Journal of Oceanography, Vol. 54, pp. 143 to 150. 1998 Clockwise Phase Propagation of Semi-Diurnal Tides in the Gulf of Thailand TETSUO YANAGI 1 and TOSHIYUKI TAKAO 2 1 Research Institute for Applied Mechanics, Kyushu University, Kasuga 816, Japan 2 Department of Civil and Environmental Engineering, Ehime University, Matsuyama 790, Japan (Received 4 August 1997; in revised form 3 December 1997; accepted 6 December 1997) The phase of semi-diurnal tides (M 2 and S 2 ) propagates clockwise in the central part of the Gulf of Thailand, although that of the diurnal tides (K 1, O 1 and P 1 ) is counterclockwise. The mechanism of clockwise phase propagation of semi-diurnal tides at the Gulf of Thailand in the northern hemisphere is examined using a simple numerical model. The natural oscillation period of the whole Gulf of Thailand is near the semi-diurnal period and the direction of its phase propagation is clockwise, mainly due to the propagation direction of the large amplitude part of the incoming semi-diurnal tidal wave from the South China Sea. A simplified basin model with bottom slope and Coriolis force well reproduces the co-tidal and co-range charts of M 2 tide in the Gulf of Thailand. Keywords: Clockwise amphidrome, natural oscillation, tide, Gulf of Thailand. 1. Introduction It is well known that the phase of tides propagates counterclockwise (clockwise) in gulfs or shelf seas such as the North Sea, the Baltic, the Adria, the Persian Gulf, the Yellow Sea, the Sea of Okhotsk, the Gulf of Mexico and so on in the northern (southern) hemisphere and such a phenomenon is well explained by the superposition of incoming and reflecting Kelvin waves (Taylor, 1920). But the phase of semi-diurnal tides in the Black Sea propagates clockwise and Sterneck (1922) explained this remarkable phenomenon in terms of the phase lag between the east-west natural oscillation forced by the east-west component of tide-generating force and the north-south one forced by the northsouth component of tide-generating force in the Black Sea, neglecting the Coriolis force. After high water along the east coast, the high water occurs along the south coast of the Black Sea because the natural oscillations in east-west and north-south directions have the phase difference of π/2. The counterclockwise phase propagation of diurnal tides (K 1 and O 1 ) in the Black Sea is also explained by the same theory (Sterneck, 1922). The phase of diurnal tides (K 1, O 1 and P 1 ) propagates counterclockwise at the central part of the Gulf of Thailand in the northern hemisphere, as shown in Fig. 1(b) but that of semi-diurnal tides (M 2 and S 2 ) propagates clockwise there as shown in Fig. 1(a) (Yanagi et al., 1997).The observed directions of the phase propagation of diurnal and semidiurnal tides are well reproduced by numerical experiments (Yanagi and Takao, 1997). In this paper we reveal the mechanism of the clockwise phase propagation of semi-diurnal tides at the Gulf of Thailand in the northern hemisphere using a simple numerical model. 2. Gulf of Thailand The Gulf of Thailand is situated in the southwestern part of the South China Sea and the length from the shelf edge to the head of the gulf, L, is about 1,500 km; its width, B, is about 460 km and its average depth, H, is about 40 m (Fig. 2). Because the phase speed of the long wave in the gulf C = gh is about 20 m s 1, the wavelength of the semidiurnal tidal wave l M2 = CT M2 (T M2 is the semi-diurnal period) is 890 km and that of diurnal one l K1 is about 1,700 km. These are nearly one half of and the same as the length of the gulf L, respectively. The inertia period T i (=2π/f, f; the Coriolis parameter = 2ωsinφ, ω; the angular velocity of the earth s rotation, φ; the latitude = 9 N in this case) of the gulf is 76.6 hours. The Rossby deformation length λ (= gh /f ) of the gulf is 870 km. Semi-diurnal and diurnal tidal periods are much shorter than the inertia period and the width of the gulf is narrower than the Rossby deformation length in the Gulf of Thailand. These facts suggest that the tidal phenomena are not seriously affected by the Coriolis force in the Gulf of Thailand. 3. Numerical Model The horizontal two-dimensional momentum and continuity equations for tide and tidal current of a homogeneous fluid under Cartesian coordinates are as follows; Copyright The Oceanographic Society of Japan. 143
