An analytical study on M 2 tidal wave in the Taiwan Strait with the extended Taylor ’ s method

The tides in the Taiwan Strait (TS) are featured by large M2 amplitudes. The extended Taylor’s method is employed in this study to provide an analytical model for the M2 tide in the TS. The strait is idealized as a rectangular basin with a uniform depth, but the Coriolis and friction forces are retained in the governing equations. The observed tides at the 10 northern and southern openings are used as open boundary conditions. The obtained analytical solution, which consists of a stronger southward propagating Kelvin wave, a weaker northward propagating Kelvin wave, and two families of Poincaré modes trapped at the northern and southern openings, agrees well with the observations in the strait. The superposition of two Kelvin waves can basically represent the observed tidal pattern, including an anti-nodal band in the central strait, and the cross-strait asymmetry (greater amplitudes in the west and smaller in the east) of the anti-nodal band. The superposition of 15 Poincaré modes can further improve the model result in that the cross-strait asymmetry can be better reproduced. In order to explore the formation mechanism of the northward propagating wave in the TS three experiments are carried out, including the deep basin south of the strait. The results show that the southward incident wave can be reflected to form a northward wave by the abruptly deepened topography south of the strait, but the reflected wave is slightly weaker than the northward wave obtained from above analytical solution, in which the southern open boundary condition is specified with observations. 20 The forcing at the Luzon Strait can strengthen the northward Kelvin wave in the TS, and thus is of secondary importance to the M2 tide in the TS.


Introduction
The Taiwan Strait (TS) is the sole passage connecting the East China Sea (ECS) and South China Sea (SCS).The strait is about 350 km long, 200 km wide and mostly located on the continental shelf with a mean depth of about 50 metres.The 25 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 M2 amplitudes.The greatest amplitude by tidal gauge observation along the western Ocean Sci.Discuss., https://doi.org/10.5194/os-2017-61Manuscript under review for journal Ocean Sci. Discussion started: 4 August 2017 c Author(s) 2017.CC BY 4.0 License.
encountering the sharply deepened bottom topography south of the strait, and suggested that the reflected wave was the main component of the northward propagating wave, and the contribution of the SCS was negligible.Yu et al. (2015) made an extensive numerical study on the formation of the M2 tide in the strait with special focus on the asymmetric nature in the cross-strait direction.15 The existing studies almost all employed data analysis and numerical modeling, except that some simple dynamical analyses were performed using one-dimensional solutions to explain the model results in Jan et al. (2002) and Yu et al. (2015).The purpose of the present study is to establish two-dimensional analytical models using the extended Taylor's method.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, and thus enable us to estimate the strengths of the southward and the 20 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 steep topography versus the incident wave from the LS in the formation of the northward Kelvin wave in the TS.
Taylor's problem is a classical tidal dynamic problem.Since his pioneering work Taylor's method has been subsequently developed and applied to many sea areas (e. g., Roos and Velema, 2011, Table 1).In the previous applications all the studied 25 basins have a closed end that can almost perfectly reflect the incident tidal wave closely retaining the phase of the tidal elevation.In contrast, the topographic step south of the TS acts as a permeable interface which can only partially reflect the incident wave, and furthermore, the elevation phase of the reflected wave is changed by nearly 180° at the step.Therefore, the strait is also a locality of particular interest for the application of the Taylor's method.Solution method used in the present study is basically the same as Fang et al. (1991), but with minor correction and generalization as done in studies of Jung et al. (2005) and Roos and Velema (2011).The analytical method initiated by Taylor and developed afterward is called the extended Taylor's method in this paper.

Governing equations and boundary conditions 10
The governing equations used in this study are as follows: where  represents time; (, ) the Cartesian coordinates; ( ̃,  ̃) the velocity components in (, ) directions;  ̃ the tidal elevation; ℎ the water depth, assumed uniform;  the frictional coefficient, taken constant;  = 9.8 ms −2 is the acceleration due to gravity;  is the Coriolis parameter, also taken as a constant due to smallness of the study area.When 15 one cosine wave is considered, ( ̃,  ̃,  ̃) can be expressed as where  is the angular frequency of the wave, i≡ √−1.For this wave Eqs.
Eqs. ( 8) are matching conditions accounting for sea level continuity and volume transport continuity respectively.The 5 individual equations ( 5) to ( 8) or their combination may be used as boundary conditions on the cross sections.Strictly speaking, Eq. ( 6) is valid only in the frictionless case, if friction is considered it contains an error of the order of  2 , and there is a phase difference between u and  (Fang and Wang, 1966).In the present study we still use the form of Eq. ( 6) due to smallness of the value of .

