Climate-scale changes of the semidiurnal tide over the North Atlantic coasts from 1846 to 2018

. We investigated the long-term changes of the principal tidal component M 2 over the North Atlantic coasts, from 1846 to 2018. We analysed 9 tide gauges with time series starting no later than 1920. The longest is Brest with 165 years of observations. We carefully processed the data, particularly to remove the 18.6-year nodal modulation. We found that M 2 variations are consistent at all the stations in the North East Atlantic (Newlyn, Brest, Cuxhaven), whereas some discrepancies 5 appear in the North West Atlantic. The changes started long before the XX th century, and are not linear. The trends vary from a station to another; they are overall positive, up to 0.7 mm/yr. Since 1990, the trends switch from positive to negative values. Concerning the possible causes of the observed changes, the similarity between the North Atlantic Oscillation and M 2 variations in the North East Atlantic suggests a possible inﬂuence of the large-scale atmospheric circulation on the tide. We discuss a possible underlying mechanism. A different spatial distribution of water heights from one year to another, depending 10 on the low-frequency sea-level pressure patterns, could impact the propagation of the tide in the North Atlantic basin. However, the hypothesis is at present unproven. It is based on the dif- 130 ference of normalized average winter sea-level pressure between Lisbon (Portugal) and Stykkisholmur/Reykjavik (Iceland). The normalization consists of removing the long-term mean (1864–1983) and dividing by the long-term standard deviation. Other also affected. Results show that retrieved SONEL analysis program MAS by the French Hydrographic Ofﬁce (SHOM). The authors warmly thank P. Woodworth for his helpful comments.

Time series beginning After 1960After 1920After -1960After 1900After -1920 Before 1900 < 80 years of data > 80 years of data The main characteristics of the 9 selected stations are synthetised in Table 1. Among them, only Brest and Halifax started in 60 the XIX th century, respectively in 1846 and 1896 (Table 1, column 2). The number of years with data for each station varies between 85 and 165 years, Brest being the longest time series (Table 1, column 3).

