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The Two-Dimensional Seiches of the Baltic Sea

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The Two-Dimensional Seiches of the Baltic Sea _____________________________________o_c_E_A_N_O_L_O_G_IC_A_A_C_T_A __ 19_7_9_-_v_o_L_._2_-_N_o_4 __ ~------ Two-dimensional seiches Numerical solution The two-dimensional seiches Baltic Sea Oscillations bidimensionnelles Solution numérique of the Bal tic Sea · Mer Baltique Ch. Wübber, W. Krauss Institut für Meereskunde, Düsternbrooker Weg 20, 23/Kiel, RFA. Received 6/4/79, in revised form 22/6/79, accepted 2/7/79. ABSTRACT Periods and structures of the gravitational free oscillations in the Baltic Sea are determined from theoretical calculations. The calculations take into account the bathymetry and shape of the Baltic and the earth's rotation. The method for determining the free oscilla­ tions is based on a difference approximation of the inviscid quasistatic shallow water equations. The seiches are produced by a forcing function which acts on the water body during an initial time. The resulting oscillations after the force is switched off are analyzed at each grid point and displayed in form of co-range and co-tidallines. The calculations were carried out for the entire Baltic Sea and for the Baltic proper. A large variety of oscillations is possible. The results are compared with those of the channel theory. It is shown that the eigenoscillations of the Baltic Sea are strongly modified by the Coriolis force. Rotation couverts all modes into positive amphidromic waves; only the interior parts of the Gulfs and of the Western Baltic exhibit standing waves. The periods of the oscillations are reduced by earth rotation, if they are longer than the inertial period, and prolongated, in case they are shorter. The Gulf of Riga, neglected in computations based on channel theory, is of considerable importance for the seiches of the Baltic Sea. Oceanol. Acta, 1979, 2, 4, 435-446. RÉSUMÉ Oscillations bidimensionnelles dans la mer Baltique. On détermine· à partir de calculs théoriques les périodes et les structures des oscillations libres de gravité dans la Mer Baltique. Ces calculs prennent en compte la bathymétrie et la géographie de la Baltique et la rotation de la terre. La méthode utilisée pour déter­ miner les oscillations libres est basée sur une approximation en différences finies des équations non visqueuses des ondes longues. Les seiches sont produites par une force qui agit sur la masse liquide durant un temps donné. Les oscillations obtenues, après avoir enlevé cette force, sont analysées à chaque nœud de la grille de calcul et présentées sous la forme de cartes d'égales amplitudes et d'égales phases. Les calculs sont réalisés pour toute la Mer Baltique et pour la Baltique proprement dite. Une grande variété d'oscillations est possible. Les résultats sont comparés avec ceux donnés par la théorie du canal. On montre que les oscillations propres de la Mer Baltique sont fortement modifiées par la force de Coriolis. La rotation transforme tous les modes en ondes amphi­ dromiques positives; seule la partie intérieure des golfes et la partie ouest de la Baltique présentent des ondes stationnaires. Les périodes des oscillations sont diminuées par la rotation de la terre, si elles sont plus grandes que la période d'inertie, et augmentées dans le cas où elles sont plus courtes. Le Golfe de Riga, négligé dans les calculs basés sur la théorie du canal, est d'importance co~~idérable pour les seiches de la Mer Baltique. Oceanol. Acta, 1979, 2, 4, 435-446. INIRODUCfiON feature in all tide gauge records (Lisitzin, 1959), especially , at the Finnish coast. The oscillations, however, turn out Owing to the ,very strong response of the Baltic Sea to be heavily damped. Typically, we find only two or . to wind action, longitudinal oscillations are a dominant three oscillations. This may be due to the complicated · 0399-1784/1979/435/ S 4.00/Cl Gauthier-Villars. 435 CH. WÜBBER, W. KRAUSS configuration of the Baltic Sea (Fig. 1 and 2), to the • the Baltic proper forks into two basins, the Gulf division into severa! sub-basins (Fig. 2) and to the of Finland and the Gulf of Bothnia, which are of different open connections to the North Sea which allow water length and depth. Consequently, a simultaneous reflection 5 3 1 exchange up to 10 m • s- through the Belts (Hardtke, at both ends is impossible. One, therefore, needs to 1978). Furthermore, the periods are by no means weil distinguish between oscillations of the system W estero separated. A wide variety of oscillations seems to belong Baltic-Gulf of Finland and Western Baltic-Gulf of to the eigenfunctions. The tide gauge records of the Bothnia; Baltic Sea have been studied by severa! authors during the past decades, notably by Neumann, 1941; Magaard • the main axis of the Baltic is S-shaped which leaves and Krauss, 1966; Lisitzin, 1974. Observations indicate sorne arbitrariness to select the cross sections; that the dominant oscillation Westero Bal tic-Gulf of Finland has a period of about 26-27 hours. This, however, • the separation into two systems, mentioned above, is an artifice on a rotating earth. The Coriolis force may vary between 26 and 32 hours. The second mode always tends to push water into that system which is of this system seems to be centered around 17-19 hours, not under consideration. Only the system Western but severa! 