Swell hindcast statistics for the Baltic Sea

The classic characterisation of swell as regular, almost monochromatic, wave trains doesn’t necessarily accurately describe swell in water bodies shielded from the oceanic wave climate. In such enclosed areas the locally generated swell waves still contribute to processes at the air and seabed interfaces, and their presence can be quantified by partitioning wave components based on their speed relative to the wind. We present swell statistics for the semi-enclosed Baltic Sea using 20 years of swell partitioned model data. The swell significant wave height was mostly under 2 m, and in the winter (DJF) the 5 mean significant swell height was typically less than 0.4 m; higher swell was found at limited nearshore areas. Swell waves were typically short (under 5 s), with mean periods over 8 s being rare. In open-sea areas the average ratio of swell energy (to total energy) was mostly below 0.4 – significantly less than in World Ocean. Certain coastal areas were swell dominated over half the times, mostly because of weak winds (U < 5 ms−1) rather than high swell heights. Swell dominated events with a swell height over 1 m typically lasted under 10 h. A cross-correlation analysis indicates that swell in the open sea is mostly generated 10 from local wind-sea when wind decays (dominant time lag roughly 15 h). Near the coast, however, the results suggest that the swell is partially detached from the local wind-waves, although not necessarily from the weather system that generates them. Namely, the highest swell typically arrives with a roughly 10 hour delay after the low-pressure system has already passed.

Further, the spectral swell partitioning used in WAM classifies energy as swell if the following criteria is fulfilled (Bidlot, 2001): 90 where u * is the friction velocity, c is the phase speed of the wave component and Φ is the wind direction. The swell spectrum, S swell (ω, θ), is therefore made up of the spectral bins S(ω, θ) that fulfills the above criteria, being 0 elsewhere.
The swell significant wave height is defined by integrating the swell spectrum and the swell mean period is defined using the inverse moment Following Semedo et al. (2011) we define the swell energy weight as This dimensionless variable takes values between 0 (no swell present) to 1 (all energy is classified as swell). We classify the sea state as swell dominated if W S >0.5, i.e. over half of the energy is considered swell.

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The wind-sea significant wave height fulfills

Statistics
The seasonal ice cover of the Baltic Sea complicates the definition of wave statistics (Tuomi et al., 2011). When ice is present, two of the possible type of statistics are: ice-free statistics (Type F) and ice included statistics (Type I). In Type F the statistic 105 (e.g. mean value) is calculated using only the times when the grid point is ice-free. In Type I the ice time is included in the calculations by assuming that the ice cover blocks the waves, i.e. by setting H s = 0. We use Type I statistics for the significant swell height, H swell s . The swell energy weight, W S , is only defined for H s > 0, which is why we use Type F statistics for this parameter.

Swell height
In the Baltic Proper the mean swell significant wave height (H swell s ) was mostly below 0.4 m during the winter (DJF) (Fig. 1a).
Small areas close to the coasts near Klaipėda, Kaliningrad, and Gotland showed larger values -up to ca 0.8 m. These coastal areas in the Baltic Proper also had the largest mean swell height (0.3-0.4 m) during the summer (JJA) (Fig. 1b).
In the smaller sub-basins the mean swell heights were heavily influenced by the seasonal ice cover. The Baltic Sea starts 115 freezing from the Bay of Bothnia and the eastern Gulf of Finland, where the ice cover can last even until May (SMHI and FIMR, 1982). As a result, the mean swell height in the Gulf of Finland and Bay of Bothnia was actually lower for the winter season compared to the summer. In the Gulf of Riga and the Bothnian Sea the mean swell heights were similar for both the summer and the winter season. Björkqvist et al. (2018) found that most over 7 m wave events takes place between November and January. These high 120 waves turned into swell after the wind decayed, leading to the 99 th percentiles of the significant swell height being around 2 m (winter) and below 1.4 m (summer) in larger parts of the Baltic Proper ( Fig. 1c-d). The highest swell height in the northern Baltic Proper (5.5 m) occurred after the storm Rafael in December 2004, and this event shows a rapid re-classification of wind-sea energy to swell energy (Fig. 2).
The ice cover affected the 99 th percentiles significantly less compared to the mean values. The 99 th percentile for the winter 125 months exceeded 1 m in the entire Bothnian Sea, with the lowest values found in the southeastern part of the basin (Fig. 1c).
The mean heights exceeded 0.8 m also during the summer season (Fig. 1d). In the Bay of Bothnia and the Gulf of Finland the 99 th percentiles were roughly similar both for the winter and summer season.

