Pole tide in the Baltic Sea

ISSN 0001�4370, Oceanology, 2014, Vol. 54, No. 2, pp. 121–131. © Pleiades Publishing, Inc., 2014.Original Russian Text © I.P. Medvedev, A.B. Rabinovich, E.A. Kulikov, 2014, published in Okeanologiya, 2014, Vol. 54, No. 2, pp. 137–148.

121

1. INTRODUCTION

The sea level surface of the ocean is under constantchange due to various astronomical factors [7]. Asearly as 1765, Leonard Euler assumed that the Earth,rotating around its axis, has to experience a slight wob�bling, i.e. free nutations with periods, according toEuler’s calculations, having to be around 10 months[2]. In the late 19th century, the American scientistSeth Chandler, based on astronomical latitudinalobservations, discovered the Eulerian nutations of theEarth’s axis. He demonstrated that their actual periodis longer than that predicted by Euler, and is around14 months [8]. These nutations became known as theChandler Wobble.

Euler additionally assumed that the sea level in theocean has to oscillate under the influence of free nuta�tions. The respective wave, which is basically similar tolong�period tidal oscillations, was called the pole tideby G. Darwin [9]. According to I.V. Maximov, the poletide in the ocean is “one of the most exciting globalgeophysical phenomena” [2]. In the early 20th cen�tury, the static theory of the pole tide was developed,which described the dependence of the height of sealevel oscillations on the Earth’s latitude and the ampli�tude of the pole axis rotation [16]. The amplitude ofthe static pole tide can be presented as

(1)

where g is the gravity acceleration; k ≈ 0.30 and h ≈

0.61 are the Love numbers (see, for example, [5]); ΔUis the variation of the Earth’s centrifugal potential :

(2)

(1 ),UH k hg

ΔΔ = + −

2 21 2 sin 2 ,U aΔ = − ω Δθ θ

where ω is the mean rotation rate and a is the meanradius of the Earth; θ is the colatitude; and Δθ is themagnitude of the radius vector of the Earth polemotion relative to the mean position. It follows from(2) that the pole tide has maximum amplitude at 45° Nand S but decreases towards the poles and equator.

The actual motions of the Earth’s rotation pole arerather complicated and subjected to noticeable long�term variations [2, 4]. This means that the ChandlerWobble can be considered as an amplitude�modulatedprocess with a temporal scale variability (Δθ) of 10–15 years [4] and the Δθ value itself changing from 0.05to 0.40′′ (arc seconds). For the typical value of Δθ =0.22′′, which approximately corresponds to theEarth’s nutation amplitude of 6.8 m, according to (1),the maximum magnitude of the pole tide at 45° N issupposed to be = 0.8 cm [2, 12]. Taking intoaccount that the accuracy of typical coastal tide gaugesis ±1 cm, we can say that our possibility of identifyingthe pole tide is at the edge of the gauge resolution.However, as follows from the long�term measurementsof sea level oscillations, the actual observed pole tide insome areas of the World Ocean is substantially higherthan follows from static theory (see, for example, [2,11, 17]). From this point of view, the Baltic Sea is ofparticular interest. According to early investigations,the highest amplitudes of the pole tide are observed inthe Gulf of Bothnia, the Baltic Sea, where they can beas large as 4.5–5 cm [3, 12, 13, 15]. The Gulf of Both�nia is located between 60° and 66° N; the theoreticalmagnitude of the static tide at these latitudes is 0.6–0.7 cm. Hence, the observed amplitudes are 6–8 timeshigher than the theoretical amplitudes. The mecha�nism of an anomalously high pole tide in the Gulf of

maxHΔ

Pole Tide in the Baltic SeaI. P. Medvedev, A. B. Rabinovich, and E. A. Kulikov

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

Received April 5, 2013

Abstract—The pole tide, which is driven by the Chandler Wobble, has a period of about 14 months and typ�ical amplitudes in the World Ocean of ~0.5 cm. However, in the Baltic Sea the pole tide is anomalously high.To examine this effect we used long�term hourly sea level records from 23 tide gauges and monthly recordsfrom 64 stations. The lengths of the series were up to 123 years for hourly records and 211 years for monthlyrecords. High�resolution spectra revealed a cluster of neighboring peaks with periods from 410 to 440 days.The results of spectral analysis were applied to estimate the integral amplitudes of pole tides from all availabletide gauges along the coast of the Baltic Sea. The height of the pole tide was found to gradually increase fromthe entrance (Danish Straits, 1.5–2 cm) to the northeast end of the sea. The largest amplitudes—up to 4.5–7 cm—were observed in the heads of the Gulf of Finland and the Gulf of Bothnia. Significant temporal fluc�tuations in amplitudes and periods of the pole tide were observed during the 19th and 20th centuries.

