1 Introduction

Coastal zones represent interface areas between the continents and the oceans, where a significant part of the global population settles given their logistic, recreational and cultural potentials, and the availability of resources. Sandy beaches are coastal environments that vary in time and space accordingly to the depositional morphology and the hydrodynamic behaviour of the region in which they are located [1]. The gradients generated by hydrodynamic processes cause alterations in beach morphology at the same time as the acting hydrodynamic is altered accordingly to the changes induced by morphology. Given the dynamics of coastal processes, sediments are constantly being moved; therefore, they are one of the most important morphodynamic components of these beaches. Hence, the sediment budget of a system modulates the dynamic balance of the coastline in different spatial and temporal scales [2].

Longshore currents (LC) are generated by the oblique incidence of waves on the coast and flow parallel to this interface, besides having potential to transport sediments by suspended load or bedload, constituting, thus, the longshore sediment transport (LST) [3]. LST acts on beach morphodynamics in temporal scales from short to long term [4], playing a fundamental role on sediment budget, mainly in microtidal regions dominated by wave action [5]. The disturbance of this budget can be triggered by natural phenomena or by anthropic interference, and persistent LST gradients, even in low rates, can result in impacts like coastline retreat [3, 4].

The prediction of LST rates and its effects are the aim of many studies developed in oceanography and coastal engineering fields [4, 6,7,8,9,10,11,12,13,14,15]. Yet, there is no consistent agreement of which is the best methodology to make predictions. Numerical modelling (process-based and analytical) is one of the most used methods when it comes to estimate LST rates, considering the limitations imposed by the adversities of measuring it in situ due to the surf zone hydrodynamics. Furthermore, numerical models represent a systematic comprehension of longshore transport once they allow the integration and the insertion of many parameters involved in this mechanism [2, 14].

Mixed beaches are composed of both sand and gravel fractions and are commonly distributed along the shorelines worldwide [16]. Given their complex dynamic associated with sorted grains of multiple sizes, the LST mechanism of mixed beaches is historically less investigated, and so, poorly understood when compared to the progress already made towards sandy beaches [16,17,18,19,20,21]. LST rates of beaches with coarser grain sizes and steeper slopes result from a different balance of morphodynamic processes than environments with fine sand; thus, classic formulae proposed, initially, for sandy beaches may not be suitable for mixed and shingle beaches [20]. In recent years, there is a growing interest concerning the establishment of reliable methods to predict LST rates in these beaches. New formulae have been proposed, and older analytical models have been revalidated and/or calibrated [4, 16, 20,21,22,23].

Hence, this study proposed the investigation and validation of the most applicable methodology to estimate the longshore transport of non-cohesive sediments at the southern shore of Rio Grande do Sul, Brazil, along varying grain size beaches. The potential of classic analytical models was evaluated like CERC [24] and Kamphuis [25], as well as two more recent alternatives developed by Bayram et al. [26] and Van Rijn [23]. All consist of analytical formulae considering process-based models in their creation and calibration processes. In this manner, the best alternative considering the study area was identified.

Mil-Homens et al. [4] and Van Rijn [23] also compared the efficiency of the models investigated herein [23,24,25], finding underestimated rates for energetic meteoceanographic conditions and overestimated values for low energy scenarios, for all the methods evaluated. When the parameters considered in the equation of each of these three models are compared, many similarities occur, specially between [23, 25]. Van Rijn [23] derives from an older formula proposed in Van Rijn [22], in which the new equation was obtained from a trial with CERC [24], Kamphuis [25] and a process-based model. The same parameters considered in [25] were maintained, because these studies found a good response of LST rates to profile shape/beach slope, significant wave height and median grain size. However, weight factors were modified and Van Rijn [23] also added low-period swell, wind and tide effects.

Overall, this research is a cooperation with previous studies, which also examined the LST mechanism in different regions around the world [4, 5, 12,13,14,15, 20, 21, 23], aiming the advancement of more accurate longshore transport assessing tools, thus contributing to the comprehension of short-term coastline evolution, the evaluation of sediment budgets for coastal areas and the long-term stability of beach protection measures [23].

