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# M2 Seasonal Variability in Northwestern Europe: Characteristics and Drivers

Lucia Pineau-Guillou, Pascal Lazure

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## To cite this version:

Lucia Pineau-Guillou, Pascal Lazure. M2 Seasonal Variability in Northwestern Europe: Characteristics and Drivers. Geophysical Research Letters, 2025, 52 (5), e2024GL112554 (9p.). 10.1029/2024GL112554. [hal-05063272](https://hal.science/hal-05063272v1)

HAL Id: hal-05063272

https://hal.science/hal-05063272v1

Submitted on 22 May 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Geophysical Research Letters®

## RESEARCH LETTER

10.1029/2024GL112554

**Key Points:**

* $M_2$ displays a strong seasonal cycle in the North Sea with amplitude larger than 5 cm

* The seasonality is well reproduced by a barotropic model forced with the tidal potential and the atmosphere only

* The seasonal cycle is mainly due to gravitational nonlinear effects in a large part and atmospheric effects in a smaller extent

**Correspondence to:**

L. Pineau-Guillou,
lucia.pineau.guillou@ifremer.fr

**Citation:**

Pineau-Guillou, L., & Lazure, P. (2025). $M_2$ seasonal variability in northwestern Europe: Characteristics and drivers. *Geophysical Research Letters*, *52*, e2024GL112554. https://doi.org/10.1029/2024GL112554

Received 13 SEP 2024

Accepted 20 JAN 2025

© 2025. The Author(s).
This is an open access article under the
terms of the Creative Commons
Attribution License, which permits use,
distribution and reproduction in any
medium, provided the original work is
properly cited.

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## $M_2$ Seasonal Variability in Northwestern Europe: Characteristics and Drivers

Lucia Pineau-Guillou¹ and Pascal Lazure¹

¹IFREMER, Laboratoire d'Océanographie Physique et Spatiale, UMR 6523 (IFREMER, CNRS, IRD, UBO), IUEM, Brest, France

**Abstract** The $M_2$ tide displays large seasonal variability in Europe, particularly in the North Sea. The tide is there larger in summer than in winter. However, there is no consensus on the physical drivers leading to such large values, atmosphere circulation and stratification being two good candidates. We analyzed hourly sea level data from observations at 35 tide gauges in Europe. The amplitude of $M_2$ seasonal cycle is the largest in the southern North Sea, reaching typically 4–6 cm. This cycle is well reproduced by a barotropic model, forced with the tidal potential and the atmosphere only. This suggests a minor role of the stratification. We show that large seasonal cycles in the southern North Sea are first due to gravitational nonlinear effects. The atmosphere also plays a role, but locally and in a smaller extent.

**Plain Language Summary** The tide is changing over a year, the amplitude of the tide being larger in summer than in winter in the North Sea. We try here to understand what are the reasons that explain such changes over a year. For this purpose, we analyze sea level data from 35 tide gauges in Europe. We found that the largest changes occur in the southern North Sea. Such changes are well reproduced by a model, whose inputs are the tidal potential and the atmosphere only, and which does not consider the stratification of the waters. This means that the stratification only plays a minor role. We show that the tide changes over a year, first because of the gravitational tide (some nonlinear components introducing some seasonal modulations). The atmosphere also plays a role, but locally and in a smaller extent.

## 1. Introduction

The tide refers to the cyclic variations of the sea level, from “gravitational” origin (due to the gravitational forces of the Moon and Sun), but also from “radiational” origin (due to cyclic meteorological effects) (Simon, 2007, 2013). Gravitational components come directly from the tidal potential (e.g., $M_2$) or from their interactions generating harmonics (e.g., $M_4$, $M_6$). Radiational components come from seasonal meteorological cycles such as the annual cycle in atmospheric pressure (e.g., $S_a$). Gravitational and radiational contributions to tide are often difficult to disentangle, as both may occur at the same frequency, as for instance for $S_2$ (Ray, 2009). (Note that our terminology for “gravitational” and “radiational” tide is in line with Simon (2007, 2013), but slightly different from Ray (2022), who considers as “gravitational” only tidal components that are in the tidal potential—and not their nonlinear interactions. As soon as we define the gravitational tide as the sea level's response to the gravitational forcing, our approach makes sense.)

