11
ISSN 00014338, Izvestiya, Atmospheric and Oceanic Physics, 2014, Vol. 50, No. 1, pp. 92–102. © Pleiades Publishing, Ltd., 2014. Original Russian Text © S.V. Smirnov, K.M. Kucher, N.G. Granin, I.V. Sturova, 2014, published in Izvestiya AN. Fizika Atmosfery i Okeana, 2014, Vol.50, No. 1, pp. 105–116. 92 INTRODUCTION Seiches are standing free oscillations of the water mass in an enclosed or semienclosed water basin. At the present, many fullscale, laboratory, and theoreti cal studies have been carried out for seiche oscillations in different water objects around the world. For lakes in the former Soviet Union, a detailed discussion of these studies as of 1960 was given in [1], and later results can be found in [2]. Among the lakes of Russia, the largest role is played by Lake Baikal: it is a unique object due to its size (it is the deepest lake in the world and holds the largest volume of fresh water). The dimensions of the lake are as follows: length is 636 km, average width is 49.3 km, average depth is 731 m, and the maximal depth is 1636 m [3]. The bottom relief of Lake Baikal is given in Fig. 1. Seiche oscillations in Baikal are observed almost continuously throughout the whole year. Some characteristics of these oscilla tions were obtained earlier via in situ measurements, laboratory experiments on the spatial hydraulic model, and the respective theoretical calculations [1, 2, 4–9]. However, available information on the seiches in Baikal is incomplete due to the difficulties of full scale measurements and the quite rough data on the bottom relief. In this work we used modern instrumental tools and a modified technique for in situ measurements and we performed a numerical calculation of seiche oscillations in Baikal based on onedimensional, plan, and spherical models. The main aim was to investigate solutions corresponding to oscillations 277, 152, 84, 67, and 59 min in period, which were identified using in situ measurements. For the plan and spherical mod els, the undisturbed lake surface is assumed to be flat and spherical, respectively. The spatial discretizaton of the plan model is implemented on a square mesh; that of the spherical model is on an irregular triangular one. The depth values in mesh points of numerical models are obtained using triangulation on the nodes of up todate bathymetric data [10]. EXPERIMENTAL MEASUREMENTS 1.1. Instrumental Measurement Equipment The measurements of lake level were made using an LHP 110 waterpressure tensor transformer (a com ponent of the automated meteorological station). The pressure sensor is attached inside the subaqueous part of the well, which is necessary for filtering the surface winddriven waves; the well is a vertical plastic pipe Seiche Oscillations in Lake Baikal S. V. Smirnov a , K. M. Kucher b , N. G. Granin b , and I. V. Sturova c a Institute of Automation and Control Processes, Far Eastern Branch, Russian Academy of Sciences, ul. Radio 5, Vladivostok, 690041 email: [email protected] b Limnological Institute, Siberian Branch, Russian Academy of Sciences, ul. UlanBatorskaya 3, Irkutsk 664033 email: [email protected], [email protected] c Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, pr. akad. Lavrent’eva 15, Novosibirsk 630090 email: [email protected] Received October 10, 2012; in final form, December 11, 2012 Abstract—The variations in the free surface of Lake Baikal at three stations (Bol’shie Koty, Listvyanka, and Baikal’sk) are measured. A modern recording method and an advanced technique of record processing are used. Based on 1yearlong observation data, the amplitudes of seiche oscillations and their seasonal changes are analyzed. It is found, in particular, that 67min seiches are manifested in different seasons. Numerical cal culations of seiches in Lake Baikal are made with the use of uptodate bathymetric data on onedimensional, plan, and spherical models. Spatial structures of oscillations with periods of 277, 152, 84, 67, and 59 min, cor responding to the wellexpressed peaks of power spectral density, are studied. It is shown that the first four periods correspond to uninodal, binodal, trinodal, and quadrinodal longitudinal seiche modes of Lake Baikal. The periods of three solutions can correspond to the value of 59 min. The first of them is the seiche of the lake’s South Basin, and two others are characterized by significant amplitude growth in the Small Sea and Chivyrkui Bay. Keywords: seiches, Lake Baikal, shallow water DOI: 10.1134/S0001433813050125

