**3. Results**

Figure 3 shows the underwater bottom profiles, both initially and the result of 20 h storm modelling. The resulting deformations of the underwater profile are presented in Table 2. The change in all of the beach profiles was characterized by erosion in the splash area, the transition of material seaward, and its accumulation at depths of more than 1 m. However, the activity of erosion and accumulation processes on profiles with different bar locations varied.

**Figure 4.** Initial and resulting profiles after 20 h of wave action (Hs = 1.5 m, Tp = 10.5 s), location of the breaking zone, and theoretical shape of the equilibrium profile calculated for a grain size of 0.2 mm.


**Table 2.** Main changes in model profiles after 20 h of wave action.

In general, if a bar is located closer to the coast and it has reduced depth above it, several patterns can be distinguished: the rate of shore erosion increases, the thickness of the sediment accumulation layer in the underwater part of the profile decreases, and the distance of sediment transfer to the sea grows (Table 2). However, the shoreline recession (Figure 5a) on the profile with the furthest bar position (0.82 from the wavelength) was substantially less than that of the profile where the bar was located closer to the shore (0.42 from the wavelength). The coastline degradation on the profile where the bar was positioned at 0.54 from the wavelength was maximum for all of the considered profiles. The seaward transfer distance of the sediment (Figure 5b) was maximal on the profile where the bar is positioned, at 0.82 (profile 1) from the wavelength. In all cases, there was no transport of sediments beyond the bar (Figure 4).

**Figure 5.** (**a**) Shoreline regression (Sh, m) and (**b**) distance of seaward sediment transport, in terms of wavelength (x/L) independence of the initial bar location (X/L). X is the distance between the bar crest and the shoreline, x is the distance of the maximum amounts of sediments that are transported away from the shore, L is the wavelength calculated for deep-water conditions

Compared to the profile without a bar, the profile with the furthest bar location (the relative distance from the coast is 0.82 of wavelength) reduces the degree of shoreline degradation. The profile with the closest position to the bar (a relative distance from the coast of 0.42, or less than half the wavelength) worked best as a barrier against carrying the beach material seaward to depth. Compared with a barless control profile, an underwater bar at some relative distances from the coast could lead to an increase in coastline degradation.

Under the storm waves, slight deformations of bars also occurred (Figure 4): the reduction of its relative height, the deepening of the bar top (on profile 5), and a retreat of the bar crest (2–4 m) towards the sea (on profiles 1, 2, and 4). The bar shape changed with regard to its slope oriented towards the coast, which became less steep due to the filling of the bar trough. Such movements of the bar crest and the change in its asymmetry are generally consistent with the data of field observations [17].

The process of the retreat of the coastline and the transformation of the coastal zone relief occurs non-uniformly over time. Figure 6 shows the changes in the coastline retreat speed after 10 h of wave action, for profiles with maximum (4) and minimum (1) changes. The fastest shoreline retreat was observed during the first hour of the storm for all profiles: it varied from 4.5 to 6.5 m/h. Erosion activity slows down over time. After 6 h of wave action, the beach profile adapts to the specific waves occurring, and assumed a relatively equilibrium state. The shoreline regression rate became ≈0.5 m/h, and it remained approximately the same for all profiles.

**Figure 6.** The speed of shoreline regression (m/h) variations during the numerical experiment that was run on profiles 1 (black) and 4 (red).

Regardless of the initial underwater relief, an underwater terrace was is formed on for all profiles (Figure 4), representing an equilibrium profile that was close to the theoretical classical Dean's profile proposed in [27]:

$$h = A x^{2/3} \tag{6}$$

where *A* = 0.1, calculated for a sediment grain size of 0.2 mm [26]. The formation of an underwater profile of similar shape with a terrace under the influence of uniform storm waves was also observed by us in the field experiment "Shkorpilovtsy 2007" [17].

Different degrees of coastline degradation during the first hour of the storm can be explained by differences in the transformation of the storm waves along the bottom profiles. Figure 7 shows the dependencies of the change in the coastline retreat, and the sediment seaward transfer distance on the relative change in the significant wave heights, as determined by the ratio of the corresponding values before and after the bar (with coordinates on model profiles 620 and 820 m). The significant wave height slightly decreases when the waves came nearer to the shore, but the decline was more strongly expressed in profiles 4 and 5, where the bar is located near the shore. On profiles 1–3, the significant wave height fall was less than in the profile without a bar. The distance of seaward sediment transport directly depends on the significant wave height: the greater the height, the further the material is transferred (Figure 7b). The relation between the shoreline regression and the change in significant wave height was not so clear (Figure 7a).

**Figure 7.** Dependence of (**a**) the shoreline retreat (Sh, m) and (**b**) the relative distance of seaward sediment transport (x/L) on the relative change in significant wave height (ΔHs, m) occurring during wave transformation over barred profile (620–820 m, see Figure 4). L is wave length calculated for deep water conditions, x is distance of maximal sediments transport away from the shore

The shoreline retreat is influenced more by the mean wave period than by wave height. Figure 8 depicts the relation between bottom deformations and relative change of mean wave period determined by the ratio of the corresponding values before and after the bar (coordinates on model profile 620 and 820 m). The smaller the change in the mean wave period, the smaller the degradation of the coastline (Figure 8a). Such a change of parameters occurred in profile 1, with a bar being located at a distance from the shoreline of 0.82 of a wavelength. There was no clear dependence of sediment transport on the mean period.