(a) (b) Fig. 1. Co-tidal and co-range charts of M 2 (a) and K 1 (b) tides in the South China Sea. Phase is referred to 135 E (Yanagi et al., 1997). Fig. 2. Gulf of Thailand. Numbers show the depth in meters. 144 T. Yanagi and T. Takao
(a) (b) Fig. 3. Co-tidal (full line) and co-range (broken line) charts of M 2 and K 1 tides in the simplified gulf with flat bottom in the cases of no rotation (a) and rotation (b). u t + ( u )u + fk u = g η γ b 2 u u H + η + ν 2 u, () 1 η t + { ( H + η)u }= 0. ( 2) Here u is the depth averaged velocity vector, t the time, the horizontal gradient operator, k the locally vertical unit vector, η the sea surface elevation from the mean sea surface, γ b 2 (=0.0026) the bottom frictional coefficient, ν (=10 6 cm 2 s 1 )the horizontal eddy viscosity and H the local water depth. Equations (1) and (2) can be approximated by finite difference and solved by the primitive method. Numerical experiments have been conducted in basins with different Clockwise Phase Propagation of Semi-Diurnal Tides in the Gulf of Thailand 145
(a) (b) (c) Fig. 4. Same as Fig. 3 except with bottom slope. Numbers in the upper panel show the depth in meters. bottom topography, shown in Figs. 3 and 4, which simplify the horizontal and vertical geometry of the Gulf of Thailand, with a length of 1500 km, a width of 460 km and an open boundary along the eastern end. The grid size is 10 km 10 km. The uniform amplitude (15 cm for M 2 tide and 35 cm for K 1 tide) and phase lag (330 degree for M 2 and K 1 tides) are given along the eastern open boundary on the basis of observed amplitude and phase lag along the shelf edge of the Gulf of Thailand shown in Fig. 1 (Yanagi et al., 1997). The time step of the calculation is 2 minutes. The quasi-steady state is obtained in four tidal cycles after the beginning of the calculation and the harmonic analysis of sea surface elevation and current field is carried out at the fifth tidal cycle. 4. Results The calculated results in the case of a constant depth of 40 m and in the absence of the Coriolis force are shown in Fig. 3(a). Two amphidromic points of the M 2 tide and one of 146 T. Yanagi and T. Takao
the K 1 tide exist in the gulf. The northern amphidrome of the M 2 tide has a counterclockwise phase propagation, but the southern one propagates clockwise. The phase of the K 1 tide propagates clockwise in the central part of the gulf. Results in the presence of the Coriolis force are shown in Fig. 3(b). The direction of phase propagation of the M 2 tide in the southern part of the gulf changes from clockwise to counterclockwise. The position of the amphidrome and the direction of phase propagation of K 1 tide change drastically from east to west and clockwise to counterclockwise, respectively. Figure 3 suggests the existence of natural oscillation nodes where the co-tidal lines gather and the amplitude decreases, that is, the natural oscillation along the east-west direction dominates for the M 2 tide with no Coriolis force. The natural oscillation along the north-south direction dominates for the K 1 tide and those along the north-south and east-west directions couple for the M 2 tide with the Coriolis force. The results in the simplified gulf with a bottom slope are shown in Fig. 4. The amphidromic point of the M 2 tide in the northern part of the gulf shown in Fig. 3(a) disappears when the bottom slope is included, as shown in Fig. 4(a), but the result for the K 1 tide shown in Fig. 3(a) is nearly the same as that shown in Fig. 4(a). When we include the Coriolis force in Fig. 4(a), the clockwise amphidrome of the M 2 tide shifts a little northward and another counterclockwise amphidrome appears at the head of the gulf. By including the Coriolis force, the position of amphidrome and the direction of phase propagation of the K 1 tide change drastically from east to west and clockwise to counterclockwise, respectively, as shown in Fig. 