Synoptic variability of the currents in the Gotland Basin of the Baltic Sea

ISSN 0001�4370, Oceanology, 2013, Vol. 53, No. 4, pp. 385–400. © Pleiades Publishing, Inc., 2013.Original Russian Text © E.A. Zakharchuk, N.A. Tikhonova, 2013, published in Okeanologiya, 2013, Vol. 53, No. 4, pp. 437–453.

385

INTRODUCTION

In oceanology, the synoptic variability is under�stood as the heterogeneities in the oceanological fieldsthat have typical time (from a few days to months) andspatial (from dozens to thousands of kilometers) scales[10, 11, 13]. The theoretical and empirical studies car�ried out during the last decades showed that the synop�tic�scale processes in the oceans and seas are widelydiverse and make a decisive contribution to the energythat drives the seawater. These processes are generatedby the tangential wind stress, the horizontal atmo�spheric pressure gradient, and the tide�generating andArchimedean forces. The perturbing forces operatingwithin the synoptic spatiotemporal range in the oceansand seas give rise to wind and barogradient currents,synoptic eddies, and diverse types of low�frequencywaves similar to topographic Rossby and Kelvinwaves.

Up to the 1970s, the oscillations of the synoptic�scale currents in the Baltic Sea were mainly identifiedas wind currents (see the review in [3]). In the nextworks of this period, the synoptic�scale heterogene�ities in the diverse oceanological fields of the BalticSea were thought to be related to the synoptic (meso�scale) eddies and low�frequency waves.

According to the interpretation of the results ofdiverse natural observations, the eddies in the BalticSea have mainly a rounded shape, time scales fromdays to eight few dozen days, horizontal sizes fromkilometers to dozens of kilometers, the rotational

speed of the currents from 5 to 50 cm/s, and propagatealong isobaths with an average velocity of 1–3 cm/s [3,18]. The results of numerical simulations using thehydrodynamic model show that the synoptic eddies inthe central Baltic region are 22–44 km in size, propa�gate to the northeast with rates from 2.5 to 8.7 cm/s,and have a lifetime from a few days to two months [7].

Different hypotheses proposed to explain whattriggered the synoptic eddies were based on thedynamic instability of meanders of the frontal zonesand flows interacting with the bottom topography fea�tures [3].

The first wave interpretation of the synoptic pertur�bations in the current field of the Baltic Sea was pre�sumably proposed by Aitsam and Talpsepp [1, 16, 19].The current measurements during the BOSEX�77experiment [19] allowed them to distinguish 3�daylow�frequency waves propagating along the isobath.The component spectral analysis of the currents mea�sured at the cluster array of 3 current meter mooringsin the open Baltic Sea in 1980 made it possible to dis�tinguish 5.5� to 8�day oscillations of the zonal andmeridional components of the current velocity, as wellas less intense 14� to 18�day oscillations of the merid�ional component within 102�day series. These oscilla�tions were interpreted by these authors as unstablebaroclinic topographic waves [1] and trapped bottomwaves [17]. According to their estimates, the 5.5� to8�day waves are 22–25 km in length and propagatedsouthwestward, while the 14� to 18�day waves are45 km long and propagated in the west�northwest

Synoptic Variability of the Currents in the Gotland Basin of the Baltic Sea

E. A. Zakharchuk and N. A. TikhonovaSt. Petersburg Branch of the Zubov’s State Oceanographic Institute, St. Petersburg, Russia

e�mail: [email protected] September 18, 2011; in final form, May 31, 2012

Abstract—The series of long�term observations of synoptic�scale currents obtained by instrumental mea�surements at the moorings in the Gotland Basin of the Baltic Sea are analyzed. The results of the statisticalanalysis of the currents reveal their wave structure. The characteristics of the low�frequency waves receivedon the basis of the cross�spectral analysis show that, in the range of periods from 2 to 20 days, they propagatedin the southwestern, southeastern, and northwestern directions with phase speeds of 0.02–2.08 m/s and havelengths from 28 to 431 km. It is suggested that the distinguished wavelike perturbations of the synoptic�scalecurrents are related to topographic waves. The analysis of the meteorological conditions and the results of thenumerical hydrodynamic modeling of the Baltic Sea free low�frequency fluctuations led us to conclude thatthe most possible mechanism of the generation of the intense wave�like oscillations of synoptic�scale currentsin the Gotland Basin is the resonance between the anemobaric forces and the relatively slow�moving anticy�clones over the open Baltic Sea and the eigenmode of the basin.

DOI: 10.1134/S0001437013030119

MARINEPHYSICS

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OCEANOLOGY Vol. 53 No. 4 2013

ZAKHARCHUK, TIKHONOVA

direction. As was suggested in [17], the distinguishedtopographic waves may be generated by the baroclinicinstability of the background (middle) currents.

According to [21], the topographic waves mani�fested in the wave�like current oscillations in the Got�land Basin were generated by tangential wind stress.However, no special studies were carried out by theseauthors to test this hypothesis.

The relationship between the wave�like low�fre�quency oscillations of the alongshore component ofthe current velocities near the western Gotland coastand the wind velocity component was studied in [24]using instrumental measurements of the currents inthe summer and autumn of 1977. The correlation ofthe currents with the local wind seemed to be low. Asignificant correlation was found between the currentsand the wind measured near the southern terminationof Gotland Island. By the comparison of the theoreti�cal and empirical dispersion relations, the authorsidentified the low�frequency current oscillations asforced coastal�trapped baroclinic topographic waves.

Up to now, the duration of the instrumental obser�vations of the currents was too short to provide a rep�

resentative statistical study of their oscillations in thesynoptic frequency range and to estimate the peculiar�ities of the seasonal and interannual variability of thesynoptic oscillations and their correlation with thediverse meteorological characteristics.

The situation changed only in the 1990s, whenDarss Sill, Arkona Sea, and Oder Bank automated sta�tions equipped with devices for the measurement ofthe diverse meteorological and oceanological charac�teristics, including the current’s velocity and directionat different horizons, were deployed in the southwest�ern part of the Baltic Sea (http://www.io�warne�muende.de).

