Seasonal variability of radiation tide in Gulf of Riga

Diurnal oscillations of water level in Gulf of Riga are considered. It was found that there is distinct daily pattern of diurnal oscillations in certain seasons. The role of sea breeze, gravitational tides and atmospheric pressure gradient are analysed. The interference of the first two effects provide the dominant role in diurnal oscillations. The effect of gravitational tides is described both with sole tidal forcing and also in real case with atmospheric forcing and stratification. The yearly variation of the declination of the Sun and stratification leads to seasonal intensification of gravitational tides in Gulf of Riga. 5 Correlation between gravitational tide of the Sun with its radiation caused wind effects appears to be main driver of oscillations in Gulf of Riga. Daily variation of wind is primary source of S1 tidal component with a water level maximum at 18:00 UTC in Gulf of Riga. Effect of solar radiation influences also K1 and P1 tidal components which are examined, too.


Introduction
A distinct feature in Gulf of Riga is diurnal oscillations of water level despite tidal influence is negligible in Baltic sea. These 10 oscillations are especially expressed in a relatively calm and sunny spring days. The amplitude of oscillations can reach 10 cm of water level in May, see Fig. 1, that is much higher than typical amplitude of gravitational tides up to 3 cm (Medvedev et al., 2013). If we look on water level oscillations for longer period, then the resulting diurnal amplitude fades away as phase and modulation of these oscillations varies. Our aim here is to examine seasonal variations of amplitude and phase of diurnal oscillations and find the dominating reason behind these variations. 15 Moreover, observations show a distinct daily pattern in spring and summer when water level culminates in late afternoon, see Fig. 1. Therefore, there should be notable S 1 tidal component with oscillation period of exactly 24 h (Williams et al., 2018). Nature of S 1 tide is more related with diurnal changes in atmosphere caused by radiation rather than gravitational effect of the Moon and the Sun (Ray and Egbert, 2004). Therefore, S 1 is usually called as radiation or atmospheric tide.
Moreover, mechanism of radiation tide should be studied more in detail as water level oscillations may be driven by sea and 20 land breeze or atmospheric pressure oscillations, which is the case for global ocean (Ray and Egbert, 2004). Radiation caused thermal expansion of surface water can yield only amplitude less than a millimeter in Gulf of Riga. Earth nutation is another factor influencing S 1 tide but the effect is rather weak comparing with atmospheric effects, see Schindelegger et al. (2016).
Topographically trapped internal waves can be another factor influencing diurnal oscillations as in strait of Otranto (Ursella et al., 2014) or Southern California Bight (Beckenbach and Terrill, 2008) but they could be important only equatorward of 30°25 latitude. Diurnal oscillations of Gulf of Riga brings hundreds of km 3 through Irbe strait. This creates pulsating currents in the strait (Lilover et al., 1998). These pulsations do not bring notable water exchange between Baltic proper and Gulf of Riga as typical length traveled by water parcel result in ≈ 1 km much shorter than the length of Irbe strait ≈ 45 km. Several hours are required to transfer amount of water between Baltic proper and Gulf of Riga. Therefore, there is typical phase lag of oscillations in Gulf 30 of Riga around 6-7 hours (Lilover et al., 1998) as compared to Baltic proper.
Hourly water level measurements in Gulf of Riga starts from 1960 in stations of Latvia. Sixty years of observations are sufficient for analysis as it is 3 times longer as lunar precession cycle of 18.6 years. Keruss and Sennikovs (1999) showed that the water level spectrum in Gulf of Riga really contains primarily diurnal tidal components with distinct peak of S 1 tidal component, see also Medvedev et al. (2013), whereas Baltic proper has relatively weak S 1 tidal component. Similar effects 35 hold in Gulf of Finland and Curonian lagoon. Rabinovich and Medvedev (2015) associated S 1 component in Otkritoje of Curonian lagoon with sea breeze effect. Is it the case also for Gulf of Riga and Gulf of Finland? Figure 1 suggests that water level oscillations, i.e. phase and amplitude, correlates excellently with gravitational tides. Numerical reanalysis by HIROMB BOOS Model (HBM) at University of Latvia catches the oscillations very well, thus we can investigate the phenomenon by parameter studies. HBM has been used by several meteorological organisations around the Baltic sea and Copernicus Marine 40 Monitoring Service (CMEMS) for operational analysis.
The period of diurnal tidal components K 1 (23.93 h) and P 1 (24.07 h) are nearly equal with S 1 tidal component. Therefore, there should be distinct interference between the solar radiation and gravitational tides, see Rabinovich and Medvedev (2015).
Components K 1 and O 1 (25.82 h) are related to period of declination of the Moon. Components K 1 and P 1 are related to period of declination of the Sun. The influence of the Moon is higher, thus K 1 and O 1 are stronger than P 1 . K 1 has usually 45 similar amplitude as O 1 but it may depend on preferred oscillation period at given location. Position of the Moon may vary at solar noon when radiation is at peak. On the other hand, position of the Sun is principal characteristic of the solar noon. Therefore, we can expect that there can be correlation between gravitational effect of the Sun and radiation caused effects as sea breeze or atmospheric pressure variation. Locations of ERA-Interim analysis: GoR, V en and BS3 are marked by black points.
The main task of the study is to obtain and analyse the daily pattern of water level oscillations for every month in Gulf of 50 Riga. Resonance of water level oscillations in the Baltic Sea will be studied in next section. Afterwards, spectral properties of water level observations will be summarised. Next, the role of gravitational forcing by the Sun and the Moon will be discussed.
And finally, we will put radiation effect of the Sun together with gravitational forcing.