2 Physical settings

The study was conducted at the southern shore of Rio Grande do Sul (RS), Brazil, considering the region located between Mar Grosso Beach, to the north of Patos Lagoon mouth in São José do Norte city, and Concheiros Beach, in Santa Vitória do Palmar city (Fig. 1). Field work made in this research was carried out in Cassino Beach, Rio Grande. The other field datasets in this study were obtained by Perotto [27] and Fontoura et al. [2] in the following places: Cassino beach, the roots of the jetties in Patos Lagoon, Mar Grosso beach, Sarita, Verga, Albardão, Concheiros and Concheiros Sul.

Fig. 1
figure 1

Map of the study area, in which black dots represent the sampling sites of field surveys and the hexagon corresponds to the SiMCosta buoy

Full size image

The southern shore of RS is approximately 220 km long, oriented by a NE–SW axis, with an almost straight coastline with subtle wavy stretches. It is characterized by a wide adjacent continental shelf, with a varying slope (smoother to the north, next to Cassino beach, and steeper to the south, close to Albardão and Concheiros).

Cassino is a dissipative beach, presenting intermediate stages with troughs, alongshore rhythmic bars and rip currents [28,29,30,31]. Concheiros Beach has reflective characteristics; thus, it is more susceptible to wave action and erosion [28]. The remaining beaches are under intermediate morphodynamic stages, displaying variations along the coast. The diversity is, partially, due to beach slope and the varying grain size distribution. The coast is predominantly composed of fine sand from Mar Grosso Beach until the north of Sarita lighthouse, influenced by Patos Lagoon discharge. The occurrence of thicker grain sizes, such as coarse sand and biodetritic gravel, is more common to the south of the study area, from Albardão lighthouse to Concheiros Beach [28, 32]. Beach declivity increases from Mar Grosso (≈ 0.018) towards Concheiros (≈ 0.077). More details regarding grain size and beach slope data of each site under study are given in Table 1 of Sect. 3.3.

Table 1 Values of physical–chemical parameters and beach morphology data used in the analytical models
Full size table

The astronomical tide follows a microtidal semidiurnal regime [36]. However, the meteorological tide can reach levels around 1.90 m during the occurrence of high-energy events [36, 37]. Two high-pressure systems of winds influence the RS coast: the South Atlantic anticyclone and the migratory polar anticyclones. The dominant wind throughout the whole year comes from NE, but the alternation among these two systems makes NE winds more frequent during spring and summer (September to February) and winds coming from the south quadrant more often through fall and winter (April to August), concomitantly with the incidence of frontal systems and the passage of storms [36, 38].

Wave climate is the most important hydrodynamic agent in this area, and it is characterized by three types of waves: swell, sea waves and storm waves [36]. Swell waves are originated mainly from SE by the tempestuous subpolar belt in the South Atlantic, sea waves usually propagate from NE, generated by the local winds, and storm waves are less frequent, but represent the most energetic ones. Littoral drift is bidirectional along the whole state, with a net transport towards NE associated with the intensification of littoral drift and LST during the passage of storms and highly energetic events [36,37,38,39,40,41].

3 Material and methods

3.1 Field surveys

Field surveys were similar for all the author’s mentioned herein, with a few differences regarding wind and wave data collection. The samples of non-cohesive sediments transported by longshore currents were collected with streamer traps developed by Kraus [9], installed in the surf zone. The trap is a metallic structure, 1.80 m high, and it displays a set of 10 capture nets (nylon, 63 µm mesh). The nets are numbered from 1 to 10, bottom to top, and have different lengths: 1 to 6 = 60 cm, 7 to 8 = 70 cm and 9 to 10 = 110 cm. The size variation aims to optimize operational efficiency according to considerations made in [2, 9].

Samples were collected during a 3–5-min interval, since after this period bottom flow is altered due to the presence of the trap, misrepresenting, thus, its natural dynamic. In each field survey, two to four sampling stations were set in the same transverse profile. LST rates, represented by QT, are estimated according to a set of equations proposed by Wang [42] as described by Eqs. 1, 2 and 3, where \(\Delta\)FI = sediment flux between two adjacent nets (kg s−1); FI+1 and FI-1 = fluxes through the upper and lower nets (kg s−1), respectively; ZI+1 and ZI-1 = vertical width of the opening of the upper and lower nets (m), respectively; \(\Delta\)ZI = distance between two adjacent nets (m); I = total flux passing through all capture nets (kg h−1 m−2); n = number of nets mounted on the trap = 10; i = identifier number of each streamer trap on the surf zone; Ii = sediment flux measured at trap array i (kg h−1 m−2); and Ai = surf cross-sectional area between traps i and i + 1 (m2).

