Clockwise Phase Propagation of Semi-Diurnal Tides in the Gulf of Thailand

Journal of Oceanography, Vol. 54, pp. 143 to 150. 1998

1 2 TETSUO YANAGI and TOSHIYUKI TAKAO

1Research Institute for Applied Mechanics, Kyushu University, Kasuga 816, Japan 2Department 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 and S ) propagates clockwise in the central part of Keywords: 2 2 ⋅ the Gulf of Thailand, although that of the diurnal tides (K1, O1 and P1) is counterclock- Clockwise wise. The mechanism of clockwise phase propagation of semi-diurnal tides at the Gulf of amphidrome, ⋅ natural oscillation, 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 tide, ⋅ Gulf of Thailand. 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 M2 tide in the Gulf of Thailand.

1. Introduction phase propagation of semi-diurnal tides at the Gulf of It is well known that the phase of tides propagates Thailand in the northern hemisphere using a simple numeri- counterclockwise (clockwise) in gulfs or shelf seas such as cal model. the North Sea, the Baltic, the Adria, the Persian Gulf, the Yellow Sea, the Sea of Okhotsk, the Gulf of Mexico and so 2. Gulf of Thailand on in the northern (southern) hemisphere and such a phe- The Gulf of Thailand is situated in the southwestern nomenon is well explained by the superposition of incoming part of the South China Sea and the length from the shelf and reflecting Kelvin waves (Taylor, 1920). But the phase of edge to the head of the gulf, L, is about 1,500 km; its width, semi-diurnal tides in the Black Sea propagates clockwise B, is about 460 km and its average depth, H, is about 40 m and Sterneck (1922) explained this remarkable phenomenon (Fig. 2). Because the phase speed of the long wave in the gulf in terms of the phase lag between the east-west natural Ð1 C = gH is about 20 m s , the wavelength of the semi- oscillation forced by the east-west component of tide-gen- diurnal tidal wave l = CT (T is the semi-diurnal pe- erating force and the north-south one forced by the north- M2 M2 M2 riod) is 890 km and that of diurnal one l is about 1,700 km. south component of tide-generating force in the Black Sea, K1 neglecting the Coriolis force. After high water along the east These are nearly one half of and the same as the length of the π coast, the high water occurs along the south coast of the gulf L, respectively. The inertia period Ti (=2 /f, f; the ω φ ω Black Sea because the natural oscillations in east-west and Coriolis parameter = 2 sin , ; the angular velocity of the φ ° north-south directions have the phase difference of π/2. The earth’s rotation, ; the latitude = 9 N in this case) of the gulf λ counterclockwise phase propagation of diurnal tides (K1 is 76.6 hours. The Rossby deformation length (= gH /f ) and O1) in the Black Sea is also explained by the same theory of the gulf is 870 km. (Sterneck, 1922). Semi-diurnal and diurnal tidal periods are much shorter The phase of diurnal tides (K1, O1 and P1) propagates than the inertia period and the width of the gulf is narrower counterclockwise at the central part of the Gulf of Thailand than the Rossby deformation length in the Gulf of Thailand. in the northern hemisphere, as shown in Fig. 1(b) but that of These facts suggest that the tidal phenomena are not seriously semi-diurnal tides (M2 and S2) propagates clockwise there affected by the Coriolis force in the Gulf of Thailand. as shown in Fig. 1(a) (Yanagi et al., 1997).The observed directions of the phase propagation of diurnal and semi- 3. Numerical Model diurnal tides are well reproduced by numerical experiments The horizontal two-dimensional momentum and con- (Yanagi and Takao, 1997). tinuity equations for tide and tidal current of a homogeneous In this paper we reveal the mechanism of the clockwise fluid under Cartesian coordinates are as follows;

143 Copyright  The Oceanographic Society of Japan. (a)

(b)

Fig. 1. Co-tidal and co-range charts of M2 (a) and K1 (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 M2 and K1 tides in the simplified gulf with flat bottom in the cases of no rotation (a) and rotation (b).