Seiche oscillations in Lake Baikal

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Page 1: Seiche oscillations in Lake Baikal

ISSN 0001�4338, Izvestiya, Atmospheric and Oceanic Physics, 2014, Vol. 50, No. 1, pp. 92–102. © Pleiades Publishing, Ltd., 2014.Original Russian Text © S.V. Smirnov, K.M. Kucher, N.G. Granin, I.V. Sturova, 2014, published in Izvestiya AN. Fizika Atmosfery i Okeana, 2014, Vol. 50, No. 1, pp. 105–116.

92

INTRODUCTION

Seiches are standing free oscillations of the watermass in an enclosed or semienclosed water basin. Atthe present, many full�scale, laboratory, and theoreti�cal studies have been carried out for seiche oscillationsin different water objects around the world. For lakesin the former Soviet Union, a detailed discussion ofthese studies as of 1960 was given in [1], and laterresults can be found in [2]. Among the lakes of Russia,the largest role is played by Lake Baikal: it is a uniqueobject due to its size (it is the deepest lake in the worldand holds the largest volume of fresh water). Thedimensions of the lake are as follows: length is 636 km,average width is 49.3 km, average depth is 731 m, andthe maximal depth is 1636 m [3]. The bottom relief ofLake Baikal is given in Fig. 1. Seiche oscillations inBaikal are observed almost continuously throughoutthe whole year. Some characteristics of these oscilla�tions were obtained earlier via in situ measurements,laboratory experiments on the spatial hydraulicmodel, and the respective theoretical calculations [1,2, 4–9]. However, available information on the seichesin Baikal is incomplete due to the difficulties of full�scale measurements and the quite rough data on thebottom relief.

In this work we used modern instrumental toolsand a modified technique for in situ measurementsand we performed a numerical calculation of seicheoscillations in Baikal based on one�dimensional, plan,and spherical models. The main aim was to investigatesolutions corresponding to oscillations 277, 152, 84,67, and 59 min in period, which were identified usingin situ measurements. For the plan and spherical mod�els, the undisturbed lake surface is assumed to be flatand spherical, respectively. The spatial discretizaton ofthe plan model is implemented on a square mesh; thatof the spherical model is on an irregular triangular one.The depth values in mesh points of numerical modelsare obtained using triangulation on the nodes of up�to�date bathymetric data [10].

EXPERIMENTAL MEASUREMENTS

1.1. Instrumental Measurement Equipment

The measurements of lake level were made using anLHP 110 water�pressure tensor transformer (a com�ponent of the automated meteorological station). Thepressure sensor is attached inside the subaqueous partof the well, which is necessary for filtering the surfacewind�driven waves; the well is a vertical plastic pipe

Seiche Oscillations in Lake BaikalS. V. Smirnova, K. M. Kucherb, N. G. Graninb, and I. V. Sturovac

a Institute of Automation and Control Processes, Far Eastern Branch, Russian Academy of Sciences,ul. Radio 5, Vladivostok, 690041e�mail: [email protected]

b Limnological Institute, Siberian Branch, Russian Academy of Sciences, ul. Ulan�Batorskaya 3, Irkutsk 664033e�mail: [email protected], [email protected]

c Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences,pr. akad. Lavrent’eva 15, Novosibirsk 630090

e�mail: [email protected] October 10, 2012; in final form, December 11, 2012

Abstract—The variations in the free surface of Lake Baikal at three stations (Bol’shie Koty, Listvyanka, andBaikal’sk) are measured. A modern recording method and an advanced technique of record processing areused. Based on 1�year�long observation data, the amplitudes of seiche oscillations and their seasonal changesare analyzed. It is found, in particular, that 67�min seiches are manifested in different seasons. Numerical cal�culations of seiches in Lake Baikal are made with the use of up�to�date bathymetric data on one�dimensional,plan, and spherical models. Spatial structures of oscillations with periods of 277, 152, 84, 67, and 59 min, cor�responding to the well�expressed peaks of power spectral density, are studied. It is shown that the first fourperiods correspond to uninodal, binodal, trinodal, and quadrinodal longitudinal seiche modes of LakeBaikal. The periods of three solutions can correspond to the value of 59 min. The first of them is the seiche ofthe lake’s South Basin, and two others are characterized by significant amplitude growth in the Small Sea andChivyrkui Bay.