20 hours have also been observed. Neumann Baltic-Gulf of Finland may allow separa te eigenfunctions (1941) also detected a period of about 39 hours by because of the narrow entrance to the Gulf of Bothnia careful visual inspection of the records; it was inter­ which is very shallow in most parts of the Aland Sea. preted as the basic mode for the system Western Baltic­ Only a trench (Alands deep), 40 km wide, is a real Gulf of Bothnia. This oscillation, however, was strongly opening to the Baltic proper. damped and transformed into· shorter ones. Spectral analysis of one year records of tide gauges around the Nevertheless, the periods computed by the channel Baltic did not shed any light into the problem (Magaard, approximation are in the range of observed ones. These Krauss, 1966). periods are listed in Table 1 according to Krauss and Magaard (1962) for a Baltic Sea closed at Fehmam Belt. The traditional theory of free oscillations of elongated basins is based on the "channel hypothesis" which In contrast to previous computations, the present study approxima tes the difficult two-dimensional problem with aims to determine the two-dimensional seiches of the a simple one-dimensional problem by integrating over Baltic Sea. The free oscillations of a two-dimensional cross sections and assuming that no cross oscillations water body on a rotating plane consist of two distinct occur. The method was first applied to the Baltic Sea types; the gravitational modes (oscillations of the first by Neumann (1941) and further used by Krauss and class) and the rotational modes (oscillations of the Magaard (1962). With respect to the Baltic Sea the second class). This paper is concemed with the gravita­ method bas severa! shortcomings : tional modes only (seiches). Figure 1 Figure 2 Stereographical projection of the Ba/tic Sea and numerical grid. Depth contours of the Ba/tic Sea. Depth contour interval is 40 m. Centre of projection: 60°N, 20•E. Grid scale 10 km. Units indicated in 10 m. 436 THE TWO-DIMENSIONAL SEICHES OF THE BALTIC SEA Table 1 Note that the equations contain no frictional terms Periods of the Ba/tic Sea based on one-dimensional seiches theory which is essential for an accurate spectral determination (hours). of the eigenfunctions. Mode The problem (1)-(4) is solved as initial value problem. The initial values are given by the state of rest System 2 3 4 5 6 V=O, V=O. (5) Western Baltic-Gulf of Finland 27,4 19,1 13,0 9,6 8,1 6,9 The forcing function P 0 is prescribed as Western Baltic-Gulf of Bothnia 39,4 22,5 17,9 12,9 9,4 7,3 ( 1-cos; t) F(x, y) 0 General methods to compute the eigenoscillations of P (x, y, t)= for (6) a natural basin of sorne arbitrary shape on a rotating 0 earth have been developed by Rao and Schwab (1976) O~t~T 0 , and Platzman (1972). Platzman's method is based on 0 else resonance iteration whereas the method of Rao and Schwab requires the numerical construction of two sets where F (x, y) is chosen in such a way asto force distinct of orthogonal functions. Both methods are rather com­ modes (typically, we use eosine functions in length plicated. For an oblong basin like the Baltic Sea where direction of the Baltic). After time T 0 (typically 30 hours) the basic structure of the modes must be similar to the pressure field is switched off. During 0 ~ t ~ T 0 those on a non-rotating earth, it seems to be much the sea level changes from 0 to Ç (x, y, T 0) and may be more efficient to use a simpler approach. In the present described at time t = T 0 by a sum of eigenfunctions case the procedure is as follows: the linearized momentum equations and the equation of continuity are solved Ç(x, y, T0)= LAn(x, y) cos (ron T0 -9n(x, y)). (7) n numerically as an initial value problem subject to a forcing function. As forcing function we prescribe an Because the model is inviscid and P0 =0 for t > T 0 , air pressure field whose space dependency is a crude these oscillations are undamped. We continue to compute approximation of the expected basic modes. The forcing Ç (x, y, t) numerically until t = T 1 , where function is switched off after sorne hours and, conse­ quently, the set-up decomposes into a sum of eigen­ T 1 - T 0 = 1024 hours.
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