Swell prevalence
We quantified the prevalence of swell using the swell energy weight, W S , which is the fraction of swell energy with respect to 130 the total energy (Eq. 5). This parameter is undefined for the ice-time (see Sect. 2.3), and the statistics were therefore calculated for the ice-free time only (Type F).
During the winter (DJF) the mean swell weight was 0.2-0.3 in the larger parts of the Baltic Proper, being below 0.2 in large parts of the sub-basins (Fig. 3a). During the summer months (JJA) the mean swell weights were roughly 0.1 higher compared to the winter, exceeding 0.3 almost in the entire Baltic Sea (Fig. 3b). These are low values compared to the World Ocean,

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where the mean swell weight exceeds 0.5 (Semedo et al., 2011). Nevertheless, coastal sections with a mean swell weight over 0.5 were found in every sub-basin of the Baltic Sea, with the exception of Gulf of Riga. In the Baltic Proper the highest swell weights were along the eastern coastlines. Nonetheless, also a short coastal section in southeastern Gotland had mean swell weights exceeding 0.6. We note that the higher swell weights during the summer are not indicative of higher absolute swell heights, as evident from Sect. 3.1 (Fig. 1).
In addition to the average swell weight, we also calculated the probability that the sea-state in any given location was swell dominated (defined as W S >0.5). During the winter (DJF) the sea state in larger parts of the Baltic sea had a 20-30 % probability of being dominated by swell (Fig. 3c), which is low compared to 75-100 % in the World Ocean (Semedo et al., 2011) and the North Sea (ca. 40 % Semedo et al., 2015). During the summer (JJA) the Baltic Sea main basin had a 40-50 % probability of being swell dominated (Fig. 3d). Such a general difference between the seasons was also identified for the 145 North Sea and Norwegian Sea by (Semedo et al., 2015). The regions that were most often swell dominated (over 70 % of the times) were closely the same nearshore areas that had the highest mean swell weight. Longer coastal sections that were swell dominated more often than not could be found in all sub-basins except the Gulf of Riga.
We next focus on six locations from the largest sub-basins of the Baltic Sea.  Table 1). The swell dominated cases were often characterized by a low wind 155 speed (under 5 ms −1 ). For example: when the wind exceeded 5 ms −1 , the sea state outside of Östergarnsholm, Gotland was swell dominated only 14 % of the times, and practically swell free (W S ≤ 0.1) 73 % of the times.
In an absolute sense, the swell height was almost always below 2 m at all locations, with heights over 1 m being rare outside the Baltic Proper (Table 1). High (over 1 m) dominant (W S > 0.5) swell typically didn't persist for long (not shown). The median duration of such events was between 2-8 h at the six locations. High dominant swell was most prevalent at Klaipėda, 160 where the longest case lasted 86 h.

Swell direction
The seasonally averaged mean swell directions did not reveal anything unexpected; the dominant direction in the southern Baltic Proper was from southwest, following the geometry of the basin up north. Waves were strongly refracted towards the coast, especially in the southern Baltic Sea where the waves refract into the Bay of Gdańsk. The swell direction was 165 perpendicular to the coast at the exposed areas on the eastern coasts of the Baltic Proper. Typical swell periods in the Baltic Sea are short (see Sec. 3.4), but they are still, on average, longer than the wind-sea. The relatively shallow coastal areas therefore refract swell waves more strongly compared to wind-sea waves. In the Bothnian Sea the averaged swell directions  were towards the coast, with the averaged direction not being clearly defined in the middle of the basin -a pattern already identified by Semedo et al. (2014). There was no major differences in the directionality of swell between winter and summer.
170 Table 1. Exceedance probabilities of swell energy weight (WS) and swell significant wave height (H swell s ) at six locations (see Fig. 3a).
Statistics are ice-free and ice-included respectively (Type F & I, Tuomi et al., 2011 The misalignment between the swell direction and the wind direction is roughly similar for all the six locations, namely small angles are most common (Fig. 4). The distribution at Kalajoki (f) in the Bay of Bothnia forms an exception, which we surmise is because the ice-cover changes the fetch geometry; the data are thus essentially not from a single population. Windswell angles are also expected to be affected by the slanting fetch, especially in the Gulf of Finland where the wind-wave angle can be up to 50°even for wind-sea (Pettersson et al., 2010).

