For the North-East Atlantic, the variations of normalized M2 amplitude are presented in Fig. a.

The first result is that since 1910, the variations show similar patterns at all the stations; M2 amplitude decreases up to the 1960s, then increases, and decreases again since the 1990s. This suggests that these changes are probably due to large-scale processes, rather than local effects due to changes in the environment (e.g. harbour development, dredging, siltation) or instrumentation errors. The similar patterns between Brest and Cuxhaven may be surprising, as Cuxhaven is located in the North Sea, and not in the open Atlantic Ocean, and far away from Brest, around 1300 km. This indicates that the spatial scale of the processes responsible for these changes must be at least as large as the North-East Atlantic. Different authors have noticed the increase of tidal range from 1960 to 1990 in the southern North Sea. found a gradual increase during the period 1955–1980 at all the stations of the Dutch coast (five stations including Hoek van Holland) and the German coast (seven stations). found a significant increase in M2 amplitude at Cuxhaven since around the mid-1950s. Note that Cuxhaven is located in the German Bight; shallow depths and the shape of the coastline may induce some amplification. Variations in M2 at Cuxhaven could therefore be sensitive to local effects, such as the migration of the underwater channels and the evolution of the tidal flats . Moreover, Cuxhaven is located in the Elbe estuary, and some river engineering works, such as narrowing and deepening, may induce tidal amplification .

Before 1910, normalized M2 values are higher at Brest than at Delfzijl. The construction of dykes that have gradually closed the harbour of Brest since the end of the 19th century may have altered the tide at Brest. The high values before 1910 may be due to local changes, in addition to large-scale changes. To go further, the potential role of these successive constructions needs to be investigated . made a first attempt to evaluate the influence of reducing the width of access to the harbour but did not take into account a potential role of dredging, for which we have no information. This example underlines the complexity of interpretation of the variations when changes of local and large-scale origin occur at the same time. Note that in the following, we focus mainly on the 20th century, as most of the stations start after 1900 (15 out of 18 stations).

The second result is that there is no obvious linear trend in M2 variations, but rather break or change points, M2 increasing and then decreasing, depending on the periods considered. Overall, M2 decreases from 1910 until 1960, increases again until 1980–1990, to finally decrease since 1990; note that the curve flattens between 1920 and 1940. already noticed these variations at Brest and Newlyn and suggested a long-period oscillation of around 140 years, rather than a steady secular trend. A careful analysis of the harmonic development of the tidal potential showed that no tidal component could explain this oscillation. Similarly, no linear combination of tidal harmonic components could explain it . This indicates that these variations are not due to an astronomical component. However, in contrast to Brest, M2 at Delfzijl stays flat between 1880 and 1920. The decrease observed at Brest between 1880 and 1920 may be due to harbour development and/or dredging (see above). This underlines the importance of sea level data archaeology, for research studies related to long-term changes .

The third result is that changes in M2 have not the same order of magnitude at each station (see Fig.  in Appendix  for time series of M2). Note that Fig.  represents normalized M2, i.e. removing the average and dividing by the standard deviation. The order of magnitude of unnormalized M2 changes is roughly the same at Brest and Newlyn (standard deviations of 0.9 and 0.8 cm respectively, Table , column 5), but more than three times larger at Cuxhaven (standard deviation of 3.7 cm), and even larger at Delfzijl (standard deviation of 7 cm). This suggests that the North Sea may be more sensitive to the processes responsible for these changes. Note also that the environmental setting of Cuxhaven and Delfzijl in the Elbe and Ems estuaries, respectively, could introduce some amplification .

For the North-West Atlantic, the variations of normalized M2 amplitude are presented in Fig. b and c. The first feature is that M2 amplitude varies differently in the North-West and in the North-East Atlantic. The second is that there are discrepancies between stations, even when close to each other (e.g. Atlantic City and Lewes). We split the stations into two groups, in order to facilitate the detection of patterns, each being consistent in terms of trends: one with positive trend (group 1 in Fig. b), the other one with negative or no trend (group 2 in Fig. c).

