4.1.2. Bar Evolution due to Wave Parameters and Non-Linearity

The initial steps to analysis of the annual and seasonal bar evolution were to examine the dependence between bar crest locations in 2009–2010 and wave parameters such as significant wave height Hs, peak period Tp and wave steepness Hs/Lp (Figure 4).

**Figure 4.** Intra-annual variations of: (**a**,**b**) monthly Q99 of significant wave heights Hs and peak periods Tp for 2009–2010 vs. annual monthly Q99 of the same parameters for 1948–2010 at depth 17 m; (**c**) wave steepness of transformed monthly Q99 waves for 2009–2010 at depth 4.4 m; (**d**) average monthly positions of the outer bar crests for 2009–2010.

The analysis comprised an estimate of 2009–2010 wave conditions in comparison to the regional wave climate and whether either of the years was more energetic than the others. To this end, monthly Q99 of Hs and Tp for 2009–2010 were compared to the same annual monthly Q99 for 1948–2010 (Figure 4a,b). Results show that in general, 2009–2010 wave parameters do not exceed the climatic ones. Exceptions concerning wave height are higher estimates for April 2009 and March 2010, and a commensurate value for September 2010. Wave periods follow the same tendency plus higher estimates for June 2010. As for the comparison between the years 2009 and 2010, it appears that on average wave heights are commensurable (1.5 m in 2009 vs. 1.61 m in 2010), but a bit longer wave periods (7.1 s) were present in 2010 as oppose to those in 2009 (6.9 s).

The comparison also shows that bar crest position depends on variations of all wave parameters throughout the year. It keeps the same displacement pattern during both years and a clear dependence is seen not only on wave height, but also on wave period and steepness. Winter storms with larger wave heights (periods) and Hs/Lp ≈ 0.04 erode the profile moving the sand bar offshore (Figures 3 and 4d). Summer storms with lower heights (periods) and Hs/Lp < 0.03 are not capable to induce shoreward bar migration, which is done in the autumn when steepness again increases up to 0.04. Thus, as per Figure 4d in November–February 2009 (same for 2010) the bar crests are found at distances 195–200 m, while at the end of the season the bar is moved offshore (≈220 m) and remains stable

during March–July 2009 and March–April 2010. Shoreward bar's shift toward its winter location was initiated by storms in August–October 2009 and August 2010.

It is of interest to further examine which values of wave steepness (Hs/Lp) correspond to the positions of the bar crest in the coastal zone. For that purpose, crest offshore distances were set against steepness values of the transformed monthly Q99 waves for 2009–2010 (Figure 5).

**Figure 5.** Dependence of steepness Hs/Lp of transformed monthly Q99 waves (depth 4.4 m) for 2009–2010 on (**a**) the offshore distances of the outer bar crest, black dashed line—wave steepness threshold, and (**b**) the mean slope tanβ, red dashed line—criterion for identification of non-linear wave transformation scenarios [tanβ = 7(Hs/Lp)3/2]; black symbols—2009 values, grey symbols— 2010; circles—seaward bar crest positions, triangles—shoreward bar crest locations, stars—bar crest displacements.

According to the presented results steepness 0.03 could be introduced as a threshold to support differentiation between bar's stable position and its relocations within the nearshore area. Thus, for Hs/Lp < 0.03, as in summer months May–August 2009 and August 2010, the sand bar was stable and located offshore, while for Hs/Lp ≥ 0.035–0.04 the bar crest was found either offshore (≈220 m) or closer to the coastline, or migrating within distances 170–210 m (Figures 3 and 4). Additionally, wave steepness for January– February 2009 (same for 2010) was higher than for November–December 2009 due to lower wave heights at the end of 2009.

As next, it was necessary to examine for which steepness values the bar remains stable or is subjected to displacement. Another parameter affecting wave steepness and non-linear evolution of waves in the coastal zone is the mean slope (tanβ). Depending on the Irribaren number, non-linear wave transformation may proceed by four characteristic scenarios [49]. The scenarios are distinguishable for the periodicity of wave energy exchange between the first and the second non-linear harmonics, as the slope affects the number of periodic cycles and the growth of the second harmonics, which in turn influence the sediment transport along the profile [60].