Tide is changing at centennial scale, for non astronomical reasons (Haigh et al., 2022; Ray & Talke, 2019; Woodworth, 2010). Over the last century, tidal constituents typically changed from 1 to 10 cm (Müller et al., 2011; Pineau-Guillou et al., 2021; Ray & Talke, 2019). The causes of observed changes may be local (e.g., deepening of channels, Familkhalili & Talke, 2016) or from large-scale origin. Many physical processes are possible drivers, such as changes in water levels due to sea level rise or ground movements (Idier et al., 2017; Pickering et al., 2017; Schindelegger et al., 2018), changes in the atmospheric circulation (Challis et al., 2023; Pineau-Guillou et al., 2021), changes in the ocean stratification that will modify internal tides and bottom friction (Müller, 2012; Opel et al., 2024), changes in the radiational forcing (Ray, 2009) or changes in the extent of sea-ice cover (Haigh et al., 2022).

Tide is also changing at seasonal scale (Müller et al., 2014; Ray, 2022; Tazkia et al., 2017). The main difference between centennial and seasonal changes, is that seasonal changes can be due to astronomical constituents (Ray, 2022), although these are rarely mentioned as drivers (compared to physical drivers such as meteorological forcing, river discharges, ice coverage or stratification, e.g., Müller et al., 2014). Various studies already
<figure><img src="image_5.png" /><figcaption>**Figure 1.** (a) *M*₂ computed over 2004–2014 from observations (colored dots) and model (filled contours), (b) monthly *M*₂ at Dunkerque observed (orange) and modeled (blue), and (c) *M*₂ seasonal cycle at Dunkerque observed (orange) and modeled (blue). The seasonal amplitude is defined as half of the range of the seasonal cycle, the range being the difference between the maximum and minimum *M*₂ over the year. The seasonal phase is defined as the month where the *M*₂ amplitude is maximum. The shaded areas correspond to the ±1σ confidence interval.</figcaption></figure>

investigated *M*₂ seasonal variability at global and regional scale, for example, Müller et al. (2014) globally, Tazkia et al. (2017) in the Bay of Bengal, Devlin et al. (2018) in southeast Asian waters, Huess and Andersen (2001) and Gräwe et al. (2014) in the North Sea. Despite all agree that tide displays a strong seasonality, with amplitude larger than 5 cm in the North Sea (Gräwe et al., 2014), and up to 25 cm in Bangladesh (Ray, 2022), there is no consensus on the physical drivers leading to observed changes in certain regions. In particular in the North Sea, some studies invoke a major role of the atmospheric circulation (Huess & Andersen, 2001), and consider the seasonality as a barotropic phenomenon, whereas others suggest a predominant role of the stratification (Gräwe et al., 2014; Müller et al., 2014) considering that the seasonality is largely a baroclinic process. Recently, Opel et al. (2024) who investigated the role of stratification on tide globally, reported that possibly, competing processes might be at work in the North Sea.

The present paper aims at understanding the physical drivers of *M*₂ seasonal variability in Europe (English Channel and the North Sea). We first analyze sea level observations at 35 tide gauges to characterize the amplitude and phase of *M*₂ seasonal cycle. We then run a barotropic model, to understand which physical drivers lead to large amplitudes, particularly in the southern North Sea.

Note that we here focus on the seasonality of the amplitude of *M*₂, and not its phase. This work is therefore a first step, as changes in phase may also be significant (1.5° in average), leading to potential elevations of a few cm.

## 2. Data and Method

### 2.1. Sea Level Observations

We used 35 tide gauges (Figure 1a) from Global Extreme Sea Level Analysis (GESLA-3) sea level data set (Caldwell et al., 2015; Haigh et al., 2022; Woodworth et al., 2017). This data set provides 5,119 records of high-frequency (mainly hourly) sea level observations around the world. We selected the coastal stations located in the North-East Atlantic (from 45°N to 60°N and from 15°W to 10°E) with at least 50 years of data. We discarded tide gauges that were not representative of the open ocean (3 stations in the Netherlands, Den Oever buiten, Kornwerderzand buiten and Harlingen), without recent data (2 stations, Southampton record in the UK ends in 1990 and Ijmuiden Noordersluis in the Netherlands ends in 1983) or with doubtful data (1 station, Hellevoetsluis in the Netherlands, reported as "not research quality data" in the Permanent Service for Mean Sea Level website). This led to consider 35 tide gauges. Hourly sea level data were extracted over the period 2004–2014 (period of our study) at these 35 stations.
## 2.2. Ocean Numerical Model

