2.1 Governing equations and boundary conditions

The governing equations used in this study are as follows:

where t represents time; (x,y) are the Cartesian coordinates; ($\stackrel{\mathrm{̃}}{u},\stackrel{\mathrm{̃}}{v}$) are the depth-averaged velocity components in the (x,y) directions, respectively; $\stackrel{\mathrm{̃}}{\mathit{\zeta }}$ is the tidal elevation; h is the water depth, assumed uniform; γ is the frictional coefficient, taken as a constant; g=9.8 ms⁻² is the acceleration due to gravity; and f is the Coriolis parameter, also taken as a constant due to the smallness of the study area. The set of equations in Eq. (1) are two-dimensional linearized shallow water equations on an f plane with the momentum advection neglected. The equations are the same as those used in the work of Taylor (1922), except that the bottom friction is incorporated, as in Fang and Wang (1966) and Rienecker and Teubner (1980). When a monochromatic wave is considered, $\left(\stackrel{\mathrm{̃}}{\mathit{\zeta }},\stackrel{\mathrm{̃}}{u},\stackrel{\mathrm{̃}}{v}\right)$ can be expressed as follows:

where $\left(\mathit{\zeta },u,v\right)$ are complex amplitudes of $\left(\stackrel{\mathrm{̃}}{\mathit{\zeta }},\stackrel{\mathrm{̃}}{u},\stackrel{\mathrm{̃}}{v}\right)$, respectively, σ is the angular frequency of the wave, and $i\equiv \sqrt{-\mathrm{1}}$. For this wave, the set of equations in Eq. (1) reduce as follows:

in which $\mathit{\mu }=\frac{\mathit{\gamma }}{\mathit{\sigma }}$ and $\mathit{\nu }=\frac{f}{\mathit{\sigma }}$.

Considering a rectangular basin with two parallel sidewalls of length L and with a width B, we placed the x axis along a sidewall and the y axis perpendicular to the x axis and pointing to the other sidewall. Thus, the basin is confined by $x=\mathrm{0},L$ and $y=\mathrm{0},B$, respectively. The boundary conditions along the sidewalls are taken as follows:

within $x\in (\mathrm{0},L)$. Along the cross sections, x=0 and x=L, various choices of boundary conditions are applicable depending on the problem concerned:

if the cross section is a closed boundary;

if the free radiation in the positive/negative x direction occurs on the cross section;

if the tidal elevation is specified as $\widehat{\mathit{\zeta }}$ along the cross section;
and/or

if the cross section is a connecting boundary of two basins A and B, each with a different uniform depth of h_A and h_B.

Eq. (8) show the matching conditions accounting for sea level continuity and volume transport continuity, respectively. The individual conditions in Eqs. (5) to (8), or their combination, may be used as boundary conditions at the cross sections y=0 and B. The relationship between u and ζ shown in Eq. (6) is based on the solution for progressive Kelvin waves in the presence of friction, which will be given in Eqs. (9) and (10) below. If μ ≪ 1, a simpler equation, $u=\pm \sqrt{\frac{g}{h}}\mathit{\zeta }$, can be used to replace Eq. (6).

2.2 General solution and collocation method

The governing set of equations in Eq. (3) have only the following four forms satisfying the sidewall boundary condition of Eq. (4) (see Fang et al., 1991; an error in the equation for β in their paper has been corrected here. Note that the error occurred during the preparation of their manuscript and the correct expression was used in their computations):

and

where

in which $k=\mathit{\sigma }/\phantom{\mathit{\sigma }c}c$ is the wave number, with $c=\sqrt{gh}$ being the wave speed of the Kelvin wave in the absence of friction. In Eq. (16), s_n has two complex values for each n, and here, we choose the one that has a positive real part. To satisfy equations in Eq. (3), ($A_n,B_n,C_n,D_n)$ and $(A_n^{\prime },B_n^{\prime },C_n^{\prime },D_n^{\prime })$ should be as follows:

Equations (9) and (10) represent Kelvin waves propagating in the −x and x directions, respectively; Eqs. (11) and (12) represent two families of Poincaré modes trapped at the cross sections x=0, L, respectively. Coefficients $(a,b,{\mathit{\kappa }}_n,{\mathit{\lambda }}_n)$ are related to amplitudes and phases of Kelvin waves and Poincaré modes. These coefficients are to be determined by boundary conditions.