$$\Delta F_{I} = \left( {\frac{{\frac{{F_{I + 1} }}{{Z_{I + 1} }} + \frac{{F_{I - 1} }}{{Z_{I - 1} }}}}{2}} \right)\Delta Z_{I}$$
(1)
$$I = \mathop \sum \limits_{i = 1}^{n} F_{I} + \mathop \sum \limits_{1 = 1}^{n - 1} \Delta F_{I}$$
(2)
$$Q_{{\text{T}}} = \sum \frac{{I_{i} + I_{i + 1} }}{2}A_{i}$$
(3)

The field work followed a standard procedure: (1) beach profile—profiles were taken using a Nikon Nivo 2C total station and a surveying reflection prism. Profiles were used to identify the location of sampling stations, bathymetric features, mean waterline position (mean swash zone position), breaker point depth hb (defined as the vertical distance between the seabed and the waterline in the breaking point) and beach slope, represented by the tangent of the beach inclination angle tanβ; (2) wind measurement—wind velocity and direction were measured with an anemometer, a windsock and a compass; (3) longshore current—currents were measured with a drifting buoy, made with PET bottles containing sand, a chronometer and marking stakes (Lagrangian measurement). Velocity is measured as a function of the time that the drifter takes to travel 30 m in the surf zone. This procedure is triplicated, and an average of the three values obtained is made. (4) Last is the longshore sediment transport step, in which streamer traps are used to sample sediments under longshore transport.

Sediment samples were washed with fresh water to remove the salts, dried in a stove (80–100 °C), and then, the grain size analysis was performed with a precision scale and a set of stainless steel sieves (interval ¼ φ in the Wentworth [31] scale).

3.2 Wave data

Authorial wave data were provided by the Sistema de Monitoramento da Costa Brasileira (SiMCosta), collected with a directional waverider buoy moored adjacent to Cassino beach (32° 17′ 43.03″ S 52° 1′ 29.95″ W), 17 km apart from the coastline. Perotto [27] and Fontoura et al. [2] measured wave parameters in the breakpoint through continuous filming of wave fields with posterior laboratory analysis. All wave data were converted from its initial acquisition form to a pattern considering the transformative processes associated with wave propagation in shallower waters. The conversion was made through linear regression, using significant wave height HS (in meters), peak wave period TP (in seconds) and wave propagation direction DP (in degrees) simulated in the Simulating WAves Nearshore (SWAN) model by [30], for the isobaths of 2.5 m and 19 m. In order to obtain greater data accuracy, all of the converted wave components referred to a 4 h interval regarding the time in which field surveys were carried out. The surf zone width LSZ was also estimated through linear regression using data from images obtained with the Argus video-imaging system [43, 44]. The regression was made among the LSZ measured through Argus images, HS and TP [45]. The correlation coefficients were r = 0.95, r = 0.98, r = 0.96 and r = 0.39, respectively, for HS, TP, DP and LSZ (Fig. 2). Although the LSZ coefficient seems low, it is significant for a 95% confidence interval with a significance level of 0.05.

Fig. 2
figure 2

Linear regression graphs referring to the relation between 19- and 2.5-m isobaths for HS, TP and DP (a–c), and to the relation between LSZ and HS (d)

Full size image

3.3 LST rates

The LST rates were estimated using 4 analytical models. The model described by CERC [24] presents an equation based on the principle that LST rates are proportional to the longshore component of wave energy (Eq. 4). The LST rate QL corresponds to the immerse volume in m3 s−1, and it considers a dimensionless coefficient K—determined initially through linear regression using field data collected by Komar and Inman [8], researched later by other authors [4, 17, 18, 46,47,48,49,50]: gravity acceleration g (m s−2), seawater density ρH2O (kg m−3), sediment particles density ρs (kg m−3), porosity \(\phi\) (dimensionless), HS (m), breaker index k (in accordance with the ratio between HS and water depth) and DP (degrees).

$$Q_{{\text{L}}} = \frac{{K . \left( {\sqrt g } \right)}}{{\left( {16.\sqrt k } \right).\left( {\rho_{S} - \rho_{{{\text{H}}_{2} {\text{O}}}} } \right).\left( {1 - \phi } \right).\left( {H_{{\text{S}}}^{\frac{5}{2}} } \right).\left( {\sin 2.D_{{\text{P}}} } \right)}}$$
(4)

