Tidal oscillations in the Baltic Sea

ISSN 0001�4370, Oceanology, 2013, Vol. 53, No. 5, pp. 526–538. © Pleiades Publishing, Inc., 2013.Original Russian Text © I.P. Medvedev, A.B. Rabinovich, E.A. Kulikov, 2013, published in Okeanologiya, 2013, Vol. 53, No. 5, pp. 596–609.

526

1. INTRODUCTION

Sea level oscillations in the Baltic Sea are influ�enced by various processes. Long�period oscillationsare determined by changes in salinity and water tem�perature, water exchange with the North Sea throughthe Danish straits, and river flow. The mesoscale sealevel oscillations in the Baltic Sea, reaching a height ofseveral meters in some regions, are associated withstorm surges caused by cyclone motions over the sea.The tidal oscillations in the Baltic Sea level are formedby free tidal waves penetrating from the North Sea andforced tidal waves excited directly in its water area[15]. Despite the relatively small amplitudes of tides inthe Baltic Sea, their accurate assessment is crucial forthe understanding of the overall dynamics of the sea.Tides produce regular periodic oscillations of the sealevel and currents, which are permanently present inthe Baltic Sea; all the other processes superimposetidal background. The mechanism responsible for theformation of tides in the Baltic Sea and in the Gulf ofFinland in particular, is in many ways similar to themechanism of the formation of storm surges in thisregion; so tidal study enables us to get important infor�mation about the formation of the St. Petersburgfloods [1].

Tidal oscillations in the Baltic Sea have been stud�ied for over 140 years, but the nature of their anoma�lous features in this body of water still remains unclear[13, 14, 23]. Defant [9] and Magaard, and Krauss [16]summarized the data obtained by various researchersand constructed tidal maps for the main diurnal andsemidiurnal harmonics across the entire Baltic Sea.

These studies describe the tidal oscillations in the Gulfof Bothnia off the coasts of Sweden and Finland in suf�ficient detail, but relatively little attention is paid totides off the coast of the Russian Federation and Balticstates. In most studies conducted earlier, except [13,14], relatively short series of observational data notexceeding one year were used. The accuracy of thesecalculations is relatively poor because of low tide levelsin the Baltic Sea compared to the noise level. There�fore, calculated amplitudes and phases of tidal constit�uents based on various series of observations werestrongly different. In this work, the analysis is based onlong�term series of hourly values of the sea level from35 stations with 25 of them providing series of observa�tions for more than 15 years. The study was conductedby means of spectral and harmonic analyses of tides[20]. The use of long�term observational series made itpossible to significantly increase the accuracy of thecalculated tidal constituents, while their spectral anal�ysis revealed a number of interesting effects, in partic�ular, showing the significant role of radiational tides insome areas of the Baltic Sea.

In the past decade, papers on numerical simulationof tidal oscillations in the Baltic Sea began to appear(see, e.g., [18, 22]). The tidal harmonic constantsobtained in this study made it possible to verify thenumerical results and to evaluate the accuracy of tidalmaps in the Baltic Sea.

2. OBSERVATIONS

In this paper, we used the data from 35 Russian,Latvian, Estonian, Lithuanian, Swedish, Finnish, and

Tidal Oscillations in the Baltic SeaI. P. Medvedev, A. B. Rabinovich, and E. A. Kulikov

Institute of Oceanology, Russian Academy of Sciences, Moscow, Russiae�mail: [email protected]

Received October 15, 2012; in final form, February 27, 2013

Abstract—Long�term hourly data from 35 tide gauge stations, including 15 stations in the Gulf of Finland,were used to examine tidal sea level oscillations of the Baltic Sea. High�resolution spectral analysis revealedthe well�defined fine structure of tidal peaks with diurnal peaks at most stations being higher than semidiur�nal. At some stations (e.g., Narva, Daugava, and Wladyslawowo), high frequency radiational tidal peaks withperiods multiple of the solar day (3, 4, 5, 6, and 8 cpd) were detected; the respective oscillations are supposedto be caused by seabreeze winds. Harmonic analysis of tides for individual yearly sea level series followed byvector averaging over the entire observational period was used to estimate the amplitudes and phases of 16tidal constituents. The maximum tidal oscillations of 17–19 cm were found to be observed in the Gulf of Fin�land and, first of all, in Neva Bay (in the head of the gulf). Diurnal or mixed diurnal tides are predominant inalmost the entire Baltic Sea. The comparison of the observed tides with those theoretically computed showedthat the existing numerical models of the main tidal harmonics generally quite accurately reproduce thestructure of the tides in the Baltic Sea except for some regions of the Gulf of Bothnia.

DOI: 10.1134/S0001437013050123

MARINEPHYSICS

OCEANOLOGY Vol. 53 No. 5 2013

TIDAL OSCILLATIONS IN THE BALTIC SEA 527

Polish coastal tide gauges. Their locations are shownin Fig. 1 and their characteristics are given in Table 1.All observational series had 1—hour sampling. Thedata used were referenced to universal time (Green�wich). The sea level for all stations was referenced tozero of the Baltic height system (0 BS).

The series of observations were carefully checked;shifts and spikes were eliminated, short gaps (shorterthan one day) were interpolated. The series containinglong gaps were excluded from the analysis.

