Analysis of historical wave climate simulations
First, the historical wave climate in the MS driven by each of the RCM was compared to the hindcast data developed and validated by Mentaschi et al. (2013) and Mentaschi et al. (2015). Attached in the A are the comparison maps for Hs (Figs. 19, 20, and 21) and Tm (Figs. 22, 23, and 24), which show the different spatial patterns of the biases models-hindcast. The maps also show the mean biases averaged over the MS. In case of the Hs statistics, the mean biases are between [− 15.3%,− 1.8%], [− 6.8%,7.7%], and [− 15.9%,3.8%] for \(H_{s}^{mean}\), \(H_{s}^{p90}\), and \(H_{s}^{\max \limits }\), respectively. When the annual statistics of Tm are considered, the mean biases attain values between [− 0.9%,6%] (\(T_{m}^{mean}\)), [− 2.7%,5.6%] (\(T_{m}^{p90}\)), and [− 5.6%,6%] (\(T_{m}^{\max \limits }\)). Table 2 reports the spatial correlations between the hindcast and the RCM data, computed through Pearson’s coefficient. The high values of the spatial correlation indicate that the spatial patterns are adequately captured by each model, for all the examined parameters and statistics. Altogether, these results show the ability of the models to reproduce reasonably well the wave dynamics on the historical period.
Table 2 Spatial correlation between the hindcast and RCM time series of annual statistics averaged over the whole MS Analysis of trends on the future wave climate
First, the trends were computed on the ensemble time series of each wave parameter for each grid point. Trend analysis could be also performed directly on the series related to each single ensemble member, averaging the trend metrics at a second time. Such approach will be discussed in the second part of this section.
Figure 2 summarizes the values of b related to Hs and Tm. First of all, it can be noticed how for all the examined statistics the vast majority of highlighted trends is negative, indicating an average reduction in the expected future wave parameters. According to the boxplots, positive values of b (i.e., upward trends) lie always above the upper bars, which denote the 75th percentile of the whole series. Moreover, the highest positive trends are generally less pronounced than the negative ones, especially as far as annual maxima data are concerned. Besides, the boxplots of Fig. 2 also reveal that the variability of b among the grid nodes increases with the order (mean, p90, and maxima) of the annual statistic considered. This can be clearly noticed by looking at the whiskers outside the upper and lower quartiles of the boxes, and leaves room for a further consideration: annual maxima data are more dispersed with respect to the annual lower percentiles, since the latter are influenced by mild sea states which, being more frequent, matter the most. This affects the estimations of b which show a higher variability for the annual maxima than for the other two statistics.
The aforementioned considerations are confirmed by the spatial distribution of b in the MS, as reported in Fig. 3. In this case, positive and negative values of b were linearly scaled in the [0,1] and [− 1,0] ranges, respectively, to better compare the trends location rather than the trends magnitude. To this end, we used the general formula:
$$ b_{s} = L + \left[\frac{b-b_{\min}}{b_{\max}-b_{\min}}\right]\times(U-L) $$
(5)
where U and L denote the upper and lower values of the desired range, respectively; \(b_{\min \limits }\) and \(b_{\max \limits }\) indicate the minimum and the maximum values in the dataset of b, respectively.
Similarities can be appreciated in the spatial distribution of upward trends for \(H_{s}^{mean}\) and \(H_{s}^{p90}\) (panels A and C) and \(T_{m}^{mean}\) and \(T_{m}^{p90}\) (panels B and D), where positive values of b mainly characterize two areas in the Aegean Sea and close to the strait of Gibraltar. On the other hand, the most negative trends for \(H_{s}^{mean}\) and \(H_{s}^{p90}\) are located around the Balearic islands, in the southern part of the Ionian Sea and in the South-East Mediterranean basin around Cyprus. As for Tm, the lowest values of b (i.e., the strongest downward trends) are found in the west coast of Greece and Crete. The weakest trends for annual mean and annual 90th percentile series of both Hs and Tm are located in the Adriatic Sea, the North Tyrrhenian Sea, the Aegean Sea (but a small area characterized by positive trends, as previously highlighted) and east of the Tunisia coastline. When annual maxima are taken into account, the spatial distribution of expected trends becomes much more irregular (panels E and F for \(H_{s}^{\max \limits }\) and \(T_{m}^{\max \limits }\), respectively). Positive trends are scattered over isolated spots in the Adriatic Sea, the Ionian Sea and off the south coast of Tunisia, while no broad areas uniformly characterized by significantly low values of b can be detected.