The collocation method is convenient when determining the coefficients $(a,b,{\mathit{\kappa }}_n,{\mathit{\lambda }}_n)$. The calculation procedure can be as follows. First, we truncate the family of Poincaré modes, Eqs. (11) and (12), at the Nth order, so that the number of undetermined coefficients for Poincaré modes is 2N and the total number of undetermined coefficients (plus those for Kelvin waves) is thus 2N+2. To determine these unknowns, we take equally spaced N+1 dots, called collocation points, located at $y=\frac{B}{\mathrm{2}(N+\mathrm{1})},\frac{\mathrm{3}B}{\mathrm{2}(N+\mathrm{1})},\mathrm{\dots },\frac{\left(\mathrm{2}N+\mathrm{1}\right)B}{\mathrm{2}(N+\mathrm{1})}$ on both cross sections x=0 and L. At these points, one of the boundary conditions given by Eqs. (5)–(8) should be satisfied. This yields 2N+2 equations. By solving this system of equations, we can obtain 2N+2 coefficients $(a,b,{\mathit{\kappa }}_n,{\mathit{\lambda }}_n)$. Since the high-order Poincaré modes decay from the boundary very quickly as can be seen from large s_n in Eqs. (11) and  (12), it is generally necessary to retain only a few lower order terms.

3.1 Model configuration and solution

In this section, we will first establish an idealized analytical model for the TS. The strait is idealized as a rectangular basin with two sidewalls roughly along the China mainland and Taiwan coastlines (as shown in Fig. 1). The width and length of the model domain are taken as B=200 km and L=330 km, respectively. The depth is taken as h=52 m, a mean depth calculated based on ETOPO1. We place the origin of the coordinates at the northernmost corner of the rectangle, the x axis along the mainland coast, and the y axis in an offshore direction. The axis of the strait is toward the south to southwest. However, to keep it short, we will hereafter simply use “south” to refer to “south to southwest”, and similarly for other directions. The Coriolis parameter f is taken as $\mathrm{0.594}\times {\mathrm{10}}^{-\mathrm{4}}$ s⁻¹, corresponding to a latitude of $\mathit{\phi }=\mathrm{24}{}^{\circ }$ N. The angular frequency of the M₂ tide is $\mathrm{1.4052}\times {\mathrm{10}}^{-\mathrm{4}}$ s⁻¹. The friction coefficient γ can be estimated from the relation $\mathit{\gamma }=C_{\mathrm{D}}\left(\frac{\mathrm{8}}{\mathrm{3}\mathit{\pi }}\right)\frac{U}{h}$, in which C_D and U represent the drag coefficient and amplitude of the M₂ tidal current, respectively (e.g. Chapter 8 of Dronkers, 1964). In this study, we take C_D=0.0026 and U=0.5 m s⁻¹ based on the numerical results of Fang et al. (1984), and then $\mathit{\mu }=\mathit{\gamma }/\mathit{\sigma }$ is approximately equal to 0.15. From these parameter values, we can obtain the wavelength of the M₂ Kelvin wave as 1009 km. Since the basin width is smaller than half of the Kelvin wavelength, the Poincaré modes can only exist in a bound form (Godin, 1965; Fang and Wang, 1966). The e-folding length of decay of the lowest Poincaré mode is approximately 63 km; that is, the amplitude of this mode reduces to approximately 37 % relative to its maximum value at a distance of 63 km away from the boundary. Equivalently, it may also reduce to approximately 20 % relative to its maximum value at a distance of 100 km. The length scales of decay for higher order Poincaré modes are even shorter.

In this study, the families of Poincaré modes are truncated at N=19 and 20 collocation points set along both the northern and southern open boundaries. The boundary condition (Eq. 7) is employed with the values of $\widehat{\mathit{\zeta }}$ equal to the observed harmonic constants from the global tide model DTU10 (Cheng and Anderson, 2011).