The sea level oscillations in the Gulf of Finland areof the greatest interest to this study. In order to exam�ine these oscillations, hourly observational series fromfifteen stations were used (Fig. 1). On the southerncoast of the gulf, there are the Estonian stations Narva,Suurpea, and Tallinn; while, on the northern coastthere are the Finnish tide gauge stations Hanko, Hel�sinki, and Hamina and the Russian stations Vyborg,Primorsk, Gogland, and Moshchniy. The mostdensely located tide gauge stations are in Neva Bay(see Box 1a), where the sea level variability wasrecorded by five tide gauges, i.e., Shepelevo,Lomonosov, Kronstadt, Nevskaya Ustievaya, and theGorniy Institute. The hourly series of sea level obser�vations obtained at Gorniy Institute station is thelongest used in this study (31 years).

3. SPECTRAL ANALYSIS

The spectral analysis of sea level records makes itpossible to evaluate the distribution of wave energy invarious frequency bands and to estimate the fre�quency�selective properties of the examined waterbodies. The spectra were calculated using the FastFourier Transform. Following the recommendationsof [10], the spectral Kaiser–Bessel window with aselected window length of N = 8766 h (i.e. one astro�nomical year) and with a half�window overlap wasused to improve the quality of the calculations and toreduce the Gibbs effect. Accordingly, the spectral res�olution was Δf ≈ 0.00274 cpd = 1 cpy.

As an example, Fig. 2 shows the spectra of sea leveloscillations from four stations located in differentparts of the Baltic Sea, i.e., Landsort (26) located inthe open part of the Baltic Sea, Skulte (2) in the Gulfof Riga, Hanko (19) at the entrance to the Gulf of Fin�land, and Kronstadt (10) in the head of the Gulf ofFinland (the numbers in the parentheses correspondto the station numbers in Fig. 1 and Table 1). Thelengths of the observational series for these stations aredifferent. The number of degrees of freedom (ν) (i.e.,56 for Landsort, 16 for Skulte, 62 for Hanko, and58 for Kronstadt) and therefore, the confidence inter�vals for the spectra vary respectively.

The major properties of all spectra are similar. Mostof the energy is concentrated at low frequencies, and

N

S

10

60 62 64 66

15

29 30

28

2726

2021

19

25

24

Sweden

B a l t i c S e a

G u l f o f B o t h n i a

Finland

Neva Bay

RussiaEstonia

Latvia

Lithuania

Poland

Danish

straits

Gulf ofGdansk

Gulf of Finland

N

S

Gulf ofRiga

4

3

21

56

7

18

17

1314

35333432

1615

8

23

22

31

9

10

11

12

60.2

60.0

29.5 30.0

20 25 30

Fig. 1. Location of the stations used in this study. The numbers in the figure correspond to the station numbers listed in Table 1.The Neva Bay area is shown in the inset.

528


MEDVEDEV et al.

quite fast decay is observed at higher frequencies. Thegeneral power law of the spectra decay is f –2, where f isthe spectral frequency. This law, indicated in Fig. 2 bya straight line, is typical for the long�wave spectra inthe open ocean [6]. However, in the Baltic Sea for spe�cific frequency bands this law is distorted. This isapparently because the basin is almost completelyenclosed. In particular, at frequencies from 0.8 to

1.6 cpd, there is a wide “bump”, which is probablyrelated to the fundamental natural period of 27 h dom�inant in the Baltic Sea [15]. The individual features ofthe spectra at specific stations are also determined by theeigen mode structures of the respective water bodies (thegulfs of Riga, Finland, Bothnia, etc.). For example, inthe Gulf of Finland, at frequencies from 0.05 to 2 cpd,there is a marked increase of the sea level energy

Table 1. Characteristics of the stations used for the analysis of sea level oscillations in the Baltic Sea

No. Station Name Latitude (N) Longitude (E) Country Beginning End

1 Daugava 57.059 24.026 Latvia 01.01.78 31.12.82

2 Skulte 57.300 24.400 Latvia 01.01.78 31.12.82

3 Pärnu 58.400 24.500 Estonia 01.01.78 31.12.82

4 Ristna 58.900 22.100 Estonia 01.01.78 31.12.82

5 Tallinn 59.450 24.800 Estonia 01.01.78 14.05.95

6 Suurpea 59.600 25.700 Estonia 01.01.78 14.05.95

7 Narva 59.450 28.050 Estonia 01.01.77 31.12.91

8 Shepelevo 60.000 29.100 Russia 01.01.89 31.06.06

9 Lomonosov 59.900 29.800 Russia 01.01.92 31.12.06

10 Kronstadt 60.000 29.800 Russia 01.01.92 31.12.06

11 Nevskaya Ustievaya 59.900 30.300 Russia 01.01.92 31.06.04

12 Gorniy Institute 59.930 30.283 Russia 01.01.77 22.06.07

13 Vyborg 60.700 28.700 Russia 01.01.92 31.12.06

14 Primorsk 60.350 28.617 Russia 26.03.80 30.04.87

15 Moshchniy 60.000 27.830 Russia 01.10.79 31.08.83

16 Gogland 60.083 26.983 Russia 01.01.92 20.11.94

17 Hamina 60.567 27.183 Finland 01.10.92 30.09.08

18 Helsinki 60.150 24.967 Finland 01.10.92 30.09.08

19 Hanko 59.817 22.983 Finland 01.10.92 30.09.08

20 Föglö 60.050 20.730 Finland 01.10.92 30.09.08

21 Rauma 61.130 21.480 Finland 01.10.92 30.09.08

22 Mantyluoto 61.600 21.480 Finland 01.10.92 30.09.08

23 Kaskinen 62.380 21.220 Finland 01.10.92 30.09.08

24 Spikarna 62.363 17.531 Sweden 01.01.92 31.08.08

25 Forsmark 60.400 18.200 Sweden 01.01.92 31.08.08

26 Landsort 58.750 17.867 Sweden 01.01.92 31.08.04

27 Landsort Norra 58.767 17.867 Sweden 14.10.04 31.08.08

28 Visby 57.633 18.283 Sweden 01.01.92 31.08.08

29 Simrisamn 55.550 14.350 Sweden 01.01.92 31.08.08

30 Eland 57.366 17.097 Sweden 01.01.92 31.08.08

31 Wladyslawowo 54.800 18.417 Poland 01.01.92 31.08.08

32 Baltiysk 54.600 19.900 Russia 01.01.92 31.12.06

33 Otrkrytoe 54.900 21.066 Russia 01.01.92 31.12.06

34 Pionerskiy 54.900 20.200 Russia 01.09.98 31.10.04

35 Klaipeda 55.700 21.133 Lithuania 01.01.98 31.12.06



104

103

102

101

100

Hanko

(c)