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For the 99 th percentile of swell heights the misalignment roughly follows that of the full data set at GoF and Bothnian Sea

Swell periods
A majority of the swell waves at the six locations had a mean period below 5 s (Fig. 5) Long swell -with a mean period above 8 s -was almost non-existent in GoF, Bothnian Sea, and Kalajoki. Such long swell waves were rare also in the Baltic Proper, constituting only 1-2 % of all swell cases at Klaipėda, Gotland, and NBP.

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The correlation of the wind-sea and swell significant wave heights were negative in the open sea, but positive for coastal areas (Fig. 6). The negative correlation of the open-sea areas was indicative of decaying or turning winds that caused existing wind-sea to be reclassified as swell. An example is the highest swell case at NBP during the storm Rafael (Fig. 2), where high waves were rapidly classified as swell when the wind turned as the cyclone passed. To study the correlation in more detail we determined the cross-correlation between swell and wind-sea heights (Fig. 7). The cross-correlation structure for open-sea  local wind-sea) and open-sea waves generated by that system. Nonetheless, coastal locations also inherit a similar correlation structure to that of the open-sea waves during decaying winds, and the final connection between swell and wind-sea surely

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The energy that the wave model partitions as swell in the open sea seems to, for the most parts, be better classified as old windsea. Namely, locally generated wind waves turn in to swell when the wind speed decays or the wind turns. While these waves fulfill the swell criteria, they might differ from a classical concept of swell as ordered, almost monochromatic and directional, waves generated by a distant storm. Our results show that even without remote swell the amount of swell -as defined using Eq. 5 -can still be significant (Fig. 3). Although differentiating between different types of swell is complicated, this kind of 215 old wind-sea is bound to be present also in swell statistics compiled for the World Ocean.
In the Baltic Sea the median duration of swell events (with a height over 1 m) was under 10 h. Near the shore, however, one event at Klaipėda lasted for 86 h (between 07 December and 10 December 2017). Such long events are not residuals of high wind waves combined with a decaying wind, since the longest fetch is below 600 km. Rather, a correlation analysis (Fig. 7) revealed that the origin of the partitioned swell energy is different near the shore. Our interpretation is that the open-sea waves 220 -generated by the same atmospheric system -arrive to the coast already during the growth stage of the local wind sea, but are classified as swell because of the lower wind speed near the coast.  (Soomere et al., 2008;Björkqvist et al., 225 2017;Björkqvist et al., 2020), thus having a deep water group speed above 28 km h −1 . Nonetheless, the more severe cyclones that were analyzed by Post and Kõuts (2014) had average translation speeds of 53-59 km h −1 . The difference in arrival time (500 km fetch) between 10 s waves and a 55 km h −1 cyclone would then also be around 9 h. The delay times estimated by these simple calculations are thus in the same order as the dominant 7 h and 11 h time lags found for the two coastal locations (Fig. 7), but further in-depth studies are required to further investigate this matter quantitatively.

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Our tentative classification of open sea and coastal areas through the correlation of wind-sea and swell is well motivated from the point of view of air-sea interaction studies. In such studies a detachment of the wind sea and the swell (negative correlation) is preferred, since it allows for more opportunities to study air-sea interaction processes without the results being tainted by simultaneous swell. This detachment of wind-sea and swell was identified in a study into air-sea momentum transfer during a long, well defined, swell case at the Östergarnsholm weather station outside of Gotland (Smedman et al., 1999). The