Figure 3Tidal system charts for the M₂ constituent: (a) Present analytical model; (b) Observed distribution based on DTU10; (c) Contribution of Kelvin waves; (d) Contribution of Poincaré modes. Solid lines represent Greenwich phase-lag (in degrees); dashed lines represent amplitude (in metres).

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The obtained analytical solution of the M₂ constituents is shown in Fig. 3a. For comparison, the observed M₂ tidal system chart based on DTU10 is also shown in Fig. 3b. Although the complicated bottom topography and the irregular coastlines are greatly simplified, the analytical model still agrees well with the observation. The observed tidal regime features significantly greater amplitudes along the mainland coast than along the Taiwan coast, showing the cross-strait asymmetry. The phase-lags near the mainland coast increase from north to south, showing a progressive wave nature, while those near the middle Taiwan coast have only small changes, showing a standing wave nature. That is, along the Taiwan coast the wave propagates southward in the northern area and propagates northward in the southern area. The largest amplitudes are roughly along the cross section of 150 km and appears as an anti-nodal band. The phase-lags in this band range from 80 to 90^∘. These features have all been reproduced in the analytical model.

3.2 Kelvin waves and Poincaré modes

To reveal the relative importance of the Kelvin waves and Poincaré modes in the model, the superposition of two Kelvin waves is given in Fig. 3c and that of the Poincaré modes is given in Fig. 3d. The contribution of the Poincaré modes is observed to be much smaller than that of the Kelvin waves. The tidal system chart constructed using only superposed Kelvin waves (Fig. 3c) resembles the complete model (Fig. 3a) and the observation (Fig. 3b) quite well, though the inclusion of the Poincaré modes improves the model to a certain degree. From Fig. 3a we can see that the difference between the highest amplitude on the west sidewall and that on the east sidewall in the anti-nodal band is approximately 0.4 m, while the corresponding difference shown in Fig. 3c is approximately 0.2 m. Thus, approximately half of the cross-strait asymmetry is explained by the superposition of two oppositely propagating Kelvin waves, with the southward one being stronger than the one moving northward. Here, both the Coriolis force and the weaker northward wave are the major factors. The superposition of Poincaré modes in this band has an amplitude of approximately 0.1 m on both sides and has nearly the same phase-lag as the superposed Kelvin wave on the west and a nearly opposite phase-lag to the superposed Kelvin wave on the east. Therefore, the superposed Poincaré modes play a role to increase the amplitudes in the west and reduce the amplitudes in the east and hence enhances the asymmetry. The superposed Poincaré modes make nearly the same contribution to the cross-strait asymmetry as the superposed Kelvin wave.

From the comparison, we find that the amplitude variation along the northern boundary in Fig. 3c is less than that in Fig. 3a. This shows that near the boundary, the Poincaré modes are of a certain importance. The existence of the Poincaré modes is related to the fact that the M₂ tide is from the Pacific Ocean; its amplitude increases from the deeper outer shelf toward the shallower inner shelf. This amplitude variation cannot be completely represented by the superposed Kelvin wave at a uniform depth, and superposed Poincaré modes are necessary to compensate for their difference. The situation at the southern boundary is similar. The distribution of the superposed Poincaré modes in the anti-nodal band is clearly related to those at the northern and southern openings (Fig. 3d). Yu et al. (2015) suggested that the orientation of the topographic step south of the strait was not perpendicular to the strait axis but has an angle. This might cause the reflected wave to propagate toward the mainland coast and thus amplify the tides there. The present solution indicates that the obliqueness of the topographic step south of the TS may also play a role in the formation of the cross-strait asymmetry, as suggested by Yu et al. (2015), but it seems not to be a controlling factor.