From 1997 to 2006, the SW�1, NE, SW, Z, and SEstations equipped with a device for current measure�ments were deployed for a long time by Germanresearchers in the framework of the experimentsMeso�Scale Dynamics (MESODYN, 1997–2000)[20] and Deep Rim Currents in the Eastern GotlandBasin (RAGO, 2006–2008) [24]. The continuousmeasurements of the currents at the stations variedfrom 326 days to 6 years (Fig. 1).

65°

64°

63°

62°

61°

60°

59°

58°

57°

56°

55°

54°

10° 12° 14° 16° 18° 20° 22° 24° 26° 28° 29°

H, m250

240

230

200

150

100

75

50

30

10

0

NE

ZSE

SW

SW1

57.60

57.55

57.50

57.45

57.40

57.35

57.30

57.25

57.20

57.15

57.10

57.05

57.00

56.9519.6 19.8 20.0 20.2 20.4

110

5070

130

150

210

190

230

110

110

130

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150

150

150

90

210

21 0

210

210

190

19019

0

190

190

230

170

170

170

170

170

170

Fig. 1. Position of the cluster array of current meter moorings.


SYNOPTIC VARIABILITY OF THE CURRENTS IN THE GOTLAND BASIN 387

The initial three�year period of the current mea�surements at the NE station was analyzed in [20].Using the component method of spectral analysis,these authors distinguished the energy�carrying max�ima of the current oscillations within periods of 100–150, 40–65, 25–37, 10, and 3–5 days. It was notedthat the oscillations with periods of 100–150 days arerecognizable only in the alongshore component of thecurrent velocities, whereas 40–65 and 25–37 daycycles were also found in the meridional and zonalcomponents. According to [20], the 10�day currentoscillations are related to the zonal fluctuations of thewind flows, while the 3� to 5�day oscillations are cor�related with the natural synoptic period of the weatherrhythms.

However, the results of the statistical analysis of thecurrents and the estimates of their relation with thewind [1, 16, 20, 24] are hotly debatable from themethodical point of view. According to the theoreticalconcepts, the correlation and spectral functions of thevector processes are the main points of the tensorfunctions and comprise four components for the two�dimensional case. During the component analysis ofthe correlation�spectral structure of the vector pro�cess, each element of the corresponding tensor is con�sidered separately; i.e., a link within a single charac�teristic—a correlation or spectral tensor matrix—isartificially broken down. Moreover, all four compo�nents of the above mentioned tensors depend on thechoice of the coordinate system; i.e., they are notinvariants. Additionally, the estimates of the disper�sion ellipses of the oscillations of the synoptic�scalewind velocities over the Baltic Sea [6] reveal theirextremely low compression. Therefore, it is impossibleto estimate the statistical characteristics of the windand its relationship with the currents using only onecomponent.

These disadvantages can be overcome by the use ofthe vector�algebraic method of the analysis of random

processes [2, 12], in which the matrix elements of theprobabilistic characteristics are merged in the form ofa well�founded algorithm for the calculation of thetensor invariants, thus providing insight into the kine�matic properties of the analyzed vector process. Inaddition, this method gives much more informationon the vector process. Therefore, the statistical analy�sis of the currents at the individual horizons of themoorings, the study of their spatiotemporal structure,and the estimates of the relationships between the cur�rents and the meteorological characteristics were car�ried out in this work using the vector�algebraic methodof the analysis of random processes.

DATA AND METHODS

The velocities and directions of the currents at5 moorings were measured in the deep�seated and bot�tom layers of the Central Baltic region in the GotlandBasin (Fig. 1, Table 1). The observations were carriedout using a mechanical RCM�7/8 Aanderaa currentmeter having an accuracy of ±1.0 cm/s at station SW�1and an electronic RCM�9 Aanderaa device (accuracyof ±0.3 cm/s) at the other moorings. The daily meansof the velocities and the directions of the currents atthe indicated moorings are available at the site of theLeibniz Institute for Baltic Sea Research in Warneme�unde (Germany): http://www.io�warnemuende.de.

The current measurements obtained at the moor�ings should be considered as complex, nonstationary,and heterogeneous vector probabilistic processescharacterizing by moment functions of the first twoorders, as well as by spectral densities and probabilitydistributions. In terms of the probabilistic characteris�tics, such vector process should be invariant withrespect to the coordinate system accepted during theanalysis. Therefore, the probabilistic characteristics ofthe synoptic�scale currents and their correlation withthe diverse meteorological parameters were studied

Table 1. Information on the measurements of the currents by moorings

Mooring N E Start of the measurements

End of the mea�surements

Local depth, m

Horizon, m

Duration, days

SW�1 57°05′ 19°45′ Aug. 30, 1997 Sept. 14, 1998 220 170 381

NE 57°23′ 20°19′ Aug. 31, 1997 Oct. 30, 2005 224 204 2253

May 1, 2000 Oct. 30, 2005 224 174 2009

May 9, 2002 Oct. 19, 2005 224 219 529

March 23, 2004 Oct. 30, 2005 224 219 587

Sept. 28, 2006 March 30, 2008 224 219 550

Sept. 28, 2006 March 30, 2008 224 174 550

SW 57°07′ 19°52′ May 8, 2006 March 29, 2007 217 167, 197, 212 326

SE 57°15′ 20°15′ May 8, 2006 March 29, 2007 221 171, 197, 212 326

Z 57°19′ 20°09′ Nov. 2, 2005 Oct. 30, 2007 185 125 728

388



using the vector�algebraic method of the analysis ofrandom processes [12].

Using this method, we estimated the absolute value and the directions ϕ° of the mean vector of theaverage daily currents (the vector of the average trans�fer), as well as the diverse invariants of the tensor of the

standard deviation , where I1(0) = λ1(0) + λ2(0)is the linear invariant of the current dispersion tensordefinable via the half�length of the main axes λ1(0) andλ2(0) of the dispersion ellipse and the orientation α° ofthe large axis of the dispersion ellipse relative to thegeographical coordinate system, and I1(0) is the totaldispersion of the process. The compression of the dis�persion ellipse was determined from the invariantχ(0) = λ2(0)/λ1(0). The dimensionless coefficient

is the indicator of the current’s stabilitydenoting the relations between the constant and variablecomponents of the flow. At r > 1, the intensity of the oscil�lations in the flow predominates over that of the meantransfer; i.e., the currents are unstable. In contrast, thelower the r value, the more stable the flow is.