The model proposed by Kamphuis [25] allows to estimate LST rates QL as a function of wave parameters, fluid and sedimentary characteristics, and beach profiles (Eq. 5). It has similarities with Eq. 4 (i.e. dimensionless coefficient K, ρH2O, g, HS, DP); however, it calculates the rates in kg s−1 and includes TP (s), beach slope tan β (dimensionless) and median grain size D50 (m).

$$Q_{{\text{L}}} = K\rho_{{{\text{H}}_{2} {\text{O}}}} \left( {\frac{g}{2\pi }} \right)^{1.25} H_{{\text{S}}}^{2} T_{{\text{P}}}^{1.5} \tan \beta^{0.75} D_{50}^{ - 0.25} \sin \left( {2D_{{\text{P}}} } \right)^{0.6}$$
(5)

Bayram et al. [26] define the QL rate (m3 s−1) as a product of the sediment concentration in suspension and the acting longshore current (Eq. 6). This method differs from the others mentioned before especially because it considers the mean longshore current velocity VLC (m s−1) in the surf zone. It is important to notice that the influence of waves, winds and tides is included in this parameter, because the total velocity is considered, not exclusively the parcel generated by the wave-induced longshore flux. Besides that, this method assumes that suspension load is the dominant transport mechanism in the surf zone because the sediment deposited on the bottom is suspended by the continuous action of waves. Hence, this model suggests a transport coefficient ε proportional to wave efficiency in keeping grains in suspension calculated through Eq. 7, considering the particle settling velocity WS (m s−1) instead of D50. The wave-energy flux energy F (Eq. 8) is also considered:

$$Q_{{\text{L}}} = \frac{\varepsilon }{{\left( {\rho_{{\text{S}}} - \rho_{{{\text{H}}_{2} {\text{O}}}} } \right).\left( {1 - \phi } \right).g. W_{{\text{S}}} }}.F\overline{{V_{{{\text{LC}}}} }}$$
(6)
$$\varepsilon = \left( {9 + 4 . \frac{{H_{{\text{S}}} }}{{W_{{\text{S}}} . T_{{\text{P}}} }}} \right) . 10^{ - 5}$$
(7)
$$F = E.C_{{\text{G}}} .\cos D_{{\text{P}}}$$
(8)

where E = wave energy (J m−2) and CG = wave group velocity in shallow waters (m s−1).

Van Rijn [23] proposes to estimate the QL rate in kg s−1 using a simple general equation (Eq. 9). This method is derived from a previous trial [43] with a detailed process-based model, and in this more recent version, it is calibrated and revalidated, making QL a function of a mobility coefficient M obtained through Eq. 10:

$$Q_{{\text{L}}} = 0.0006 . M$$
(9)
$$M = K_{{{\text{swell}}}} {.} \rho_{{{\text{s}} }} {.}\left( {\tan \beta } \right)^{0.4} .\left( {D_{50} } \right)^{ - 0.6} .\left( {H_{{\text{S}}} } \right)^{2.6} .V_{{{\text{LC}}\;{\text{ total}}}}$$
(10)
$$V_{{{\text{LC}}\,{\text{total}}}} = V_{{{\text{LC}}\;{\text{wave}}}} + V_{{{\text{ LC}} \;{\text{wind}}}}$$
(11)
$$V_{{{\text{LC}}\;{\text{wave}}}} = 0.3.\left( {gH_{{\text{S}}} } \right)^{0.5} .\sin \left( {2D_{{\text{P}}} } \right)$$
(12)
$$V_{{{\text{LC}}\;{\text{wind}}}} = 0.01. p_{1} .v_{1} + 0.01. p_{2} .v_{2}$$
(13)

The mobility coefficient M includes beach profile effects through beach slope, represented by tan β, and through median grain size D50, aiming a wider applicability of this method when it comes to different morphodynamic stages. It also contemplates the effect of low-period swell waves (TP = 9–12 s, HS = 0.70–2.0 m), indirectly considering TP, since these waves can produce significantly larger transport rates when compared to wind waves of the same height [23].