Oscillation Baltic proper -Gulf of Riga
The observations in Fig. 1 show that Gulf of Riga in connection with Baltic proper prefers to oscillate with nearly constant 55 period of about 1 day but semi-diurnal components are effectively filtered out. Thus, there should be resonant nature of these oscillations that could be figured out by a numerical model, see Webb (2014). Otsmann et al. (1999) associates diurnal oscillations in Gulf of Riga and Väinameri sea with free Helmholtz oscillations of connected water bodies of Baltic proper, Gulf of Riga and Väinameri sea neglecting frictional and Coriolis forces. Assuming that the main connection of the Baltic sea is by Irbe strait the cyclic frequency is given by: where A Irbe and L Irbe are area of cross-section and length of Irbe strait, respectively, A Gulf is area of the Gulf of Riga and g is gravitational acceleration. This gives about 24 hour period. Quarter-wavelength resonance theory can sometimes fail to give correct resonance pattern, see Cui et al. (2019). Instead, let us obtain the resonance pattern numerically by small harmonic Baltic proper has its own response to perturbations on western boundary with one peak at 16 hours. We are interested to resonance pattern in Gulf of Riga relative to perturbations in Baltic proper in order to see oscillation associated with exchange of water through Irbe strait. Therefore, let us draw resonance pattern in Gulf of Riga with respect to Visby station (Gotland) representing Baltic proper. Gotland is located close to amphidromic point of Baltic sea, thus it has rather weak resonance 75 amplitudes both in semi-diurnal and diurnal range.
As can be seen in Fig. 3, Gulf of Riga has distinct peak with eigen period of 26-27 hours. That is a bit higher than analytic estimate 24 hours by Otsmann et al. (1999). But adding a Coriolis force generally increases period for longer seiche modes, see Leppäranta and Myrberg (2009). The amplitude and phase is almost equal in all stations around Gulf of Riga. This means that Gulf of Riga oscillates as a single body. The phase lag with respect to Baltic proper is around 6 hours for diurnal oscillations that 80 agrees with estimates by Lilover et al. (1998), see Fig. 4 where phase difference with respect to Ventspils station is shown. The highest amplitude is in Pärnu and the lowest in Kolka. As Virtsu station is close to Väinameri sea, then it has a semidiurnal peak associated with Väinameri oscillations. Väinameri sea have both semidiurnal and diurnal tidal components equally pronounced, see Otsmann et al. (1999). Seiches at Gulf of Finland have a period of 28 hours consistent with other numerical estimations   without providing details about temporal variability, e.g., average amplitudes for each month. There are several sources of seasonal variability: stratification, ice build-up, sea and land breezes, climatology of wind, river inflow, distance to the Sun, etc. Let us examine the daily pattern of oscillation in monthly scale. We can expect that due to solar radiation, highest daily 105 temperature variations should be in June and lowest in December. However, Fig. 5 suggests that highest daily variations occur in late spring and the lowest in early autumn. That could not be described just by solar radiation. Thus, we need to account the gravitational tides as will be done in the next section. The time of maximum water level in Gulf of Riga is late afternoon