DOI: 10.1134/S0001437014020179

MARINE PHYSICS

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Bothnia of the Baltic Sea remains unclear [2, 14, 19].Moreover, it is unclear whether this anomaly is a spe�cific feature of the Gulf of Bothnia itself, or an inher�ent property of other areas of the Baltic Sea, in partic�ular, the Gulf of Finland.

Long�term series of sea level observations in theBaltic Sea have become available in recent years. Thismakes it possible to extensively investigate the spatialvariability of the pole tide over the entire area of the seaand to explore the nature of anomalous features of thepole tide in this basin. Medvedev et al. [6], use long�term hourly observations of sea level oscillations from35 tide gauges, including 15 in the Gulf of Finland, fora thorough analysis of tides in the Baltic Sea. Specifi�cally these data, supplemented by hourly observationsat several additional stations and monthly sea leveldata from 64 stations, were used in the present study toexamine the structure and specific features of the poletide in the Baltic Sea.

2. OBSERVATIONS

From 35 tide gauge stations with hourly recordsdescribed in [1, 6] we selected 23 that had the longestseries (typically 15–17 years) and the best quality (seetable). At Gorniy Institute, the station located at theentrance of Neva Bay (see Fig. 1), it was as long as31 years. In addition, we analyzed the data from Göte�borg and Stockholm (Sweden) obtained from the Uni�versity of Hawaii Sea Level Center (UHSLC)(http://uhslc.soest.hawaii.edu). The observationalperiods were 123 years at Stockholm and 40 years atGöteborg.

For a more detailed description and examination ofthe pole tide in the Baltic Sea, we collected and exam�ined mean monthly sea level data series from varioussites in the Baltic Sea; the data were obtained from thePermanent Service for Mean Sea Level (PSMSL) inLiverpool, UK. In total, we acquired information from61 stations located in the Baltic Sea, as well as in theDanish Straits and Kattegat, which connect the Balticand North seas (Fig. 1 and table). For comparison, wealso used three North Sea stations (Delfzijl, Cux�haven, and Esbjerg). All the measurements werereduced to the same time scale (GMT). The sea levelwas adjusted to the zero of the Baltic System ofHeights (0 BSH). The monthly mean data series wereconsiderably longer than the hourly series. For 22 sta�tions, we could construct series of monthly observa�tions longer than 100 years and for Stockholm amonthly series 211 years long (see table). The sea leveldata were thoroughly checked for errors and spikes;the gaps in the records were filled with interpolateddata.

It is evident from Fig. 1 that the combined use ofthe stations with hourly and monthly data enables usto cover the entire Baltic Sea, including the majorgulfs. As a result, the list of stations for the Gulf of Fin�

land included 13 stations and for the Gulf of Bothnia22 stations.

3. SPECTRAL PROPERTIES OF THE POLE TIDE IN THE BALTIC SEA

In contrast to the astronomical tide, the pole tide isnot a strictly deterministic harmonic process. Asemphasized by Munk and McDonald [4], the pole tideperiod does not remain uniform in time but varieswithin the limits of ±4%. Actually, the period andamplitude of the Chandler Wobble’s motion areinversely proportional; the increase in period corre�lates with decrease in amplitude of the motion. In gen�eral, the pole tide may be regarded as a stochastic nar�rowband signal in a predetermined frequency range.

Spectral analysis based on the fast Fourier trans�form was used to examine the pole tide in the BalticSea [10]. In order to achieve the high spectral resolu�tion (Δf ≈ 0.06687 cycles per year), the width of thespectral window for most of the stations with hourlysea level data was set to N = 131 072 h (for GorniyInstitute, N = 262 144 h, Δf ≈ 0.03344 cycles per year(cpy); for Stockholm, N = 999 944 h, Δf ≈ 0.008766 cpy).In this way this, we obtained spectra with the highestpossible spectral resolution.

Figure 2 shows the low�frequency sea level oscilla�tions with periods ranging from 100 to 500 days forfour stations located in various regions of the BalticSea: (a) Stockholm in the open Baltic Sea; (b) Kaski�nen in the Gulf of Bothnia; (c) Tallinn at the entranceto the Gulf of Finland; (d) Kronstadt at the head of theGulf of Finland.

In the low�frequency range, the most energeticspectral peak corresponds to the solar annual har�monic Sa, whose amplitude widely varies in the BalticSea from 4 to 20 cm [6, 12]. At certain stations, thesolar semi�annual harmonic Ssa reaches noticeableamplitudes of 5–10 cm. For example, the amplitude ofthe semi�annual constituent is comparable with theannual constituent at Kronstadt and Kaskinen.