The obtained analytical solution enables us to see the magnitudes and characteristics of both the southward and northward Kelvin waves. These two oppositely propagating waves, which correspond to Eqs. (9) and (10), respectively, are displayed separately in Fig. 4a and b. From Fig. 4a, we see that the phase-lag of the southward wave increases from north to south. The amplitude deceases from north to south due to friction and from west to east due to the Coriolis effect. The characteristics of the northward wave are the opposite. The area mean amplitude of the southward wave is 1.18 m, while that of the northward wave is 0.84 m, smaller than the former by 0.34 m. Along the western sidewall, the amplitudes of the southward wave range from approximately 1.4 to 1.6 m, while those of the northward wave range from approximately 0.6 to 0.7 m; thus, the superposition of the waves is dominated by the former and appears as a southward progressive wave. Around the cross section x≈150 km, the phase-lags of the southward and northward waves are nearly equal, between 80 and 90^∘. Thus, the superposed tides here have the greatest amplitudes equal to the sum of the amplitudes of these two waves, exceeding 2.1 m, as already seen in Fig. 3c. Along the eastern sidewall, however, the differences in amplitudes of the southward and northward waves are much smaller, and thus the superposition of the waves tends to appear as a standing wave. Around the point x≈150 km, the phase-lags of the southward and northward waves are also nearly equal. Thus, the amplitude of the combined tide is also relatively large, equal to the sum of the amplitudes of these two waves; but now it is only slightly greater than 1.9 m, which is smaller than the corresponding value at the western sidewall.

Figure 4Southward (a) and northward (b) propagating Kelvin waves. Solid lines represent Greenwich phase-lag; dashed lines represent amplitude (in metres).

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4.1 Reflection of the incident wave from the East China Sea at the topographic step

Three experiments have been carried out to explore the formation mechanism of the northward Kelvin wave in the TS. The first experiment (denoted as Ex. 1) has the model geometry shown in Fig. 5a. The TS is represented by area A, with the width and depth equal to the above single area model. Since the topographic step is located away from the southern boundary of the single area model domain (Fig. 1), we extend the length of the area to 400 km. Area B represents the deep basin south of the topographic step, and the water depth of the deep basin is taken as 1000 m, as was done in Jan et al. (2002, 2004). The purpose of this experiment is to examine the effect of the topographic step in reflecting the incident wave from the ECS. The experimental design for area A is similar to that of Roos and Schuttelaars (2011): a southward Kelvin wave is specified to be identical to the single basin solution, as shown in Fig. 4a in the preceding section. The Poincaré modes trapped at the cross section x=0 are neglected, while those trapped at the cross section x=400 km are retained. The matching condition (Eq. 8) is applied at the connecting boundary of areas A and B, and the radiative condition (Eq. 6) is used at the southernmost opening.

Figure 5b displays the solution of Ex. 1. It can be seen that the basic pattern of the tidal regime is similar to that of the single area model solution shown in Fig. 3c. In particular, there is again an anti-nodal band near x=150 km, though the amplitudes in this band produced by this experiment are smaller than those given in Fig. 3c. The smallest amplitudes appear along the connecting cross section, showing that a nodal band exists there. Therefore, the anti-node is located approximately 250 km away from the topographic step. The wavelength of the M₂ tide in a channel of a uniform depth of 52 m is equal to 1009 km, and so the distance between the anti-node and the topographic step is equal to one-quarter of the wavelength. This result further implies that if the channel were 500 km long, resonance would occur. However, Taiwan Island is approximately 380 km long and is not able to support a resonance for the M₂ constituent. In fact, the resonant period of the TS is 13.5 h, according to the experiments performed by Cui et al. (2015), which is almost the same as one of the resonant periods of the ECS (13.7 h, obtained by Cui et al., 2015). This means that the tidal response in the TS is not independent, but rather closely related to the tides in the ECS.

The southward and northward Kelvin waves obtained from Ex. 1 are shown in Figs. 5c and d, respectively. Comparison of these figures with Figs. 4a and b indicates that in area A, the southward wave is identical, but the northward wave from Ex. 1 is weaker. For the area x=0 to 330 km and y=0 to 200 km, the area mean amplitude of the northward Kelvin wave is 0.57 m, which is smaller than the single area model value by 32 %. In area B, the amplitudes of the transmitted southward Kelvin wave are approximately 0.4 m, and those of the northward wave are negligible. An important difference in the co-phase-lag distributions is that Figs. 3a–c show a northward propagation along the southern part of the eastern sidewall, while Fig. 5b does not have such a feature. This is because in the single area case, the amplitudes of the northward Kelvin wave are greater than those of the southward Kelvin wave in this area (Fig. 4a, b), while in Ex. 1, this situation does not occur (Fig. 5c, d).