General solution and collocation method 10
The governing equations (3) have only the following four forms satisfying the sidewall boundary condition (4) (see Fang et al., 1991, an error in the equation for β in their paper has been corrected here): where (14) 20 in which  =   ⁄ is the wave number of the Kelvin wave in absence of friction and  = √ℎ is the wave speed.In Eq. ( 1−i  2 ;   has two complex values and here we choose the one that has a positive real part.In order to satisfy Eqs. ( 3 ′ = −  (21) 5 Eqs. ( 9) and ( 10) represent Kelvin waves propagating in the − and  directions respectively; Eqs. ( 11) and ( 12) represent two families of Poincaré modes trapped at the cross sections  = 0,  respectively.(, ,   ,   ) are coefficients related to 10 amplitudes and phases of Kelvin waves and Poincaré modes.These coefficients are to be determined by boundary conditions.
The collocation method is convenient in determining the coefficients (, ,   ,   ).The calculation procedure can be as follows.First, we truncate the family of Poincaré modes, Eqs. ( 11) and ( 12), at the N-th order, so that the number of undetermined coefficients for Poincaré modes is 2, the total number of undetermined coefficients (plus those for Kelvin 15 waves) is thus 2 + 2. To determine these unknowns we take  + 1 equally spaced dots, called collocation points, located on both the cross sections  = 0 and .At these points one of the boundary conditions given by Eqs. ( 5)-( 8) should be satisfied.This yields 2 + 2 equations.By solving this system of equations we can obtain 2 + 2 coefficients (, ,   ,   ).Since the high order Poincaré modes decay from the boundary very fast (e. g., Godin, 1965), it is generally necessary to retain only a few lower order terms.20

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 25 based on ETOPO1.We put the origin of the coordinates at the northernmost corner of the rectangle,  axis along the mainland coast, and  axis in offshore direction.The axis of the strait is toward the south by southwest.But for short we will hereafter simply use "south" to refer "south by southwest", and similarly for other directions.The Coriolis parameter   The obtained analytical solution of M2 constituents is shown in Fig. 3a.For comparison the observed M2 cotidal chart 10 based on DTU10 is also shown in Fig. 3b.It can be seen that 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 has the features that the amplitudes are significantly greater 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, the wave 15 propagates southward in the northeast area and propagates northward in the southeast area.A highest amplitude band is present roughly along the cross section of  ≈150 km, appearing 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.

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 20 waves is given in Fig. 3c, and that of Poincaré modes is given in Fig. 3d.It can be observed that the contribution of the Poincaré modes is much smaller than that of Kelvin waves.The cotidal chart constructed using superposed Kelvin wave alone (Fig. 3c) resembles the complete model (Fig. 3a) and the observation (Fig. 3b) quite well, though the Poincaré modes can improve the model to a certain degree.From Fig. 3a we can see that the difference between highest amplitude on the Ocean Sci.Discuss., https://doi.org/10.5194/os-2017-61Manuscript under review for journal Ocean Sci. Discussion started: 4 August 2017 c Author(s) 2017.CC BY 4.0 License. to 0.7 m, so that the superposition of them is dominated by the former and appears as a southward progressive wave.Around the cross section  ≈150 km, the phase-lags of the southward and northward waves are nearly equal, between 80°and 90°, thus the superposed tides have greatest amplitudes, equal to the sum of the amplitudes of these two waves, which exceeds 2.1 m as already seen in Fig. 3c.Along the eastern sidewall, however, the differences of amplitudes of southward and northward waves are much smaller, so that the superposition of them tends to appear as a standing wave.Around the point 30  ≈150 km, the phase-lags of the southward and northward waves are also nearly equal, thus the amplitude of 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,

Formation mechanism of the northward Kelvin wave in the Taiwan Strait 5
There are two possible origins for the northward Kelvin wave in the TS.One is reflection of the southward wave at sharply deepened topography; another is an incident wave from the Luzon Strait propagating toward the TS.In the following we examine their respective contributions by using the extended Taylor's models.