Keywords: seiches, Lake Baikal, shallow water

DOI: 10.1134/S0001433813050125

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SEICHE OSCILLATIONS IN LAKE BAIKAL 93

55

54

55

52

53

0

104

°E109108107

105 106 107 108 109 °E

54

53

52

106105104

°N°N

600

500

400

300

200

100

B

L K

Aka

dem

iche

skii

Rid

ge

Small S

ea

Olkhon Is. Barguzin

Bay

Chi

vyrk

uiB

ay

Posol’skaya bank

0

200

400

600

800

1000

1200

1400

1600

Fig. 1. The bottom relief map of Lake Baikal (200�m depth resolution gray�scale). The scheme of a lake with the middle line is inthe inset (tick marks at the line are drawn with a 20�km step).

160 mm in diameter attached to the pier. The upperend located above water level is opened. The lower endis closed, but has several holes for water supply. Thetime constant of the measurement system is about 150 s.The sensor is in impermeable casing, while its measur�ing chamber is connected with the water medium. Thepipe with a measuring cable is attached to the casing;this pipe ends above the water level to compensate theinfluence of atmospheric pressure changes. Analog�

to�digital signal transformation is implemented viaADS1256 delta�sigma ADT (Texas Instruments).Measurement accuracy is within 0.5 mm; resolution is0.1 mm. Reference to the atomic clock of GPS satel�lites had an accuracy of up to 1 µs. The signal from thelevel sensor was digitized with a frequency of 2.6 Hz.The record of counts to memory case was made every10 s on 26 averaged values. Discretization of recordingthe meteorological parameters was 2 min. The

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SMIRNOV et al.

25

20

15

10

5

09/2909/14

08/3008/15

07/3107/010

06/0107/1606/16 10/14

10/2911/13

11/2812/13

12/2801/12

01/2702/11

02/2603/13

03/2804/12

04/2705/12

05/27

1500

1000

09/2909/14

08/3008/15

07/3107/0150006/01

07/1606/16 10/1410/29

11/1311/28

12/1312/28

01/1201/27

02/1102/26

03/1303/28

04/1204/27

05/1205/27

Ice

(а)

(b)

Fig. 2. The record of annual variation in lake level (a) and that of annual variation in amplitude of uninodal seiche (b). The hor�izontal axis indicates month/day; the vertical axis shows variation values in millimeters.

obtained data are available online at the server of Lab�oratory for Hydrology and Hydrophysics, Limnologi�cal Institute, Siberian Branch, Russian Academy ofSciences (http://hydro.lin.irk.ru).

1.2. Analysis of Level Data for a Year�Long Period

An analysis of variations in the lake level for theperiod from June 2010 to June 2011 has been carriedout on the basis of data from three stations. The loca�tions of these observation points are shown in Fig. 1;hereinafter, the stations are denoted with letters: B forBaikal’sk, L for Listvyanka, and K for Bol’shie Koty.The record of the year�long variations in the lake levelat Listvyanka station is presented in Fig. 2a. The spec�tral analysis of this record was implemented after low�frequency filtering, which allowed us to remove thevariations in level within a year. The resulting spec�trum (Fig. 3) demonstrates the clearly expressed peaksof power spectral density at tidal and seiche frequen�cies. The oscillations with periods of 24.27 and 12.42 hare of tidal origin, while those of 277, 152, 84, and67 min are the first four modes of longitudinal seiche;the 59�min one is probably the seiche of the SouthBasin, and the 11�min one is the transverse seiche.Note that all the mentioned periods do not depend onseason. The values of periods of seiche oscillationsagree well with the previously published ones. Thepresence of a 66�min seiche oscillation was shown forthe first time in [8] and [9], where observations of lakelevel at a station in Listvyanka settlement were carried

out in the framework of complex studies of LakeBaikal in winter period in the presence of ice. In thispaper we present a further investigation in this field(the number of stations was increased to three and theanalysis of the year�long observation series was imple�mented). There is no 67�min seiche in [5] and [7],where the periods of first four seiches have the follow�ing values: 278.4, 153, 87.7, and 60 min. Small differ�ences in values can be explained by the use of moreaccurate instruments and analysis methods in thepresent work.