We used the global ocean model TUGO (Lyard et al., 2006). This model in barotropic mode resolves the classical shallow water continuity and momentum equations. This numerical model is particularly adapted to investigate tide, as it is used to develop the FES2014 global ocean tide atlas (Lyard et al., 2021) and to correct satellite altimeter data from tide and atmospheric effects (Carrère & Lyard, 2003). Note that there is no stratification in the model (barotropic mode). We used the default TUGO model configuration, with the FES2014 unstructured grid (Lyard et al., 2021). The grid resolution varies from 15 km offshore to 2 km in nearshore areas (see Figure 5 in Pineau-Guillou et al. (2020) for a grid illustration). The model is forced by the tidal potential and by the winds and atmospheric pressure. Those atmospheric data come from Twentieth Century Reanalysis Version 3 (Slivinski et al., 2019), a 3 hr and 1° atmospheric data set (0.5° for the native grid) over 1836–2015.

We conducted simulations over 2004–2014 (period of our study). Outputs are the hourly sea levels at the 35 tide gauges (nearest grid point in the native grid), and hourly sea levels on a 0.05° grid (around 5 km resolution). We conducted two simulations: one forced with the tidal potential only (referred as “tide only” on figures), the other forced with the tidal potential and the atmosphere (referred as “tide + atm” on figures).

## 2.3. Computation of $M_2$ Seasonal Cycle

To compute the $M_2$ seasonal cycle at the 35 stations, we processed the 2004–2014 hourly sea level data (observed and modeled). In a first step, we computed $M_2$ each month, using Utide tidal analysis software (Cioda, 2011). Nodal modulations were carefully empirically removed (more details in Appendix A from Pineau-Guillou et al. (2021)). Only months with sufficient data (more than 75%) were considered. We then obtain monthly $M_2$ time series over 2010–2014 (see Figure 1b for an example at Dunkerque, France). In a second step, we averaged all the monthly values, to estimate the seasonal cycle over a year and its associated standard deviation (see Figure 1c for an example, still in Dunkerque). The seasonal cycle is characterized by its seasonal amplitude and phase, defined as follows. The seasonal amplitude is half of the range of the seasonal cycle, the range being the difference between the maximum and minimum $M_2$ over a year. The seasonal phase is the month where the $M_2$ amplitude is maximum over a year (see Figure 1c for an illustration).

## 2.4. Computation of Tidal Atlases

To compute atlases of harmonic constituents on the whole area ($M_2$ on Figure 1a and other constituents on Figure 4), we analyzed simulations over 2004–2014 with the Tidal Toolbox (Allain, 2021), which has been specifically developed for this purpose.

Finally, note that the average $M_2$ (i.e., computed over the whole period 2004–2014) may differ between the model and observations (see the discrepancies on Figure 1a, Root Mean Square Error of 14.7 cm). For this reason, $M_2$ from the model was bias corrected ($M_{2\text{mod}}(t) = M_{2\text{mod}}(t) - \overline{M_{2\text{mod}}(t)} + \overline{M_{2\text{obs}}(t)}$), when plotted on figures (Figures 1b, 1c, 2e, and 3b). This ensures an easier visual comparison between the observed and modeled seasonal cycles.

# 3. Results and Discussion

## 3.1. $M_2$ Seasonal Cycle

We analyzed the 2004–2014 observed and modeled sea level data, to compute the $M_2$ seasonal cycle amplitude (observed Figure 2a, modeled Figure 2b) and phase (observed Figure 2c, modeled Figure 2d). Some examples of $M_2$ seasonal cycle over a year are displayed at 4 stations (Figure 2e), whose name are labeled on Figure 2d to show their locations. The main result is that the model correctly reproduces the seasonal cycle, whereas it is only forced by the tidal potential and the atmosphere (no stratification). More details are given in the following.