4(b). The results with a constant and a variable Coriolis parameter are nearly the same, as shown in Figs. 4(b) and 4(c), and they are qualitatively the same as the observed one shown in Fig. 1, that is, the counterclockwise phase propagation with large amplitude at the head of the gulf, the clockwise one and the large amplitude along the eastern Malay coast at the central part of the gulf, the counterclockwise one with large amplitude along the southern coast of the Indo-China peninsula and along the northern coast of Borneo at the southeastern part of the gulf for the M 2 tide and the counter-clockwise phase propagation with large amplitude along the head of the gulf for K 1 tide at the central part of the gulf. We conducted other numerical experiments with horizontal eddy viscosities of 10 5 cm 2 s 1 and 10 7 cm 2 s 1 and bottom drag coefficients of 0.001 and 0.004, but the results are nearly the same as Figs. 3 and 4. Numerical experiments without the nonlinear term give similar results (not shown here). 5. Discussions We investigate the mechanism of clockwise phase propagation of semi-diurnal tide and counterclockwise phase propagation of diurnal tide at the central part of the Gulf of Thailand on the basis of the calculated results with simple models shown in Figs. 3 and 4. The natural oscillation periods T w N-S i along the northsouth direction in the western part of the simplified gulf with the constant depth shown in Fig. 3 and those along the eastwest direction in the northern part T n E-W i are calculated by the following equation for a closed basin, T i = 2L b i gh ( 3) where i denotes the mode number, L b (=880 km along the north-south direction and 460 km along the east-west direction) the length of the basin and g (=9.8 m s 2 ) the gravitational acceralation. The natural oscillation periods along the north-south direction in the western part are T w N-S 1 = 24.7 hours, T w N-S 2 = 12.4 hours and T w N-S 3 = 8.2 hours, those along the east-west direction in the northern part are T w N-S 1 = 12.9 hours and T w N-S 2 = 6.5 hours. The natural oscillation periods T s E-W i along the eastwest direction in the southern part of the simplified gulf shown in Fig. 3 are calculated by the following equation for a semi-closed basin, T i = 4L b ( 2i 1) gh ( 4) where L b (=880 km) denotes the bay length in the southern part of the simplified gulf shown in Fig. 3. The natural oscillation periods along east-west direction in the southern part are T E-W s 1 = 49.4 hours, T E-W s 2 = 16.5 hours, and T E-W s 3 = 9.9 hours. Therefore only the semi-diurnal tide may resonate in the whole gulf (T N-S w 2 = 12.4 hours, T E-W n 1 = 12.9 hours, and T E-W s 3 = 9.9 hours) and its oscillation mode is schematically shown in Fig. 5(a), though the diurnal tide can resonate only along the north-south direction in the western gulf. The natural oscillation along the east-west direction in the western part dominates for the M 2 tide in the case without the Coriolis force, as shown in Fig. 3(a). This is due to the fact that the incoming M 2 tidal wave energy cannot transmit through the square bend of the L-shaped channel when kb = mπ (k = 2π/l and l = the wavelength, B = the width of the channel, and m = positive integer), which was discussed theoretically by Momoi (1974). In this case the wavelength l M2 is 890 km, the width B 460 km and kb becomes nearly π. The incoming M 2 tidal wave mainly reflects at the western wall in the southern part and only the natural oscillation along the east-west direction is dominant. The direction of phase propagation of the natural oscillation is mainly governed by the propagation direction of Clockwise Phase Propagation of Semi-Diurnal Tides in the Gulf of Thailand 147