Figure 3 allows assessing where trends characterized by different magnitude are most likely expected to take place, but it does not provide any information about their significance. To this end, the reliability of the b estimates was further evaluated looking at their 90% confidence interval, and coupling this information with the values of pMK computed on the respective time series, as explained in Sect. 2. Results related to the Hs annual statistics are presented in Fig. 4, while results related to the Tm annual statistics are presented in Fig. 5. The locations showing a change of sign between the upper and the lower confidence intervals of b are underlined in the leftmost side of the figures.
As regards \(H_{s}^{mean}\) and \(H_{s}^{\max \limits }\), the areas characterized by non-significant trends are similarly located. In particular, attention is posed to the strait of Gibraltar and the Aegean Sea, being characterized by widespread marked areas (panels A and C of Fig. 4). A majority of the time series within such areas are in turn characterized by close-to-1 values of pMK, indicating the absence of a significant trend. A similar result characterizes the upward trends of \(H_{s}^{\max \limits }\) (panels E and F of Fig. 4). In fact, positive values of b in the Adriatic and in broad areas to the east of Tunisia and the Ionian Sea, are associated to change of sign between the respective b− and b+. A clear correlation exists also when negative trends are considered, that is, the values of b closest to 0 occur jointly with high values of pMK and no significant trends (e.g., in the southeast Mediterranean basin, the areas to the east of the Balearic islands, and in the North Tyrrhenian Sea).
As regards the time series of Tm (Fig. 5), most of the positive trends in the Aegean Sea are still found to be not significant in case of annual mean (panels A and B) and annual 90th percentile (panels C and D). An exception is the area close to the Strait of Gibraltar, where positive trends of \(T_{m}^{mean}\) are significant, while in case of \(T_{m}^{p90}\) a majority of the locations are characterized by negligible trends. For both the statistics considered, trends in the North-Tyrrhenian east of Corsica and Sardinia are generally not relevant, and these are indeed related to low values of b (reference is made to panels B and D of Fig. 3). When the time series of \(T_{m}^{\max \limits }\) are considered (panels E and F), the locations showing not significant trends are concentrated in the Adriatic Sea, the Ionian Sea, the Aegean Sea, and the southeast basin of the Mediterranean Sea.
Given that the wave model was only forced by the RCM wind data, the trends previously highlighted are to be related to expected variations in the atmospheric circulation patterns. The MS is an enclosed basin at the mid-latitudes; thus, there is no environmental forcing affecting the wave climate as much as the wind (for instance, swells generated elsewhere or variation in the ice coverage). Overall, the results presented so far indicate a widespread decrease in the future wave heights and periods over the MS, and this seems to be consistent with expected decreases in the surface wind speed in this area, as shown for example by Mori et al. (2013) and Casas-Prat et al. (2018) (even though the latter considered a shorter period for the future projection, i.e., 2081–2100).
It can be clearly noticed that, when annual maxima are taken into account, the number of locations where trends are not significant increases dramatically (panels E and F in Figs. 4 and 5). This is due to the fact that the most negative trends are concentrated in isolated spots (as shown in panels E and F of Fig. 3); thus, it is difficult to detect homogeneous behaviors at basin level, compared with \(T_{m}^{mean}\) and \(T_{m}^{p90}\). Moreover, annual maxima refer to single instantaneous values per year, resulting in noisier datasets. Even though the methodology of Sen (1968) and Theil (1992) is sound with respect to possible outliers, this affects the computation of the confidence intervals, often leading to not significant trends.
In addition to the analysis so far described, as a compulsory step in the evaluation of the significance of the projected changes, we looked at the consistency of the trends related to the each single ensemble member. To this end, we computed the number of members resulting in concordant/discordant values of b for all the time series analyzed. Such number ranges from 7−, meaning that all the models present negative trends, to 7+ (all the models presenting positive trends). Results are shown in Fig. 6.
A small stripe of fully consistent positive trends is found in the South Aegean Sea for \(H_{s}^{mean}\) and \(H_{s}^{p90}\), which exactly overlap with the locations characterized by significant upward trends according to panels A, B, C, and D of Fig. 4. When annual maxima are considered, there are instead no locations characterized by positive trends according to all the members. On the contrary, broad areas of fully consistent negative trends are placed in the South Tyrrhenian, and off the southern coastlines of Spain up to the westmost side of the Mediterranean Sea, consistent with results reported in panels E and F of Fig. 4. In case of \(T_{m}^{mean}\), a remarkable correlation can be noticed between the areas close to the Strait of Gibraltar, being characterized by significant positive trends according to both the approaches. Extending the analysis to \(T_{m}^{p90}\), a wide area east of Sardinia shows poor agreement among the ensemble members (panels C and D of Fig. 6), and this is exactly corresponding to the areas characterized by not significant trends underlined in panels A, B, C, and D of Fig. 5. Again, results related to the time series of annual maxima show a much more irregular spatial distribution (see panels E and F of Fig. 6).