The relative magnitudes of the incident and the reflected and transmitted Kelvin waves can be evaluated by comparing their amplitudes along the connecting cross section at x=400 km. The sectional mean amplitudes for the incident, reflected, and transmitted waves, H_i, H_r, and H_t, are 1.06, 0.64, and 0.40 m, respectively (Figs. 5c, d). Thus, the ratios H_r∕H_i and H_t∕H_i are equal to 0.61 and 0.37, respectively. The corresponding values based on the theory ignoring the earth's rotation can be calculated from $\frac{H_{\mathrm{r}}}{H_{\mathrm{i}}}=\frac{\mathrm{1}-\mathit{\rho }}{\mathrm{1}+\mathit{\rho }}$ and $\frac{H_{\mathrm{t}}}{H_{\mathrm{i}}}=\frac{\mathrm{2}\mathit{\rho }}{\mathrm{1}+\mathit{\rho }}$ with $\mathit{\rho }=\sqrt{h_{\mathrm{A}}/h_{\mathrm{B}}}$ (e.g. Dean and Dalrymple, 1984, p. 144). Substitution of the present model depths into these equations yields $H_{\mathrm{r}}/H_{\mathrm{i}}=\mathrm{0.63}$ and $H_{\mathrm{t}}/H_{\mathrm{i}}=\mathrm{0.37}$. This indicates that the magnitude of the reflected waves in the two-dimensional case with the earth's rotation being taken into account is smaller than that based on the theory with the earth's rotation being ignored.

Figure 5Model domain and boundary conditions of Ex. 1 (a); solution of Ex. 1 (b); southward Kelvin waves (c); northward Kelvin waves (d). Solid lines represent Greenwich phase-lag (in degrees); dashed lines represent amplitude (in metres).

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4.2 Influence of the shelf region southwest of the Taiwan Strait

From Fig. 1, we can see that there is a narrow shelf along the mainland coast. To simulate the effect of the narrow shelf on the tides in the TS, we performed a second experiment, numbered Ex. 2. In this experiment, the deep basin has moved 60 km eastward, allowing the tides in the shallow basin to freely radiate southward as shown in Fig. 6a. The radiative condition (Eq. 6) is retained along the southernmost opening. The results of Ex. 2 are given in Fig. 6. It can be seen that the tides in area A have only small changes, though the deep basin has moved 60 km eastward. Observable changes can only be found in area B where the tidal amplitudes are slightly reduced.

Figure 6Same as Fig. 5, but for Ex. 2.

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4.3 Influence of the Luzon Strait forcing

The purpose of performing a third experiment, numbered Ex. 3, is to consider the tidal input from the LS. The major difficultly in including the LS input in Taylor's model for the TS is that the LS has a meridional orientation, while Taylor's model does not allow any part of the sidewalls to open. Here, we will use a rather crude model to solve this issue. We use the same model domain as Ex. 2, but the radiative boundary condition (Eq. 6) is retained only for the west segment of the southernmost opening, and the boundary condition (Eq. 7) is applied to the remaining east segment of the opening. From Fig. 1, we can see that the cross section from the mainland shelf to the LS is much longer than the width of the LS. Thus, in our model, we take the lengths of the west and east segments to be 120 and 80 km, respectively, as shown in Fig. 7a. In addition, from Fig. 2, we observe that the tidal amplitude along the LS is roughly 0.2 m, and the phase-lag is approximately 310^∘. Since a significant portion of the incident wave from the LS propagates toward the SCS deep basin (e.g. Fang et al., 1999; Yu et al., 2015), we use a 0.1 m amplitude and 310^∘ phase-lag as an open boundary condition for the east segment of the southernmost opening in Ex. 3. The model results are given in Fig. 7b to d. From Fig. 7b, we can see that the amplitudes of the tide in area A now become greater than the results of Ex. 2 (Fig. 6b), and a northward propagating character can be seen in the south-eastern portion of area A. These improvements can be attributed to the increased amplitudes of the northward Kelvin wave (Figs. 6d, 7d).

Figure 7Same as Fig. 5, but for Ex. 3.

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