In order to study the synoptic variability of the cur�rents, the seasonal oscillations (the year wave and itsovertones (a half�year, one�third of a year, and a quar�ter of a year) were omitted from the initial series oftheir daily means using the Fourier transform andleast�squares fit procedure. The obtained residualseries of currents were processed using dispersionanalysis with allowance for the nonstationarity of theprocess. The variations of the linear invariant of thetensor of the variance I1(0)(t) with time (t) were esti�mated. For the calculation of I1(0)(t), the period of thequasi�stationarity was taken to be 30 days, while thesliding interval was 1 day. The two�dimensional den�sity of their probability distribution was calculated foreach series of the daily means of the current velocityvectors. This parameter characterizes the scatter of theabsolute values and the directions of the current veloc�ities at each of the considered horizons. The currentdirections and velocities were divided into 12 grada�tions, while the probabilities (in percentages) ofassigning of directions and velocities to certain grada�tions were then reduced to its center.

The correlation and spectral functions of the timeseries of the two�dimensional currents measured at themoorings are tensor functions, while those at the fixedshift are second�rank tensors consisting of four com�ponents. The vector–algebraic analysis of the currentswas based on the principle of the merging matrix ele�ments of the probabilistic characteristics in form of awell�founded algorithm of the calculation of thediverse invariants of the correlation and spectral ten�sors, which expose the set of kinematic properties ofthe analyzed vector process.

In order to estimate the spectral structure of theoscillations of the synoptic�scale currents, two invari�

vm

(0)1I

v(0)1r I m=

ants of the spectral tensor function, I1(ω) and D(ω),were calculated. Linear invariant I1(ω) of the spectraldensity tensor equals the matrix trace of the spectraltensor function and, according to [2], describes thefrequency distribution of the absolute intensity of theoscillations of the collinear components of the currentvelocities in any direction. The confidence intervals ofthe estimated spectral density for I1(ω) were calcu�lated as the sum of the confidence intervals of thespectra of the velocity current components along the Хand Y axes, which were estimated using technique [5].

Invariant D(ω) of the spectral density tensor isdetermined as the difference between the nondiagonalcomponents of its matrix, describes the frequency dis�tribution of the oscillation intensities of the orthogo�nal component of the current velocities in any direc�tion, and serves as an indicator of the current velocityrotation vector at the given frequency, while its signdetermines the predominant direction of the rotation(“+” clockwise, and “–” counterclockwise).

In order to estimate the statistical relationsbetween the synoptic current oscillations and themeteorological processes according to technique [2,12], we performed a cross�correlation analysisbetween perturbations of the synoptic�scale currentsand the different meteorological parameters, whichwere obtained from the atmospheric pressure (Pa) and

wind field massif developed on the basis of thereanalysis of the meteorological data [23]. Using thismassif, the series of daily means of the horizontalatmospheric pressure gradients (Grad Pa) and the tan�

gential wind stress (where с is a dimen�sionless coefficient, and ρ0 is the air density) that aresynchronous with currents were calculated for thedeployment point of the mooring. Then, two invari�ants of the cross�correlation tensor function KVU(θ, t)(1) were calculated for two pairs of the hydrometeoro�logical parameters, GradPa and the currents (V), as

well as for and with allowance for the nonstation�arity of the processes [12]. These are the linear invari�

ant (θ, t) and the rotation indicator ΩVU(θ, t),where V and U are vector processes, while θ is the timeshift; υ1 is the latitudinal component of the vector pro�cess V(t); υ2 is the longitudinal component of the vec�tor process V(t); u1 is the latitudinal component of thevector process U(t); and u2 is the longitudinal compo�nent of the vector process U(t).

KVU(θ, t) = (1)

The linear invariant equals the trace of thematrix of the correlation tensor function KVU(θ, t) andcharacterizes the community of the intensities of the col�linear variations of the vector processes V(t) and U(t).

( ),W��

0c W Wτ = ρ

� ��

τ

�

V��

1VUI

1 1 1 2

2 1 2 2

( , ), ( , ).

( , ), ( , )u u

u u

K t K t

K t K tυ υ

υ υ

θ θ⎛ ⎞⎜ ⎟θ θ⎝ ⎠

( )1 ,VUI tθ



The rotation indicator ΩVU(θ) is the differencebetween the nondiagonal components of the matrix ofthe correlation tensor function KVU(θ) and designatesthe community of the orthogonal variations in the V(t)and U(t) processes. Note that, if ΩVU(θ) > 0, the U(t) pro�cess is clockwise rotated relative to the V(t) process withinthe given time range, and vice versa if ΩVU(θ) < 0.

After and ΩVU(θ, t) are normalized to thelinear invariant of the dispersion tensor, their normal�ized values and were calculated [2]. Asfor estimating I1(0)(t), the period of the quasi�station�arity for calculating and was taken to be30 days.

The cross�correlation analysis was applied to esti�mate the relationship in the frequency region betweenthe synchronous series of current oscillations at theclose horizons of different moorings, as well asbetween the currents and anemobaric forces.

The function of the cross�spectral density between the two vector processes, V(t) and U(t), wasdetermined as a Fourier transform of the tensor func�tion KVU(θ) (1):

(2)

According to the technique reported in [2, 12], four

invariants of the cross�spectral density tensor

ψ(ω), f(ω) and two invariants of the coher�

ence tensor were computed. The

invariant characterizes the common moduleof the collinear variation intensities of the currentvelocities at two points of the ocean in the frequencyregion, while the invariant ψ(ω) is the phase lag of thecorresponding harmonics of the time series V(t) and

U(t) relative each other. The invariant

( )1 ,VUI tθ

1( , )Ir tθ ( , )r tΩ θ

1( , )Ir tθ ( , )r tΩ θ

( )VUS ω

1( ) ( ) .2

iVU VUS e K dt

∞

− ωθ

−∞

ω = θ

π∫

( )1 ,VUI ω

( ) ,VUω�

( )col2 ,F ω ( )orth

2F ω

( )1VUI ω

( )VU

ω�

describes the common module of the orthogonal vari�ation intensities of the current velocities at two pointsof the ocean in the frequency region, while the invari�ant f(ω) denotes the phase lag of the correspondingharmonics of the time series V(t) and U(t).