The first group (with positive trends) consists of nine stations (Fig. b). Three outcomes can be highlighted. The first is that M2 amplitude has increased overall since 1900. However, between 1980 and 1990, all the stations slightly decrease, and since 1990 they have increased again. The second outcome is that the rate of increase is very different from one station to another (keeping in mind that M2 is normalized by standard deviation in Fig. ). Portland is increasing 1.4 times faster than Charleston (standard deviations being respectively of 1.82 and 1.33 cm) and 28 times faster than Key West (standard deviation being only 0.36 cm at Key West). The large increase in Portland may be explained by some amplification in the Gulf of Maine. In many semienclosed basins, resonance leads to tidal amplification . In the Gulf of Maine, reported that the tides in the Gulf are in resonance, with a natural resonance frequency close to the N2 tide . Tides may be then very sensitive to any changes in the environment (e.g. basin configuration – shape, depth – but also external forcing). The third outcome, and probably the most interesting one, is related to the values of M2 at Eastport, Portland and New York in the 1860s, estimated from and , and represented (after normalization) as stars in Fig. b. These values are not consistent with the positive linear trends observed since 1900, which provides some consistency with the hypothesis formulated from the analyses of the data prior to the 20th century in Fig. a: long-term variations introduce some breaks or change points, M2 increasing and then decreasing, depending on the periods considered. The decrease observed between the 1870s and 1920s at the four stations (Brest, Eastport, Portland, New York) suggests a possible large-scale signal, in addition to local processes.

The second group (with negative or no trend) consists of four stations (Fig. c). Two points can be highlighted. The first is that M2 decreases overall for Halifax, Newport and Lewes. This is less clear for Atlantic City, which is quite noisy and shows no significant trend. The second point is that at Halifax, M2 values in 1896–1897 are higher than those after 1920. This suggests that the decrease may have started before the 20th century. However note that at Halifax, there is a long gap in the data recording (1898–1919), which raises the possibility of an instrumentation origin in the observed decrease of the M2 amplitude.

We estimated the trends for M2 amplitude at each station, using linear regression. We computed the trends over two periods: 1910–2018, which corresponds roughly to the whole period of data (only five stations start before 1910), and 1990–2018, which corresponds to recent decades. Some tests showed that the later results were not very sensitive to the start date (moving 1990 to 1985 or 1995). The trend uncertainties were estimated considering the noise content in the time series using SARI software . The noise was modelled as a white plus power-law noise, whose spectral index was found to be close to -1 (flicker noise). The results are summarized in Table  (columns 7 and 8) and Figs.  and .

The trends estimated since 1910 vary significantly from one station to another (Fig. ). They are positive overall (up to 2.5 mm/yr at Wilmington), which is consistent with previous findings . They are slightly negative at three stations (Halifax, Newport, Lewes), and one station shows no trend (Atlantic City). The estimates are statistically consistent with those found previously by different authors (e.g. 0.14 ± 0.09 mm/yr at Newlyn compared to 0.19 ± 0.03 mm/yr in , 0.56 ± 0.06 mm/yr in Portland, compared to 0.59 ± 0.04 mm/yr in ). Note that our error bars are larger, because we considered the noise content in the time series as a white noise plus power law noise (we obtained the same error bars considering white noise only). In the North-East Atlantic, the trends are consistent with each other (in terms of sign), which is not surprising as the stations vary similarly (Fig. a).

The largest trends since 1910 are mainly observed in semi-closed basins: Wilmington in the Cape Fear River estuary, Delfzijl in Ems estuary, Cuxhaven in Elbe estuary, and Eastport and Portland in the Gulf of Maine. This suggests a possible amplification due to resonance effects (e.g. Gulf of Maine) and/or propagation in shallow waters (e.g. Cuxhaven), in addition to local effects. The stations located in estuaries or in a harbour with a channel may have been subject to dredging. Channel deepening increases the water depths, which reduces the effective drag and leads to tidal range amplification. This effect may be particularly large in estuaries and may explain the larger trends at Wilmington and Delfzijl. Finally, the shifting locations of amphidromic points could also play a role . In the North Sea, different authors show a possible migration of the present-day amphidromes, under a 2 m sea-level rise scenario .

The trends estimated since 1990 are quite different from those estimated since 1910 (Figs.  and ), with more stations with negative trends: 9 stations out of 18 have post-1990 negative trends, whereas only 3 stations out of 18 have post-1910 negative trends (Table , columns 7 and 8). In the North-East Atlantic, they all switch from positive to negative trends. This underlines (1) some spatially coherent changes in recent decades and (2) the difficulty in estimating long-term trends from short records (i.e. less than 30 years), especially if the data are noisy (interannual variability) and the underlying processes non-linear (change points).

The trends have to be interpreted very carefully as the M2 variations are not linear and may increase or decrease depending on the years; as a consequence, the estimated trends depend strongly on the period considered to estimate it. The interannual variability also plays an important role, and when substantial, trends can vary depending on the computational period. For example, at Cuxhaven, the large interannual variability leads to a large uncertainty on the trend computed since 1990 (-0.47±0.78 mm/yr).