∂u γ 2 u u horizontal gradient operator, k the locally vertical unit vec- + ()u ⋅∇ u + fk × u =−g∇η − b + ν∇2u,1() ∂ + η tor, η the sea surface elevation from the mean sea surface, t H 2 6 γb (=0.0026) the bottom frictional coefficient, ν (=10 cm2sÐ1)the horizontal eddy viscosity and H the local water ∂η +∇×{}()H + η u = 0.() 2 depth. ∂t Equations (1) and (2) can be approximated by finite difference and solved by the primitive method. Numerical Here u is the depth averaged velocity vector, t the time, ∇ the 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 time step of the calculation is 2 minutes. The quasi-steady the horizontal and vertical geometry of the Gulf of Thailand, state is obtained in four tidal cycles after the beginning of the with a length of 1500 km, a width of 460 km and an open calculation and the harmonic analysis of sea surface elevation boundary along the eastern end. The grid size is 10 km × 10 and current field is carried out at the fifth tidal cycle. km. The uniform amplitude (15 cm for M2 tide and 35 cm for K1 tide) and phase lag (330 degree for M2 and K1 tides) are 4. Results given along the eastern open boundary on the basis of The calculated results in the case of a constant depth of observed amplitude and phase lag along the shelf edge of the 40 m and in the absence of the Coriolis force are shown in Gulf of Thailand shown in Fig. 1 (Yanagi et al., 1997). The Fig. 3(a). Two amphidromic points of the M2 tide and one of