The observed amplitude of the seasonal cycle is of 3.3 cm in average at the 35 stations (Figure 2a), and ranges from 0.5 to 6.7 cm depending on the station. The eastern and northern Europe display small amplitudes (minimum of 0.5 cm at Stavanger, Norway), whereas the eastern English Channel and the southern North Sea display large amplitudes, typically more than 4 cm (maximum of 6.7 cm at Delfzijl, Netherlands). In these areas of large values, the seasonal cycle is maximum in June (Figure 2b).
<figure><img src="image_6.png" /><figcaption>**Figure 2.** *M*<sub>2</sub> seasonal amplitude from (a) observations and (c) model. *M*<sub>2</sub> seasonal phase from (b) observations and (d) model. (e) *M*<sub>2</sub> seasonal cycle at Le Havre, Dover, Dunkerque, and Cuxhaven. Note that these four stations are labeled on panel (d). At the underlined stations on panels (a, c), the main frequencies are annual and semi-annual, when applying a Fast Fourier Transform analysis on monthly *M*<sub>2</sub> time series (see Figure 1b for an example of monthly *M*<sub>2</sub> time series).</figcaption></figure>

Overall, the model well reproduces the amplitude of the seasonal cycle (Figure 2c), with similar high values in the eastern English Channel (e.g., 4.8 cm at Le Havre, vs. 4.6 cm for the observations) and in the southern North Sea (e.g., 5.6 cm at Cuxhaven against 5.5 cm for the observations). However, the amplitude is overall underestimated

<figure><img src="image_7.png" /><figcaption>**Figure 3.** (a) Group *M*₂ at Cuxhaven obtained from harmonic analysis of the model with tide only (green) and with tide and atmosphere (blue) over 2004–2014, using the Tidal Toolbox. The same for observations (gray diamonds). *M*₂ is modulated semi-annually by *MSK*₂ and *MKS*₂ and annually by *M(SK)*₂ and *M(KS)*₂. The names are in line with Simon (2007, 2013) (see Table 1). Note that γ₂ is schematically represented, but ignored in the Tidal Toolbox due to its very low value in the tidal potential (b) *M*₂ seasonal cycle from the model at Cuxhaven, with a fit of an annual and semi-annual cycle.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Figure 4.** Amplitude of $MKS_2$, $M(KS)_2$, and $Sa$ obtained from harmonic analysis of the model with (a) tide only and (b) tide and atmosphere over 2004–2014.</figcaption></figure>

by the model (average amplitude of 2.5 cm against 3.3 cm for the observations), particularly at Delfzijl (Netherlands) in the southern North Sea (3.1 cm for the model against 6.7 cm for the observations). Note that this station, located in the Ems estuary, is surrounded by a chain of islands, which makes it difficult to model. The model also well reproduces the phase of the seasonal cycle (maximum in June in the English Channel and the North Sea), except for some stations. Note that for stations with small amplitude (less than 1 cm, see the small dots on Figure 2c) the phase is not relevant, as there is almost no seasonal cycle.

Beyond the amplitude and phase, the whole $M_2$ seasonal cycle over a year is generally correctly modeled, as illustrated on Figure 2e for 4 stations with large amplitude (more than 4 cm). Again, the maximum in June is correctly reproduced by the model. However, a small shift of around 15 days appears sometimes between the model and the observations, in March and November (see for example Le Havre on Figure 2e). This is possibly due to missing processes in the model (see the discussion further).

## 3.2. Physical Drivers

The model, which is only forced by the tidal potential and the atmosphere, correctly reproduces the observed seasonal cycle. This suggests that these two processes (tidal potential and atmosphere) explain most of the seasonal cycle. We remind that there is no effect of stratification, as we use a barotropic model. In the following, we will analyze more deeply the $M_2$ monthly time series, to understand the origin of the $M_2$ seasonal cycle.

We first applied a Fast Fourier Transform on the monthly $M_2$ time series (observed and modeled) at each of the 35 stations, to detect the frequencies of the $M_2$ seasonal cycle (see Figure 1b for an example of monthly $M_2$ time series). We found that monthly $M_2$ time series major frequencies are annual and semi-annual in southern North Sea (see the underlined stations on Figures 2a and 2c). Interestingly, these stations are also those with the largest seasonal amplitude. In fact, at these stations, the seasonal cycle can be correctly represented with an annual and a semi-annual fit, as illustrated at Cuxhaven (Germany) on Figure 3b. But what are the origins of these annual and semi-annual components?