The invariants of the coherencetensor give the measure of the collinear and orthogo�nal time variations of two vector processes by the com�parison of the modulus of the eigen and cross oscilla�tions in the given frequency region.

RESULTS AND THEIR INTERPRETATION

Figure 2 demonstrates plots of the time variationsof the velocity vectors of the daily mean currents at theNE mooring, as well as the current rose. Their maxi�mum velocities exceed 20 cm/s. The current roseshave a symmetrical ellipsoid shape, which is bestexpressed in the higher horizons.

Table 2 presents estimates of the mean and differ�ent invariants of the tensor of the standard deviation.It is seen that the currents in the deep�seated and near�bottom layers travel on the average in the northerndirection with insignificant velocities varying from 0.9to 7.7 cm/s. The directions and velocities of the aver�age annual currents show insignificant variations intime and by the vertical. An exception is station SE,where the estimates of the linear invariant of the stan�

dard deviation tensor of the current velocities ( )are 1.5 and sometimes 2 times higher than , whichindicates the unstable character of the currents in thisarea of the Baltic Sea. The velocity vectors have anelliptical distribution with an insignificant degree ofcompression and extension from the north to thesouth, from the north to the northeast to the south�southwest, and from the northwest to the southeast. Atthe SW�1, NE, SE, and Z moorings (2005–2006), thedirections of the principal axes of the dispersionellipses are close to the vector of the average transfer,

( )col2 ,F ω ( )orth

2F ω

(0)1I

vm

сm/s10

0–10

сm/s2010

0

сm/s10

0

1999 2000 2001 2002 2003 2004 2005 2006 2007 2008Time, years

(a)

(b)

(c)

Fig. 2. Time variations of the velocity vectors of the daily mean currents and their roses at different horizons of the NE mooring:174 m (a), 204 m (b), and 291 m (c).

390



being almost normal to it at moorings Z (2006–2007)and SW.

The maximum velocities of the seasonal currentsestimated using the Fourier transform varied from 2.8to 4.0 cm/s depending on the year and reached valuesof 11–22 cm/s in the residual series of current veloci�ties. These results, together with the mean estimatespresented in Table 2, indicate that the synoptic oscil�lations make the main contribution to the current dis�persion within the range from days to several years.

Figure 3 illustrates the time variations in the esti�mates of the linear invariant dispersion tensor of thesynoptic current oscillations. There are well expressedseasonal variations in the synoptic current dispersion,most frequently with the autumn–winter maximum(up to 25–63 cm2 s–2) and the summer minimum (2–10 cm2 s–2). An exception is 2003, when the maximumvalues of the dispersion were noted in the summer. The

interannual variations of the maxima and minima ofthe current dispersions are also significant.

The estimates of the two�dimensional densities ofthe probability distribution of the synoptic currentvelocity vectors reveal mainly a two�mode structure(Fig. 4). The difference between the modes by thedirection is close to 180°, which is typical of the waveprocess.

Figure 5 demonstrates the spectra of the synoptic�scale currents estimated from the annual implementa�tion. These results show that the maximum repetitionof the significant peaks of the spectral density isobserved for the oscillations with periods of 7 and8 days (5 events of the estimated 10 spectra), 2.5 days(4 events), 3.7, 4.4, 6.0, and 12 days (3 events each).The spectral structure of the oscillations of the synop�tic currents significantly varied in the different years.In 2001, 2003, and 2005, the maximum spectral den�sity at station NE was found in the low�frequency syn�

Table 2. Probability characteristics of the currents calculated from the daily mean series of the current velocities measuredat moorings

The moorings and the years of their

operationHorizon, m

Mean Invariants of the tensor of the mean square deviation

ϕ° α°, ±180° r

cm/s deg cm/s cm/s cm/s deg

SW�1, 1997–1998 170 3.2 174 5.07 3.94 3.20 21 0.84 1.58

NE, 2001 174 3.8 359 5.00 4.19 2.73 4 0.65 1.32

NE, 2002 174 2.7 347 4.48 3.86 2.28 5 0.59 1.66

NE, 2003 174 4.1 350 5.49 4.15 3.58 14 0.86 1.34

NE, 2004 174 2.9 6 4.18 3.66 2.01 16 0.55 1.43

NE, 2005 174 2.9 7 3.98 3.52 1.87 20 0.53 1.38

NE, 2007 174 3.3 2 3.54 2.98 1.91 11 0.64 1.06

NE, 2000 204 3.4 2 3.94 3.39 2.02 14 0.60 1.17

NE, 2001 204 3.2 4 4.90 4.29 2.37 12 0.55 1.52

NE, 2002 204 2.2 349 4.63 4.13 2.09 14 0.50 2.11

NE, 2003 204 4.0 350 5.26 4.43 2.83 3 0.64 1.32

NE, 2004 204 3.4 1 4.89 4.45 2.02 24 0.46 1.44

NE, 2005 204 3.0 7 4.40 4.01 1.83 20 0.46 1.44

NE, 2007 204 3.6 5 3.76 3.30 1.79 20 0.54 1.05

NE, 2003 219 3.0 351 4.25 3.80 1.90 9 0.50 1.44

NE, 2004 219 2.9 358 4.13 3.75 1.74 18 0.46 1.42

NE, 2005 219 2.8 0 4.23 3.83 1.79 17 0.47 1.49

NE, 2007 219 3.3 355 3.76 3.30 1.79 20 0.54 1.12

SE, 2006–2007 171 7.7 16 5.25 4.97 1.68 17 0.34 0.68

Z, 2005–2006 125 0.9 355 3.13 2.45 1.95 –38 0.80 3.36

Z, 2006–2007 125 0.9 324 3.89 3.13 2.32 34 0.74 4.46

SW, 2006–2007 167 1.3 96 4.79 4.17 2.37 30 0.57 3.71

vm (0)1I (0)1λ (0)2λ (0)χ



optic range within periods of 48–67 days, whereas themaximum energy of the synoptic oscillations at thisstation and at station SE in 2000, 2002, and 2007 wasobserved within the range of 6–13 days. The maxi�mum values of the spectral density at mooring SW1 in1997–1998 and at mooring SW in 2006–2007 wererecorded in the high�frequency synoptic range withinperiods of 2.4–4.1 days.