The main equation (Eq. 10) considers the mean longshore current velocity VLC total in m s−1, just like the analytical model proposed by Bayram et al. [26]. However, it allows to distinguish the velocities induced by the waves VLC wave from the tide-wind-induced velocities (Eqs. 11, 12, 13). The study area is under a microtidal regime; therefore, only the wind was considered as a influence factor over the LST, and so the tide-wind-induced velocity is represented by VLC wind (Eq. 13), in which p1 = percentage of time with positive flow; p2 = percentage of time with negative flow; v1 = representative velocity in positive longshore direction due to wind (m s−1); v2 = representative velocity in negative longshore direction due to wind (m s−1). VLC WIND = 0.1 m s−1 as suggested by Van Rijn [23] for microtidal conditions.

LST rates estimation requires the awareness of many specific characteristics of the study area. Table 1 contains the values attributed to each parameter considered in the four models described above, except for the values referent specifically to the day of field surveys (i.e. waves, winds and longshore currents). It also displays information regarding data origin.

3.4 Performance verification

Analytical models were verified through different statistical methods as follows: (1) Bias—This calculation considered the resampling method through Jackknife [51], a nonparametric technique that reduces the estimator variance; (2) Root Mean Square Error (RMSE)—since LST rates extend through several orders of magnitude, logarithmic values (base 10) were considered for this statistical measure as proposed in [4]. When RMSE = 1, it means that the modelled values are approximately ten times bigger or smaller than the sampled ones. The remaining are (3) Scatter Index (SI); (4) Pearson Correlation; (5) Spearman’s Rank Correlation; and (6) Student’s t Test.

4 Results

4.1 Field surveys

This research comprehended seventeen field surveys in distinct beach locations, among the years of 2002, 2003, 2004 and 2018, under different meteoceanographic conditions. The predicted and observed results Qin situ can be seen in Fig. 3. The comparison between Qin situ and the predicted results allows to observe that the method proposed by Kamphuis [25] tends to underestimate the LST occurring in the system, representing the most unrealistic rates, varying in one order of magnitude for most of the cases. Predictions according to the model proposed by Bayram et al. [26] also resulted in underestimated rates for most of the scenarios. When it comes to CERC [24] and Van Rijn [23], no tendencies of under- or overestimation were observed, except for high LST scenarios (20/02/2003, 05/09/2003, 12/12/2004—Sarita, 07/08/2018), in which the estimates were lower than the observed in situ.

Fig. 3
figure 3

Bars for LST rates comparing the predicted and observed data for each author. Notice that for Perotto [21], all field surveys occurred in the same date, and Fontoura et al. [2] rates are not arranged chronologically because they are disposed according to their sampling sites, as displayed in Table 2

Full size image

4.2 Wave and wind data

The conversion of wave parameters resulted in significant changes in HS and DP, but not in TP, which was almost constant for all the scenarios. Waves reduced around 33.9% of their original height, and the direction changed approximately 21.9%, also with a reduction trend from the angle found at the 19-m isobath. The final converted values are in Table 2.

Table 2 Meteocenographic conditions for each field survey. The sampled data are used as input to the analytical models and as a source to the performance verification of the models
Full size table

Multiple linear regression was applied aiming to identify the meteoceanographic agents with more influence over longshore currents and LST rates. The statistical tests were reproduced using different combinations between wind and wave variables (Table 3). It is important to notice that some of the displayed regressions consider wind direction instead of velocity, as seen in others, because only the best multilinear responses are shown in this table. Waves have more influence over the LST, for all parameters: LC velocity (r = 0.73), LC direction (r = 0.96) and LST rate (r = 0.89). Wind is only statistically significant over LC velocity (r = 0.70) and direction (r = 0.74). Although waves are more influent, when wave and wind components are combined as predictor variables, at least 95% of the LST is explained statistically, for all components.

Table 3 Multiple linear regression coefficients relating LC velocity, LC direction and LST rates with wave and wind parameters
Full size table

4.3 Analytical modelling

The statistical results (Table 4) demonstrated that the model proposed by Van Rijn [23] is the most applicable to estimate LST rates in the southern region of Brazil. Van Rijn [23] was the only method that showed good results for all verification tests applied, being statistically significant in confidence levels of 95% (α = 0.05) and 99% (α = 0.01), for both correlations (Pearson and Spearman). It is the only model in which t-test has enough evidence to support that the predicted rates correspond to the observed values, and it is also the model with the lowest SI and RMSE.