**Figure 8.** Dependence of (**a**) shoreline retreat (Sh, m) and (**b**) the relative distance of seaward sediment transport (x/L) on the relative change in the mean wave period (ΔTmean, s) occurring during wave transformation over a barred profile (620–820 m, see Figure 4). L is the wavelength calculated for deep-water conditions, and x is the distance of maximal sediment transportation away from the shore.

The modelling of profiles with five different bar locations shows that a growth of significant height in the low-frequency waves after passing a bar leads to a decline of coastline degradation (Figure 9a). This relation is close to linear, except for profile 5, where the change in the low-frequency wave height is the same, but the shoreline regression rate and the distance of seaward sediment transport (Figure 9b) are different. This could be caused by the different natures of low-frequency waves. Waves of low-frequency bands can include infragravity waves (IGW) of different kinds: bound long waves and break point-forced long waves [6]. Previous investigations have shown show that various IGW affect the coast in two different ways: a) bound long waves protect the shore, b) the break-point forces long waves, leading to an intensification of near-shore erosion [28].

**Figure 9.** Dependence of (**a**) the shoreline retreat (Sh, m) and (**b**) the relative distance of seaward sediment transport (x/L)over a on relative change in mean wave period (ΔTmean, s) occurring during wave transformation over a barred profile (620–820 m, see Figure 4). L is the wavelength calculated for deep-water conditions, x is the distance of the maximal sediments transported away from the shore.

The presence of the underwater bar changed the symmetry of the waves. The wave skewness, after passing the bar (the x-coordinate on the profile was 820 m), remained almost similar ≈ 1.6 for all profiles. High values of the skewness coefficient show that breaking in XBeach model is probably described as spilling, because in observations and laboratory experiments plunging breaking waves have skewness less than 1 [29]. Waves breaking by spillage are symmetric relative to the vertical axis, they have sharp crests and flat troughs, while plunging breaking waves have steep fronts [30].

The wave asymmetry coefficient behaves differently. Figure 10 shows the wave asymmetry coefficient in a nearshore area (the 820 m coordinate on the profile) and its relation with shoreline regression and seaward sediment transport distance. According to the model data, the more asymmetric waves are after the bar, the less the coastline degrades.

**Figure 10.** Dependence of (**a**) shoreline retreat (Sh, m) and (**b**) the relative distance of seaward sediment transport (x/L) on wave asymmetry (As) after wave transformation (near the shore, at the x-coordinate 820 m). L is the wavelength calculated for deep-water conditions, and x is the distance of maximal sediments transported away from the shore.

The most asymmetric waves behind the bar were observed on profiles where the first wave breaking occurred, between the coast and the bar, for example, on profiles with the bar's distant location (profiles 1 and 2), as well as on a profile without a bar (profile 0, see Figure 3). When the bar was located closer to the shore, and accordingly, the depth decreased above it, the waves broke both at the top of the bar, and near the shore. In this case, the waves broke over the bar farther from the shore than on the profiles 0–2, and the second breaking zone was closer to the coastline. The presence of two breaking points leads to a significantly greater shoreline regression. The exception for this is the profile 5, where the bar is located at the closest distance to the coast line. The bar prevents seaward sediment transport, so that the shoreline degradation is reduced.

An Explanation of wave asymmetry impact on sediment transport w discussed in detail in [30]. Cross-shore sediment transport is defined by the balance of wav- induced sediment transport directed to the shore, and the undertow, which moves sediments seaward. According to Bailard's formulation of wave-induced sediment transport discharge [31] depends on highest statistical moments of near bottom velocity:

$$q = \frac{1}{2} f\_w \rho \left( \frac{\varepsilon\_b}{\tan \Phi} \overline{u \left| u \right|^2} + \frac{\varepsilon\_s}{\mathcal{W}\_s} \overline{u \left| u \right|^3} \right) \tag{7}$$

where u = u(t): the instantaneous near-bottom velocity.

Formula (7) was adopted by Leontiev [6] for calculations through amplitudes of first and second nonlinear harmonics of near bottom-velocity and cosines of phase lag between them (bi-phases):

$$
\overline{\left|u\right|^2} = \frac{3}{4} \mu\_m^2 \mu\_{2m} \cos \beta,\\
\overline{\left|u\right|^3} = \frac{16}{5\pi} \mu\_m^3 \mu\_{2m} \cos \beta \tag{8}
$$

where *um* and *u*2*m* – amplitudes of first and second harmonics, β- phase shift between the first and second nonlinear harmonics (bi-phases).

As it was revealed in [31], the wave asymmetry coefficient *A*s was linear, depends on bi-phase:

$$As = 0.8\beta$$

Thus, according to (8 and 9), the shoreward wave induced-sediment discharge depends on the cosine of the bi-phase, or As magnitude. Maximum of it will occur when bi-phase (and accordingly *A*s) is zero, because cosine will has maximal value. A decrease of wave-induced sediment discharge due to wave asymmetry will lead to increase in the role of the undertow in sediment transport. Therefore, sediments will move more seaward, and the shoreline will retreat more, due to the erosion in nearshore zones (see Figure 10b).