In the context of the present study, the main inter�est is in the broad spectral maximum with the periodranging from 410 to 440 days, which corresponds tothe Р14 pole tide. In the sea level spectra for Kaskinen,Tallinn, and Kronstadt, this peak “expands” becauseof the low spectral resolution (Figs. 2b–2d). In con�trast, the spectral peak of the pole tide at Stockholm,the station with the longest series of hourly data(123 years), becomes narrower and divides into three

individual components of 417 ( ), 434 ( ), and 443

( ) days (Fig. 2a). In addition, the spectral density ofthe main peak at the period of 434 days, is more thantwo times higher than the adjacent local maxima of thepole tide. Also, there is a relatively well expressed spec�tral peak at a frequency of 1.78 cpy (~205 days), whichis probably an overtone of the pole tide frequency. Ingeneral, it is quite difficult to determine the pole tide

214P 0

14P1

14P


POLE TIDE IN THE BALTIC SEA 123

Station, observation period, sampling, and the integral amplitude ( )

No. Station name LatitudeN

LongitudeE Country Observation

periodSampling

(h, month) , cm

1 Tallinn 59.45 24.80 Estonia 1978–1995 1 h 6.0

2 Narva 59.45 28.05 Estonia 1977–1991 1 h 7.0

3 Shepelevo 60.00 29.10 Russia 1992–2006 1 h 4.4

4 Lomonosov 59.90 29.80 Russia 1992–2006 1 h 4.8

5 Kronstadt 60.00 29.80 Russia 1992–20061835–2005

1 h1 month

4.73.9

6 Gorniy Institute 59.93 30.28 Russia 1977–2007 1 h 3.9

7 Primorsk 60.35 28.62 Russia 1921–1939 1 month 4.3

8 Vyborg 60.70 28.73 Russia 1992–20061889–1944

1 h1 month

4.83.5

9 Hamina 60.57 27.18 Finland1992–20081928–2010

1 h1 month

4.74.6

10 Soderskar 60.12 25.42 Finland 1866–1936 1 month 3.0

11 Helsinki 60.15 24.97 Finland 1992–20081879–2010

1 h1 month

4.43.9

12 Hanko 59.82 22.98 Finland 1992–2008 1 h 4.0

13 Russaro 59.77 22.95 Finland 1866–1936 1 month 3.0

14 Jungsfrusund 59.95 22.37 Finland 1858–1934 1 month 3.0

15 Turku 60.43 22.10 Finland 1922–2010 1 month 4.1

16 Uto 59.78 21.37 Finland 1866–1936 1 month 2.9

17 Föglö 60.03 20.38 Finland 1992–20081923–2010

1 h1 month

3.94.4

18 Lemström 60.10 20.02 Finland 1889–1936 1 month 1.7

19 Lypyrtti 60.60 21.23 Finland 1858–1936 1 month 3.0

20 Lyuokki 60.85 21.18 Finland 1858–1936 1 month 3.1

21 Rauma 61.13 21.43 Finland 1992–20081933–2010

1 h1 month

4.14.4

22 Mäntyluoto 61.60 21.48 Finland 1992–20081910–2010

1 h1 month

4.24.4

23 Kaskinen 62.35 21.22 Finland1992–20081926–2010

1 h1 month

4.34.1

24 Vaasa 63.08 21.57 Finland 1883–2010 1 month 4.0

25 Pietarsaari 63.70 22.70 Finland 1914–2010 1 month 4.1

26 Raahe 64.67 24.40 Finland 1922–2010 1 month 4.5

27 Oulu 65.03 25.42 Finland 1889–2010 1 month 3.8

28 Kemi 65.67 24.52 Finland 1920–2010 1 month 4.5

29 Kalix 65.70 23.10 Sweden 1974–2011 1 month 4.3

30 Furuögrund 64.92 21.23 Sweden 1916–2011 1 month 4.0

31 Ratan 63.99 20.90 Sweden 1892–2011 1 month 3.7

32 Draghällan 62.37 17.53 Sweden 1898–1967 1 month 3.8

33 Spikarna 62.36 17.53 Sweden 1992–2008 1 h 4.0

34 Nedre Gävle 60.68 17.17 Sweden 1896–1986 1 month 3.5

35 Björn 60.63 17.97 Sweden 1892–1976 1 month 3.6

36 Forsmark 60.40 18.20 Sweden 1992–2008 1 h 3.8

A

A

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MEDVEDEV et al.

Table. (Contd.)