Table 4 Statistical parameters referring to the analytical models performances
Full size table

Although CERC [24] has a t value that rejects the hypothesis that the predicted rates are similar to the observed ones, the correlation coefficients r and rs are significant for the confidence intervals of 95% (α = 0.05) and 99% (α = 0.01). Moreover, this model results showed the lowest bias and low values of SI and RMSE. In this way, CERC [24] was considered the second most adequate method for the estimation of longshore transport rates in the study area. Neither Kamphuis [25] nor Bayram et al. [26] performed adequately for the region under study.

5 Discussion

5.1 Wave and wind data

Waves have more influence over the LST overall, and this can be clearly seen through the correlation coefficients, specially for QT, for which the wind itself is not statistically significant. Although waves presented the highest correlation coefficients for all the variables under consideration, their influence can be altered by wind interference, as seen specially during field surveys. That means they are statistically sufficient when isolated to explain entirely the LST mechanism, but not enough to describe real scenarios. Wind parameters are relevant to the direction and velocity of longshore currents. Thus, the combination of wave and wind components as predictor variables explained more than 95% of the three properties evaluated. This demonstrates that LST is a direct reflex of the concomitant action of both agents, in a way longshore currents are set up, predominantly, by the characteristics of its generation wave, and the wind corroborates with the dynamics involved in the flux, influencing, hence, the amount of sediment mass being transported and in the resulting LST.

Beyond the statistics, the influence of winds was observed during the current author’s field surveys: in 31 March 2018, the opposite incidence directions between the waves and the wind resulted in a low longshore sediment transport (forces were almost cancelling each other), and in 7 August 2018, the action of both components coming from the same quadrant resulted in an intensified LST rate, the highest value among all the period under investigation. In that way, it can be affirmed that the wind has potential to reduce or increase wave action over LC generation and, consequently, over LST, except when its intensity is too low or absent.

Besides being the most accurate model overall, Van Rijn [23] performed better for the authorial measurements than for the other datasets as seen in Fig. 3. It is supposed that the differences between wave data acquirement influenced the simulated rates. Thus, predictions were more accurate when wave parameters were sampled with SiMCosta buoy and later converted, rather than by observations and filming in the beach field work as made by Perotto [27] and Fontoura et al. [2]. Moreover, the conversion of wave parameters was relevant for this study considering that longshore currents are mostly influenced by nearshore breaking waves. During wave propagation from deep waters to 15-m and 18-m isobaths, wave transformation processes start, and as they gradually approach shallower waters, HS and DP change their values as an adjustment caused by energy conservation through these processes (i.e. refraction, diffraction and shoaling). DP is modified in a way the crests tend to become parallel to the bathymetric contours. Therefore, NE and SW waves gradually disappear, focusing their directions on E and SE quadrants in the southern coast of Rio Grande do Sul [2], quadrants which are associated with the most energetic scenarios, and so the highest LST rates. The alterations occurred in DP are mainly attributed to refraction, and the variations occurred in HS represent, partially, the redistribution of the energy flux seeking its conservation. Furthermore, wave celerity and length diminish progressively towards the coast at the same time wave height tends to increase (these values will keep changing until the wave breaks, dissipating its energy), and the period remains nearly constant [3].

5.2 LST rates

The predictability skill of each analytical model depends on the parameters involved in the equations for the calculation of LST rates. Accordingly, different analytical formulae resulted in distinct values for the same meteoceanographic conditions.

Van Rijn [23] was proven to be applicable in sandy and mixed beach systems, given its good performance among all the different beaches studied herein. Notwithstanding, the models elected as most adequate tend to underestimate the observed rates during high-energy scenarios, like the other two tested. The underrates can be partially attributed to the inaccuracy of the coefficients involved in the equations, as discussed by Van Rijn [23] through the suggestion of a new formula with calibrated factors, and also related to the lack of sampled data during high-energy events, since these values are usually not considered in model validation and calibration, and it is when the highest LST rates occur [2]. Moreover, wind influence is only considered in Van Rijn [23] and Bayram et al. [26] calculations through the LC velocity. Considering the wind’s potential to alter longshore currents, and so the LST, as seen in the multilinear regressions (Table 3), the underrates associated with high-energy conditions may be attributed to the negligence of relevant wind components in the analytical models. This is exemplified by 07 August 2018 scenario, date in which the biggest LST value was observed, but none of the models were able to fully predict it, and the wind velocity was expressive (around 7 m s−1 with 8 m s−1 gusts). Bayram et al. [26] contemplate wind effects, but it does not consider beach slope and D50 (sediment size is indirectly considered through WS). Given these two parameters are very relevant for LST, jointly with wave properties [4, 14, 23], and considering that WS does not change the net longshore transport, as computed by [14, 26], predictions presented many inaccuracies. In that way, further investigations are required to validate a model with a more accurate wind effect, and to calibrate the Van Rijn [23] equation, increasing, possibly, its accuracy.