O1

M2

K1 S2

f –2

95%

10–1

10–2 10–1 100 101

ν = 62

104

103

102

101

100

Kronstadt

(d)

O1

M2

K1

S2

f –2

95%

10–1

Frequency, cpd10–1 100 101

ν = 58

10–2

104

103

102

101

100

Landsort

(а)

O1M2

K1

S2

f –2

95%

10–1

10–2

103 102 101

ν = 56

104

103

102

101

100

Skulte

(b)

O1

M2

K1

S2

f –2

95%

10–1

10–2

103 102 101

ν = 16

Period, hS

pect

ral d

ensi

ty, c

m2 /c

pd

Fig. 2. Spectra of sea level oscillations at (a) Landsort, (b) Skulte, (c) Hanko, and (d) Kronstadt. The peaks corresponding to themain tidal harmonics are indicated. The number of degrees of freedom in calculated spectra ( ) is specified, and the correspond�ing confidence intervals are shown. The spectral window length is N = 8766 h (one astronomical year).

ν

toward the head of the gulf. At these frequencies, thespectral energy at the entrance of the Gulf of Finland(at Hanko, Fig. 2c) is approximately an order of mag�nitude smaller than at the head of the gulf (e.g., atKronstadt, Fig. 2d). This increase of the spectrumeastward is probably due to the structure of eigen oscil�lations in the Gulf of Finland with the nodal line nearthe entrance and the antinodal point at the head of thegulf (see, e.g., [1, 12]). At frequencies higher than 2cpd, the difference in the spectral energy for the sta�

tions located at the entrance to the gulf and at its headis insignificant.

One of the most important features of the sea levelspectra in the Baltic Sea are narrow and sharp peakscorresponding to the frequencies of the main tidal har�monics: diurnal K1 (period of 23.93 h) and O1 (25.82 h)and the semidiurnal M2 (12.42 h) and S2 (12.00 h).Although the amplitudes of tidal oscillations in theBaltic Sea are relatively small, because these oscilla�tions are very regular and deterministic, the spectra ofthe sea level oscillations at all stations have well�

530


MEDVEDEV et al.

defined tidal peaks. The presence of these tidal peaksin sea level spectra in the Baltic Sea was also indicatedby other researchers (see, e.g., [12,17]).

The specific feature of tidal oscillations of the Bal�tic Sea level, which is clearly seen in the spectra, is themarked predominance of the amplitudes of diurnaltidal harmonics over semidiurnal. This feature seemssurprising taking into account that in the AtlanticOcean and the associated marginal seas semidiurnaltides significantly prevail [20]. The strongest domi�nance of diurnal tides is observed in the eastern part ofthe Baltic Sea and, above all, in the Gulf of Finland(see the spectrum of sea level oscillations at Kronstadt,Fig. 2d). At the same time, at Hanko located at theentrance to the gulf (see Fig. 1), the sea level spectralpeaks at semidiurnal frequencies (M2 and S2) arehigher than at diurnal frequencies (Fig. 2c). Thisappears to be associated with the presence of the eigenmode nodal line for near�diurnal frequency [1,12].

4. RADIATIONAL TIDES

In addition to spectral peaks associated with majordiurnal and semidiurnal tidal constituents, spectralpeaks at frequencies of 3, 4, 5, 6, 7, and 8 cpd, multi�ples of the solar day, are well defined in the sea levelspectra at some stations, in particular at Narva, Wla�dyslawowo, and Daugava (Fig. 3) (for clarity, not alogarithmic, as in Fig. 2, but a linear scale is used inFig. 3 for the frequency axis). Typically, the long�wavespectra at frequencies higher than tidal are rathermonotonic, which is distorted at frequencies ofapproximately of 3, 4, 5, and 6 cycles per lunar day [6,20]. These spectral peaks are associated with highertidal harmonics, which are formed in shallow waterareas as a result of the nonlinear interaction of majorgravitational tidal constituents associated mainly withmovements of the Moon [20]. In the Baltic Sea, thetides are weak and there are no conditions to generatenonlinear shallow–water harmonics. In the consid�ered case, the precise diurnal cyclicity suggests thatthese oscillations are associated not with the Moon butwith the Sun and are caused not by gravitationaleffects but by the solar radiational effects on the sealevel. Munk and Cartwright called the movementsdirectly or indirectly related to the radiation of the Sunas radiational tides [19]. Thus, the radiational har�monics are annual (Sa), semiannual (Ssa), and diurnal(S1) species. The corresponding sea level oscillationsare formed not under the influence of gravitationalforcing but due to the solar radiation [20]. Radiationexposure also affects the semidiurnal frequency. Asshown by Zetler [26] and Wunsch [24], the radiationalharmonic S2 is typically about 16% of the gravitationalS2. Radiation tides are formed by the combined effectof various periodic factors associated with Sun radia�tion. The most important of them are (1) oscillationsof the air temperature and associated sea surface tem�perature, (2) atmospheric tides, and (3) seabreeze

winds. The prevalence of one of these factors dependson the specific physical and geographical conditions inthe area of the observations [24]. The effect of high�frequency radiational species S3, S4, S5, S6, S7, etc.,was found by Kulikov and Rabinovich [2, 3 ] based onthe data of the second Soviet–American expedition tomeasure tsunamis in the open ocean and wasexplained by the asymmetry (day/night) of the solarradiation on the atmosphere and ocean surface.

