreline orientation alongshore, impact the effects induced by cross-shore processes. Based on the results, situations are observed where this component of sediment transport leads to gains in beach width, but there are also situations where longshore sediment transport diminishes the benefits induced by cross-shore processes in terms of beach width.

The results demonstrate the potential of the proposed method to combine models for medium-term projections, allowing for the interpretation of beach–dune dynamics and the evaluation of the importance of longshore- and cross-shore-sediment-transport processes. Also, the outcomes of the suggested model allow for the evaluation of diverse morphological characteristics that are important for coastal management purposes. The focus of the present study was to assess solely the dune–berm interaction while controlling the various variables involved in the modelling processes. The model, however, has the potential to combine other longshore and cross-shore sediment transport processes involved in coastal evolution, such as dune recovery due to aeolian sediment transport, and berm-sandbar interaction, enabling the development of real-word case studies. It also has the capacity to consider real situations of wave climates and related seasonality.