Data processing
Harmonic analysis was performed to compute the M 2 amplitude. We used the MAS program (Simon, 2007(Simon, , 2013, developped by the French Hydrographic Office (SHOM). This program gives results similar to T_Tide harmonic analysis toolbox (Pawlowicz et al., 2002), largely used in the scientific community. For instance, Pouvreau et al. (2006) found non-significant differences on the yearly amplitudes of M 2 at Brest over the period 1846 to 2005 using T_Tide or MAS. Hourly time series were analysed yearly. We processed only years with at least 180 days, considering that six months was long enough to compute correctly M 2 (Pouvreau et al., 2006). This constraint resulted in excluding between 1 and 9 years, depending on the station (Table 1, columns 3 and 4). Note that M 2 is affected by a seasonal variation of a few percent (Huess and Andersen, 2001; 70 but would lead to exclude more years.
We carefully retrieved the nodal modulation of M 2 amplitude (Simon, 2007(Simon, , 2013. Here is a short description of the method. The M 2 component is subject to a 18.6-year modulation, when poorly separated from a neighboring component. Indeed, M 2 75 is very close in terms of frequency to another component (m 2 ) whose Doodson number differs only from the 5 th figure (255 555 and 255 545 for M 2 and m 2 , respectively). This 5 th figure corresponds to N , the opposite mean longitude of the Moon ascending node -hence the "nodal" term -whose period is 18.6 years. Note that there is also another component close to M 2 , whose Doodson number differs only from the 5 th figure (255 565), but it is negligible as its amplitude in the tidal potential is only 0.05% of M 2 , whereas m 2 amplitude is 3.7 % of M 2 (Simon, 2007(Simon, , 2013. With one year of hourly data, 80 the two components M 2 and m 2 are not correctly separated with a harmonic analysis (at least 18.6 years are necessary). As a consequence, M 2 amplitude is modulated by m 2 . However, we can estimate this modulation, and remove it. The harmonic formulation is expressed as a sum of harmonic components where h(t) is the sea level height at time t, V i (t) is the astronomical argument (computed from Doodson number) and a i , κ i 85 the amplitude and phase shift of each component. Considering that M 2 and m 2 are very close in terms of frequency, we can assume that their phase shift are similar (κ M 2 κ m2 ). As their difference of astronomic arguments is the M 2 and m 2 contributions to the total water level may be expressed as where f nod , the nodal modulation, is the ratio of the amplitude of m 2 and M 2 . As M 2 and m 2 are very close in terms of 90 frequency, f nod is generally considered as close to the ratio of their amplitude in the tidal potential, A m2 and A M 2 The opposite of the mean longitude of the Moon ascending node is simply expressed as a function of time (p . 116 in Simon The tidal program we used (MAS) corrected M 2 applying the usual 3.7% nodal modulation (Eq. (3)). However, this value may vary significantly from a station to another; Ray (2006) reported values ranging from 2.3 % to 3.6 % in the Gulf of Maine. Here, we computed directly f nod from the observed data, proceeding as follows.
(2) We detrended the obtained signal removing the last Intrinsic Mode Function (IMF) of an Empirical Mode Decomposition (EMD) (Huang et al., 1998); note that the EMD is an analysis tool which partitions a series into 'modes' (i.e. IMFs), the last one being the trend of the signal. (3) We fitted a function a m2 cos(N + π) on this detrended signal to estimate a m2 , N being expressed as in Eq. (4). (4) We finally computed f nod as the ratio between m 2 and M 2 amplitudes (Eq. (3)). Figure 2 (a) shows an example of estimate of M 2 modulation at Newlyn: the fit leads to a nodal 105 modulation of 3.3 %. Note that this value is consistent with Woodworth (2010) (3.2 %), whereas Woodworth et al. (1991) gave a slightly different value (2.8 %). Figure 2 (b) shows the impact of this value rather than the default one: oscillations of 18.6 years are clearly reduced. Note that in this study, the m 2 amplitude -and then the nodal correction -could have been computed from the full time series harmonic analysis, as records are longer than 18.6 years. However, the method presented here to compute the nodal correction, can be applied even for time series shorter than 18.6 years.
Only the value at Charleston differs significantly -3.0 % in our study compared to 3.7% in Müller (2011).  1910,2010] σ M2 [1910,2010] the average M 2 and standard deviation σ M2 over the 1910-2010 period being given in Table 1 (column 6). The idea is to scale the data, in order to compare all the stations together.

Climate indices
We investigated the correlation between secular changes in the tide and climate indices, such as the North Atlantic Oscillation (NAO) or the Arctic Oscillation (AO) -also called Northern Annular Mode (NAM). Climate indices are related to the distribution of atmospheric masses. They are based on the difference of average sea-level pressure between two center of actions (i.e. stations), at large time scale (e.g. monthly, seasonal, annual).

125
The NAO is the major pattern of weather and climate variability over the Northern Hemisphere (Hurrell, 1995;Hurrell and Deser, 2009). Variations of NAO are essential, as they drive the climate variability over Europe and North America (Hurrell et al., 2003). We used the wintertime (December to March) Hurrell station-based NAO Index (retrieved from https://climatedataguide.ucar.edu/climate-data/hurrell-north-atlantic-oscillation-nao-index-station-based). It is based on the dif-130 ference of normalized average winter sea-level pressure between Lisbon (Portugal) and Stykkisholmur/Reykjavik (Iceland).
The normalization consists of removing the long-term mean  and dividing by the long-term standard deviation.
The NAO index covers the period 1864-2019, with yearly values.
The Artic Oscillation (AO) is another index which resembles to NAO index. It is defined as the first EOF of northern hemi-135 sphere winter sea-level pressure data Wallace, 1998, 2000;. The AO index is highly correlated with the NAO. We used the wintertime Hurrell AO index (retrieved from https://climatedataguide.ucar.edu/climatedata/hurrell-wintertime-slp-based-northern-annular-mode-nam-index). The AO index covers the period 1899-2019.
To remove the interanual variability and estimate low frequency variations, climate indices were filtered with a 9-year median 140 filter.