the large amplitude part of the incoming wave from the open boundary. When the M 2 tidal wave enters from the eastern open boundary into the gulf, the large amplitude part propagates along the southern wall in the southwestern part because the southern wall is situated at the anti-node of the natural oscillation along the north-south direction. Therefore the phase around the amphidrome in the southern part of the gulf propagates clockwise as shown in Figs. 3(a) and 5(a). The phase in the northern part propagates counterclockwise because the high water in the central part of the western gulf (shown by ➁ in Fig. 5(a)) occurs after the high water along the western coast (shown by ➀ in Fig. 5(a)) due to the natural oscillation mode there. When we include the Coriolis force in Fig. 3(a), the principal oscillation mode does not change. However the incoming M 2 tidal wave has the characteristic of a Kelvin wave, a part of its energy can transmit through the square bend of the L-shaped channel and it is reflected not only at the western wall but also at the northern wall in the western gulf. Therefore the natural oscillations along the northsouth and east-west directions couple in the western gulf as shown in Fig. 3(b). The phase propagation direction becomes counterclockwise at the northern and southern amphidromes, as shown in Fig. 3(b), because the amplitude of the incoming and reflected Kelvin tidal wave is large at the right-hand side of its propagation direction. When the simplified gulf has a bottom slope, the natural oscillation along the east-west direction in the northern part does not change, but those along the north-south direction in the western part and along the east-west direction in the southern part couple and the natural oscillation along the deepest bottom line dominates. The natural oscillation periods T di along the deepest bottom line in the gulf shown in Fig. 4 are calculated by Eq. (4) with L b = 1200 km and H = 40 m (average depth): T d1 is 67.3 hours, T d2 = 22.4 hours, T d3 = 13.5 hours and T d4 = 9.6 hours. Therefore only the semi-diurnal tide may also resonate in the whole gulf (T n E-W 1 = 12.9 hours and T d3 = 13.5 hours) as shown in Fig. 5(b), though the diurnal tide can resonate only along the deepest bottom line (T d2 = 22.4 hours). The incoming tidal wave has the characteristic of an edge wave in this case, and the dispersion relation is as follows (Ursell, 1952), (2π/T n ) 2 = g(2π/l n )sin(2n + 1)φ (5) where T denotes the period, n the mode number, l the wavelength and φ the angle of bottom slope. When we give φ = 0.016 degree from the bottom slope shown in Fig. 4, we understand that n = 0, T 0 = 12.4 hour and l 0 = 877 km satisfy the dispersion relation (5). This means that the incoming M 2 tidal wave from the eastern open boundary acts as an edge wave with its energy mainly along the coast. The propagating edge waves along the northern and southern walls are (a) (b) Fig. 5. Schematic representation of oscillation mode in the simplified gulf with flat bottom (a) and sloping bottom (b). reflected at the northern and western walls in the western gulf and the natural oscillations along the east-west direction and along the deepest bottom line are generated as shown in Figs. 4(a) and 5(b). The propagating M 2 tidal wave along the northern wall decreases its amplitude along the eastern wall in the western gulf because it corresponds to the node of natural oscillation along the deepest bottom line. On the other hand, the amplitude decreasing ratio of the propagating M 2 tidal wave along the southern wall is not so large as that of another propagating wave along the northern wall. Therefore the phase propagation direction of the central amphidrome of the M 2 tide becomes clockwise, as shown in Fig. 4(a). With the Coriolis force added to Fig. 4(a), the M 2 tidal wave principally behaves as an edge wave because the effect of Coriolis force can be neglected in the following dispersion relation of inertiogravitational edge wave (Kajiura, 1958; Reid, 1958) due to the high ratio of 2π/( f T n ) (which is about 6). (2π/T n ) 3 {f 2 + (2n + 1)g(2π/l n )tanφ}(2π/t n ) + g(2π/l n ) tanφ f = 0. (6) The co-tidal and co-range charts of the M 2 tide with the Coriolis force shown in Fig. 4(b) are nearly the same as those without the Coriolis force shown in Fig. 4(a), except that new amphidromes appear in the northwestern and southeastern parts and the amplitude along the northern wall is greater than the southern wall. The decisive mechanism governing the propagating direction of the amphidromes in 148 T. Yanagi and T. Takao