Using the criterion for realization of the scenarios [49] a dependence was sought between the wave steepness and the mean slope (Figure 5b). As per criterion, for mean slope tanβ > 7(Hs/Lp)3/2 waves in the nearshore zone transform by either of two scenarios, i.e., amplitudes of the second harmonics grow only very close to the shore (S1) or remain small along the entire coastal zone (S3). Results in (Figure 5b) state that for waves in May–August 2009 and in August 2010 scenarios S1 and S3 are applicable. According to the hindcast data these waves have Hs/Lp < 0.03, the average monthly Hs varies within 0.67–0.86 m and the average monthly Tp is between 5.1–5.6 s. Based on 2007 field data, the influence of scenario S3 on the morphology of the submerged profile was explored in [60] on a case of a wave regime with significant height 0.6 m and peak period ≈ 5 s measured at

the pier's end. These results confirm the validity of a wave parameters' range introduced above. Moreover, for this scenario only the inner bar profile was subjected to deformations and significant shoreward relocation, while changes on the outer bar were minor. Thus, absence of large second harmonics and spatially small periods of energy exchange dictate weak non-linearity of waves over the outer bar, which contributes to its relative stability in the spring-summer months.

On the other hand, for tanβ < 7(Hs/Lp)3/2 wave transformation proceeds according to one of the other two scenarios, i.e., amplitudes of the second harmonics reach their maxima within the coastal zone (S2) or second harmonics already have large amplitudes upon entering the coastal zone (S4). Such highly non-linear scenarios are valid for data pairs in January–April 2009 and September 2009–April 2010 (Figure 5b). According to hindcast data, steepness of these waves varies between 0.035–0.04, the average monthly Hs: 1.48–2.12 m and the average monthly Tp: 6.7–9.1 s. Once again on the basis of 2007 field data, the influence of scenario S2 on the evolution of the submerged beach profile was studied in [60,62] for a wave regime with significant height 1.1 m and period 7 s measured at the pier's end. They established that at the length of each full period of energy exchange between the first and the second non-linear harmonics erosion occurs on the seaward bar's slope with sediments being transferred to its shoreward front either for the inner or the outer bar. Furthermore, if the contribution of the undertow and the transition from S2 to S3 is considered, such periodic energy exchange can lead to bar's significant shift toward the shore and changes in its symmetry [60].

To draw a more precise conclusion about the waves capable to act on the bar's stability or displacement, threshold values for Hs and Tp were determined by setting the monthly Q99 Hs and Tp (depth 4.4 m) against the bar crest offshore distances. This resulted in definition of Hs ≈ 1 m and Tp ≈ 6 s. Having this in mind, it was assumed that waves responsible for cross-shore bar migration transform according to scenarios S2 or S4 and have steepness >0.03, Hs > 1 m and peak periods over 6 s.

Therefore, it might be suggested that the summer profiles and the relevant bar's stability are aided and contributed by the influence of weakly non-linear waves (scenarios S1 or S3), while winter and transitional profiles, as well as bar's migration are governed by strong non-linearity of waves represented by scenarios S2 or S4.

Next in line was to examine whether any combination of wave parameters render influence on the bar's movement. To this end, we considered the distribution of bar crest positions against the average monthly data of Hs, Tp and Hs/Lp (depth 4.4 m) for January 2009–August 2010. In addition, a reference bar crest distance of 196 m was determined, based on the most frequent profile measurements in winter 2009 (Figure 3a). The comparison showed that in January–April 2009 and September 2009–April 2010 for similar wave steepness values ≈ 0.04 the bar crest was localized either close or away from the shore with regard to the reference line (Figure 3a). As for the other wave parameters, it became evident that to a narrow interval of steepness values (0.035–0.04) corresponded a wide range of wave heights: 1.48–2.12 m, respectively periods: 6.7–9.1 s. Despite the time scale discrepancy in the compared data, it was concluded that regardless of the similar steepness values waves that cause the bar crest to shift seaward are higher and with longer periods than those moving it in the opposite direction, which is especially valid for January–April 2009 (2010)—Figures 3a and 4. However, these general findings do not justify in completion the observed intra-annual bar evolution. This implied the necessity to investigate the influence of individual storms and their characteristics on the cross-shore bar crest relocations during January–April 2009 and September 2009–April 2010.