The MSL rise could partly explain M2 changes. Simulations show that MSL rise can result in an change of M2 up to ±10 % of the rise . show that the sign of the observed M2 trend is correctly reproduced at 80 % of the tide gauges on a global scale, but their simulated trends tend to differ from observations by a factor of 3 to 5; i.e. their simulations underestimate the M2 response to MSL rise in terms of magnitude. conclude that “magnitudes of observed and modeled M2 trends are within a factor of 4 (or less) from each other in nearly 50 % of the considered cases”. The large discrepancies between the simulations and the observations strongly suggest that MSL rise is not the only process that may explain M2 changes – other large-scale processes, in addition to local processes, may also play a role.

Figure shows the annual MSL, after removing the average over the period 1910–2010 and filtering with 9-year time windows. The correlations between M2 and MSL indicate that M2 varies strongly with MSL (see Sect. ). However, M2 variations show some variability in the North-East Atlantic (Fig. a), which may not be explained with MSL rise alone.

Processes other than MSL rise may impact the tide (see Sect. ), such as the atmospheric circulation and the ocean stratification. Ocean and atmosphere are fully coupled, and air–sea fluxes are responsible for the exchange of momentum, water (evaporation and precipitation budget) and heat at their interface. Among the wide range of possible interactions, two mechanisms have been explored for their ability to modify the tide: (1) the momentum flux (wind stress) and the gradient of sea level pressure which act on the barotropic tide and (2) the water and heat fluxes which induce changes in both temperature and salinity distribution in the ocean. The latter effect acts on the stratification, which in turn could impact the tide in two different ways. The first way is the internal tide generation which transfers energy from barotropic and baroclinic motion and modifies surface tidal expression . However, in the present study, most of the observations come from coastal stations sheltered by wide continental shelves which dampen internal waves. More important is the second way: the stratification acts on the eddy viscosity profile by modifying current profiles and bottom drag over continental shelves, which in turn modifies the M2 surface expression .

Here, we focus on the effect of the atmospheric circulation on the tide. We used pressure indices (NAO and AO) that are relevant to represent atmospheric circulation. The NAO index represents the difference of normalized sea level pressure between the Azores high pressure system and the Iceland low pressure one . It indicates the redistribution of atmospheric masses between the subtropical Atlantic and the Arctic . In the North-East Atlantic, the similarity between the variations of the low-frequency winter NAO index and those of M2 (Fig. ) suggests a possible impact of large-scale atmospheric circulation on the tide. The NAO index varies from positive to negative phases. Filtering the interannual variability, the NAO index tends overall to decrease between 1910 and 1970, then increase until 1990, and once again decrease. In the same way, M2 amplitude tends to decrease up to 1960, then increase until 1990, and once again decrease. These similar patterns raise a possible connection between NAO and M2 variation, already mentioned by on the basis of qualitative criteria. In the following, we provide quantitative insights into the possible influence of NAO.

We computed the correlations (r value) between normalized M2 and climate indices, NAO and AO (Fig. ). M2, NAO and AO are filtered using the same time window (9 years). The correlations are computed since 1910, to have similar periods for all the stations. The correlations are considered as significant only if the p value is lower than 0.05 (95 % significance level). (Note that other statistics to measure the degree of association between the M2 and NAO (AO) quantities would be worth exploring, for instance, nonlinear association using Spearman’s correlation coefficient. In this respect, our study should be regarded as a first step that identifies sites worth considering in future investigations, especially investigating causal relationships with physics-based modelling.) The results are the following: (1) for NAO, 14 stations out of 18 show significant correlation. Note that at Brest, the correlation is significant since 1910, but not since 1864 (the NAO index used in this study starts only in 1864). This can be explained by the M2 larger amplitude over all the 19th century, which decreases between 1890 and 1910 (Fig. a), possibly due to harbour development and construction of dykes (see Sect. ). (2) In the North-East Atlantic, all the stations are positively correlated with NAO. (3) The strongest correlations (i.e. greater than 0.5) are in the northern part of the North Atlantic, with strong positive correlations at Cuxhaven and Hoek van Holland and strong negative correlation at Halifax (-0.55). (4) For AO, we found similar, but overall larger, r values. This is not surprising as these two indices are closely related.

To go further in the relative contribution of MSL and NAO in M2 variability, we fitted two linear regression models on M2 variations. In the following, M2, MSL and NAO are filtered over 9-year time windows and normalized. At all the stations, we fitted M2 variations with a MSL linear regression model (model 1) and a MSL and NAO multiple linear regression model (model 2). Models 1 and 2 may be expressed as 2Model1=α1MSL3Model2=αMSL+βNAO.