The tendencies observed presented a positive proportionality between HS, beach slope and LST rates, and a negative proportion with D50 [4, 14, 23, 26], likewise the trends occurring in the results herein. Albardão and Concheiros Sul scenarios exemplify the effect of D50, in which the suspended load transport is diminished because grain size increases. D50, tan β, HS and DP at the breaker line are the key parameters for predicting LST rates according to [23]. Waves are also the most influent agent in the current study; however, the most significant regression responses are associated with TP (influencing wave type and energy), instead of DP as postulated in [23], although wave propagation direction was still very significant in this research. This may be related to the importance of swell effect, computed indirectly through peak wave period, and highlighted by the good performance of Van Rijn [23].

Bergillos et al. [13] evaluated the performance of empirical equations (i.e. Inman and Bagnold [7], CERC [24], Kamphuis [25] and Van Rijn [23]) through RMSE and through a ratio between the predicted and the observed rates. Their results pointed that all four equations tend to overestimate LST rates when D50 is based on the sandy fraction of bimodal beaches, but Van Rijn [23] is still the most accurate model overall. When gravel is considered, [23] efficiency is improved and the model presents the lowest RMSE associated, highlighting, once more, that grain size is a key parameter for estimating LST rates.

Concheiros Beach has a bimodal sediment distribution (D50 = 0.2102 and D50 = 0.707 mm according to [28]), and like Bergillos et al. [13], the most accurate LST rates were found in this study when the gravel fraction was under consideration. Therefore, the agreement with [13] results sustain, once again, Van Rijn [23] as the most adequate model for the southern coast of RS, since the best model elected by them is the same found herein, and they also looked for a model applicable in sandy, mixed and shingle beaches. The current study area is predominantly composed of fine sandy beaches, but the southernmost stretches are composed of coarse sand and biodetritic gravel (i.e. Sarita, Albardão, Concheiros). Thus, Van Rijn [23] applicability is proven to be efficient on coasts with a variable grain composition.

LST behavioural response will be different for each beach morphodynamic stage and meteoceanographic component in intensity, in agreement proportion, and in different time and space scales (may these components be related to wind, waves, tide, sedimentology, geomorphology and/or others). Furthermore, the contribution of each property able to influence the LST mechanism is not fully understood, and their weight factors still need to be investigated, specially when it concerns new tests with datasets around different world regions [12,13,14, 16, 20, 21, 52,53,54,55]. Therefrom, studies referring to each specific study area may be conducted, considering the main characteristics of the beach system under study, so the most applicable methodology can be chosen and the prediction of LST rates can be more accurate.

6 Conclusions

The observed LST rates demonstrated that this phenomenon varies as a function of daily meteoceanographic conditions, specially referring to wave and wind components. The predictability skill of each model is dependent on the parameters considered in its main equation, and the factors of influence each one has over the LST. Therefore, the results obtained through distinct analytical formulae were discrepant for the same beach under the same meteoceanographic scenario.

Wave parameters, median grain size and beach slope were proven to strongly influence longshore sediment transport, as found by previous studies, corresponding to the choice of best model overall. During high-energy scenarios, predictions were underestimated for all analytical models considered herein. Van Rijn [23] results were slightly closer to the observed rates, when compared to the others, and this may be attributed to its inclusion of swell factor and wind-induced longshore velocity, demonstrating the importance of wind in LST predictions.

Results herein can be compared with available literature in other regions around the world, but it is important to notice that LST behavioural response and model’s effectiveness will be different for each specific study area, especially in what concerns the beach morphodynamic stage and proportionality feedback between longshore transport and system component. Hence, the method prediction accuracy depends greatly on the suitability of the dataset with respect to the analytical model being used.

Overall, this study demonstrates that every effort towards the assessment of LST is still needed since it has been studied for many decades by scientists from all around the world, and it is not fully comprehended yet, still presenting inaccuracies.