The annual variation in the uninodal seiche is shownin Fig. 2b. The typical amplitude changes from minimalvalues of 3–4 mm to maximal ones of 15–20 mm last forabout 2 weeks. Note that there is no significant differ�ences between the amplitudes of uninodal seiche inthe period when the lake is covered with ice and pro�tected from wind influence (January 12–April 20) andthose in the rest of the year. A comparison of ampli�tudes for oscillations having seiche frequencies,recorded at three points, is carried out using the exam�ple of level record fragments from November 9–24,2011. To detect oscillations with certain frequencies,narrow�band digital filters were used. Graphs showingfiltered oscillations with periods of 277, 152, 84, and67 min are given in Figs. 4a–4d, respectively; the let�ters K, B, and L denote stations. The most obvioussimilarity of all three graphs for the uninodal mode isseen in Fig. 4a. Since this seiche oscillation withperiod of 277 min is dominating in amplitude, the

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SEICHE OSCILLATIONS IN LAKE BAIKAL 95

graphs of level changes for all three stations differ insig�nificantly. Small differences in shape of oscillations inFig. 4b can be explained, for example, by the influence ofwind surge. The shapes of oscillations in Figs. 4c and 4dhave significant differences, while the similarity of graphsfor stations B and L is seen only in portions with relativelyhigh amplitudes. This is explained by the fact that, as theperiod of oscillations decreases, amplitudes decrease aswell; so more pronounced are the differences in shape ofoscillations caused by the effects of wind and atmo�spheric pressure. At Bol’shie Koty station, which is closerthan two others to the nodal lines, the decrease in ampli�tude of seiche oscillations is expressed more intensivelythan at other stations.

2. DESCRIPTION OF CALCULATION METHODS

Numerical methods of calculating seiches in natu�ral water objects can be divided into two groups [11]:solving the eigenvalue problem and solving theCauchy problem. In the present work, the firstapproach is used.

2.1. Initial Equations

Let us write the shallow�water equations system,linearized relative to the normal state of rest, in termsof spherical coordinate system [12, 13]:

(1)

(2)

, ,cos

g gU VfV fUt a t a

∂η ∂η∂ ∂− = − + = −

∂ φ∂λ ∂ ∂φ

cos1 0.cos

HVHUt a

⎛ ⎞∂η ∂ φ∂+ + =⎜ ⎟∂ φ ∂λ ∂φ⎝ ⎠

Hereinafter, a is the average radius of the earth; λ andφ are the geographic latitude and longitude, respec�tively; t is the time; g is the free�fall acceleration; H isthe depth of undisturbed water layer; f is the Coriolisparameter, f = 2Ωsinφ, Ω is the angular speed of theearth’s rotation; U and V are the components of veloc�ity vector along the λ and φ directions, respectively;and η is the free surface elevation. Shallow�waterapproximation [14] is applicable for describing large�scale motions in Lake Baikal, because the ratio of itsaverage depth to its average width is small (<0.015). Atthe solid vertical boundary, the impermeability condi�tion is set:

(3)

where nλ and nφ are the components of normal to theboundary Γ. Then it is assumed that all time variablesobey the harmonic law in time in the form exp iσt,where σ is the frequency of oscillations.

2.2. Plan Model

Since the present work considers motions withinthe water reservoir of a limited horizontal extent, it isreasonable to introduce the rectangular coordinatesystem 0xyz with the vertical axis z. Under the condi�tions that the depth and characteristic horizontalextent of the water reservoir are significantly less thanthe earth’s radius [14], and under the assumption thatthe earth’s rotation can be neglected, the plan model’s

0,n U n Vλ φΓ Γ+ =

80

50

70

40

60

30

20

11 min25 min277 min12.24 h24.27 h10

596784 min152 min

Po

wer

/fre

quen

cy

Period (h, min)

Fig. 3. Power spectral density of lake�level variations.