Figure 3a shows the sea level spectrum at Daugava.The peaks at frequencies of 3, 4, 5, and 6 cpd are quitepronounced and comparable with those of the majortidal constituents. The radiational tides at Narva(Fig. 3b) are relatively weaker and have their mainpeaks at frequencies of 3, 4, and 5 cpd. At Wladysla�wowo, broad radiational peaks are observed at fre�quencies of 3, 4, 5, 6, 7, and 8 cpd (Fig. 3c). Appar�ently, the main factor forcing radiational harmonics atsome stations of the Baltic coast are the seabreezewinds. This specific factor strongly depends on theparticular observational area, while the variability ofthe air/water temperature and the atmospheric tidesdo not have such a selective nature. Note that becauseof the closeness of the frequencies of the diurnal radi�ational (S1) and the gravitatinal (K1 and P1) tidal har�monics, their separation is quite a challenge, but it ispossible to resolve them based on long series of observa�tions (see the next section). It should be emphasized thatalthough the amplitudes of the radiational tidal sea leveloscillations are relatively small (except of seasonal oscil�lations), tidal currents caused by radiational effects, inparticular by seabreeze winds, can be quite significantin the surface layer and reach 1–2 knots and more(see, e.g., [25]).

5. HIGH RESOLUTION SPECTRA

Long series of tide gauge observations make it pos�sible to provide the detailed tidal spectroscopy andeven to “split” individual tidal harmonics [19]. Suchan analysis is especially useful in areas with strong tidaloscillations (see, e.g., [7, 11]). However, even in areaswith relatively weak tides, in particular in the BalticSea, the high�resolution spectra can reveal certainproperties of the tidal formation imperceptible in theordinary spectra. For this analysis, we selected two sta�tions with long�term and high�quality observationalseries, i.e., Gorniy Institute (31) and Tallinn (18). Thelength of the spectral window for both data sets wasN = 87660 h = 3652.5 days, the spectral resolution wasΔf ≈ 0.000274 cpd, and the number of degrees of free�dom was ν = 10 for the first station and ν = 4 for thesecond. Most attention was paid to the diurnal andsemidiurnal frequency bands (Fig. 4).

The most interesting features of sea level oscilla�tions in the Baltic Sea found as a result of this analysisare as follows:

(1) The background long–wave noise level in thediurnal frequency range was significantly higher (by



more than an order of magnitude) than in the semidi�urnal frequency range.

(2) The spectral peaks corresponding to the majortidal harmonics were clearly above the noise level andsignificantly exceeded the 95% confidence spectrallevel. This is not only for the semidiurnal frequencyband, where the noise level is low, but also for the diur�nal band, where it is high.

(3) The spectral peaks corresponding to the diurnaltidal harmonics dominate. The amplitude of the О1

harmonic (the main lunar diurnal harmonic with aperiod of 25.82 h) is even higher than of К1 (the luni�solar declinational harmonic with a period of 23.93 h),although the theoretical ratio of these harmonics is:

0.71.

(4) In contrast, the ratios of four major semidiurnalharmonics (М2, S2, N2, and K2) is in good agreementwith the theoretical ratio.

One of the interesting effects revealed as a result ofthe high�resolution spectral analysis was the manifes�

1 1O KH H ≈

104

103

102

101

100

10–1

10–2

Wladyslawowo

O1

M2

K1

S3 95%

ν = 64

S4 S5 S6 S7

S2

0 1 2 3 4 5 6 7 8Frequency, cpd

104

103

102

101

100

10–1

10–2

NarvaO1

M2

K1

S3 95%

ν = 56

S4 S5S6 S7

S2

104

103

102

101

100

10–1

10–2

(а)

O1

M2

K1

S3

95%

ν = 16

S4 S5S6

S7

S2

(b)

(c)

DaugavaS

pect

ral d

ensi

ty,

cm2 /c

pd

Fig. 3. Spectra of sea level oscillations at (a) Daugava, (b) Narva, and (c) Wladyslawowo. The main tidal peaks, including thepeaks caused by the high–frequency radiational harmonics, are indicated.

532


MEDVEDEV et al.

tation of the radiation harmonic S1. Its frequency islocated between the frequencies of gravitational har�monics К1 and Р1 and was indistinguishable in theconventional spectra. At Gorniy Institute, the magni�tude of S1 is only slightly inferior to the P1 harmonic,and, at Tallinn, it even exceeds it. In general, theresults of the spectral analysis show the clear preva�lence of tidal harmonics in comparison with the back�ground noise level in the Baltic Sea. Because of this, itis possible to calculate tides directly by means of theharmonic analysis (see, e.g., [20]).

6. HARMONIC ANALYSIS OF THE TIDES

In this paper, the least squares method of harmonicanalysis [5, 20] was used to estimate mean tidal ampli�tudes and phases. Using this method, the tidal constit�uents for 35 stations in the Baltic Sea were computed.Yearly series of observations without significant gapswere used in this analysis. Short gaps were interpo�lated. The results of the calculations for individualyears were followed by vector averaging over the entireobservational period [8]. In total, 16 tidal constituentswere calculated, i.e., two seasonal (Sa and Ssa), sixdiurnal (Q1, O1, P1, S1, K1, J1), four semidiurnal (N2,M2, S2, and K2), and four high–frequency shallow–water harmonics (MK3, MO3, M4, and MS4).