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Östergarnsholm tower is an established part of the Baltic Sea research infrastructure, and our results show that the location where the Gotland wave buoy is moored is in the open sea when using this correlation-based criterion.
The sea state in larger parts of the Baltic Sea was swell dominated for about a third of the hindcast period (Fig. 3b). The winter average swell weight was below 0.3 (Fig. 3a), which is similar to the winter average for the Black Sea (Van Vledder and Akpınar, 2016), but lower than in the North Sea (ca. 0.4-0.6, Semedo et al., 2015) and the Norwegian Sea (ca. 0.6-240 0.7, Semedo et al., 2015). In the World Ocean the average swell weight around the equator can exceed 0.9 during the winter (Semedo et al., 2011). The swell weights in the Baltic Sea during the summer months are slightly higher than during the winter (vice versa is true for the actual swell heights), which is also the case for the North Sea and the Norwegian Sea (Semedo et al., 2015). The Baltic Sea swell climate differs from the World Ocean by typically being very short (under 5 s, Fig 5). In other words, our quantitative results show that high and long swell is not persistent in the Baltic Sea (Table 1 and Fig. 3c&d), which 245 confirms the existing qualitative understanding of the Baltic Sea wave community.
Our results differ slightly from those of Semedo et al. (2014) that were based on the 10 km resolution NORA10 product (Reistad et al., 2011). The largest differences can be seen for the swell weights near the coast, simply because NORA10 can't capture the nearshore areas. In the southeastern part of the Baltic Proper (near Kaliningrad) the results of Semedo et al. In coastal areas swell typically drives wave-induced sediment transport, since long swell waves reach deeper than shorter wind waves. In our results swell was dominant roughly 70 % of time in long nearshore areas. The mean swell periods were 255 typically short, being comparable to those of the mean wind wave periods, and the wind wave heights can exceed the swell wave heights near the shore. Therefore, the physical significance of swell partitioning for coastal processes in the Baltic Sea In remote sensing applications the sea state affects Synthetic Aperture Radar (SAR) back-scatter and the signal of microwave sensors (e.g. Quilfen et al., 2004;Hwang and Plant, 2010;Stopa et al., 2017). Stopa et al. (2017) linked the accuracy of the wind speed retrieved from SAR-data to both the total significant wave height and the swell height. Although the authors found that the swell height was not the main explaining factor in the accuracy of winds retrieved form scatterometers, Wang 265 et al. (2014) showed that altimetry-based wind estimates were significantly more accurate when swell was absent. New highresolution altimeters, such as Sentinel-3, have also brought a concern that swell can affect existing algorithms used to estimate wave heights, especially since the altimeter waveform is particularly noisy when long waves are present (Moreau et al., 2018).
Furthermore, the presence of well-developed swell has been used to explain high-frequency signals in sea-level anomaly products derived using SAR-altimetry (Rieu et al., 2021 that setting the criterion for swell as c/U = 1 instead of 1.2 might be more appropriate. However, these results were derived using omnidirectional spectra -thus including slower components because of the directional spread -and are therefore not directly applicable for partitioning directional spectra. Indeed, reducing the constant from 1.2 to 1 in Eq. 2 would lead to even more energy being flagged as swell during the growth stage. The coastal geometry in the Baltic Sea is challenging for spectral partitioning, and we surmise that the small artefacts of the partitioning were caused by a slight misalignment between the wave 285 and wind direction, perhaps made worse by the 15 degree directional resolution of the spectrum. Nonetheless, this issue seems to be a harmless artefact, and the relative simple structure of the swell in the Baltic Sea open-sea areas doesn't warrant in-depth studies into more elaborate swell partitioning schemes. One approach to classify the entire sea state (as opposed to a single wave component) as swell is to use a lower threshold value for the inverse wave age, c p /U (e.g. 1.2, Semedo et al., 2011). Nonetheless, the peak period near complex Baltic Sea 290 shorelines might not be well defined or carry the meaning we normally attach to it (Björkqvist et al., 2019). We therefore defined the sea state to be swell dominated if over half of the energy was classified as swell (W S >0.5); this metric is well defined even in complex coastal conditions, although the 1 nmi wave product used in this paper was too coarse to reliably study waves in these types of areas. Our probabilities for swell dominated sea states (Fig. 3c&d) mostly coincide with those of Semedo et al. (2014) (calculated using the c p /U definition), meaning that these two definitions have a general agreement, negative, because swell was mostly created simply by a decaying wind, turning the existing wind-sea waves to swell in the Environment Monitoring Service, https://catalogue.marine.copernicus.eu/documents/QUID/CMEMS-WIND-QUID-012-002-003-005.