The correlations between M2 and model 1 (MSL) and model 2 (NAO and MSL) are presented in Fig. . We checked if there was correlation between NAO and MSL at the stations: there is no correlation at six stations, and r value is between 0.2 and 0.6 at eight stations; see Fig.  and discussion below. The results are the following: (1) M2 varies at first order with MSL (Fig. ). (2) The introduction of the NAO (model 2) allows increasing the predictive performance of the model, beyond the inherent effect of adding an additional regression parameter. Indeed, on average, the Akaike information criterion (AIC) is 99.9 for model 2, instead of 112.7 for model 1. On average, the r2 value is 0.67 for model 2 instead of 0.61 for model 1. At some stations, the increase is quite large. For example at Cuxhaven, the r2 value jumps from 0.42 to 0.64 between model 1 and 2. (3) The ratio βα+β represents roughly the relative contribution of the NAO compared to the total effect of MSL and NAO (Fig. ), as MSL and NAO are normalized. We found a significant contribution at some stations (e.g. more than 30 % at Cuxhaven and Halifax), whereas it is negligible at others (e.g. only 5 % at Portland). A total of 8 stations out of 18 show large NAO contribution (>20 %). The North-East Atlantic seems to be more sensitive to the NAO. Note that the interpretation of the results is tricky when MSL–NAO correlation is significant (orange bars in Fig. ). For example, at Hoek van Holland, the relative NAO contribution is very small, mainly because MSL and NAO are highly correlated (r=0.59). Figure  shows M2 variations along with the predictions from the two models, at all four stations where the NAO contribution is significant (βα+β>0.25), and the correlation between M2 and model 2 is large enough (r>0.3). At Cuxhaven, Halifax and Key West, model 2 (MSL- and NAO-dependent) naturally captures the M2 variations better than model 1 (MSL-dependent); at Brest, the improvement is less significant. The trend switch observed since 1990 in the North-East Atlantic could be partly explained by the influence of the NAO on the tide.

These results suggest that a NAO-related mechanism may explain part of the variability of M2. As mentioned by , “it is shown that sea‐level, sea surface temperature and Arctic ice thickness are correlated with the NAO index. Thus, changes in the dynamics of the atmosphere could affect both M2 and S2 tides by processes discussed under (1), (2) and (3).” An underlying mechanism linked with (2) – sea surface temperature – could be changes in the ocean stratification. This is one of main possible hypotheses invoked in to explain secular changes in M2 amplitude in the Gulf of Maine; this is also the main hypothesis in and to explain seasonal modulation of M2 in the North Sea. The relationship between the NAO index and stratification is complex and spatially variable across the North-East Atlantic . In the North Sea, the sea surface temperatures are positively correlated with NAO , while subsurface temperatures show no significant correlation with NAO . Stratification could therefore be (positively) correlated with NAO, but a dedicated study, outside the scope of this paper, would be necessary. Another underlying mechanism linked this time with (1) – sea level – could be the difference of spatial distribution of water level, due to different sea-level pressure and wind stress patterns. This is the hypothesis invoked in Huess and Andersen (2011) to explain the seasonal modulation of M2 in the North Sea. They ran a barotropic model, forced with tides only and with both tides and meteorological fields; their results show that the M2 seasonal modulation is better captured when the model is forced with both tides and meteorological fields rather than with tides only. Figure a shows the average sea-level pressure during the period 1850–2015, derived from the Twentieth Century Reanalysis (20CR) . A positive NAO winter (e.g. 1989) corresponds to a situation with a stronger pressure gradient than average, between the two pressure systems of Azores and Iceland (Fig. c). By contrast, a negative NAO winter (e.g. 1969) corresponds to a weaker gradient pressure than usual (Fig. b). (We define winter here as December–February.) This way, from one year to another, the large-scale atmospheric masses are distributed differently, and as a consequence, the water volumes are also distributed differently in the North Atlantic. In a situation of NAO+, the surface waters are pushed onshore by westerly winds, moving from Iceland to the European coasts of France, Spain and Portugal. Figure  shows the redistribution of the sea-level pressure, between two years with high and low NAO indices (here 1989 and 1969). Note that this is an extreme situation, as these years have strong positive and negative indices. Assuming an inverse barometer response of sea level, the changes in terms of water level may vary from more than 24 cm in the northwestern part of the area to around -12 cm in the region that includes most of the North-East Atlantic tide gauges considered in this study. This variation of a few tens of centimetres is probably negligible offshore but may have some impact on tide propagation along the continental shelves and in shallow waters. It could also shift slightly the amphidromic points. Assuming that these changes have a similar impact (in terms of magnitude) on M2 as MSL changes, that is, ±10 % in shallow waters according to recent simulations , we find that they can yield centimetric changes in M2 amplitude. In other words, their order of magnitude is roughly in agreement with the changes observed in M2 (Table 1). However, it is difficult to disentangle the effects of stratification and meteorological forcing (sea-level pressure and wind stress) in M2 changes, and possibly both mechanisms coexist. Dedicated simulations should be conducted to assess the effects of atmospheric forcing on M2 variability.