The following expressions (see [20]) were used toestimate errors in the calculated tidal amplitudes ( )and phases ( ):

(in radians), (1)

where is the variance of the noise (backgroundoscillations) in the frequency range of Δω (diurnal orsemidiurnal), T is the length of the series used for thecalculations (in this case, Т = 1 year), and H is theamplitude of the corresponding tidal harmonic. From(1), it is clear that the amplitude error is absolute anduniform for all harmonics in the particular frequencyrange, while the phase error is relative; i.e., the smallerthe amplitude of the corresponding harmonic, thegreater the phase error.

According to the spectral analysis (Section 5), for

Gorniy Institute, = 23.1 cm2 for the

diurnal range and = 2.2 cm2 for the

semidiurnal range, where is the mean spectrumvalue for the respective frequency range (Fig. 4a), and

= 0.2 cpd is the width of the range. For Tallinn, the

similar variances are significantly smaller, i.e., =

5.1 cm2 and = 0.31 cm2 (Fig. 4b). The calcu�lated error values for diurnal amplitudes at these stationsare εH(1) = ±0.56 cm (Gorniy Institute) and ±0.26 cm(Tallinn), and for semidiurnal amplitudes these valuesare εH(2) = ±0.17 cm and ±0.06 cm, respectively.

Hε

Gε

2

,HT

Δωσε =

Δω

21 H

GH T H

Δωσ εε = =

Δω

2Δωσ

2 ˆ(1) (1)xSΔωσ = Δω

2 ˆ(2) (2)xSΔωσ = Δω

ˆ (1, 2)xS

Δω

2 (1)Δωσ

2 (2)Δωσ

The values of the phase error (εG) are inversely pro�portional to the amplitudes of the corresponding har�monics. For Gorniy Institute, these errors (based onthe data from Table 2) are ±65° (K1), ±71° (O1), ±30°(M2), and ±67° (S2), and, for Tallinn, these values are±63° (K1), ±57° (O1), ±31° (M2) and ±70° (S2). Forthe other stations, the amplitude and phase errors aregenerally similar. Thus, the errors of the calculatedamplitude values are relatively low, while for phasesthey can be significant. The results of the calculationsof tidal amplitudes and phases for various years con�firm this tendency, i.e. relatively consistent amplitudevalues for major diurnal and semidiurnal constituents(within ±0.3 cm) and rather large discrepancy inphases (±8–30°). Nevertheless, the averaging of thecalculation results over a large number of years can sig�nificantly improve the accuracy because the error

decreases as , where n is the number of indepen�dent yearly series of observations used for the calcula�tions.

The harmonic deviation and the errors were deter�mined for all series of observations and for all specificharmonics at individual stations. For stations in theGulf of Finland, the amplitude error for major diurnalharmonics O1 and K1 is about 14% of the estimatedvalue. For Baltiysk, the error value for the O1 har�monic is 25% of the amplitude. The phase error formajor diurnal harmonics is 8°. The amplitude andphase of constituent M2 changes insignificantly fromyear to year. The amplitude error is less than 5% of itsvalue, the phase error is ~2°–4°.

As an example, the averaged values of tidal ampli�tudes and phases calculated for six stations located invarious parts of the Baltic Sea are given in Table 2. Theannual (Sa) and semiannual (Ssa) seasonal harmonicsclearly dominate. Thus, the amplitude of the Sa har�monic at Daugava reaches 19.4 cm (together with Ssa,the overall variation is almost 0.5 meters!). At Tallinn,the annual harmonic amplitude reaches 15 cm and atGorniy Institute it is 12.3 cm. Significant seasonal sealevel changes are also observed at other stations. Thesemiannual harmonic Ssa at almost all stations isweaker than the annual, although there are exceptions(Baltiysk and Landsort). The dominating contributionof seasonal oscillations to the overall energy budget ofsea level oscillations in the Baltic Sea was noted byLisitzin [15]. The interannual variability of annual andsemiannual harmonics is high: the Sa and Ssa ampli�tudes vary from year to year with a standard deviationfor most of the stations of 3 and 5 cm, respectively. Thephases of seasonal harmonics are also strongly vari�able, i.e., the difference in the Sa phases for two con�secutive years can be up to 80°. In general, the analysisof seasonal oscillations in the Baltic Sea is an indepen�dent issue which requires a special study.

1 n



Most attention was paid to the analysis of the majordiurnal and semidiurnal tidal harmonics. The ampli�tudes of the diurnal harmonics O1 and K1 are approx�imately equal and reach their maximum values at thehead of the Gulf of Finland (about 3 cm at Kronstadtand Gorniy Institute), as well as in the Gulf of Riga(up to 1.8 cm). The amplitudes of constituents Р1 andS1 have similar values (0.2–1.5 cm) and are approxi�mately equal to 0.33–0.5 of the amplitude of К1. Themaximum amplitudes of the main semidiurnal М2

harmonic (about 2 cm) are observed at the head of the

Gulf of Finland (Kronstadt, Gorniy Institute andother stations in the area).

