Study on the Tidal Dynamics of the Korea Strait Using the Extended Taylor Method

The Korea Strait (KS) is a major navigation passage linking the Japan Sea (JS) to the East China Sea and Yellow Sea. Almost all existing studies on the tides in the KS employed either data analysis or numerical modelling methods; thus, theoretical research is lacking. In this paper, we idealize the KS-JS basin as three connected uniform-depth rectangular areas 10 and establish a theoretical model for the tides in the KS and JS using the extended Taylor method. The model-produced K1 and M2 tides are consistent with the satellite altimeter and tidal gauge observations, especially for the locations of the amphidromic points in the KS. The model solution provides the following insights into the tidal dynamics. The tidal system in each area can be decomposed into two oppositely travelling Kelvin waves and two families of Poincaré modes, with Kelvin waves dominating the tidal system. The incident Kelvin wave can be reflected at the connecting cross-section, where abrupt increases 15 in water depth and basin width occur from the KS to JS. At the connecting cross-section, the reflected wave has a phase-lag increase relative to the incident wave by less than 180°, causing the formation of amphidromic points in the KS. The above phase-lag increase depends on the angular frequency of the wave and becomes smaller as the angular frequency decreases. This dependence explains why the K1 amphidromic point is located farther away from the connecting cross-section in comparison to the M2 amphidromic point. 20

first conducted a comprehensive study of the tides in the seas adjacent to Japan using data from the tidal stations along the coast and gained a preliminary understanding of the characteristics of the tides, including amphidromic systems in the KS. Since then, many researchers have investigated the tides in the strait via observations (Odamaki, 1989a;Matsumoto et al., 2000;Morimoto et al., 2000;Teague et al., 2001;Takikawa et al., 2003) and numerical simulations (Fang and Yang, 1988;Kang et al., 1991;Choi et al., 1999;Book et al., 2004). The results of these studies show consistent structures 5 of the tidal waves in the KS. Fig. 2 displays the distributions of the K1 and M2 tidal constituents based on the global tidal model DTU10, which is based on satellite altimeter observations (Cheng and Andersen, 2011). The figures show that the amplitudes of the diurnal tides are smaller than the semidiurnal tides. The peak amplitude of the semidiurnal tide appears on the south coast of South Korea, and lower amplitudes occur along the southern shore of the strait from the ECS to the JS. Distinguishing features include (1) K1 and M2 amphidromic points in the strait that appear in the northeast part of the KS close to the southern 10 coast of the Korean Peninsula; and (2) the M2 amphidromic point appears further northeast and closer to the shelf break relative to the K1 tide.
However, almost all previous studies have employed either data analysis or numerical modelling methods; thus, theoretical research is lacking. In particular, the existence of amphidromic points in the northeast KS for both diurnal and semidiurnal tides has not been explained based on geophysical dynamics. In this paper, we intend to establish a theoretical model for the https://doi.org /10.5194/os-2020-86 Preprint. Discussion started: 28 September 2020 c Author(s) 2020. CC BY 4.0 License.  https://doi.org /10.5194/os-2020-86 Preprint. Discussion started: 28 September 2020 c Author(s) 2020. CC BY 4.0 License.

The extended Taylor method and its application to multiple rectangular areas
The Taylor problem is a classic tidal dynamic problem (Hendershott and Speranza, 1971). Taylor (1922) first presented an analytical solution for tides in a semi-infinite rotating rectangular channel of uniform depth to explain the formation of amphidromic systems in gulfs and applied the theory to the North Sea. The classic Taylor problem was subsequently improved by introducing frictional effects (Fang and Wang, 1966;Webb, 1976;Rienecker and Teubner, 1980) and open boundary 5 conditions (Fang et al., 1991) to study tides in multiple rectangular basins (Jung et al., 2005;Roos and Schuttelaars, 2011; as well as to solve tidal dynamics in a strait (Wu et al., 2018).
The method initiated by Taylor and developed afterwards is called the extended Taylor method (Wu et al., 2018). This method is especially useful in understanding the tidal dynamics in marginal seas and straits because the tidal waves in these sea areas can generally be represented by combinations of the Kelvin waves and Poincaré waves/modes (e. g., Taylor, 1922;10 Fang and Wang, 1966;Hendershott and Speranza, 1971;Webb, 1976;Fang et al., 1991;Carbajal, 1997;Jung et al., 2005;Roos and Schuttelaars, 2011;Wu et al., 2018).

Governing equations and boundary conditions for multiple rectangular areas
A sketch of the model geometry is shown in Fig. 3, and it consists of a sequence of rectangular areas with length , width and uniform depth ℎ for the th rectangular area (denoted as Areaj, =1, …, ). For convenience, the shape of the study 15 region shown in Fig. 3 is the same as that for the idealized KS-JS basin, which will be described in the next section. In particular, Area1 represents the KS, which is our focus area in this study.