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SMIRNOV et al.

equation system and impermeability condition at thelateral boundary can be written as follows:

(4)

where u and are the components of velocity vectoralong the directions x and y, respectively; ζ is the freesurface elevation, ζ = Zexp iσt; Z is the amplitude ofelevation; and nx and ny are the components of normalto the shoreline γ. The problem described by Eq. (4)

, , 0,

0,x y

u Hu Hg gt x t y t x y

n u nγ γ

∂ζ ∂ζ ∂ζ ⎛ ⎞∂ ∂ ∂ ∂= − = − + + =⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

+ =

v v

v

v

can be reduced to one equation for Z(x, y) with therespective boundary condition:

(5)

To solve the problem described by Eq. (5), the finite�difference method was used [16]; this method requirescalculating the eigenvalues and eigenvectors of thesquare matrix, whose order equals the number ofnodes in the calculated area. The Cartesian horizontalcoordinate system starts in the point λ0 = 100° andφ0 = 50°. In order to minimize the number of nodes in

( )2

, 0.Z Z Z ZH Hx x y y g n γ

⎛ ⎞∂ ∂ ∂ ∂ σ ∂+ = − =⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

2

0

–2

Nov. 23, 2011Nov. 18, 2011Nov. 13, 2011

5

0

Nov. 23, 2011Nov. 18, 2011Nov. 13, 2011

5

0

Nov. 23, 2011Nov. 18, 2011Nov. 13, 2011

2

0

–2

Nov. 23, 2011Nov. 18, 2011Nov. 13, 2011

2

0

–2

Nov. 23, 2011Nov. 18, 2011Nov. 13, 2011

5

0

–5Nov. 23, 2011Nov. 18, 2011Nov. 13, 2011

5

0

–5

Nov. 23, 2011Nov. 18, 2011Nov. 13, 2011

20

0

–20Nov. 23, 2011Nov. 18, 2011Nov. 13, 2011

5

0

–5

Nov. 23, 2011Nov. 18, 2011Nov. 13, 2011

20

0

–20Nov. 23, 2011Nov. 18, 2011Nov. 13, 2011

5

0

–5

Nov. 23, 2011Nov. 18, 2011Nov. 13, 2011

20

0

–20Nov. 23, 2011Nov. 18, 2011Nov. 13, 2011

–5

–5

K

L

B

(а) (b)

(c) (d)

K

L

B

K

L

B

K

L

B

Fig. 4. Filtered variations with periods of 277 (a), 152 (b), 84 (c), and 67 min (d) for the Bol’shie Koty (K), Listvyanka (L), andBaikal’sk (B) stations. Vertical axis indicates values of the lake level in millimeters.

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SEICHE OSCILLATIONS IN LAKE BAIKAL 97

the calculated area, the x and y axes of the coordinatesystem are rotated 37° clockwise relative to the initialzonal and meridional directions.

2.3. One�Dimensional Model

In the case of a long and narrow lake, a one�dimen�sional model can be used, for example, the Defantmethod described in detail, with its calculation algo�rithm, in [15]. The middle line of the lake, usually

defined based on thalweg, is assumed to be an x axiswith the origin at one of the lake ends. The problem forone�dimensional model can be written as follows:

(6)

where L is the length of middle line; B and S are widthand area of the lake’s cross section perpendicular tothe middle line. In the present work, the middle line is

[ ]( ), ( ) ,

(0) ( ) 0,

uS xu g B xt x t x

u u L

∂∂ζ ∂ζ∂= − = −

∂ ∂ ∂ ∂

= =

8

6

–10

–8

–6

4

2

–4

–2

6005004003002001000

10

x, km

a1 b1

c1

a2

b2

c2

d1

d2

d1c1

b1

Fig. 5. Water�level distribution along the middle line for the first four modes in terms of the one�dimensional model (dashed linesa1–d1) and the respective modes in terms of the two�dimensional spherical model (solid lines a2–d2).