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 10 first experiment (denoted as Ex. 1) has a 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.The area B represents the deep basin south of the topographic step, and the water depth of the deep basin is taken 1000 m as done in Jan et al. (2002 and2004).The purpose of this experiment is to examine the effect of the topographic step in reflecting the incident wave from the ECS.As in the 15 single basin solution, the open boundary condition ( 7) is used at the northern opening with values of  ̂ equal to the observations taken from the global tidal model DTU10.The matching condition ( 8) is applied at the connecting boundary of areas A and B, and the radiative condition ( 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. 3a.In particular, there is also an anti-nodal band near x = 150 km though the amplitudes in 20 this band produced by this experiment are smaller than those given in Fig. 3a.The smallest amplitudes appear along the connecting cross section, showing a nodal band exists there.Therefore, the anti-node is located about 250 km away from the topographic step.The wavelength of M2 tide in a channel of uniform depth 52 m is equal to 1009 km, so the distance between Ocean Sci.Discuss., https://doi.org/10.5194/os-2017-61Manuscript under review for journal Ocean Sci. Discussion started: 4 August 2017 c Author(s) 2017.CC BY 4.0 License.
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 about 380 km long, and is not able to support a resonance for M2 constituent.In fact, the resonant period of the TS is 13.5 h according to the experiments made by Cui et al. (2015), which is almost the same as one of the resonant periods of the ECS (13.7 h), meaning that the tidal response in the TS is not independent, but rather closely related to the tides in the ECS. 5 The southward and northward Kelvin waves obtained from Ex. 1 are shown in Figs.5c and 5d respectively.Comparison of these figures with Figs.4a and 4b 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, 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 about 0.4 m, and those of northward wave are negligible.An important difference in co-phase-lag distributions is that Figs. 10 3a-c show a northward propagation along the south part of the eastern sidewall, while Fig. 5b does not have such a feature.This is because in single area case the amplitudes of the northward Kelvin wave are greater than those of the southward Kelvin wave in this area (Figs. 4a,4b), while in Ex. 1 this situation does not occur (Figs.5c, 5d).
The relative magnitudes of the incident, 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 15 transmitted waves, Hi, Hr, and Ht, are 1.06, 0.63 and 0.40 m respectively (Figs. 5c, 5d).Thus the ratios Hr/Hi and Ht/Hi are equal to 0.60 and 0.38 respectively.The corresponding values based on the theory ignoring earth's rotation can be calculated with ρ = √ℎ  /ℎ  (e. g., Dean and Dalrymple, 1984, p. 144).Substitution of the present model depths into these equations yields Hr/Hi = 0.63 and Ht/Hi =0.37.This indicates that the magnitude of the reflected waves in two-dimensional case with the earth's rotation taken into account is smaller than that based on the theory with the earth's 20 rotation ignored.

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 5 tides in the TS we carry out the second experiment, numbered Ex. 2. In this experiment the deep basin is moved 60 km eastward, allowing the tides in the shallow basin radiate freely southward as shown in Fig. 6a.The radiative condition ( 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 is moved 60 km eastward.Observable changes can only be found in area B where the tidal amplitudes are slightly reduced.

Influence of the Luzon Strait forcing
The purpose of making the third experiment, numbered Ex.3, is to consider tidal input from the LS.The major difficultly in including the LS input in the Taylor's model for the TS is that the LS has a meridional orientation, while the Taylor's model 5 does not allow to open any part of sidewalls.Here we will use a rather crude model to simulate this issue.We still use the same model domain as Ex. 2, but the radiative boundary condition ( 6) is retained only for the west segment of the southernmost opening, and the boundary condition ( 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 km and 80 km respectively as shown in Fig. 7a.In addition, from Fig. 2 we 10 observe that the tidal amplitude along the LS is roughly about 0.2 m, and the phase-lag is about 310°.Since a significant part of 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 Figs.7b to 7d.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 southeastern portion of area A. 15 These improvements can be attributed to the increased amplitudes of the northward Kelvin wave (Figs.6d, 7d).