For half of the events, the directions of the largeaxes of the spectral ellipses at the frequency of theenergy�carrying maximums in the mooring deploy�ment were close to the isobath directions. For theother events, the long axes of the spectral ellipses devi�ate from the isobath at angles from 20° to 70°. In somecases, the spectral ellipses at the frequencies of the sig�nificant peaks have a very high compression degree,and the shape of the tensor curves at the energy�carry�ing maximum frequencies is close to a circle only inseveral cases. The estimated spectral density at theenergy�carrying maxima of the synoptic range for thelinear invariant I1(ω) is two and more times higherthan the rotation indicator D(ω), which indicates thehigher contribution of the collinear velocity variationsin the synoptic variability of the currents as comparedto their orthogonal oscillations. However, the esti�mates of D(ω) may occasionally approach I1(ω),which indicates that the rotational velocity changes atcertain frequencies and makes a significant contribu�tion to the synoptic current variability. For instance,

such a feature was observed for the 7�day oscillations in2000 and for the 3.4�day oscillations in 2004. Thevelocity vector with 6�to 13�day oscillations showsmainly clockwise rotation and rotates mainly counter�clockwise for the other periods.

The results of the cross�spectral analysis of the cur�rent oscillations at the close horizons of the SE, Z, andNE moorings are shown in Fig. 6. The moorings SEand Z are separated by the shortest distance of 9.5 km,and the SE and NE moorings, by the longest distanceof 15.3 km. There is high coherence in the individualfrequency ranges. The phase difference between thehighly coherent current oscillations at different pointsof the water basin indicates that they propagate inspace as progressive waves. Using the phase difference,let us estimate approximately the velocities, the direc�tion of propagation, and length of these waves with thefollowing known approach. The wave period is knownfrom the spectral analysis. The distance between themoorings is also known. The phase differencesbetween the fluctuations of the same period at twomoorings estimated from the cross�spectral analysiscan be used to determine the time required for the cur�rent field perturbation to travel the distance betweenthe moorings. Dividing the distance between these sta�tions by this time yields the velocity of the perturbationpropagation from one to another station. Since thedeployed moorings form a triangle, the same calcula�tions for the other pairs of stations yield the phase dif�

I1(0), (cm/s)2

70

60

50

40

30

20

10

0

01.2

000

07.2

000

01.2

001

07.2

001

01.2

002

07.2

002

01.2

003

07.2

003

01.2

004

07.2

004

01.2

005

07.2

005

01.2

006

07.2

006

01.2

007

07.2

007

01.2

008

Months, years

Fig. 3. Time variations of the estimates of the linear invariant of the variance tensor of the synoptic�scale current velocity oscilla�tions in the Gotland Basin area of the Baltic Sea at the horizons of 174 m (fine line), 204 m (bold line) and 210 m (dashed boldline) at the NE mooring and at 125 m (thin dashes) at Z mooring.

392



ference pattern for the waves with different periodsand their phase speed, propagation direction, andlength. The obtained characteristics of the low�fre�quency waves are shown in Table 3. It is seen that thelow�frequency waves in the studied Baltic regionpropagate in the southwestern, southeastern, andnorthwestern directions with phase speeds of 0.02–2.08 m/s and have a length from 28 to 431 km.

The obtained wave characteristics within periodsfrom 3 to 20 days are well consistent with the parame�ters of the low�frequency waves that were estimatedpreviously to the southeast of Gotland Island [1, 17,19] and indentified as topographic waves.

To verify the hypothesis of the forced low�fre�quency waves of anemobaric origin, we carried out across�correlation analysis between the synoptic cur�rents and the meteorological characteristics withallowance for the nonstationarity of the process using

the technique reported above. The analysis results areshown in Fig. 7. The correlations between the tangen�tial wind stress and the currents (Figs. 7a, 7b), as wellas between the horizontal atmospheric pressure gradi�ent and the currents, (Figs. 7a, 7b) are low andincrease to absolute values of 0.5–0.6 only withinindividual time periods when the current oscillationsshow a 2� to 8�day lag relative to the meteorologicalprocesses. The comparison of the results of the cross�correlation analysis with Fig. 2 shows that the periodsof high correlation of the currents with theanemobaric forces occur in the spring and autumn,i.e., when the dispersion of the synoptic�scale currentswas low. This raises doubts that the most intense syn�optic current oscillations were generated only by thelocal influence of the anemobaric forces and does notexplain the revealed temporal variability of the synop�tic current dispersion in the central Baltic region.

300270240210180150120

906030

0

NE, 174 m

1

23456

1

1

1

1

2

2

2

3

3

3

4

4

4

4

5

5

315285255225195165135105

754515

NE, 219 m1

356 1

1

1

2

2

2

3

3

3

4

4

4

5

562

78

345

285255225195165135105

754515

2 4 6 8 10 12

SW, 167 m1

2 4 6 8 10 12

14 16Velocities, cm/s

315

1

1

1

1

1

22

2

2

2

22

3

3

33

3

4

300270240210180150120

906030

0

NE, 204 m1

5

345

285255225195165135105

754515

SE, 171 m

345

285255225195165135105

754515

2 4 6 8 10 12

Z, 125 m

2 4 6 8 10 16

Velocities, cm/s

315

3

3

3

33

3 3

2

2

2

1

11

11

11

4

4

2

5

2

315

12 14

1

11

1

1

1

1

1

2

21

2

2

2

3

33

4 4

4

35

2 4 6 8 10 12 2 4 6 8 10 12

5

54 4

4

4

3

3

3

3

3

2

2

2

2

6 7

1

1

1

1

Dir

ecti

on,

deg

Fig. 4. Two�dimensional densities of the probability distribution of the velocity vectors of the synoptic�scale currents with indi�cations of the mooring stations and horizons.