No. Station name LatitudeN

LongitudeE Country Observation

periodSampling

(h, month) , cm

37 Stockholm 59.32 18.08 Sweden 1889–20111801–2011

1 h1 month

3.63.6

38 Nedre Södertalje 59.20 17.62 Sweden 1869–1970 1 month 2.6

39 Landsort 58.74 17.87 Sweden 1887–2005 1 month 3.4

40 Visby 57.63 18.28 Sweden 1992–20081916–2011

1 h1 month

3.43.4

41 Öllands Norra Udde 57.37 17.10 Sweden 1992–20081887–2011

1 h1 month

3.33.3

42 Kungsholmsfort 56.11 15.59 Sweden 1887–2011 1 month 2.8

43 Simrishamn 55.55 14.35 Sweden 1992–2008 1 h 2.5

44 Ystad 55.42 13.82 Sweden 1887–1981 1 month 1.9

45 Klagshamn 55.52 12.89 Sweden 1929–2011 1 month 2.5

46 Varberg 57.10 12.22 Sweden 1887–1981 1 month 1.8

47 Göteborg 57.68 11.79 Sweden 1967–20061889–2011

1 h1 month

1.91.9

48 Smogen 58.35 11.22 Sweden 1911–2011 1 month 2.1

49 Hirtshals 57.60 9.96 Denmark 1892–2011 1 month 2.5

50 Frederikshavn 57.44 10.55 Denmark 1894–2011 1 month 1.7

51 Aarhus 56.15 10.22 Denmark 1888–2011 1 month 1.2

52 Hornbaek 56.09 12.46 Denmark 1898–2011 1 month 2.1

53 Cobenhavn 55.71 12.60 Denmark 1889–2011 1 month 2.0

54 Fredericia 55.56 9.75 Denmark 1889–2011 1 month 1.7

55 Korsor 55.33 11.14 Denmark 1897–2001 1 month 1.6

56 Slipshavn 55.29 10.83 Denmark 1896–2011 1 month 1.3

57 Rodbyhavn 54.66 11.35 Denmark 1955–2011 1 month 1.9

58 Gedser 54.57 11.93 Denmark 1898–2011 1 month 1.9

59 Travemünde 53.97 10.88 Germany 1856–1986 1 month 1.3

60 Wismar 53.90 11.47 Germany 1848–2010 1 month 1.4

61 Warnemünde 54.18 12.08 Germany 1855–2010 1 month 1.5

62 Swinoujscie 53.92 14.23 Poland 1811–1999 1 month 2.3

63 Kolobrzeg 54.18 15.55 Poland 1951–1999 1 month 2.9

64 Ustka 54.58 16.87 Poland 1951–1999 1 month 3.0

65 Wladyslawowo 54.80 18.42 Poland 1992–20081951–1999

1 h1 month

3.03.3

66 Gdansk 54.40 18.68 Poland 1951–1999 1 month 3.2

67 Baltiysk 54.60 19.90 Russia 1992–2006 1 h 3.8

68 Kaliningrad 54.70 20.48 Russia 1926–1986 1 month 3.1

69 Klaipeda 55.70 21.13 Lithuania 1898–2011 1 month 3.4

70 Liepaja 56.53 20.98 Latvia 1865–1936 1 month 2.6

71 Daugavgriva 57.05 24.03 Latvia 1872–1938 1 month 3.4

72 Esbjerg 55.46 8.44 Denmark 1889–2011 1 month 3.3

73 Cuxhaven 53.87 8.72 Germany 1843–2010 1 month 2.7

74 Delfzijl 53.32 6.93 Netherlands 1865–2011 1 month 2.3

A



66°N

64°

62°

60°

58°

56°

54°

10° 14° 18° 22° 26° 30° E

Sweden

Finland

Estonia

Latvia

Gu

l f o

f

AlandIslands

BA

LT

IC

Ka

t t

Denmark

DanishStraits

Poland

Belarus

Lithuania

Germany

30

31

32

33

12

3 4

5

6

789

101112

13

14

15

1617

18

1920

21

22

23

24

25

26

27

2829

34 35

36

3738

39

40

41

42

43

44

45

46

47

48

4950

51 52

5354 55

57 58

59 6061

6263

6465

6667

68

69

7071

56

G u l f o f

Gulf ofRigaS

EA

F i n l a n d

Bo

t hn

i a

eg

at

Russia

Fig. 1. Location of tide gauge stations (the numbers correspond to the numbers in the table).

overtones in the spectrum of the sea level oscillationsdue to the very small amplitude of the main harmonic(434 days) and the high noise level. An additionalspectral peak at a frequency of 1.22 cpy (300 days) wasfound in the low�frequency part of the sea level spec�trum at Stockholm. This harmonic, with a 300�dayperiod, was also recognized in the spectra of about20 stations in the Baltic Sea. At present, the reason forformation of this peak remains unknown.

For a more comprehensive description of the poletide in the Baltic Sea, we performed a spectral analysisof monthly mean series of sea level observations at

61 stations. Figure 3 shows the low�frequency part ofthe sea level spectra (periods of 300–500 days) for (a)Stockholm, (b) Oulu, (c) Swinoujscie, and (d) Kro�nstadt. The spectral resolution varied with the serieslength and the spectral window width. The highestspectral resolution was achieved at Stockholm: Δf ≈0.0047393 cpy, N = 2532 months. For the remainingstations, we have Δf ≈ 0.005291 cpy, N = 2268 months(Swinoujscie); Δf ≈ 0.0087591 cpy, N = 1370 (Oulu);and Δf ≈ 0.0058508 cpy, N = 2050 months (Kron�stadt). The number of degrees of freedom at all sta�tions was set at the minimum ( = 2).ν

126


MEDVEDEV et al.