(a) (b) Fig. 6. Co-tidal (full line) and co-range (broken line) charts of M 2 and K 1 tides in the simplified gulf with flat bottom (a) and sloping bottom (b) without rotation. Fig. 4(b) is considered to be the same as that in Fig. 4(a) mentioned earlier. The phase of diurnal tide, which resonates only along the north-south direction in the western gulf, propagates clockwise without the Coriolis force as shown in Figs. 3(a) and 4(a). This is due to the fact that the amplitude of the propagating K 1 tidal wave along the northern wall becomes small at the northeastern square bend of the gulf because it is a node of natural oscillation along the north-south direction. On the other hand, it propagates counterclockwise along the northern and eastern walls with the Coriolis force due to the Kelvin wave s characteristic of a K 1 tidal wave, as shown in Figs. 3(b), 4(b), and 4(c). This is because the effect of the Coriolis force on the K 1 tidal wave is larger than that on the M 2 tidal wave due to its longer period. These facts suggest that the phase of diurnal tide may Clockwise Phase Propagation of Semi-Diurnal Tides in the Gulf of Thailand 149
propagate clockwise in the gulf, which has a suitable horizontal scale and depth for resonance of a diurnal period. We carried out another numerical experiment in the simplified gulf with different dimensions from Figs. 3 and 4 without the Coriolis force, using the same parameters. The results are shown in Fig. 6. The phase of diurnal tide propagates clockwise in the central part of the gulf with bottom slope, as shown in Fig. 6(b), though the phase propagation of the semidiurnal tide is very complicated. Systematic natural oscillations along the east-west direction are generated for the M 2 tide without bottom slope, as shown in Fig. 6(a). This is due to the fact that the incident M 2 tidal wave energy cannot transmit through the square bend as discussed earlier, and only the natural oscillation along the east-west direction dominates. In the case with bottom slope, the energy of the incident M 2 tidal wave transmits through the square bend as the edge wave and the natural oscillation along the northsouth direction couples with that along the east-west direction as shown in Fig. 6(b). The results of the K 1 tide shown in Figs. 6(a) and 6(b) are nearly the same as those of the M 2 tide shown in Figs. 3(a) and 4(a), respectively. In this simplified gulf, only the diurnal tide can resonate at the lower mode because T n E-W 1 = 25.0 hours, T w N-S 2 = 23.9 hours, T s E-W 3 = 19.1 hours and T d4 = 21.2 hours in the whole gulf. These results suggest that the phase of diurnal tide may propagate clockwise at some coastal seas in the northern hemisphere with suitable horizontal and vertical scales for the resonance of a diurnal period. Acknowledgements The authors express their sincere thanks to Drs. H. Takeoka and K. Ichikawa for their fruitful discussions and two anonymous reviewers for their useful comments on the first draft. A part of this study was supported by the research fund of the Ministry of Education, Science, Culture and Sports, Japan. References Kajiura, K. (1958): Effect of Coriolis force on edge waves (II) Specific examples of free and forced waves. J. Mar. Res., 16, 145 157. Momoi, T. (1974): A long wave in an L-shaped channel. J. Phys. Earth, 22, 395 414. Reid, R. O. (1958): Effect of Coriolis force on edge waves (I) Investigation of the normal modes. J. Mar. Res., 16, 109 144. Sterneck, R. V. (1922): Schematische Theorie der Gezeiten des Schwarzen Meeres. S.B. Akad. Wiss. Wien. (Math.-Naturwiss, K1.), 131, 81. Taylor, G. I. (1920): tidal oscillations in gulfs and rectangular basins. Proc. Lond. Math. Soc., 20, 148 181. Ursell, F. (1952): Edge waves on a sloping beach. Proc. Roy. Soc., A214, 79 97. Yanagi, T. and T. Takao (1997): A numerical experiment on the tide and tidal current in the South China Sea. 9th JECSS-PAMS Proceedings. Yanagi, T., T. Takao and A. Morimoto (1997): Co-tidal and corange charts in the South China Sea derived from satellite altimetry data. La mer, 35, 85 94. 150 T. Yanagi and T. Takao