I1(ω); D(ω), (cm2/s2) days1210

86420

–2–4

0 1 2 3

18 11

4.94.0

3.7

2.8

2.4SW1, 170 m; 1997–1998

1086420

–2–4

0 1 2 3

NE, 204 m; 2000

35 1210 7

4.74.4

2.5

20

0 1 2 3

1086420

–2–4

0 1 2 3

NE, 204 m; 2002 138

5.44.7

3.6NE, 204 m; 2001

15

10

5

0

60

1210

8

7

4.63.7

2.3

1086420

–2–4

0 1 2 3

NE, 204 m; 2003 129

7 6

2.4

1086420

–2–4

0 1 2 3

NE, 204 m; 2004 27

139

7 4.2

3.4 2.5

1086420

–2–4

0 1 2 3

NE, 204 m; 2005 67 11

4.43.7

1086420

–2–4

0 1 2 3

SW, 167 m; 2006–2007

8

6

4.13.4 3.1 2.5

2.73.3 1086420

–2–4

0 1 2 3

NE, 204 m; 2007

8 64.4 2.53.5

7

1086420

–2–4

0 1 2 3

SW, 171 m; 2006–2007

8

3.2

161412

ω, rad/days

Fig. 5. Linear invariants I1(ω) (solid line) and rotation indicators D(ω) (dashed line) of the spectral tensor function of the vectorsof the daily mean current velocities at the moorings operating in the Gotland Basin of the Baltic Sea. The numerals above thesignificant peaks of the spectral density show the periods in days.

The extremely rarely observed relationshipbetween the synoptic�scale currents and the meteoro�logical characteristics in the central Baltic regionallows us to propose a resonance mechanism of theirgeneration. This resonance may exist between theanemobaric forces in the atmospheric cyclones and

anticyclones spreading above the Baltic Sea and thelow�frequency eigenmode of the Baltic Sea.

To verify this hypothesis, the three�dimensionalhydrodynamic model of the Baltic Sea developed byO.A. Andreev and A.V. Sokolov at the St. Petersburg

394



T,

day

s2.

0

2.2

2.5

2.9

3.3

4.0

5.0

6.7

10.0

20.0

01

23

45

SE

× Z

2

–18

0–

900

9018

0

00.

20.

40.

60.

81.

0

01

23

45

SE

× N

E

–18

0–

900

9018

0

00.

20.

40.

60.

81.

0

01

23

45

Z2 ×

NE

–18

0–

900

9018

0°

00.

20.

40.

60.

81.

0

(а)

T,

day

s2.

0

2.2

2.5

2.9

3.3

4.0

5.0

6.7

10.0

20.0

01

23

45

SE

× Z

2

–18

0–

900

9018

0

00.

20.

40.

60.

81.

0

01

23

45

SE

× N

E

–18

0–

900

9018

0

00.

20.

40.

60.

81.

0

01

23

45

Z2 ×

NE

–18

0–

900

9018

0°

00.

20.

40.

60.

81.

0

(b)

|I1V

U(ω

)|,

(сm

/s)2 d

ays

f(ω

)

F2 co

l(ω

)

2.0

2.2

2.5

2.9

3.3

4.0

5.0

6.7

10.0

20.0

01

23

4

SE

× Z

2

–18

0–

900

9018

0

00.

20.

40.

60.

81.

0

01

23

4

SE

× N

E

–18

0–

900

9018

0

00.

20.

40.

60.

81.

0

01

23

4

Z2 ×

NE

–18

0–

900

9018

0°

00.

20.

40.

60.

81.

0

(c)

2.0

2.2

2.5

2.9

3.3

4.0

5.0

6.7

10.0

20.0

02

46

8

SE

× Z

2

–18

0–

900

9018

0

00.

20.

40.

60.

81.

0

02

46

8

SE

× N

E

–18

0–

900

9018

0

00.

20.

40.

60.

81.

0

02

46

8

Z2 ×

NE

–18

0–

900

9018

0°

00.

20.

40.

60.

81.

0

(d)

|I1V

U(ω

)|,

(сm

/s)2 d

ays

f(ω

)

F2 co

l(ω

)



Branch of the Zubov State Oceanographic Institutewas numerically simulated. The model is based on theknown equation system of the geophysical hydrody�namics in the Boussinesq approximation, the hydro�statics, the seawater’s incompressibility, and thehypothesis of semi�empirical turbulence theory. Adetailed description of the model and the results of itsverification are given in [8, 14, 22]. Using mathematicmodeling, the free barotropic oscillations of the BalticSea in the synoptic range of spatiotemporal scales wereestimated. The problem of the barotropic free oscilla�tions in the Baltic Sea, as for other basins [4, 15], wassolved by integrating the hydrodynamic equations sys�tem as an inhomogeneous boundary problem. Theriver run off and the exchange with the Northern Seawere omitted. The spatial step of the model was twomiles. The horizontal viscosity coefficient was taken tobe zero. The conditions of the nonleaking were takenat the solid side boundaries. The counting durationwas taken as one year.

The free oscillations in the barotropic sea weregiven by the natural meteorological conditions notedin the autumn of 1994 in the form of several sequentialcyclones and anticyclones. For these purposes, we

used the wind and atmospheric pressure fieldsobtained using the Swedish revision of the initialmeteorological information on northwestern Europefrom 1979 to 2000 with 3�hour temporal discontinuityand a spatial resolution of one degree (the web site isheloios.oce.gu.se of the Swedish Marine GeophysicalInstitute; e�mail: [email protected]). Then, theanemobaric forces were terminated and the free baro�tropic fluctuations were calculated with the subse�quent study of their spectral structure.

The numerical simulations yielded the sea level andcurrent fields in the Baltic Sea, which then wererewritten as time series in grid nodes. The analysis ofthe sea level and current series showed that the freesea�level oscillations very rapidly attenuate and com�pletely disappear already at the end of the first 7–10 days, whereas the free fluctuations usually revealcomparatively slower attenuation and are clearlytraced over the entire range of the calculations. Similarfeatures of the attenuation of the free low�frequencyoscillations of the sea�level and currents were also dis�covered by us in other regions of the Baltic Sea [9].