Sea level pressure
We explored the gridded seasonal sea-level pressure reconstruction from 1750 to 2002, covering eastern North Atlantic, Europe and the Mediterranean area (Küttel et al. (2009), https://www.ncdc.noaa.gov/data-access/paleoclimatology-data). This 5°X5°g ridded dataset is based on ship logs and instrumental pressure series. We computed the mean winter (December to February) 145 sea-level pressure over the period 1850-2002. We averaged from 1850 rather than 1750 to be consistent with tide gauges temporal coverage. We also computed yearly anomalies, i.e. removing the average sea-level pressure.

M 2 variations
For the North East Atlantic, the variations of normalized M 2 amplitude are presented Figure 3 (a). The first result is that the 150 variations between Newlyn, Brest and Cuxhaven are very similar. This suggests that these changes are probably due to largescale processes, rather than local effects due to changes in the environment (e.g. harbor development, dredging, siltation) or instrumentation errors. The high correlation between Brest/Newlyn and Cuxhaven may be surprising, as Cuxhaven is located  Cuxhaven time series starting only in 1915 and 1918, respectively, do not allow to confirm at large-scale the decrease observed at Brest from 1880 to 1920. This underlines the importance of sea level data archaelogy, for research studies related to long-165 term changes Ray and Talke, 2019;Bradshaw et al., 2015Bradshaw et al., , 2020. The third result is that changes in M 2 have not the same order of magnitude at each station, even if trends are similar. Note that Figure 3 represents normalized M 2 , i.e. removing the average and dividing by the standard deviation. The order of magnitude of (not normalized) M 2 changes are roughly the same at Brest and Newlyn (standard deviations of 0.9 and 0.8 cm, Table 1, column 6), but more than three times larger at Cuxhaven (standard deviation of 3.7 cm). This suggests that Cuxhaven may be more sensitive to the processes 170 responsible for these changes and/or that the environmental setting of Cuxhaven in a semi-closed basin could introduce some amplification (e.g. resonance effects, propagation in shallow waters).
For the North West Atlantic, the variations of normalized M 2 amplitude are presented on Figure 3 (b) and (c). We split the stations in two groups, in order to facilitate the detection of patterns. The first feature is that M 2 amplitude varies differently 175 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 in two groups, each being consistent in terms of trends: one with globally positive trend, the other one with globally negative or no trend.
The first group in the North West Atlantic consists of Portland, Charleston and Key West (Figure 3 (b)). Three outcomes 180 can be highlighted. The first is that M 2 amplitude globally increases since 1900. However, between 1980 and 1990, the three stations slightly decrease and since 1990, only Portland is still increasing significantly. The second outcome is that the rate of increase is very different from a station to another: 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 of 0.36 cm at Key West). The very slow increase at Key West is due to a small tidal amplitude (i.e. only 17.5 cm of mean amplitude for M 2 , see 185 Table 1, column 6). The large increase in Portland may be explained by some amplification in the Gulf of Maine. Ray and Talke (2019) reported that the tides in the gulf are in resonance, with a natural resonance frequency close to the N 2 tide (Garrett, 1972;Godin, 1993). Tides may be then very sensitive to any changes in the environment (e.g. basin configuration -shape, depth -but also external forcing). The third oucome, and probably the most interesting one, is the value of M 2 at Portland in 1864-1865 (134.1 cm), estimated from Ray and Talke (2019), and represented (after normalization) as a blue star on Figure   190 3 (b). This value is not consistent with the positive linear trend observed at the three stations since 1900, which confirms the hypothesis formulated from Brest analysis: climate-scale variations show some breaks or change points, M 2 increasing and then decreasing, depending on the periods considered.
The second group in the North West Atlantic consists of Halifax, Charleston and Key West (Figure 3 (c)). Two points can be 195 highlighted. The first is that M 2 globally decreases for Halifax and Lewes, particularly since 1980. This trend is less clear for Atlantic City, which is quite noisy and shows no significant trend. The second point is that at Halifax, M 2 values in 1896-1897 are higher than those after 1920. This suggests that the decrease may have started before the XX th century.