These annual and semi-annual modulations of $M_2$ are due to neighboring components in $M_2$ group (see the example of Cuxhaven, based on a harmonic analysis of the model over 2004–2014, Figure 3a, in complement to Figure 1 from Ray (2022)). Note that in the present paper, the constituent names are in line with Simon (2007, 2013) and current international conventions (from International Hydrographic Organization) and displayed
<table><caption>**Table 1**

Harmonic Constituents Characteristics in the M₂ Group</caption><thead><tr><th>Names</th><th>Other names</th><th>q (°/h)</th><th>Doodson</th><th>Argument</th><th>Origin</th></tr></thead><tbody><tr><td rowspan="3">MSK<sub>2</sub></td><td>MSK<sub>2</sub></td><td>28.901967</td><td>2 535 555</td><td>2τ - 2h</td><td>Interaction q<sub>MSK2</sub> = q<sub>M2</sub> + q<sub>S2</sub> - q<sub>K2</sub></td></tr><tr><td>No name</td><td>28.901967</td><td>2 535 555</td><td>2τ - 2h</td><td>Interaction q<sub>noname</sub> = q<sub>M2</sub> - q<sub>Ssa</sub></td></tr><tr><td>OP<sub>2</sub></td><td>28.901967</td><td>2 535 557</td><td>2τ - 2h + π</td><td>Interaction q<sub>OP2</sub> = q<sub>O1</sub> + q<sub>P1</sub></td></tr><tr><td>γ<sub>2</sub></td><td>GAM<sub>2</sub></td><td>28.911251</td><td>2 537 557</td><td>2τ - 2h + 2p + π</td><td>Tidal potential (a = 0.00300)</td></tr><tr><td rowspan="3">M(SK)<sub>2</sub></td><td>M(SK)<sub>2</sub></td><td>28.943036</td><td>2 545 554</td><td>2τ - h - π/2</td><td>Interaction q<sub>M(SK)2</sub> = q<sub>M2</sub> + (q<sub>S1</sub> - q<sub>K1</sub>)</td></tr><tr><td>MA<sub>2</sub></td><td>28.943036</td><td>2 545 555</td><td>2τ - h</td><td>Interaction q<sub>MA2</sub> = q<sub>M2</sub> - q<sub>Sa</sub></td></tr><tr><td>α<sub>2</sub> or H<sub>1</sub></td><td>28.943038</td><td>2 545 567</td><td>2τ - h + p<sub>1</sub> + π</td><td>Tidal potential (a = 0.00345)</td></tr><tr><td rowspan="2">M<sub>2</sub></td><td>M<sub>2</sub></td><td>28.984104</td><td>2 555 555</td><td>2τ</td><td>Tidal potential (a = 1)</td></tr><tr><td>KO<sub>2</sub></td><td>28.984104</td><td>2 555 557</td><td>2τ + π</td><td>Interaction q<sub>KO2</sub> = q<sub>K1</sub> + q<sub>O1</sub></td></tr><tr><td rowspan="3">M(KS)<sub>2</sub></td><td>M(KS)<sub>2</sub></td><td>29.025173</td><td>2 565 556</td><td>2τ + h + π/2</td><td>Interaction q<sub>M(KS)2</sub> = q<sub>M2</sub> + (q<sub>K1</sub> - q<sub>S1</sub>)</td></tr><tr><td>MB<sub>2</sub></td><td>29.025173</td><td>2 565 555</td><td>2τ + h</td><td>Interaction q<sub>MB2</sub> = q<sub>M2</sub> + q<sub>Sa</sub></td></tr><tr><td>β<sub>2</sub> or H<sub>2</sub></td><td>29.025171</td><td>2 565 545</td><td>2τ + h - p<sub>1</sub></td><td>Tidal potential (a = 0.00305)</td></tr><tr><td rowspan="3">MKS<sub>2</sub></td><td>MSK<sub>2</sub></td><td>29.066242</td><td>2 575 555</td><td>2τ + 2h</td><td>Interaction q<sub>MKS2</sub> = q<sub>M2</sub> + q<sub>K2</sub> - q<sub>S2</sub></td></tr><tr><td>No name</td><td>29.066242</td><td>2 575 555</td><td>2τ + 2h</td><td>Interaction q<sub>noname</sub> = q<sub>M2</sub> + q<sub>Ssa</sub></td></tr><tr><td>δ<sub>2</sub></td><td>29.066242</td><td>2 575 555</td><td>2τ + 2h</td><td>Tidal potential (a = 0.0015)</td></tr></tbody></table>

Note. Names (column 1) are in line with Simon (2007, 2013). Constituents with other names (column 2) can not be separated from the main name (column 1), as their frequencies are the same, because they only differ by the 6th Doodson number (mean longitude of the solar perigee $p_1$, whose period is 20,9 centuries) or by the 7th Doodson number (multiple of $π/2$). For the origin of the constituents (column 6), a is the relative amplitude of the component compared to $M_2$ in the tidal potential.