Measured and computed periods for four seiche modes

Modes T1, min T2, min T3, min T4, min

Measurements 277 152 84 67

Measurements, [5, 7] 278.4 153 87.7 –

One�dimensional model (with the thalweg from [10] as the middle line)

260.1 140.7 81.5 62.7

One�dimensional model with a calculated middle line 270.4 146.8 81.6 63.9

Plan model 270.7 160.7 83.7 64.06

Spherical model 276.96 151.58 84.25 67.38

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SMIRNOV et al.

constructed under the assumption that the verticalsurface passing through the middle line should splitthe areas of vertical cross sections perpendicular to themiddle line into equal parts. To calculate the middleline, the iteration routine was constructed. Note thatsplitting the cross sections into equal parts at two sitesis impossible due to relief complexity. One site islocated in the area of the Posol’skaya bank; the secondis in the area of the Akademicheskii submarine ridge.The solid curve in Fig. 1 denotes the middle line con�structed on the basis of the suggested algorithm; thedashed line is the thalweg after [10].

2.4. Numerical Spherical Model

Implementation of the model is generally analo�gous to that described in [17] for the f�plane case. Thenumerical solution for η is sought in the form ofη(λn, φn) = ηnexp iσt. Grid nodes ηn are located in thevertices of spherical triangles; nodes Um and Vm are inthe centers of circles circumscribed around triangles.Every node ηn corresponds to the element of thespherical surface. Three portions of boundariesbetween vertices are located inside the triangle. Everyportion is an arc from the velocity node to the middleof the triangle side. The boundary of element ηn is

composed of arcs lying within triangles that have com�mon vertex with coordinates (λn, φn). At the boundaryof the computational region, the boundary of elementincludes the portions of the triangle sides. Differen�tial�difference analogs of Eqs. (1) and (2) are builtwith the preservation of volume and energy taken intoaccount. These equations can be reduced to an equa�tion for function ηn. At constant Coriolis parameter f,coefficients in equations for ηn contain variable σraised to a power of no more than 3. The system ofthese equations can be written in matrix form:

(7)

where N is the number of nodes ηn; A is the square matrixof 3N order. When solving the eigenvalue problem fromEq. (7), linear algebra packages are used [18, 19].

The construction of a triangular mesh is made withMercator projection applied. The shoreline is mappedonto the surface of cylindrical projection; equilateraltriangles of the set size, adjacent to the boundary, arelocated in the internal zone. The rest of the internalpart is filled with acute triangles whose sides graduallyincrease with a growth in distance from the boundary

22

, , ,

, 1 ,

n n n N n

n N n

A

n N

+

+

σξ = ξ ξ = η ξ = ση

ξ = σ η ≤ ≤

8

6

–10

–8

–6

4

2

–4

–2

6005004003002001000

10

x, km

a

b

c

d

e

f

a

Fig. 6. Water�level distribution along the middle line: (a) for the fifth mode of the one�dimensional model (dashed line); for themodes of the two�dimensional spherical model with periods of (b) 58.87 (solid line), (c) 57.59 (dashed and dotted line), and(d) 58.42 (dashed and double�dotted line); (e) 59.82 (double�dashed and pointed line); and (f) 60.72 min (dotted line).

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SEICHE OSCILLATIONS IN LAKE BAIKAL 99

to a certain value. After inverse mapping onto sphere,we obtain a mesh constructed of spherical triangles.

3. CALCULATION OF SEICHE OSCILLATIONS IN LAKE BAIKAL

Based on the one�dimensional model, the calcula�tions were made on mesh having a 5 km interval and134 cross sections. The periods of first five modes are270.4, 146.8, 81.6, 63.9, and 53.7 min. Lake�level dis�tribution along the middle line for the first four modesis denoted by dashed lines a1–d1, respectively, inFig. 5; that for the fifth mode is denoted by the dashedline a in Fig. 6. To compare, the periods of first fivemodes with the thalweg from [10] applied are 260.1,140.7, 81.5, 62.7 and 52.6 min. The square mesh of theplan model with the horizontal step 5 km contains1304 nodes of the computational region. The calcu�lated periods of the first four modes are given in table.The shapes of lake level surfaces for the first threemodes are shown in Figs. 7b, 7d, and 7f. The solutions

of one�dimensional and plan models are normalizedin a way that lake�level values fall within the rangefrom –10 to 10.