Estimated trends
We estimated the trends for M 2 amplitude at each station, using linear regression. We computed the trends over two periods:  The trends estimated from 1910 vary significantly from a station to another (Figure 4). They are globally positive (up to 0.7 mm/yr at Cuxhaven), which is consistent with previous findings (Araújo and Pugh, 2008;Ray, 2009;Woodworth, 2010;205 Müller et al., 2011;Ray and Talke, 2019). They are slightly negative at two stations (Lewes, Halifax), and one station shows no significant trend (Atlantic City). The estimates are statistically consistent with those previously found by different authors (e.g. 0.15 ± 0.02 mm/yr at Newlyn compared to 0.19 ± 0.03 mm/yr in Araújo and Pugh (2008), 0.56 ± 0.03 mm/yr in Portland, compared to 0.59 ± 0.04 mm/yr in Ray and Talke (2019)). In the North East Atlantic, the trends are consistent, which is not surprising as the stations vary similarly (Figure 3).

210
The trends estimated since 1990 are quite different from those estimated since 1910 (Figures 4 and 5), with more stations with negative trends: 6 stations (Atlantic City, Lewes, Charleston, Brest, Newlyn, Cuxhaven), instead of 2 stations (Halifax, Lewes). In the North East Atlantic, they switch from positive to negative trends. This underlines (1) some recent changes in the latest decades (Müller, 2011;Ray and Talke, 2019) (2) the difficulty to estimate long-term trends from short records (i.e. less 215 than 30 years), especially if the data are noisy (interannual variability) and the underlying processes non-linear (change points).
Note that the largest trends are observed in semi-closed basins (Cuxhaven in the North Sea, and Portland in the Gulf of Maine). This suggests a possible amplification due to resonance effects.

220
The trends have to be interpreted very carefully. The M 2 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. Mean sea level rise could partly explain M 2 changes, but is not sufficient to explain alone the secular changes in tide (Ray and Talke, 2019). Simulations show that mean sea level rise impact M 2 up to ±10% of the rise (Pickering et al., 2017;Idier et al., 2017). Changes are often of the same sign than mean sea level rise, but sometimes opposite. Figure 6 shows the annual mean sea levels at all the stations, after removing the average over the period 1910-2010 (Table 1, column 5). Mean sea level is rising steadily over all the XX th century, which is not always in line with the changes observed in M 2 amplitude, particularly in the 230 North East Atlantic (Figure 3 (a)). Moreover, global simulations with mean sea level rise suggest that M 2 could increase in the western part of the English Channel (i.e. Brest and Newlyn), and decrease in the southern part of the North Sea (i.e. Cuxhaven) (Pickering et al., 2017). Once again, this is not not supported by our observational results, as M 2 varies the same way at these three stations.

235
Note that mean sea levels obtained from tide gauges include a solid Earth component as they are referenced to the land.
Consequently, if the land is subsiding, mean sea level as observed with a tide gauge will increase (Wöppelmann and Marcos, 2016). Estimates of vertical land motion from SONEL (www.sonel.org, Santamaría-Gómez et al. (2017) ) show that the stations considered here are quite stable in the North East Atlantic (i.e. vertical land movements smaller than 0.5 mm/yr), but slightly falling in the North West Atlantic (i.e. trends between -1 and -2 mm/yr), with an exception in the Gulf of Maine, where 240 land tends slightly to rise. Note that these trends are computed on relatively short periods (i.e. generally < 15 years), making it difficult to infer robust trends over the last century.  Table 1, column 5)

Possible link with climates indices
Other processes than mean sea level rise may impact the tide (see section 1). Here, we focus on atmospheric circulation and The NAO index represents the difference of normalized sea level pressure between the Azores high pressure system and the Iceland low pressure one (Hurrell, 1995). It indicates the redistribution of atmospheric masses between the Subtropical Atlantic 260 and the Arctic (Hurrell and Deser, 2009). As the AO is highly correlated with the NAO (Figure 7), in the following, we focus only on the NAO index. In the North East Atlantic, the similarity between the variations of the low-frequency winter NAO index (Figure 7) and those of M 2 (Figure 3 (a)) suggests a possible impact of large-scale atmospheric circulation on tide. The NAO index varies 265 from positive to negative phases. Filtering the interannual variability, NAO tends globally to decrease between 1910 and 1970, then increase until 1990, and once again decrease. The same way, M 2 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 M 2 variations. Yet, this hypothesis is at present unproven. It was tentatively proposed by Müller (2011), without providing any description of the physical mechanism, however. In the following, we develop further this idea.