Table 1 (column 1). Other names may appear in the literature (column 2), depending on the tidal analysis software or the authors. For example, the following names refer to the same tidal frequency: $M(SK)_2$ in the Tidal Toolbox (Allain, 2021), $MA_2$ in Ray (2022) or $H_1$ in Utide software (Codiga, 2011). Surprisingly, $MSK_2$ does not appear in Utide software, and signs at $γ_2$ frequency. Note that the nodal constituent $m_2$ is not mentioned in Table 1 (or displayed on Figure 3a), because it introduces an 18.6 years modulation of $M_2$ which is not in the scope of the present paper. Figure 3a illustrates that 1 year of data is required to correctly separate $M_2$ from its two neighboring components $M(SK)_2$ and $M(KS)_2$ (as their frequencies differ from 1 year), and 6 months is needed to correctly separate $M_2$ from $MSK_2$ and $MKS_2$. As a consequence, with monthly analysis, $M_2$ is modulated by these neighboring components, annually and semi-annually.

To understand the origin (tide or atmosphere?) of these neighboring components, we conducted a harmonic analysis of the model forced with tide only or forced with both tide and atmosphere. Amplitudes are displayed at Cuxhaven on Figure 3a, and on the whole study area on Figure 4 (note that only $M(KS)_2$ and $MKS_2$ are displayed, atlases being very similar for $M(SK)_2$ and $MSK_2$ due to the symmetry of the $M_2$ group, see Figure 3a). We found that the semi-annual modulation components $MSK_2$ and $MKS_2$ are clearly of gravitational origin. Indeed, the simulation with tide only or with both tide and atmosphere give similar amplitudes (2 cm at Cuxhaven Figure 3a and up to 6 cm elsewhere Figures 4a and 4b). The picture is different for the annual modulation components $M(SK)_2$ and $M(KS)_2$. These components are small in the simulation with tide only (0.5 and 0.3 cm at Cuxhaven respectively, Figures 3a and 4a), but they are enhanced when the atmosphere is added (1.2 and 1.6 cm at Cuxhaven respectively, Figures 3a and 4b). The atmosphere acts through the annual component $S_a$, a radiational component only due to the atmosphere (amplitude close to 0 for the model with tide only Figure 4a, but up to 9 cm for the model with both tide and atmosphere Figure 4b). Once generated, this $S_a$ component interacts with $M_2$, and the resulting components (sometimes referred as $MA_2$ and $MB_2$ in the literature) are exactly at the frequency of $M(SK)_2$ and $M(KS)_2$, which explains the enhancement of these components with the atmosphere forcing.

To go further, the reader can refer to Table 1, which details the origin of the components of the $M_2$ group and underlines the complexity to disentangle the drivers. At the same frequency, components of different origin overlap. (Note that components in the column entitled “Other names” can not be separated from components in the column “Names,” as their frequencies are the same, see the figure caption for more details). At the same
frequency, components may come directly from the tidal potential, or result from interactions between purely gravitational components, or between gravitational and radiational components. For example, at the frequency of $M(KS)_2$, three components overlap: (a) $\beta_2$ which comes directly from the tidal potential, (b) $M(KS)_2$ which comes from the interaction between $M_2$, $K_1$ and $S_1$, (c) $MB_2$ which comes from the interaction between $M_2$ and $S_a$ (as explained previously). Note that the picture is even more complex, as the atmosphere not only play a role in $MB_2$, but also in $M(KS)_2$, as $S_1$ is partly radiational (but very small, $S_1$ is smaller than 0.8 cm in the simulation with tide only, and smaller than 1.7 cm in the simulation with tide and atmosphere). Finally, the contribution that comes directly from the tidal potential ($\alpha_2$, $\beta_2$, $\gamma_2$, and $\delta_2$) is quite small, that is, always smaller than 0.3% of $M_2$ (see the value of $a$ in Table 1), which means for example, around 0.5 cm at Cuxhaven. Note that the gravitational $M(KS)_2$ (Figure 4a middle panel) mainly comes from the tidal potential (i.e., linear origin), as $\beta_2$ alone would lead to amplitudes close to those of $M(KS)_2$ (around 0.5 cm). Finally, note that only part of possible nonlinear interactions are listed in Table 1, as many others may occur at a given frequency (e.g., $M_2 + S_2 - T_2$ matches with $M(KS)_2$ frequency).