146 T. Yanagi and T. Takao the K1 tide exist in the gulf. The northern amphidrome of the propagation of diurnal tide at the central part of the Gulf of M2 tide has a counterclockwise phase propagation, but the Thailand on the basis of the calculated results with simple southern one propagates clockwise. The phase of the K1 tide models shown in Figs. 3 and 4. N-S propagates clockwise in the central part of the gulf. Results The natural oscillation periods Tw i along the north- in the presence of the Coriolis force are shown in Fig. 3(b). south direction in the western part of the simplified gulf with The direction of phase propagation of the M2 tide in the the constant depth shown in Fig. 3 and those along the east- E-W southern part of the gulf changes from clockwise to coun- west direction in the northern part Tn i are calculated by terclockwise. The position of the amphidrome and the the following equation for a closed basin, direction of phase propagation of K1 tide change drastically from east to west and clockwise to counterclockwise, re- = 2Lb () spectively. Ti 3 igH 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 where i denotes the mode number, Lb (=880 km along the direction dominates for the M2 tide with no Coriolis force. north-south direction and 460 km along the east-west direc- The natural oscillation along the north-south direction tion) the length of the basin and g (=9.8 m sÐ2) the gravita- dominates for the K1 tide and those along the north-south tional acceralation. The natural oscillation periods along the and east-west directions couple for the M2 tide with the north-south direction in the western part are N-S N-S N-S Coriolis force. Tw 1 = 24.7 hours, Tw 2 = 12.4 hours and Tw 3 = 8.2 The results in the simplified gulf with a bottom slope hours, those along the east-west direction in the northern N-S N-S are shown in Fig. 4. The amphidromic point of the M2 tide part are Tw 1 = 12.9 hours and Tw 2 = 6.5 hours. E-W in the northern part of the gulf shown in Fig. 3(a) disappears The natural oscillation periods Ts i along the east- when the bottom slope is included, as shown in Fig. 4(a), but west direction in the southern part of the simplified gulf the result for the K1 tide shown in Fig. 3(a) is nearly the same shown in Fig. 3 are calculated by the following equation for as that shown in Fig. 4(a). When we include the Coriolis a semi-closed basin, force in Fig. 4(a), the clockwise amphidrome of the M2 tide shifts a little northward and another counterclockwise 4L T = b ()4 amphidrome appears at the head of the gulf. By including the i − ()2i 1 gH Coriolis force, the position of amphidrome and the direction of phase propagation of the K1 tide change drastically from east to west and clockwise to counterclockwise, respectively, where Lb (=880 km) denotes the bay length in the southern as shown in Fig. 4(b). The results with a constant and a part of the simplified gulf shown in Fig. 3. The natural oscillation periods along east-west direction in the southern variable Coriolis parameter are nearly the same, as shown in E-W E-W part are Ts 1 = 49.4 hours, Ts 2 = 16.5 hours, and Figs. 4(b) and 4(c), and they are qualitatively the same as the E-W observed one shown in Fig. 1, that is, the counterclockwise Ts 3 = 9.9 hours. Therefore only the semi-diurnal tide may resonate in phase propagation with large amplitude at the head of the N-S E-W the whole gulf (Tw 2 = 12.4 hours, Tn 1 = 12.9 hours, and gulf, the clockwise one and the large amplitude along the E-W eastern Malay coast at the central part of the gulf, the Ts 3 = 9.9 hours) and its oscillation mode is schematically counterclockwise one with large amplitude along the south- shown in Fig. 5(a), though the diurnal tide can resonate only ern coast of the Indo-China peninsula and along the northern along the north-south direction in the western gulf. The natural oscillation along the east-west direction in coast of Borneo at the southeastern part of the gulf for the M2 tide and the counter-clockwise phase propagation with large the western part dominates for the M2 tide in the case without the Coriolis force, as shown in Fig. 3(a). This is due amplitude along the head of the gulf for K1 tide at the central part of the gulf. to the fact that the incoming M2 tidal wave energy cannot We conducted other numerical experiments with transmit through the square bend of the L-shaped channel π π horizontal eddy viscosities of 105 cm2sÐ1 and 107 cm2sÐ1 and when kB = m (k = 2 /l and l = the wavelength, B = the width bottom drag coefficients of 0.001 and 0.004, but the results of the channel, and m = positive integer), which was dis- are nearly the same as Figs. 3 and 4. Numerical experiments cussed theoretically by Momoi (1974). In this case the wavelength l is 890 km, the width B 460 km and kB be- without the nonlinear term give similar results (not shown M2 π here). comes nearly . The incoming M2 tidal wave mainly reflects at the western wall in the southern part and only the natural 5. Discussions oscillation along the east-west direction is dominant. We investigate the mechanism of clockwise phase The direction of phase propagation of the natural os- propagation of semi-diurnal tide and counterclockwise phase cillation 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 (a) boundary. When the M2 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 (b) 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 M2 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 north- Fig. 5. Schematic representation of oscillation mode in the south and east-west directions couple in the western gulf as simplified gulf with flat bottom (a) and sloping bottom (b). 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 reflected at the northern and western walls in the western the right-hand side of its propagation direction. gulf and the natural oscillations along the east-west direc- When the simplified gulf has a bottom slope, the tion and along the deepest bottom line are generated as natural oscillation along the east-west direction in the north- shown in Figs. 4(a) and 5(b). The propagating M2 tidal wave ern part does not change, but those along the north-south along the northern wall decreases its amplitude along the direction in the western part and along the east-west direc- eastern wall in the western gulf because it corresponds to the tion in the southern part couple and the natural oscillation node of natural oscillation along the deepest bottom line. On along the deepest bottom line dominates. The natural oscil- the other hand, the amplitude decreasing ratio of the propa- lation periods Tdi along the deepest bottom line in the gulf gating M2 tidal wave along the southern wall is not so large shown in Fig. 4 are calculated by Eq. (4) with Lb = 1200 km as that of another propagating wave along the northern wall. and H = 40 m (average depth): Td1 is 67.3 hours, Td2 = 22.4 Therefore the phase propagation direction of the central hours, Td3 = 13.5 hours and Td4 = 9.6 hours. Therefore only amphidrome of the M2 tide becomes clockwise, as shown in the semi-diurnal tide may also resonate in the whole gulf Fig. 4(a). E-W (Tn 1 = 12.9 hours and Td3 = 13.5 hours) as shown in Fig. With the Coriolis force added to Fig. 4(a), the M2 tidal 5(b), though the diurnal tide can resonate only along the wave principally behaves as an edge wave because the effect deepest bottom line (Td2 = 22.4 hours). of Coriolis force can be neglected in the following disper- The incoming tidal wave has the characteristic of an sion relation of inertiogravitational edge wave (Kajiura, edge wave in this case, and the dispersion relation is as 1958; Reid, 1958) due to the high ratio of 2π/( fáTn) (which follows (Ursell, 1952), is about 6).