As a further step, b was computed on the series of the 2006–2100 annual statistics averaged over the entire MS (i.e., across all the hindcast nodes taken into account), to derive an insight on the expected variation in the regional future wave climate. The ensemble values of regional b were carried out in 2 ways: (a) computing the slope of the ensemble values of the wave parameters. This estimate is hereinafter referred to as br1; (b) computing the slope separately for each ensemble member, and then averaging on the ensemble members. This estimate is hereinafter referred to as br2.
Figures 7, 8, and 9 show the time series of the regional Hs statistics, while Figs. 10, 11, and 12 show the regional statistics relative to Tm. In all the panels, the whole period covered by the models (1970–2100) is shown, along with the historical series computed from the hindcast developed and validated by Mentaschi et al. (2015). It can be seen from the orange lines in Figs. 7, 8, 9, 10, 11, and 12 that the hindcast series fall within the envelopes of the simulated ensembles for both Hs and Tm. Accordingly, the trends computed on the hindcast series are within the ranges provided by the estimates of b related to the ensemble members and summarized in Table 4 (see the Appendix), and serve to confirm the validity of the simulation models in producing reliable future wave climate projections. Results of br1 and br2 are reported in Table 3.
Table 3 Comparison between the values of br1 (trend on the parameters averaged across the members) and br2 (mean of the single members trends) for different parameters and annual statistics averaged over the whole Mediterranean Sea From Figs. 7, 8, 9, 10, 11, and 12, it is clear at a glance that the regional time series of the investigated parameters are characterized by a downward trend. This finding can be also verified by the ITA, performed over the mean data of the regional averaged statistics (i.e., the black lines in Figs. 7, 8, 9, 10, 11, and 12) for two different time periods, 2010–2040 and 2070–2100. Results are shown in Figs. 13 and 14. The ITA on the regional \(T_{m}^{mean}\), \(T_{m}^{p90}\), and \(T_{m}^{\max \limits }\) series reveals negative trends, as the scatters of the 2010–2040 period lie below the no-trend line. Similarly, the series of regional \(H_{s}^{mean}\) and \(H_{s}^{p90}\) show negative trends. As for the series of regional \(H_{s}^{\max \limits }\), the ITA seems to be noisier, with a couple of data being above the no-trend line, though a clear negative trend can be pointed out also in this case.
Differences in the trends according to the ITA can be better appreciated by looking at the ecdf of δ, as shown in Fig. 15. Here, it can be noticed how the δ related to the annual maxima are generally farther from the zero line with respect to the other statistics, indicating a stronger negative trend. Similarly, the 90th percentile series are characterized by a trend more negative than the mean data. This particularly applies to the series of regional Hs (panel A), while in case of Tm the ITA trends are more similar among the investigated statistics (panel B). Such finding is further supported by the results of br1 and br2, which show values increasing from the annual mean to the annual maxima.
These results show also that the differences between br1 (trends computed on the parameters averaged across the models) and br2 (trends computed as mean of the single model trends) are small. Indeed, the highest difference is observed for the time series of regional \(H_{s}^{\max \limits }\), where using br1 (br2) would result in a variation of − 24.4 cm (− 27.3 cm) between 2006 and 2100.
Finally, the mean direction \(\theta _{m}^{mean}\) was considered. The trend analysis was carried out on the time series averaged over the whole MS, as shown in Fig. 16. Here, all the ensemble members agree on a slight clockwise shift.
A more detailed analysis was performed on six locations selected among different sub-basins of the MS, plotting the \(\theta _{m}^{mean}\) series versus the respective years on a polar plot, along with the best linear fit. The investigated locations and the correspondent polar plots are shown in Figs. 17 and 18, respectively.
On average, an eastward trend seems to characterize the wave directions at locations A, D, and E, corresponding to the Gibraltar, South Mediterranean, and the Aegean basin, respectively. Conversely, no trends can be highlighted at location B (North Tyrrhenian Sea) and F (in front of the Nile Delta), while at location C scatters seem to be too dispersed to derive a reliable trend analysis, even though an anti-clockwise shift can be pointed out. This is most likely due to the local climatology of the Adriatic Sea, which is influenced by several local circulation patterns, resulting in no prevailing fetches for the point at hand.