Figure 8 demonstrates the spectrum I1(ω) of thefree low�frequency oscillations in the current field cal�

Table 3. Characteristics of the low�frequency waves calculated from the cross�spectral analysis of currents at the SE, Z, andNE moorings

Wave period, days Wave length, km Phase speed, m/s Direction of the wave propagation, degrees

2.18 380 2.02 27

2.40 431 2.08 310

2.50 248 1.15 280

2.73 106 0.45 235

2.86 67 0.27 310

3.24 42 0.15 315

5.00 21 0.05 127

5.00 30 0.07 136

10.00 22 0.03 120

11.00 28 0.03 292

20.00 38 0.02 300

Fig. 6. Estimates of the invariants (fine line) and f(ω) (dashed line) of the cross�spectral tensor function as well as the

invariant of the coherence function (bold line) of the synoptic�scale current oscillations at the close horizons of threemoorings (SE, Z, NE) in different periods: (a) June–July, 2006; (b) August–September, 2006; (c) November–December, 2006;(d) January–February, 2007.

( )1VUI ω

( )col2F ω

396



culated for the area of the SE, Z, and NE moorings onthe basis of the numerical hydrodynamic modeling.The spectrum exhibits three energy�carrying maximaat periods of 2.5, 4.3, and 7 days. These peaks arepresent as the most frequent significant energy�carry�ing maxima, as well as in the spectra of the observedsynoptic�scale current oscillations (Fig. 5).

The time series of currents calculated from thehydrodynamic model for the mooring area were sub�jected to cross�spectral analysis in the framework ofthe above described vector–algebraic method of thecurrent analysis. Two invariants of the cross�spectral

density tensor ψ(ω), and one invariant of the

coherence tensor were calculated. The mainenergy�carrying maxima of cross�spectral densitywere noted also at periods of 2.5, 4.4, and 7 days athigh coherence values and phase difference. Theseresults indicate that the free current oscillations withperiods of 2.5, 4.4 and 7 days in the studied area are

( )1 ,VUI ω

( )col2F ω

progressive waves. Using the phase lag, the velocities(V), the propagation direction (α°), and the length λof the low�frequency waves were estimated (Table 4).Depending on the period, the free low�frequencywaves propagate to the northeast and north�northeastwith phase speeds of 0.8–4.3 m/s and have a length of471–929 km. The phase speeds of the free low�fre�quency waves are too low as compared to the velocitiesof the barotropic Kelvin waves (24 m/s) estimated forthe average depth of the open sea of Н = 59 m using theknown formula = (gH)1/2, where g is the free fallacceleration. They are presumably related to the baro�tropic topographic waves.

Figure 9 demonstrates the results of the cross�spec�tral analysis between the synoptic�scale currents at theNE station during the periods of their maximumintensification (autumn–winter) and the differentmeteorological characteristics. The cross spectra showhighly coherent energy�carrying maximums at fre�quencies close to that of the Baltic Sea eigenmodes

0KC

(а)

2005

20

0420

0320

0220

0120

00

–8–4 0 2 4 6 810

(b)

–8 –4 0 2 4 6 810θ, days

(c)

–8 –4 0 2 4 6 810

(d)

–8 –4 0 2 4 6 810

rI1 (θ, t)

rΩ (θ, t)

0.70.60.50.40.30.20.10–0.1–0.2–0.3–0.4–0.5–0.6–0.7

Fig. 7. Results of the cross�correlation analysis between the synoptic�scale current velocity oscillations and the anemobaric

forces: (a) estimates between and (b) estimates between and (c) estimates between

and Grad Pa (t); (d) estimates between and Grad Pa.1( , )Ir tθ ( )V t

��

( );tτ

�

( , )r tΩ θ ( )V t��

( );tτ

�

1( , )Ir tθ ( )V t

��

( , )r tΩ θ V��



calculated for the mooring site. This result shows thatthe energy transfer from the anemobaric forces to theBaltic Sea waters during the indicated periods of themost intensification of the synoptic�scale currents(Fig. 2) mainly occur at frequencies close to the eigen�mode of the basin, which confirms the proposedhypothesis of the resonance energy transfer. This res�onance requires the equality of the velocities of theatmospheric disturbances such as the cyclones andanticyclones and the free topographic waves propagat�ing over the Baltic Sea.

In November 2001, which was marked by the larg�est dispersion of the synoptic�scale currents, the mete�orological conditions over northwestern Europe weredetermined by the passage of three cyclones fromnorthern Scandinavia to the central part of the conti�nent and two anticyclones with centers localizedsouthwest and south of the Baltic Sea. The southwest�ern anticyclone moved to the northeast, while the

southern one, to the southeast. The reanalysis of themeteorological fields made it possible to estimate thevelocities of the atmospheric cyclones and anticy�clones according to the sequential record of the coor�dinates of the cyclone centers in the different meteo�rological periods (Fig. 10). The velocities of thecyclones were sufficiently high (from 12 to 49 m/s).The anticyclone with its center located south of theBaltic Sea also was characterized by comparativelyhigh velocities (9–28 m/s). The velocities and direc�tions of the anticyclone centered southwest of the Bal�tic region within the individual segments of its trajec�tory were close to those of the free barotropic low�fre�quency waves in the Gotland Basin (Table 4). Thus, inmid�November of 2001, the most intense synoptic�scale wavelike current oscillations were observed in theCentral Baltic region. The hydrometeorological con�ditions that formed at that time may have lead to theresonance between the anemobaric forces in the NE�moving anticyclone and the eigenmodes of the BalticSea, which were presumably related to the barotropictopographic waves. This resonance generates barotro�pic and baroclinic modes of forced low�frequencywaves similar to topographic waves, which manifestthemselves in the intense wavelike synoptic�scale cur�rent oscillations recorded at the mooring stations.After the cessation of the resonance, the water massesin the synoptic frequency range reach an equilibriumstate in the form of different modes of slowly attenuat�ing free low�frequency waves. This presumablyexplains the absence of correlation between theanemobaric forces and the observed synoptic�scalecurrents in the Gotland Depression of the Baltic Sea(Fig. 7).