Likewise, for the spectra estimated from the hourlydata, the main spectral peaks are at the annual (Sa)and semi�annual (Ssa) harmonics; the spectral peakSa in Fig. 3 is chopped off for better visibility of thepeaks unrelated to the seasonal oscillations. A notice�able increase of the spectral density is observed in therange of periods 400–450 days, corresponding to thepole tide. The character of the broad maximum sub�stantially changes with the spectral resolution. Forinstance, at Oulu (Fig. 3b), at this spectral range there

are two local peaks with periods = 434 and =417 days, and the spectral density of the former is threetimes higher than the latter. Three local peaks withinthe range of the pole tide at 443, 434, and 417 days are

014P 2

14P

evident in the sea level spectrum at Kronstadt(Fig. 3d). The shorter is their period, the higher is theirspectral density. The broad maximum of the pole tideat station Swinoujscie divides into several local peaks;the highest spectral density is observed at the period of

= 443 days and, in descending order, follow thepeaks at periods of 434, 417, and 401 days. At Stock�holm, the station with the maximum spectral resolu�tion, the main peak is at 443 days and the weaker peakis at 434 days.

The long�term (>100 years) monthly observationalseries were used to evaluate the Q factor of the poletide. The calculations were done for a greater numberof degrees of freedom and, respectively, for narrower

114P

5 × 105

4 × 105

3 × 105

2 × 105

105

0500 400 300 200 100

Period, days

P14

Sa

(c)

Tallinn5 × 105

4 × 105

3 × 105

2 × 105

105

0500 400 300 200 100

P14

Sa

(d)

Kronstadt

Ssa

5 × 105

4 × 105

3 × 105

2 × 105

105

0500 400 300 200 100

P14

Sa

(a)

Stockholm5 × 105

4 × 105

3 × 105

2 × 105

105

0500 400 300 200 100

P14

Sa

(b)

Kaskinen

Ssa

Ssa

Spe

ctra

l den

sity

, cm

2 yea

r

Fig. 2. Sea level spectra at periods from 100 to 500 days at stations (a) Stockholm, (b) Kaskinen, (c) Tallinn, and (d) Kronstadt.The spectral density of the annual (Sa) harmonic is cut off at the level of 5 × 105 cm2 yr.



confidence intervals. The Q�factor, one of the mostimportant characteristics of the pole tide, describesthe attenuation rate of the oscillations. It can be esti�mated as:

, (3)

where f0 is the spectral peak frequency and Δf is itshalf�power frequency width [4]. The higher the Q�fac�tor, the slower the oscillations attenuate. The spectralanalysis of the observational series for the stationsshown in Fig. 3 with a spectral window width of N =343 months allowed us to estimate the Q�factor as 9.5

Q 0f

f=Δ

(for Stockholm) and 12.5 (for Kronstadt). Accordingto [4], the typical value of the Q�factor for the pole tideis in the range of 6–60.

The results of spectral analysis of the hourly andmonthly data series (Figs. 2, 3) show that the observedproperties of the spectral peak of the pole tide arestrongly influenced by the length of the series, i.e. bythe spectral resolution. Unlike the astronomical tides,there is no particular deterministic frequency for thepole tide. For short observational series (<30 years),the spectral peak of the pole tide appeared as a broad“fuzzy” maximum. The same maximum divides intotwo or three peaks in the spectra of long series. It is dif�

103

8 × 102

6 × 102

4 × 102

2 × 102

0480 440 400 360 320

Period, days

P14

Sa

(c)

Swinoujscie

Spe

ctra

l den

sity

, cm

2 yea

r

103

8 × 102

6 × 102

4 × 102

2 × 102

0480 440 400 360 320

P14

Sa

(d)

Kronstadt

103

8 × 102

6 × 102

4 × 102

2 × 102

0480 440 400 360 320

P14

Sa

(a)

Stockholm103

8 × 102

6 × 102

4 × 102

2 × 102

0480 440 400 360 320

P14

Sa

(b)

Oulu

Fig. 3. Sea level spectra at periods from 300 to 500 days at stations (a) Stockholm, (b) Oulu, (c) Swinoujscie, and (d) Kronstadt.The solid line indicates the high�resolution spectrum and = 2; the dashed line denotes the smoothed spectra at N = 343 and

= (a) 14, (b) 8, (c) 12, and (d) 10.ν

ν

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MEDVEDEV et al.

ficult to get a reliable estimate of several maxima of thepole tide due to the small number of degrees of free�dom and the inadequately broad confidence intervals.This division into a number of individual maxima withimproved spectral resolution is, probably, a result ofthe temporal variability in the dominant period of thepole tide. Nevertheless, it is possible to say that reason�ably consistent spectral peaks with periods of = 434and = 417 days are observed at most of the BalticSea stations.