2 π 3 2 π φ π (2π/Tn) = g(2π/ln)sin(2n + 1)φ (5) (2 /Tn) Ð {f + (2n + 1)g(2 /ln)tan }(2 /Tn) + g(2π/ln) tanφáf = 0. (6) where T denotes the period, n the mode number, l the wavelength and φ the angle of bottom slope. When we give The co-tidal and co-range charts of the M2 tide with the φ = 0.016 degree from the bottom slope shown in Fig. 4, we Coriolis force shown in Fig. 4(b) are nearly the same as those understand that n = 0, T0 = 12.4 hour and l0 = 877 km satisfy without the Coriolis force shown in Fig. 4(a), except that the dispersion relation (5). This means that the incoming M2 new amphidromes appear in the northwestern and south- tidal wave from the eastern open boundary acts as an edge eastern parts and the amplitude along the northern wall is wave with its energy mainly along the coast. The propagating greater than the southern wall. The decisive mechanism edge waves along the northern and southern walls are 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 M2 and K1 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) is a node of natural oscillation along the north-south direc- mentioned earlier. tion. On the other hand, it propagates counterclockwise The phase of diurnal tide, which resonates only along along the northern and eastern walls with the Coriolis force the north-south direction in the western gulf, propagates due to the Kelvin wave’s characteristic of a K1 tidal wave, clockwise without the Coriolis force as shown in Figs. 3(a) as shown in Figs. 3(b), 4(b), and 4(c). This is because the and 4(a). This is due to the fact that the amplitude of the effect of the Coriolis force on the K1 tidal wave is larger than propagating K1 tidal wave along the northern wall becomes that on the M2 tidal wave due to its longer period. small at the northeastern square bend of the gulf because it 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 hori- Acknowledgements zontal scale and depth for resonance of a diurnal period. We The authors express their sincere thanks to Drs. H. carried out another numerical experiment in the simplified Takeoka and K. Ichikawa for their fruitful discussions and gulf with different dimensions from Figs. 3 and 4 without the two anonymous reviewers for their useful comments on the Coriolis force, using the same parameters. The results are first draft. A part of this study was supported by the research shown in Fig. 6. The phase of diurnal tide propagates fund of the Ministry of Education, Science, Culture and clockwise in the central part of the gulf with bottom slope, Sports, Japan. as shown in Fig. 6(b), though the phase propagation of the semidiurnal tide is very complicated. Systematic natural References oscillations along the east-west direction are generated for Kajiura, K. (1958): Effect of Coriolis force on edge waves (II) the M2 tide without bottom slope, as shown in Fig. 6(a). This Specific examples of free and forced waves. J. Mar. Res., 16, 145Ð157. is due to the fact that the incident M2 tidal wave energy cannot transmit through the square bend as discussed earlier, and Momoi, T. (1974): A long wave in an L-shaped channel. J. Phys. Earth, 22, 395Ð414. only the natural oscillation along the east-west direction Reid, R. O. (1958): Effect of Coriolis force on edge waves (I) dominates. In the case with bottom slope, the energy of the Investigation of the normal modes. J. Mar. Res., 16, 109Ð144. incident M2 tidal wave transmits through the square bend as Sterneck, R. V. (1922): Schematische Theorie der Gezeiten des the edge wave and the natural oscillation along the north- Schwarzen Meeres. S.B. Akad. Wiss. Wien. (Math.-Naturwiss, south direction couples with that along the east-west direction K1.), 131, 81. as shown in Fig. 6(b). Taylor, G. I. (1920): tidal oscillations in gulfs and rectangular The results of the K1 tide shown in Figs. 6(a) and 6(b) basins. Proc. Lond. Math. Soc., 20, 148Ð181. are nearly the same as those of the M2 tide shown in Figs. Ursell, F. (1952): Edge waves on a sloping beach. Proc. Roy. Soc., 3(a) and 4(a), respectively. In this simplified gulf, only the A214, 79Ð97. diurnal tide can resonate at the lower mode because Yanagi, T. and T. Takao (1997): A numerical experiment on the E-W N-S E-W tide and tidal current in the South China Sea. 9th JECSS-PAMS Tn 1 = 25.0 hours, Tw 2 = 23.9 hours, Ts 3 = 19.1 hours Proceedings. and T = 21.2 hours in the whole gulf. d4 Yanagi, T., T. Takao and A. Morimoto (1997): Co-tidal and co- These results suggest that the phase of diurnal tide may range charts in the South China Sea derived from satellite propagate clockwise at some coastal seas in the northern altimetry data. La mer, 35, 85Ð94. hemisphere with suitable horizontal and vertical scales for the resonance of a diurnal period.

150 T. Yanagi and T. Takao