Gravitational tides
As noted in the previous section it is not enough to consider solar radiation alone to describe the seasonal variability of daily changes of the water level. Another periodic forcing is provided by gravitational tides. First, let us consider that tides have  Table 1.
In order to study real conditions, it is better to evaluate the gravitational tides with atmospheric forcing, real temperature and salinity distributions, and real boundary conditions accounted. It can be evaluated by taking the difference between real case with tidal potential and without. This is done for the period 2014.07.01-2020.01.16. The two simulations may eventually yield different distributions of temperature and salinity as tidal contribution enhances mixing of the water. Therefore, distribution of 125 temperature and salinity in non-tidal simulation is fitted to tidal one after every few months.
7 https://doi.org/10.5194/os-2020-7 Preprint. Discussion started: 27 January 2020 c Author(s) 2020. CC BY 4.0 License.   with tidal forcing only and 3 centimeters in real case. Amplitude is a bit higher in December than in June with only tidal forcing because the Sun is slightly closer to Earth in December-January. However, tidal amplitudes are smaller in winter (wellmixed conditions) than during summer (stratified conditions) and this difference is up to 10 % (Müller, 2012). December. If we look at particular day of summer or winter solstice each year then the moment of maximum water level is always ±2 hours from either solar noon or solar midnight, respectively. Thus, there is deterministic gravitational influence of the Sun despite the Moon has stronger effect on tides. In April-May the maximum water level is achieved few hours after the solar noon and vice versa in August-September. Water level influenced by gravity of the Sun can be represented by the form

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where t 0 is time moment of water level maximum at noon of the day of summer solstice with gravity forcing of the Sun only; K and P are constants characterising K 1 and P 1 components, respectively. Data in Fig. 8  suggests that daily variation of wind is much stronger in summer than in winter. Thus, radiation effects are important not only for S 1 but also K 1 and P 1 tidal components, see Medvedev et al. (2016). Now, let us check the pressure variations at various times of the day, see Fig. 11. The average surface pressure is nearly 1 mbar higher in Gulf Riga than in Baltic proper in late autumn and winter due to lower Atlantic influence. Figure 11 confirms that at the time of the daily maximum air temperature, surface pressure is falling at its fastest rate (Cook, 2012). The reanalysis data also confirms that sea breeze winds peak at the 165 maximum temperature with a small lag, less than one hour (Cook, 2012). Daily change of pressure is stronger in Baltic proper than in Gulf of Riga, i.e., the difference between these points is mainly due to pressure change in Baltic proper. Notable daily pressure variations occur between March and October. The amplitude of mean sea level pressure difference is only 30 Pa, which can result in daily amplitudes of water level of Gulf of Riga with less than 1 cm by barotropic pressure gradient. The anti phase of pressure difference, see Fig. 11, is roughly one to two hours before phase of water level in Gulf of Riga. Such a 170 small lag time is not enough to fill Gulf of Riga. Therefore, daily water level oscillations can be caused primarily by sea and land breezes.
is not as large to call the effect as "breeze" but it helps to interpret the results. Coriolis force in northern hemisphere leads to clockwise daily variation of wind direction (Miao et al., 2009). The amplitude of breeze is stronger in Gulf of Riga than in central Baltic proper as the breezes are stronger at the interface between land and sea. It is interesting that amplitude of daily sea and land breezes is weakly dependent on mean sea level pressure. Approximately the same amplitude of daily variation of wind occurs in cases of high pressure with typically light easterly winds as for low pressure situations with stronger westerly 180 winds generated by Atlantic cyclones. The phase of daily variation of water level is consistent with phase of land and sea breezes accounting time lag between these. Easterly wind cedes quickly after 12:00-15:00 UTC, see Fig. 12. Thus, the time of maximum water level in Gulf of Riga neglecting tidal influence is at about 18:00 UTC.
Further, combining the effect of sea and land breezes with gravitational tides (Fig. 8) we can explain the seasonal variability of daily water level variations in Fig. 5. Atmospheric forcing is nearly in phase with gravity of the Sun in March-May that 185 yields highest daily variation of water level. This occurs despite both the gravitational tides are stronger in June-July and the effect of daily wind variation is at peak. The reason of lower daily variations in late summer and especially September, is that sea and land breeze is almost in anti-phase with gravitational tides and they cancel each other. In December, there is almost no effect from solar radiation but there is strong influence of solar gravity when the Sun is close to antipodal position in midnight.
Let us approximate the variation of radiation tide by: where ω d , ω y are cyclic frequencies corresponding to 24 h and tropical year periods, respectively; ∆t is time interval between maximum water level created by solar radiation and gravity forcing of the Sun at day of summer solstice; A, B are constants.
The approximation assumes symmetric behaviour with respect to solstice. The product can be written as:

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Combining with gravitational approximation in Eq.
(2) the total change of water level becomes where α 1 and α 2 are phase constants and resulting amplitudes, i.e., tidal components K 1 and P 1 are ∆t is roughly about 7-8 hours in Gulf of Riga according to Fig. 8 and Fig. 5. Thus, cos(ω d ∆t) ≈ −0.4. Therefore, radiation 200 tide slightly changes the amplitudes of both components K 1 and P 1 . According to observations A ≈ 1 cm and B ≈ 0.8 cm in Skulte station that are between gravitational amplitudes of K ≈ 1.7 cm and P ≈ 0.6 cm in Eq.
(2). Thus, gravitational tides have still larger contribution in diurnal oscillations than see breeze. Nevertheless, phase difference between them is the main source of daily pattern of diurnal oscillations in Gulf of Riga from spring to autumn equinox. Results for station Skulte are summarised in Fig. 13. Despite the approximation by Eq. (5) is rather simple, it qualitatively well describes daily water level 205 variation in every season.

Conclusions
Gulf of Riga experiences distinct diurnal mode of seiches whose period is mainly determined by the hydrodynamic resistance in Irbe strait. Väinameri sea laying in the north has also semidiurnal oscillations.
Phase lag between Gulf of Riga and Baltic proper is typically about 6 hours for all locations in the Gulf of Riga. The phase 210 is almost the same in all stations around Gulf of Riga with highest water level amplitude in Pärnu and lowest in Kolka along south eastern coast.
Observations suggest presence of distinct S 1 tidal mode in Gulf of Riga. Purely gravitational tides do not cause appearance of S 1 tidal mode despite strong daily pattern both in June and December. Atmospheric influence causes appearance of not only S 1 component but influences K 1 and P 1 tidal modes.

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Real atmospheric forcing tends to minimize the effect of gravitational tides. The influence of stratification is minimal for the Gulf of Riga with respect to gravitational tides as the shallow Irbe strait is usually well mixed.
There are distinct daily mean sea level pressure variations in Gulf of Riga and Baltic proper with gradient of fraction of millibar per 100 km. This is, however, too small to cause significant daily water level oscillations in Gulf of Riga. Daily mean sea level pressure variations result from sea and land breeze effect.

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There is a distinct sea and land breeze effect in Gulf of Riga with daily amplitude of around 1 m/s in spring-summer. It results that water level in Gulf Riga culminates in late afternoon when the sea breeze is ceasing. That is not the case for autumn and winter when the daily phase of water level is related to cyclones. The interpretation of daily wind variation as sea or land breeze is only approximate as actual wind directions may be arbitrary.
Average daily water level variations are strongest in April-May when the solar gravity is in phase with sea breeze effect. In 225 contrast, average daily water level variations are lowest in September when gravitational tides are almost in anti-phase with effects of solar radiation.