4. SPATIAL DISTRIBUTIONOF THE AMPLITUDE OF THE POLE TIDE

The results of the spectral analysis enabled us toexamine the spatial distribution of the pole tide ampli�tudes in the Baltic Sea. Over 45 years ago, Maximovand Karklin estimated the polar tide amplitudes, aver�aged over the 30�year period (1900–1930), for13 ports in the Baltic Sea [3]. In the present work, thecharacter of the pole tide is described based on thedata from 71 stations. At some of them, the length ofthe data series is over 100 years. The hourly andmonthly data series given in the table were treated sep�arately. Using the spectral analysis, we identified broadpeaks in the frequency range 0.81–0.91 cpy (400–450 days) and integrated them to calculate the “inte�gral amplitude” of the pole tide for every jth station:

, (4)

where Δ f is the spectral resolution, and is thespectral density within the spectral range for the jthstation. The results of the calculations are presented inthe table and in Fig. 4.

According to the results of the analysis of thehourly sea level series, a substantial increase in the poletide amplitude begins eastward from the Danish Straits(Fig. 4a). In the region of the Aland Islands, it is 4 cm.In the Gulf of Bothnia, the pole tide amplitudeincreases from the inlet entrance to the head. Alongthe eastern coast of the gulf, the pole tide amplitudesnoticeably exceed the amplitudes on the western coast.In the Gulf of Finland, the pole tide amplitude alsoincreases towards the head of the gulf. The highestpole tide amplitudes are observed on the southerncoast of the Gulf of Finland (7 cm at Narva and 6 cmat Tallinn).

A more detailed picture of the spatial variability ofthe pole tide in the Baltic Sea was obtained from themonthly mean data series (Fig. 4b). The smallestamplitudes of the pole tide are observed in the south�western region: from 1.3–1.5 cm near the Baltic coastof Germany to 1.9 cm close to the islands in the Dan�ish Straits. In the central Baltic Sea near the AlandIslands, the pole tide is as high as 3.4–4 cm. In theGulf of Bothnia and the Gulf of Finland, the pole tide

014P

214P

1/2

ˆ ( )j j i

i

A f S f⎡ ⎤

= Δ⎢ ⎥⎢ ⎥⎣ ⎦∑

( )j iS f

if

amplitude increases slightly: 4.4 cm at Kemi andRaahe and 4.6 cm at Hamina. In the Gulf of Bothnia, thetidal amplitudes on the eastern coast are 0.5–0.8 cmhigher than on the western coast. At stations whereobservations were conducted in the late 19th to early20th centuries, the pole tide amplitudes were underes�timated in comparison with the nearby located sta�tions with more recent observations. In particular, thepole tide amplitude at Lemstrom is 2.5 times lowerthan that at Föglö (Fig. 4b). The amplitudes at thesestations are marked in Fig. 4b with empty circles.Many authors have previously emphasized that theChandler Wobble is subject to significant temporal varia�tions in amplitude and period [4]. Probably a number ofindividual peaks in the wide spectral range of the pole tideare caused by the variability of the dominant period of theprocess during the 19–20th centuries.

5. DISCUSSION

In general, the integral pole tide amplitudes esti�mated from the hourly and monthly data are in goodagreement. At some stations (Öllands Norra Udde,Visby and Helsinki), the evaluated characteristics arealmost the same, despite the relatively short observa�tion periods of the hourly data. Figure 5 shows thecomparison of the low�frequency parts of the sea levelspectra estimated from the hourly and monthly dataseries at Stockholm. We compared the hourly andmonthly spectra from the same observational period of123 years (1889–2010). For clarity, the spectral den�sity of the sea level oscillations was recalculated intothe amplitude according to (4). The difference in thespectral amplitudes is almost absent; however thenoise level of the spectrum computed from the hourlydata is much lower compared to that computed fromthe monthly data (Fig. 5). Apparently, this is due to thedifference in the methods used to interpolate the miss�ing data.

Figure 4b shows that the pole tide amplitude in theBaltic Sea increases both with longitude and latitude.In this paper, we examined the dependence of the poletide amplitude as a function of the linear distance fromthe sea entrance. To do this, the coordinate axes foreach station were converted into the Cartesian systemwith the origin at 9° E and 53° N and the X� and Y�axisdirected eastward and northward, respectively. Themultiple correlation coefficient of the amplitude withthe longitude and latitude was found to be 0.9, and theX axis of the maximum correlation (the pole tide axis)was directed from the southwest to the northeast of theBaltic Sea (this direction is marked with the dottedline in Fig. 6b). The distribution of the pole tide ampli�tudes relative to the maximum correlation axis exhibitsa fairly uniform dependence and weak scatter of thevalues in the northward and eastward directions(Fig. 6a). The amplitude increase in the meridionaldirection contradicts the static theory of the pole tide[2, 12, 16]. According to the latter, the amplitude max�



imum should take place close to 45° N and decreasepoleward, while in the longitudinal direction the poletide amplitude is supposed to be uniform.