To resume, in the southern North Sea, the seasonal cycle is mainly due to gravitational nonlinear effects (in a large part) and atmospheric effects (in a smaller extent). However, some small discrepancies between the model and observations suggest that other processes may also play a role. In the model, the semi-annual and annual cycle are both maximum in June, which explains the strong amplitude in the North Sea (see Figure 3b). Differently, in the observations, the annual cycle seems to be slightly shifted earlier in the year (maximum rather in May, not shown), the semi-annual cycle being still in June. This may explain the observed shift of around 15 days around March and November, when comparing the modeled and observed seasonal cycles (Figure 2e). These discrepancies are possibly due to physical processes that are omitted in the model. Our barotropic model only captures the atmospheric-induced annual cycle, which is likely underestimated in some areas (e.g., at Cuxhaven, see observed and modeled $M(SK)_2$ on Figure 3a), as we are missing the baroclinic-induced annual cycle, due for example, to the summer stratification (e.g., in the North Sea). Despite this, the model well catches the overall seasonal cycle, as in most of the areas, the seasonal effect is dominated by the semi-annual modulation (e.g., in the southern English Channel, where MKS2 reaches 2–3 cm, against less than 1 cm for $M(KS)_2$, see Figure 4b). Discrepancies between observations and the model may also come from some limitations of the model, including uncertainties on input data or parameterizations, but also its poor resolution (only 2 km in nearshore areas). This resolution might not be sufficient to correctly capture the fine-scale nearshore dynamics, in complex areas such as the North Sea, with tidal inlets and sand banks.

## 4. Conclusion

We analyzed hourly sea level data from observations and simulations at 35 tide gauges to characterize $M_2$ seasonal cycle. We found strong amplitudes in the southern North Sea, typically from 4 to 6 cm, $M_2$ being maximum in June. A barotropic model forced only with the tidal potential and the atmosphere correctly reproduces this cycle.

We analyzed simulations forced with tide only and with both tide and atmosphere, to understand the origin of the large amplitudes in the southern North Sea. We found that the seasonal cycle was mainly due to annual and semi-annual modulation of $M_2$ from neighboring nonlinear components. Whereas the semi-annual modulation is from gravitational origin, the annual modulation is partly from gravitational origin, and enhanced by the atmosphere. This enhancement may exceed 1 cm locally.

To conclude, Huess and Andersen (2001) partly anticipated these results almost 25 years ago, reporting that the seasonal tide was mainly a barotropic phenomena in the North Sea, due first to “non linear combination between the tidal frequencies” and with “a dependency on the meteorological field over the area.” In the present paper we went further, explaining which constituents interact and to what extent.

In future modeling studies investigating the tide seasonality, we recommend to systematically run the model with tide only, to first estimate which part of the seasonality is due to gravitational components. Once this part is determined, an analysis of the frequencies of $M_2$ monthly time series should help to understand the other drivers. Note that any change in the sea level annual cycle ($S_a$), whether due to annual cycle in atmospheric pressure, large river discharges or stratification, will enhance the seasonality.
## Data Availability Statement

The GESLA-3 sea level data set (Caldwell et al., 2015; Haigh et al., 2022; Woodworth et al., 2017) analyzed during the current study is available on the GESLA website, https://gesla787883612.wordpress.com/downloads/. The 20CR (Twentieth Century Reanalysis) Version 3 atmospheric data (Slivinski et al., 2019) are available on the 20CR website https://psl.noaa.gov/data/20thC_Rean/. Support for the Twentieth Century Reanalysis Project version 3 data set is provided by the U.S. Department of Energy, Office of Science Biological and Environmental Research (BER), by the National Oceanic and Atmospheric Administration Climate Program Office, and by the NOAA Physical Sciences Laboratory. The Utide tidal analysis software (Codya, 2011) is available on the Utide website, https://pypi.org/project/UTide/.

## Acknowledgments

This research has been supported by the French National Research Agency (ANR) Grant ClimEx (ANR-21-CE01-0004). We thank Florent Lyard from LEGOS (Laboratoire d'Etudes en Géophysique et Océanographie Spatiales, France) for providing the FES2014 unstructured grid (Lyard et al., 2021) and the TUGO software (https://sirocco.obs-mip.fr/ocean-models/tugo/).

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