The ratio of the major diurnal harmonics to themajor semidiurnal harmonics (“Form Factor”) deter�mines the type of tides (see, e.g., [20]):

(2)

For the six stations shown in Table 2, the largest valueof this ratio was observed at Daugava (F = 4.6), wherethere is a regular diurnal tide, while the lowest value

1 1

2 2

K O

M S

.H H

FH H

+

=

+

104

103

102

101

100

(b)

K1

S1

P1Q1

O1

95%

0.88 0.90 0.92 0.94 0.96 0.98 1.00 1.02 1.04

M2

S2

K2N2

95%

1.88 1.90 1.92 1.94 1.96 1.98 2.00 2.02 2.04

Tallinn

Spe

ctra

l den

sity

, cm

2 /cpd

Frequency, cpd

105

104

103

102

101

(а)

K1

S1

P1

Q1

O1

95%

0.88 0.90 0.92 0.94 0.96 0.98 1.00 1.02 1.04

M2

S2

K2

N2

95%

1.88 1.90 1.92 1.94 1.96 1.98 2.00 2.02 2.04

Gorniy Institute

Fig. 4. High�resolution spectra of sea level oscillations in the Baltic Sea level for diurnal and semidiurnal frequency bands at (a)Gorniy Institute and (b) Tallinn. The thin dashed line shows the 95% confidence levels for diurnal and semidiurnal harmonicscalculated in accordance with the χ2 distribution for each band. For better visualization, the spectra for Gorniy Institute wereraised by one order of magnitude in comparison with Tallinn.

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MEDVEDEV et al.

was observed at Baltiysk (F = 0.9), where there aremixed semidiurnal tides.

A rough estimate of the maximum possible theoret�ical tidal height can be evaluated as [23]:

(3)

According to the data presented in Table 2, the maxi�mum tides (I ~ 18 cm) are at the head of the Gulf ofFinland (Kronstadt and Gorniy Institute) and theminimum tides (I = 3.5 cm) are at the eastern part ofthe Gulf of Gdansk (Baltiysk).

The nonlinear tidal effects in the Baltic Sea areweak. According to the results of the harmonic analy�sis, the amplitudes of high–frequency shallow–waterconstituents proved to be negligible, which is consis�tent with the results of the spectral analysis (Fig. 2).

The averaged values of the amplitudes and phasesof the major tidal harmonics (О1 , К1, M2, and S2) andalso of the characteristics F and I for all 35 stations areshown in Table 3. In general, the results of the analysisof all 35 stations confirm the previous findings basedon analyses of six stations presented in Table 2:

(1) The eastern part of the Baltic Sea is dominatedby diurnal tides. This prevalence is particularly notice�able in the central part of the Gulf of Finland (Hels�

1 1 2 2K O M S2( ).I H H H H= + + +

inki, F = 8.1; Suurpea, F = 6.3) and in the eastern partof the Gulf of Riga (Pärnu, F = 6.5).

(2) The maximum relative impact of semidiurnaltides (F ~ 0.6–0.9) is in the Gulf of Gdansk (Wladys�lawowo and Baltiysk) and at the entrance to the Gulfof Bothnia (Föglö, Rauma, and Hanko). In theseareas, mixed semidiurnal tides are observed.

(3) The main diurnal harmonics K1 and O1 changewithin the Baltic Sea basin in a similar way, and theiramplitudes are approximately equal.

(4) The maximum total tide (I ~ 17–19 cm) isobserved at the head of the Gulf of Finland and, aboveall, in Neva Bay (Nevskaya Ustievaya, Lomonosov–Kronstadt, and Gorniy Institute).

7. DISCUSSION

The results of analysis of the long�term sea levelobservations in the Baltic Sea demonstrate that inmost parts of the sea (in particular, in the gulfs ofBothnia, Riga and Finland), the type of tides is eitherdiurnal (F > 3) or mixed diurnal (1.5 < F < 3.0). Only atthe entrance to the Gulf of Bothnia, in the Gulf ofGdansk, and in some western parts of the Baltic Sea domixed semidiurnal tides prevail (0.5 < F < 1.5). It is

Table 2. Amplitudes and phases of the tidal constituents for six selected stations in the Baltic Sea

HarmonicBaltiysk Daugava Kronstadt Tallinn Gorniy Institute Landsort

H, cm G, ° H, cm G, ° H, cm G, ° H, cm G, ° H, cm G, ° H, cm G, °

Sa 2.56 244 19.36 204 8.89 251 15.04 223 12.28 218 5.84 255

Ssa 4.91 276 4.74 165 8.21 237 6.26 234 5.05 215 7.76 251

Q1 0.06 218 0.20 291 0.67 307 0.34 293 0.70 323 0.09 78

O1 0.41 231 1.76 357 3.05 343 1.65 320 2.84 352 0.38 80

P1 0.21 229 0.59 61 1.22 16 0.62 340 1.52 24 0.17 74

S1 0.13 2 0.91 116 1.01 157 0.71 133 1.01 162 0.25 332

K1 0.41 263 1.70 36 3.05 4 1.49 333 3.07 12 0.53 78

J1 0.08 259 0.31 141 0.17 318 0.09 35 0.02 7 0.02 91

N2 0.18 343 0.12 33 0.43 353 0.12 191 0.45 17 0.13 97

M2 0.57 354 0.60 62 1.93 25 0.70 227 2.02 47 0.41 113

S2 0.38 12 0.16 345 0.79 36 0.31 234 0.91 56 0.15 112

K2 0.08 13 0.07 333 0.27 18 0.09 205 0.27 44 0.04 79

MO3 0.04 329 0.02 140 0.04 22 0.03 211 0.07 354 0.01 31

MK3 0.01 223 0.01 19 0.03 337 0.01 70 0.07 334 0.02 92

M4 0.02 199 0.01 126 0.01 313 0.01 305 0.05 54 0.01 42

MS4 0.00 221 0.03 265 0.01 35 0.01 9 0.02 350 0.00 13

F 0.9 4.6 2.2 3.1 2.0 1.6

I 3.5 8.4 17.6 8.3 17.7 2.9



known that diurnal tides in the Atlantic Ocean and themarginal seas of the ocean are small compared tosemidiurnal tides. For example, in the North Sea,which is connected to the Baltic Sea by the Danishstraits, the amplitudes of the semidiurnal harmonicsare approximately 20–30 times greater than those ofthe diurnal tides [20]. Consequently, either the diurnaltide is formed directly in the Baltic Sea (i.e., it is indig�

enous (locally generated) by the definition of Nekrasov[4]) or resonant amplification of small diurnal tidalwaves entering the sea occurs in the sea as a whole andin its individual gulfs(similar to the resonance amplifi�cation of the incoming tsunami and meteotsunamiwaves in some water bodies of the World Ocean [21]).The possible resonant nature of diurnal tides in theBaltic Sea is indicated by the fact that the amplitude of

Table 3. Calculated amplitudes and phases of major tidal constituents (O1, K1, M2, and S2) for all the stations. ParametersF (tidal Form Factor) and I (the characteristic tidal height) are also specified

No. StationO1 K1 M2 S2

F IH, cm G, ° H, cm G, ° H, cm G, ° H, cm G, °

1 Daugava 1.76 357 1.70 36 0.60 62 0.16 345 4.6 8.4

2 Skulte 1.69 355 1.53 41 0.48 57 0.16 60 5.0 7.7

3 Pärnu 1.56 315 1.88 357 0.29 357 0.23 326 6.5 7.9

4 Ristna 0.50 261 0.43 270 1.17 185 0.43 191 0.6 5.0

5 Tallinn 1.65 320 1.49 333 0.70 227 0.31 234 3.1 8.3

6 Suurpea 1.59 325 1.68 346 0.32 294 0.20 279 6.3 7.6

7 Narva 2.44 328 2.43 338 1.15 337 0.14 322 3.8 12.3

8 Shepelevo 2.77 334 3.09 355 1.62 7 0.54 19 2.7 16.0

9 Lomonosov 3.17 345 3.21 8 2.05 31 0.86 43 2.2 18.6

10 Kronstadt 3.05 343 3.05 4 1.93 25 0.79 36 2.2 17.6

11 Nevskaya Ustievaya 3.12 351 3.38 13 2.10 45 0.94 48 2.1 19.1

12 Gorniy Institute 2.84 352 3.07 12 2.02 47 0.91 56 2.0 17.7

13 Vyborg 2.72 2 2.54 25 1.42 62 0.48 65 2.8 14.3

14 Primorsk 2.70 349 2.35 354 1.46 25 0.54 36 2.5 14.1

15 Moshchniy 2.02 333 2.19 360 0.96 7 0.29 19 3.4 10.9

16 Gogland 3.00 338 2.30 359 0.96 1 0.30 17 4.2 13.1

17 Hamina 2.53 346 2.50 10 1.09 35 0.34 46 3.5 12.9

18 Helsinki 1.87 348 1.81 13 0.35 117 0.11 142 8.1 8.3

19 Hanko 0.81 8 0.72 38 1.22 182 0.40 201 0.9 6.3

20 Föglö 0.38 82 0.45 109 0.88 195 0.30 211 0.7 4.0

21 Rauma 0.33 141 0.44 169 0.65 266 0.26 287 0.8 3.4

22 Mantyluoto 0.40 149 0.54 174 0.58 276 0.23 292 1.2 3.5

23 Kaskinen 0.54 152 0.71 177 0.48 285 0.21 297 1.8 3.9

24 Spikarna 0.53 83 0.60 85 0.57 344 0.26 1 1.4 3.9

25 Forsmark 0.44 66 0.44 49 0.54 10 0.23 23 1.1 3.3

26 Landsort 0.38 80 0.53 78 0.41 113 0.15 112 1.6 2.9

27 Landsort Norra 0.48 67 0.70 78 0.85 112 0.35 106 1.0 4.8

28 Visby 0.29 86 0.42 84 0.50 106 0.21 103 1.0 2.9

29 Simrisamn 1.15 107 1.18 111 0.72 42 0.17 68 2.6 6.4

30 Åland 0.58 72 0.78 73 0.38 96 0.19 87 2.4 3.9

31 Wladyslawowo 0.69 187 0.41 232 1.19 345 0.55 1 0.6 5.7

32 Baltiysk 0.41 231 0.41 263 0.57 354 0.38 12 0.9 3.5

33 Otrkrytoe 0.07 14 0.31 81 0.04 351 0.10 133 2.8 1.0

34 Pionerskiy 0.96 208 0.62 256 0.80 351 0.37 324 1.3 5.5

35 Klaipeda 0.80 198 0.57 257 0.42 323 0.15 314 2.4 3.9

536


MEDVEDEV et al.