The variation of the pole tide along the X�axis, i.e.,from the entrance into the Baltic Sea to the heads ofthe gulfs, is described by a simple empirical relation�ship:

, (5)

where , , where Х

and are in kilometers and centimeters, respectively.

The mechanism for the polar tide formation in theBaltic Sea remains unclear. Attempts to explain theextraordinary high amplitudes of the pole tide in theNorth and Baltic seas, which are about 6–8 timeshigher than the typical amplitudes of the staticresponse, were undertaken in [13, 19]. The authorspaid particular attention to the eastward amplificationof the pole tide [13]. As was assumed by the authors of[13, 19, 20], such amplification can be a result of thedynamical resonance response of the water basin in theform of eastward propagating topographic Rossbywaves. An alternative hypothesis has been proposed in[18]. Based on the numerical simulation of sea levelvariability in the North Sea forced by the atmosphericpressure and wind, the authors showed that most of theenergy of the pole tide oscillations is rather well repro�duced by the model. This indicates the meteorological

ˆ( )A X aX b= +

0.002 0.0001a = ± 1.39 0.14b = ±

A

nature of the 14�month oscillations in this sea.According to the results of that study, the anomalousresponse of the sea level is primarily produced by thezonal component of the tangential wind stress uponthe sea surface. Thus, it is actually assumed that theChandler Wobble of the pole position generate the

66°

N

64°

62°

60°

58°

56°

54°

10° 30° 12° 14° 16° 18° 20° 22° 24° 26° 28°

(a)7 сm

5 сm

3 сm

1 сm

66

64

62

60

58

56

54

10° 30° E12° 14° 16° 18° 20° 22° 24° 26° 28°

(b)

Fig. 4. Distribution of the integral amplitude of the pole tide in the Baltic Sea from (a) hourly and (b) monthly sea level observa�tions.

10

8

6

4

2

0480 440 400 360 320

Period, days

Am

plit

ude,

cm

1 month1 h

Fig. 5. Comparison of the low�frequency sea level spectraestimated from hourly (black columns) and monthly (greycolumns) data from 123 years of observations at Stock�holm.

130


MEDVEDEV et al.

pole tide in the North Sea not directly but indirectly,i.e., by triggering atmospheric oscillations with therespective periods, which in turn influence the sealevel. It appears that the pole tide in the Baltic Sea canbe induced in the same way.

The results of the analysis presented in this studysupport the postulation about the anomalously highamplitudes of the pole tide in the Baltic Sea. In addi�tion, it was found that the maximum pole tide occursin the Gulf of Finland (Narva and Hamina) ratherthan in the Gulf of Bothnia, as was assumed earlier[13]. According to the monthly data series, there is anevident northeastward increase in the amplitude of the

pole tide from 1.5 to 4.6 cm in the Baltic Sea. In con�trast to the findings of Wunsch [19] based on pole tideobservations in the North Sea, where he identifiedcertain changes in phase that provide evidence of apropagating wave, the cross–spectral analysis of thesea level records at various sites in the Baltic Sea hasnot revealed a meaningful phase change. It should benoticed that the maximum pole tide amplitudes areobserved at stations adjacent to extensive shallow–water areas (the gulfs of Bothnia and Finland). Thisfact indirectly confirms the impact of the wind on theformation of the pole tide in the Baltic Sea.

(b) N

S

0 400 800 1200 1600

Distance, km

4

3.5

3

2.5

2

3

4

(а)

1

2

3

4

R2 = 0.82A = 0.002X + 1.39

5

4

3

2

1

0

Inte

gral

am

plit

ude,

cm

Fig. 6. (a) Dependence of the pole tide amplitude upon the distance along the maximum correlation axis directed from the south�west to northeast of the Baltic Sea; (b) Distribution of the integral amplitude of the pole tide in the Baltic Sea; isolines of integralamplitudes are indicated by solid lines, a dotted line denotes the direction of the maximum correlation axis (X). Station locations:(1) the Danish Straits; (2) the central Baltic Sea; (3) the Gulf of Finland; (4) the Gulf of Bothnia.