CONCLUSIONS

The statistical analysis of the multiyear observa�tions of the synoptic currents in the Gotland Deep ofthe Baltic Sea performed using the vector�algebraicmethod revealed their wave nature. This follows fromthe significant narrow�band peaks in the current spec�tra, the high coherence of the phase differencebetween the current oscillations at the close horizonsof the different moorings, and the two�mode patternof the vector distribution of the synoptic currents withthe difference between the modes according to theirdirection being close to 180°. The estimated charac�teristics of these waves found using the cross�spectralanalysis show that, within the range from 2 to 20 days,they propagate in the southwestern, southeastern, andnorthwestern directions with phase speeds of 0.02–2.08 m/s and have a length from 28 to 431 km. Theobtained characteristics of the low�frequency waveswithin the current field are well consistent with thewave parameters obtained previously from directobservations in the central Baltic and identified as the

I1(ω), (сm2/s2) days5

4

3

2

1

3ω, rad/days

7

4.3

2.5

0 1 2

Fig. 8. Spectrum of the free barotropic current oscillationsin the Gotland Basin of the Baltic Sea obtained from thenumerical hydrodynamic modeling. The numerals abovethe peak denote the periods in days.

Table 4. Characteristics of the free low�frequency waves inthe current field in the point of the NE moorings obtainedby numerical hydrodynamic modeling

T, days C, m/s λ, km α°

2.5 4.3 929 25

4.4 2.5 864 30

7.0 0.8 471 45

Note: T⎯the wave period; C⎯the phase speed; λ⎯the wavelength; α°⎯the propagation direction

398



|I1V

U(ω

)|, (

сm/s

)2 day

s

40 30 20 10 00.

51.

01.

52.

02.

53.

0

|DV

U(ω

)|,

(сm

/s)2 d

ays

40 30 20 10 00.

51.

01.

52.

02.

53.

0

|I1V

U(ω

)|, (

сm/s

)2 day

s

40 30 20 10 00.

51.

01.

52.

02.

53.

0

|DV

U(ω

)|, (

сm/s

)2 day

s

60 50 20 00.

51.

01.

52.

02.

53.

0

180 60 0

–12

0 00.

51.

01.

52.

02.

53.

0

120

–60

–18

0

ψ(ω

), d

eg18

0 60 0

–12

0 00.

51.

01.

52.

02.

53.

0

120

–60

–18

0

f(ω

), d

eg18

0 60 0

–12

0 00.

51.

01.

52.

02.

53.

0

120

–60

–18

0

ψ(ω

), d

eg18

0 60 0

–12

0 00.

51.

01.

52.

02.

53.

0

120

–60

–18

0

f(ω

), d

eg

1.0 0

0.5

1.0

1.5

2.0

2.5

3.0

F2 co

l(ω

)

0.8

0.6

0.4

0.2

Fre

quen

cy,

rad

/day

s

1.0 0

0.5

1.0

1.5

2.0

2.5

3.0

F2 o

rth(ω

)

0.8

0.6

0.4

0.2

Fre

quen

cy,

rad

/day

s

1.0 0

0.5

1.0

1.5

2.0

2.5

3.0

F2 co

l(ω

)

0.8

0.6

0.4

0.2

Fre

quen

cy,

rad

/day

s

1.0 0

0.5

1.0

1.5

2.0

2.5

3.0

F2 о

rth(ω

)

0.8

0.6

0.4

0.2

Fre

quen

cy,

rad

/day

s

(а)

(b)

(c)

(2)

(1)

Fig

. 9. E

stim

ates

of t

he

inva

rian

ts o

f th

e cr

oss�

spec

tral

ten

sor

fun

ctio

n (

a–b)

an

d th

e co

her

ence

fun

ctio

n (

c) o

f th

e sy

nop

tic�

scal

e cu

rren

ts a

nd

the

tan

gen

tial

win

d st

ress

(1)

, as

wel

l as

the

curr

ents

an

d th

e h

oriz

onta

l atm

osph

eric

pre

ssur

e gr

adie

nt

(2),

for

Nov

embe

r–D

ecem

ber

1999

(bo

ld li

ne)

, Ja

nua

ry–

Feb

ruar

y 20

01 (

bold

das

hed

lin

e),

Nov

embe

r–D

ecem

ber

2001

(th

in li

ne)

, an

d N

ovem

ber–

Dec

embe

r 20

04 (

thin

das

hed

lin

e).



barotropic and baroclinic modes of topographic waves[1, 17, 24]. Testing the hypothesis that the wavelikesynoptic current oscillations are generated byanemobaric forces showed that the correlationbetween the tangential wind stress and the currents, aswell as between the horizontal atmospheric pressuregradient and the currents, is mainly low and only inthe individual time intervals increases to absolute val�ues of 0.5–0.6, which are observed during a 2–8 daylag of the current fluctuations relative to the meteoro�logical processes. However, the periods of elevatedcorrelation values between the currents and theanemobaric forces do not coincide with the period ofthe maximum intensification of the synoptic currents.

The analysis of the meteorological conditions inthe Baltic Sea area and the numerical hydrodynamicmodeling of its free low�frequency oscillations showthat the most probable mechanism for the generationof the intense wavelike synoptic�scale current fluctua�tions in the Gotland depression is the resonancebetween the forces of the tangential wind stress and thehorizontal atmospheric pressure gradient in the anti�cyclones comparatively slowly moving over the openBaltic region and the eigenmode of the basin in thestudied area. This follows from the high coherence atthe frequencies of the eigen sea oscillations betweenthe anemobaric forces and the observed oscillations ofthe synoptic currents during their maximum intensifi�cation, as well as from the close values of the velocitiesand directions of the propagation for the free low�fre�quency waves and atmospheric anticyclones in thecentral Baltic. This resonance leads to the generationof barotropic and baroclinic modes of forced low�fre�quency waves similar to topographic waves. After itscessation, the water masses within the synoptic fre�

quency range are equilibrated in the form of differentmodes of slowly attenuating free low�frequency waves.

ACKNOWLEDGMENTS

This work was supported by the federal target pro�grams Scientific and Scientific–Pedagogical Personalfor Innovation in Russia (2009–2013) and ComplexEstimates of the State of the Drainage�Basin Area inthe System of Russia’s Seas with the Aim to Preservetheir Bioresource Potential (2012–2014).

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m/s504540353025201510

509.11 10.1111.1112.1113.1114.1115.1116.1117.1118.11

2001

1

23

4

5

Fig. 10. Velocities of the centers of the cyclones (1, 2, 5)and anticyclones (3, 4) in northwestern Europe during theperiod of the maximum dispersion of the synoptic�scalecurrent velocities (in November of 2001).

400



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Translated by M. Bogina

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Synoptic variability of the currents in the Gotland Basin of the Baltic Sea