Summary and discussion
In the present study we first establish an analytical model for M2 tide in the TS using the extended Taylor' method.The superiority of the analytical solution is that the tides can be decomposed into a southward Kelvin wave, a northward Kelvin 5 wave, and two families of Poincaré modes, providing a deeper insight into the dynamics of the tides in the area.Though the coastlines and bottom topography are greatly simplified the model-produced pattern resembles the observed tidal regime quite well.We then carry out several experiments to examine the formation mechanism of the northward propagating wave, especially the roles of the abruptly deepened bottom topography south of the TS and the tidal forcing in the LS in the formation of the northward wave.From this study we obtain the following results.10 The M2 tide in the TS can be basically represented the superposition of a southward propagating and a northward propagating Kelvin waves, with the former stronger than the latter.The superposed Kelvin waves give an anti-nodal band near the cross-strait transection roughly from Haitan Island to Taichung.The maximum amplitude on the mainland side is greater than that on the Taiwan side, showing the cross-strait asymmetry.Therefore, the observed features can be reproduced by the composition of a stronger southward propagating and a weaker northward propagating Kelvin wave.In this regard, 15 the Coriolis force and friction play essential roles.
Inclusion of the Poincaré modes into the analytical model can improve the model result: the east to west increase in amplitudes along the northern and southern openings can be better reproduced; and in particular, the Poincaré modes have 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, though the reflected wave is slightly weaker than that obtained from the analytical solution with open boundary conditions determined by observations.Inclusion of the tidal forcing at the LS can strengthen 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 TS tides.5 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 Taylor's model does not allow a forcing on the sidewalls, thus we are bound to let a part of southern opening represent the LS (Fig. 7a).In addition, we have assumed that the water depths change from 52 m to 1000 m immediately at the connecting cross section without considering the existence of continental slope there.The obliqueness of the orientation of the topography step relative to the cross-strait direction is also 10 ignored.These approximations will induce uncertainty in the result for the magnitude of reflected wave.

Figure 1 .
Figure 1.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).

Figure 2 . 5 TC
Figure 2. Cotidal chart of M2 constituent in the Taiwan Strait (based on DTU10, see Chen and Andersen, 2011).(MT-Matsu, HT-Haitan, 5 basin with two parallel sidewalls of length  and with a width of , we put the  axis along a sidewall and the  axis perpendicular to the  axis and pointing to another sidewall.Thus the basin is confined by  = 0,  and  = 0, , respectively.The boundary conditions along the sidewalls are taken  = 0 at  = 0 and  =  (4) within  ∈ (0, ).Along the cross sections  = 0 and  = , various choices of boundary conditions are applicable 25 depending on the problem concerned:  = 0, if the cross section is a closed boundary (5) Ocean Sci.Discuss., https://doi.org/10.5194/os-2017-61Manuscript under review for journal Ocean Sci. Discussion started: 4 August 2017 c Author(s) 2017.CC BY 4.0 License. the free radiation in the positive/negative  direction occurs on the cross section (6) ζ =  ̂, if the tidal elevation is specified as  ̂ along the cross section (7) and/or, ζ  =   and   ℎ  =   ℎ  , if the cross section is a connecting boundary of two basins  and .
is taken 0.594 × 10 −4 s −1 , corresponding to a latitude of  = 24°N.The angular frequency of M2 tide is 1.4052 × Ocean Sci.Discuss., https://doi.org/10.5194/os-2017-61Manuscript under review for journal Ocean Sci. Discussion started: 4 August 2017 c Author(s) 2017.CC BY 4.0 License. 10 −4 s −1 .The friction coefficient γ can be estimated from the relation γ =   (   and  represent the drag coefficient and amplitude of the M2 tidal current respectively.If we take   equal to 0.0026 and  equal to 0.5 m/s, then μ = γ/σ is approximately equal to 0.15.The families of Poincaré modes are truncated at  = 19, and 20 collocation points are set along both the northern and southern open boundaries.The boundary condition (7) is employed with the values of  ̂ equal to the observed harmonic constants from the global tide model DTU10 (Cheng and Anderson, 2011).