270
The underlying mechanism could be the difference of spatial distribution of water heights, depending on the NAO index.  (Küttel et al., 2009). A positive NAO year (e.g. 1989 corresponds to a situation with a 13 https://doi.org/10.5194/os-2020-56 Preprint. Discussion started: 15 June 2020 c Author(s) 2020. CC BY 4.0 License. stronger gradient pressure than average, between the two pressure systems of Azores and Iceland (Figure 8 (c)). By contrast, a negative NAO year (e.g. 1969) corresponds to a weaker gradient pressure than usually (Figure 8 (b)). This way, from one year to another, the large-scale atmospheric masses are differently distributed, and as a consequence, the water volumes are also differenly distributed in the Northern Atlantic. In a situation of NAO+, the waters are pushed southern, moving from Iceland to the European coasts of France, Spain and Portugal. Figure 9 shows the redistribution of the water volumes, 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 280 positive and negative indices. The impact in terms of water height may vary from -21 cm to 12 cm. This variation of a few tens of cm 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 M 2 as mean sea level changes, that is, ± 10% according to recent simulations (Pickering et al., 2017;Idier et al., 2017), we find that they can yield changes in M 2 amplitude up to a few centimeters. In other words, their order of 285 magnitude is in agreement with the changes observed in M 2 ( Table 1). The assumption is reasonable, but dedicated simulations should be conducted to confirm or discard the water volumes redistribution hypothesis. Finally, note that NAO variability results not only in sea-level pressure change, but also wind stress, air surface temperature and precipitations (Visbeck et al., 2001). Large changes in winds at the scale of the Atlantic could also play a role. Finally, we investigated the link between M 2 variations and AMO. The AMO index is defined as the average sea surface temperature in the North Atlantic, detrended to isolate the natural variability (Enfield et al., 2001). However, we did not find any 295 clear relationship. This index shows an oscillation with a period of around 70 years (Schlesinger and Ramankutty, 1994;Enfield et al., 2001). Since 1856, the lowest indices (i.e. the coldest sea surface temperature periods) were observed in 1900-1920 and 1970-1990, which is not consistent with the observed M 2 variations.

Conclusions
We investigated the long-term changes of the principal tidal component M 2 over the North Atlantic coasts. We analysed 9 tide 300 gauges with time series starting no later than 1920. The longest is Brest with 165 years of data. We carefully processed the data, particularly to remove the 18.6-year nodal modulation.
We found that M 2 variations were consistent at all the stations in the North East Atlantic (Newlyn, Brest, Cuxhaven), whereas some discrepancies appear in the North West Atlantic. The changes started long before the XX th century, and are not 305 linear. The trends vary significantly from a station to another; they are overall positive, up to 0.7 mm/yr, or slightly negative.
Since 1990, in many stations, the trends switch from positive to negative values. The significant difference between the trends since 1910 and 1990 calls for caution when interpreting trends based on short records, i.e. less than 30 years, especially if the data are noisy (interannual variability) and the underlying processes non-linear (change points).

310
Concerning the causes of the observed changes, the mean sea level rise is not sufficient to explain alone the variations. The similarity between the North Atlantic Oscillation and M 2 variations in the North East Atlantic suggests a possible influence of the large-scale atmospheric circulation on the tide. The underlying mechanism would be a different spatial distribution of water heights from one year to another, depending on the low-frequency sea-level pressure patterns, and impacting the propagation of the tide in the North Atlantic basin. In the future, dedicated modelling studies should be undertaken to confirm or discard 315 this hypothesis.
In this study, we focused only on M 2 amplitude. A similar analysis on the phase would draw a more complete picture of the M 2 variations (Müller, 2011;Woodworth, 2010;Ray and Talke, 2019). Other constituents are also affected. Results show that S 2 amplitude decreases at all the stations located in the North West Atlantic, and in contrast, tend to increase in the North East 320