The computational mesh of the spherical model isirregular; the side length for triangles near the shore�line is 30 m and that for triangles in the main part ofthe model area is about 1 km. Due to the absence ofdata on the bottom relief, the shallow Angarskii SorBay is excluded from calculation. The present studyincludes results with the earth’s rotation taken intoaccount (at f = 0.000116 s–1). The complex solutionsare normalized in a way that the imaginary componentis minimal. The computational domain of the spheri�cal model contains nodes of the shelf (small depth)zone. To compare these results with those obtained interms of the plan model, the real parts of the sphericalmodel solution are normalized in a way that the valuesfrom –10 to 10 are located in most of the computa�tional region, excluding nodes with a depth of less than10 m and nodes within the Small Sea contour (out�lined with dashed line in Fig. 1). Solution in the nodes

8

6

4

2

0

–2

–4

8 8 8

6

6 6

4

4 4

2

2 2

0

00

–2

–2

–2–2

–2–2

–4

–4–4

–4

–4 –4

–2

–6–6

–4 –2

00

0

00

–6

–4–2

22

2

2 2 –4

4

4 4

4

2 222

4 40

0

0

0

668 8

–6–8 –6

–6

–2

–2

4

(а) (b) (c) (d) (e) (f)4

Fig. 7. Level surfaces for seiche modes with periods of (a) 276.96, (c) 151.58, and (e) 84.25 min calculated on the triangular mesh,and those with periods of (b) 270.7, (d) 160.7, (f) 83.7 min calculated on the square mesh.

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IZVESTIYA, ATMOSPHERIC AND OCEANIC PHYSICS Vol. 50 No. 1 2014

SMIRNOV et al.

2

1

0

–1

0

1

432

–1

0

1

0

0

2

20

0

0

11

1

2

–1

–2

–3 12

34

–1

–1

–1

–2

–3

0

0

0

–2

–11

2

(а) (b)

Fig. 8. Level surfaces of seiche modes with the periods (a) T = 67.38 and (b) T = 58.87 min calculated on the triangular mesh.

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SEICHE OSCILLATIONS IN LAKE BAIKAL 101

53.3

53.2

53.1

53.0

107.6107.4107.2107.0106.8 109.0 109.1 109.2 109.3

53.7

53.8

53.9°N

°Е

–4

–2

0

246

12

14

16 –5

–10

0

–4 –6

–2

0

–15

–20

5

15

25

10

15

5

5

–5

10

°Е

°N

(а) (b)

Fig. 9. Seiche modes in the water areas of (a) the Small Sea, 58.42 min period, and (b) Chivyrkui Bay, 59.82 min period.

excluded from normalization can be beyond the limitsof the mentioned range. In Figs. 7 and 8, real componentsof solution are shown. In Figs. 7a, 7c, and 7e, the shapes oflevel for seiches of 276.96, 151.58, and 84.25 min in periodare shown, respectively. Figure 8 demonstrates theshapes of level for seiches with periods of 67.38 and58.87 min. The distribution of level along the middleline for the first four mentioned modes are shown inFig. 5 with solid lines a2–d2, respectively; that for thefifth mode is shown with solid line b in Fig. 6. Curvesc, d, e, and f in Fig. 6 denote the graphs of the level dis�tribution for modes with periods of 57.59, 58.42,59.82, and 60.72 min, respectively. Note that, in shal�low water sites such as Mukhor Bay, Proval Bay,Cherkalovskii Sor, and Posol’skii Sor, where the bot�tom friction of water should be likely taken intoaccount, other approaches are necessary to specify thesolutions.

The similarity of the results presented in Fig. 5shows that the positions of node lines and the distribu�tion of the seiche�oscillation intensity in differentparts of the lake are well predicted for the first fourmodes via even a one�dimensional model. The resultsfor the first mode are consistent with the data on thedistribution for the height of seiche oscillations alongthe length of Lake Baikal [2, Fig. 5.2], where the cal�culation data and those of studies on the spatialhydraulic model are compared.