N

S

0.386

0.672

1500.6

0.479

0.566

0.466

1.2106

0.5151

0.4149

0.3140

0.482

0.582

0.7187

0.4231 0.07

14

1.8356 1.7

355

1.6314

300270

0.5260

0.96208 0.8

1982.4327

1.6320

1.6324

180

120

0.3

210

30

0

90

Gulf ofRiga

0.88

1.2

0.660

1.9347

330

3337

2.5345

2.72

2.73492

333

2.1

Gulf of Finland0.6

Gulf of Bothnia

240

Neva60.2

60.0

29.5 30.0

Bay 3.1342

2.8352

2.8334 3.2

3453.1351

Gulf ofRiga

Gulf of Finland

Gulf of Bothnia

NevaBay

O1

29.0

60.2

60.0

29.5 30.029.0

1.925

1.67 2.1

31

2.047

2.145

0.5106

0.496

0.9112

0.6344

1.2345 0.8

351 0.4323

0.6354

0.04351

0.662

0.3357

0.7227

0.557

1.2182

0.7266

1.2185 0.4

117

0.3294

1.1337

0.967

1.135

0.5285

0.6276

1.462

1.525

0.961

0.510

0.4113

0.9195

330

270

120

1.2

0.3

0.60.9

150

330

120

0.6 0.9

150600

1.5

0.6

0.6

330

0.742 30

0 60

300

1.2

M2

330

Fig. 5. Tidal maps of diurnal (O1) and semidiurnal (M2) harmonics in the Baltic Sea according to the calculations and observa�tions. The solid lines are isoamplitudes, the dashed lines are cotidal lines (isophases) according to the numerical calculations [18,22]. The observed values are placed near the respective sites, the amplitude is in the numerator (in regular font) and the phase isin the denominator (in italics).



lunar diurnal harmonic O1 (having a period of 25.82 h,which is closer to the fundamental Baltic Sea period of27 h denoted in Fig. 2) exceeds the mean amplitude ofthe luni�solar declinational harmonic K1 (period of23.93 h), despite that in the tidal potential, the ampli�tude of O1 is approximately 30% smaller than that ofK1 [20]. This question requires more detailed research.

The calculations of mean tidal amplitudes andphases at 35 stations made it possible to compare theseobservations with the results of numerical modeling oftidal oscillations in the Baltic Sea. Figure 5 shows thetheoretical tidal maps of M2 and O1 according to thedata from [18, 22] and the observed amplitudes andphases for these two harmonics based on results of thepresent study. Some differences in the O1 phases areobserved on the east coast of the Gulf of Bothnia,while the O1 amplitudes are reproduced accurately.There are certain differences in this area, as well as atthe entrance to the Gulf of Bothnia, for M2. However,in general, the agreement of the calculations andobservations is good. The constructed tidal maps [18,22] correctly reproduce the main features of the tidaldynamics in the Baltic Sea.

The complicated distribution of tidal amplitudesand phases with amphidromic systems in certain largegulfs appears to be formed under the influence of eigensea level oscillations in the Baltic Sea and its gulfs. Thepresence of the amphidromic points leads to rapidphase changes in specific parts of the sea and abruptamplitude changes. The complex geometry of the Bal�tic Sea contributes to the formation of numerous localoscillation systems. In order to study them, it is neces�sary to conduct detailed numerical simulations using afine�resolution grid and additional sea level measure�ments.

8. CONCLUSIONS

In studies conducted in this paper, the long�termhourly observations of sea level changes in the BalticSea were thoroughly examined. The results of thisanalysis demonstrate that the sea level spectrum at fre�quencies higher than 0.65 cpd (i.e., at periods shorterthan 1.5 days) is formed under the significant influ�ence of the tidal forces of the Moon and the Sun; thestructure of the tidal oscillations and the sea levelresponse to the external forcing are determined by thestructure of eigen oscillations in the Baltic Sea and itsindividual water bodies. The general character of sealevel spectra in the Baltic Sea, and the frequency decayaccording to the power law f–2, is associated with nat�ural long�wave background oscillations formed underthe influence of atmospheric processes. However, thedistortion of this law and formation of a broad uplift(“bump”) at frequency range from 0.8 to 1.6 cpd withthe center frequency at about 0.9 cpd (27�hour period)at stations in the Gulf of Finland is probably associ�ated with the fundamental mode of the Gulf of Fin�land, which, according to calculations [1, 12, 15], has

approximately this particular period. It may beassumed that specifically the structure of this mode,which is similar to the Helmholtz mode (see [21]),leads to the increase in spectral density and amplifica�tion of diurnal oscillations toward the gulf head.

Despite their relatively small amplitude, tidal oscil�lations in the Baltic Sea clearly manifest themselves inthe form of sharp and distinct spectral peaks (Figs. 2,4). The peaks of major diurnal tidal harmonics О1 andК1 at most of the stations exceed the peaks associatedwith main semidiurnal harmonics М2 and S2. Thediurnal harmonics dominate over semidiurnal proba�bly because of the resonant character of diurnal tidesin the Baltic Sea. In addition, at some stations (Narva,Daugava, and Wladyslawowo), peaks with frequenciesof solar radiational harmonics appear (3, 4, 5, 6, and 7cpd) associated with the seabreeze winds at these sta�tions.

ACKNOWLEDGMENTS

We thank Drs. Sylvin Müller�Navarra (FederalMaritime and Hydrographic Agency of Germany,Hamburg, Germany) and Josef Cherniawsky (Insti�tute of Ocean Sciences, Sidney, British Columbia,Canada) for their help in adopting the results ofnumerical modeling of tides in the Baltic Sea; we alsoacknowledge Drs. V. I. Byshev and E. G. Morozov(Institute of Oceanology, Russian Academy of Sci�ences, Moscow) for reviewing the manuscript andhelpful comments.

This work was supported by the Russian Foundationfor Basic Research, Projects Nos. 12�05�00733, 12�05�00757 and 13�05�41360, and by the Ministry of Educationand Science (Project No. 11.G34.31.0007).

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3. E. A. Kulikov and A. B. Rabinovich, “Radiation tidesin the ocean and atmosphere,” Transactions (Doklady)USSR Acad. Sciences, Earth Sci. Sections, 271, 221–225 (1983).

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Translated by O. Pismenov

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Tidal oscillations in the Baltic Sea