6. CONCLUSIONS

In the present work, we evaluated the amplitudeand main features of the pole tide in the Baltic Seabased on long�term hourly and monthly sea levelrecords. The spectral analysis of long data seriesrevealed evident temporal and spatial variations in theamplitudes and periods of the pole tide. The mainpeak clearly recognizable in the spectra corresponds tothe period of 434 days. However, marked peaks withperiods of 417 and 443 days were also also quite evi�dent at the stations where the series were long enoughto get high spectral resolution. The temporal variabil�ity of the amplitudes and periods of the polar tide wasalso repeatedly denoted by other authors (see, forexample, [2, 12]). Specifically, this issue was discussedin the book of Munk and MacDonald [4].

A map of the spatial variability of the pole tide wasprepared based on estimated integral amplitudes of thetide at 71 stations. The results of analysis show theanomalous heights of the pole tide in the Baltic Sea(significantly higher than follows from the static the�ory [16]) and the obvious northeastward growth of theamplitude, i.e. from the Danish Straits (1.5–2 cm) tothe gulfs of Finland (up to 7 cm) and Bothnia (4.5 cm).

The existing theories of pole tide formation in theNorth Sea [14, 19] do not allow us to explain theanomalous character of this tide in the Baltic Sea andthe significant increase of its amplitude from theentrances to the heads of the gulfs of Finland andBothnia. Nevertheless, our results provide substantialinformation on the integral properties of this phenom�enon and give cues to answer particular questions. Theatmospheric processes of the same periods areundoubtedly one of the factors influencing the poletide in the Baltic Sea. Accordingly, the next step in theinvestigation of the 14�month sea level oscillations inthe Baltic Sea may be a joint analysis of these oscilla�tions and the fluctuations of the atmospheric pressureand wind in the region of this water basin.

ACKNOWLEDGMENTS

The authors would like to thank E.G. Morozov forhis valuable comments. We are very grateful toUHSLC and PSMSL that provided us the sea leveldata, which we used in the present study. This work wassupported by the Russian Foundation for BasicResearch (Grants 12�05�00733, 12�05�00757, and 13�05�41360) and by the Ministry of Education and Sci�ence of the Russian Federation (Contract11.G34.31.0007).

REFERENCES

1. E. A. Kulikov and I. P. Medvedev, “Variability of theBaltic Sea level and floods in the Gulf of Finland,”Oceanology (Engl. Transl.) 53 (2), 145–151 (2013).

2. I. V. Maksimov, Geophysical Forces and Ocean Waters(Gidrometeoizdat, Leningrad, 1970) [in Russian].

3. I. V. Maksimov and V. P. Karklin, “Pole tide in the Bal�tic Sea,” Dokl. Akad. Nauk SSSR 161 (3), 580–582(1965).

4. W. H. Munk and G. J. F. MacDonald, The Rotation ofthe Earth: A Geophysical Discussion (Cambridge Univ.Press, New York, 1960).

5. G. I. Marchuk and B. A. Kagan, Dynamics of the OceanTides (Gidrometeoizdat, Leningrad, 1983) [in Rus�sian].

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7. L. M. Fomin, “Influence of slow rotations of the Earthon cyclic climate variability,” Oceanology (Engl.Transl.) 43 (4), 455–464 (2003).

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10. W. J. Emery and R. E. Thomson, Data Analysis Methodsin Physical Oceanography (Elsevier, New York, 2003).

11. R. Haubrich and W. H. Munk, “The pole tide,” J. Geo�phys. Res. 64 (12), 2373–2388 (1959).

12. E. Lisitzin, Sea Level Changes (Elsevier, Amsterdam,1974).

13. S. P. Miller and C. Wunsch, “The pole tide,” Nature(London), Phys. Sci. 246 (155), 98–102 (1973).

14. W. P. O’Connor, B. F. Chao, D. Aheng, and A. Y. Au,“Wind stress forcing of the North Sea ‘Pole Tide,’”Geophys. J. Int. 142, 620–630 (2000).

15. D. T. Pugh, Tides, Surges and Mean Sea Level (Wiley,Chichester, 1987).

16. W. Schweydar, “Theorie der Deformation der Erdedurch Flutkrafte,” Verofg Preuss. Geod. Inst. NeueFolge 66, 55 (1916).

17. T. Shimizu, “The variation of the sea level and the baro�metric pressure with Chandler’s period,” in Geophysi�cal Papers Dedicated to Prof. Kenso Sassa, 1963,pp. 499–515.

18. M. N. Tsimplis, R. A. Flather, and J. M. Vassie, “TheNorth Sea pole tide described through a tide�surgenumerical model,” Geophys. Res. Lett. 21 (6), 449–452 (1994).

19. C. Wunsch, “Dynamics of the pole tide and the damp�ing of the Chandler Wobble,” Geophys. J. R. Astron.Soc. 39, 539–550 (1974).

20. C. Wunsch, “Dynamics of the North Sea pole tiderevisited,” Geophys. J. R. Astron. Soc. 87, 869–884(1986).

Translated by G. Karabashev

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Pole tide in the Baltic Sea