In the main part of lake area, the solutions of planand spherical models differ only insignificantly in thepositions of node lines and shapes of level surface. Thesimilarity of solutions is caused by the following fac�tors: the size of the water reservoir is insufficient forthe effects of the earth’s sphericity to appear; in the

case of a relatively narrow and deep reservoir, takingthe earth’s rotation into account poorly changes thesolution for seiches [14]; there is a small shelf zone,which sharply transits into a deepwater slope (exclud�ing the area of the Selenga shallow zone). The narrowshelf zone has very small influence on the dynamics ofseiche oscillations at these scales of a water reservoir,and the absence of this zone in the plan model causesan insignificant alteration. Some differences in thecalculation results can be explained by differences in adescription of the shoreline and topography of theSmall Sea, Barguzin Bay, and Chivyrkui Bay.

Antinodes of the fifth one in the one�dimensionalmodel are located close to the boundaries of the SouthBasin and Barguzin Bay, which can play the role of anadditional wall and significantly weaken the ampli�tudes of oscillations in the Central Basin and do thesame to a higher degree in the Northern Basin. Such apeculiarity occurs for a mode with a period of 58.87, asdenoted by the solid b line in Fig. 6. The oscillationwith period of 58.42 min is the second resonant modeof the Small Sea and it is given in Fig. 9a. The oscilla�tion with period of 59.82 min is one of the resonantones of Chivyrkui Bay and it is shown in Fig. 9b.

CONCLUSIONS

Based on the data from three water�level measuringstations located in the southern part of the lake, ananalysis of amplitudes of seiche oscillations in LakeBaikal and seasonal variations in seiches has been car�ried out. In the power density spectrum constructedon the basis of the year�long record of water level,well�expressed peaks are seen for oscillations with

Page 11: Seiche oscillations in Lake Baikal

102

IZVESTIYA, ATMOSPHERIC AND OCEANIC PHYSICS Vol. 50 No. 1 2014

SMIRNOV et al.

periods of 277, 152, 84, 67, and 59 min. For the one�node seiche, no significant deviations have beenrevealed for the amplitudes when the lake is coveredwith ice and protected from wind relative to the ampli�tudes in the rest of the year. It has been found thatseiches with period of 67 min can be observed in dif�ferent seasons. The shapes of level changes at threestations differ poorly for the oscillation with period of277 min, they have small differences for the152 minone, and they demonstrate similarity only in sites withrelatively big amplitudes for those of 84 and 67 min(this is explained by the effects of wind and atmo�spheric pressure).

In numerical calculations of seiches in Lake Baikalmade on the basis of up�to�date bathymetric data interms of one�dimensional, plan, and spherical mod�els, the spatial structure of oscillations with periods of277, 152, 84, 67, and 59 min has been studied. Thenumerical method includes a solution of the eigen�value problem in a linear formulation and allows oneto obtain the set of frequencies and the respectiveshapes of seiche oscillations. Calculations of seiches inLake Baikal based on two�dimensional models havebeen implemented for the first time. It has been shownthat the first four periods correspond to uninodal, bin�odal, trinodal, and quadrinodal longitudinal seichemodes of Lake Baikal. The measured and calculatedperiods of these seiche modes are summarized in tableform. The value of 59 min is close to periods of threesolutions obtained on triangular mesh. The first is 58.87min, corresponding to the seiche of the South Basin; itis distinct by a significant weakening of oscillationamplitude in the central part and even more significantweakening in the northern part of the lake. Two othersolutions (periods of 58.42 and 59.82 min) are charac�terized by significant resonant increasing of amplitudein the Small Sea and Chivyrkui Bay, respectively.

ACKNOWLEDGMENTS

This work was supported in part by the SiberianBranch of the Russian Academy of Sciences (integratedproject no. 132), the Far East Branch of the RussianAcademy of Sciences (project no. 12�II�SU�03�001),and the Russian Foundation for Basic Research (projectnos. 11�01�98510 and 13�01�12043).

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Translated by N. Astafiev