The tides in the Taiwan Strait (TS) feature large
semidiurnal lunar (

The Taiwan Strait (TS) is the sole passage connecting the East China Sea (ECS) and the South China Sea (SCS). The strait is approximately 350 km long, 200 km wide, and located mostly on the continental shelf with a mean depth of approximately 50 m. The bottom topography of the TS can be viewed as the extension of the ECS shelf in the north and becomes irregular in the south. The SCS deep basin is located south of the strait and is connected to the Pacific Ocean through the Luzon Strait (LS). An abrupt depth change is present between the TS and the SCS deep basin (Fig. 1).

The tides in the strait feature large

The existing studies almost all employed data analysis and numerical modelling, except that some simple dynamical analyses were performed using one-dimensional solutions to explain the model results by Jan et al. (2002) and Yu et al. (2015). The purpose of the present study is to establish two-dimensional analytical models using an extended Taylor method (see Sect. 2 for details). In the analytical models, the classical Kelvin waves and Poincaré modes in idealized basins are used to approximately represent the tides in the natural basin. This enables us to estimate the strengths of the southward and the northward waves to reveal the role of each classical wave in the formation of the tides in the strait and to clarify how the waves are generated. In particular, we can roughly estimate the relative importance of the reflected wave at the steep topography versus the incident wave from the LS in the formation of the northward Kelvin wave in the TS.

The Taylor problem is a classical tidal dynamic problem (Hendershott and
Speranza, 1971). Since his pioneering work, Taylor's method has been
subsequently developed and applied to many sea areas (e.g. Table 1 of Roos
et al., 2011). In the previous applications, most of the studied basins have
a closed end that can almost perfectly reflect the incident tidal wave, thus
closely retaining the phase of the tidal elevation. In contrast, the
topographic step south of the TS acts as a permeable interface that can only
partially reflect the incident wave, and furthermore, the elevation phase of
the reflected wave is changed by nearly 180

Bathymetric chart of the Taiwan Strait and its neighbouring area; the rectangle indicates the idealized model basin representing the Taiwan Strait. Isobaths are in metres (based on ETOPO1 from the US National Geophysical Center).

The

Taylor (1922) first presented an analytical solution for tides in a semi-infinite rotating rectangular channel of uniform depth to explain the existence of amphidromic systems in gulfs. His solution showed that the tide in such a channel can be represented by the superposition of an incident Kelvin wave, a reflected Kelvin wave, and a family of Poincaré modes trapped near the closed end. In 1925, Defant simplified Taylor's solution approach by applying the collocation method (see Defant, 1961, pp. 213–215). In the original version of Taylor's problem, as well as Defant's approach, the friction and open boundary condition were left out of consideration. Fang and Wang (1966) and Rienecker and Teubner (1980) extended the Taylor problem by taking friction into consideration in the governing equations. The introduction of friction can explain why the amphidromic point in the Northern Hemisphere shifts from the central axis toward the right, as seen from the closed end and looking seaward. The mechanism of the shift of the amphidromic point was also explained by Hendershott and Speranza (1971), in which the dissipation was assumed to occur at the closed end of the basin rather than during the wave propagation. Fang et al. (1991) further extended the Taylor problem by introducing the open boundary condition, enabling solutions accounting for the finite length of the basin. Jung et al. (2005), Roos and Schuttelaars (2011), and Roos et al. (2011) further extended the Taylor method to model tides in multiple rectangular basins. The solution method used in the present study is basically the same as Fang et al. (1991), but with minor correction and generalization, as was done in studies of Jung et al. (2005), Roos and Schuttelaars (2011), and Roos et al. (2011). The analytical method initiated by Taylor and developed afterward is called an extended Taylor method in this paper.

The governing equations used in this study are as follows:

Considering a rectangular basin with two parallel sidewalls of length

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

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

The collocation method is convenient when determining the coefficients

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

In this study, the families of Poincaré modes are truncated at

Tidal system charts for the

The obtained analytical solution of the

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

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

Southward

In the preceding section, we have shown that the northward Kelvin wave is weaker than the southward wave on average, but they have a similar magnitude along the Taiwan coast. In this section, we will examine the formation mechanism of the northward Kelvin wave. There are two possible origins for the northward Kelvin wave in the TS. One is the reflection of the southward wave at the sharply deepened topography and another is an incident wave from the LS propagating toward the TS. In the following, we examine their respective contributions by using the extended Taylor models.

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

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

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

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

Model domain and boundary conditions of Ex. 1

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.

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

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

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

In the present study, we first established an analytical model for the

The

Inclusion of the Poincaré modes in the analytical model improves the model results: the east to west increase in amplitudes along the northern and southern openings is better reproduced; and in particular, the Poincaré modes make approximately the same contribution as the Kelvin waves to the cross-strait asymmetry in the anti-nodal band.

The reflection of the southward wave at the abruptly deepened topography
south of the TS is a major contribution to the formation of the northward
propagating wave in the strait. However, the reflected wave is slightly
weaker than that obtained from the analytical solution with open boundary
conditions determined by the observations. Inclusion of the tidal forcing at
the LS strengthens the northward Kelvin wave in the TS and thus improves the
model result. This indicates that the LS forcing is of some (but lesser)
importance to the

The analytical solutions can help us to understand the dynamics of tidal motion in the TS, but there are some limitations. For example, the LS is located on the east side of the study area, while the Taylor model does not allow for a forcing on the sidewalls, and thus we are bound to let a part of southern opening represent the LS (Fig. 7a). In addition, we have assumed that the water depth changes from 52 to 1000 m immediately at the connecting cross section without considering the existence of the continental slope at that location. The obliqueness of the orientation of the topography step relative to the cross-strait direction is also ignored. These approximations will induce uncertainty in the results for the magnitude of the reflected wave.

The ETOPO1 data (doi: 10.7289/V5C8276M) is available online at

The authors declare that they have no conflict of interest.

This study was supported by the NSFC-Shandong Joint Fund for Marine Science Research Centers (grant no. U1406404), the National Natural Science Foundation of China (grant no. 41706031), the Basic Scientific Fund for National Public Research Institutes of China (grant no. 2014G15), and the National Key Research and Development Program of China (grant no. 2017YFC1404201). The authors sincerely thank John M. Huthnance and two anonymous referees for their constructive comments and suggestions, which are of great help in improving our study. Edited by: John M. Huthnance Reviewed by: two anonymous referees