20
Consider a tidal wave of angular frequency and typical elevation amplitude H. We assume /ℎ ≪ 1 , and the conservation of momentum and mass leads to the following depth-averaged linear shallow water equations on the plane: https://doi.org /10.5194/os-2020-86 Preprint. Discussion started: 28 September 2020 c Author(s) 2020. CC BY 4.0 License.
where and are coordinates in the longitudinal (along-channel) and transverse (cross-channel) directions; represents time; � and � represent the depth-averaged flow velocity components in the and directions, respectively, with the subscript j indicating the area number; ̃ represents the free surface elevation above the mean level; represents the frictional coefficient, which is taken as a constant for each tidal constituent in each area; =9.8 ms −2 represents the 5 acceleration due to gravity; and represents the Coriolis parameter, which is also taken as a constant based on the average of the concerned area. The equations in (1) for each j are two-dimensional linearized shallow water equations on an -plane with the momentum advection neglected. For any , the equations are the same as those used in the work of Taylor (1922) except that bottom friction is now incorporated, such as in Fang and Wang (1966), Webb (1976), Rienecker and Teubner (1980), etc. When a monochromatic wave is considered, �̃, � , � � can be expressed as follows: 10 where Re stands for the real part of the complex quantity that follows, � , , � are referred to as complex amplitudes of �̃, � , � �, respectively, i≡√−1 is the imaginary unit, and is the angular frequency of the wave. For this wave, Eq. (1) can be reduced as follows: (4) Provided that the j-th rectangular area, denoted as Areaj, has a width of , a length of , and ranges from = to = +1 ( +1 = + ) in the x direction and from y = ,1 to y = ,2 ( ,2 = ,1 + ) in the y direction, the boundary conditions along the sidewalls within ∈ [ , +1 ] are taken as follows: 20 = 0 at = ,1 and = ,2 .
Along the cross-sections, such as = , various choices of boundary conditions are applicable depending on the problem: if the cross-section is a closed boundary; if the free radiation in the positive/negative direction occurs on the cross-section; if the tidal elevation is specified as ̂ along the cross-section; and if the cross-section is a connecting boundary of the areas and + 1, with each having a different uniform depth of ℎ and 30 ℎ +1 .
(6) to (9), or their combination, may be used as boundary conditions for the cross-sections. The relationship between and shown in Eq. (7) is based on the solution for progressive Kelvin waves in the presence of friction, which will be given in Eqs. (10) and (11) below.
(3) only have the 5 following four forms satisfying the sidewall boundary condition of Eq. (5) (see, e. g., Fang et al. 1991): and 10 where , , , and , are equal to the following: and in which = ⁄ is the wave number, with = � ℎ being the wave speed of the Kelvin wave in the absence of friction.
The parameters , in Eq. (17) are of fundamental importance in determining the characteristic of the Poincaré modes. If Re( 2 − 2 ) 1/2 < / , all Poincaré modes are bound in the vicinity of the open, connecting or closed cross-sections (see 20 Fang and Wang, 1966;Hendershott and Speranza, 1971 for in absence of friction); while if Re( 2 − 2 ) 1/2 > / , the nth and lower-order Poincaré modes are propagating waves. In the present study, the inequality Re( 2 − 2 ) 1/2 < / holds for both the idealized KS and JS, so that all Poincaré modes in the present study appear in a bound form. The parameter , https://doi.org/10.5194/os-2020-86 Preprint. Discussion started: 28 September 2020 c Author(s) 2020. CC BY 4.0 License.
has two complex values for each , and here,we choose the one that has a positive real part. To satisfy the equations in Eq.

Defant's collocation approach
The collocation approach was first proposed by Defant in 1925(see Defant, 1961, and is convenient in determining the coefficients ( , , , , , ). In the simplest case, that is, if the model domain contains only a single rectangular area, then =1 and the index j has only one value: = 1, the calculation procedure can be as follows. First, we truncate each of the two families of Poincaré modes in Eqs. (12) and (13) at the 1 -th order so that the number of undetermined coefficients for two 20 families of Poincaré modes is 2 1 and the total number of undetermined coefficients (plus those for a pair of Kelvin waves) is thus 2 1 + 2. To determine these unknowns, we take equally spaced 1 + 1 dots, which are called collocation points, on both cross-sections = 1 and 2 . At these points, one of the boundary conditions given by Eqs. (6) to (8) should be satisfied, which yields 2 1 + 2 equations. By solving this system of equations, we can obtain 2 1 + 2 coefficients ( 1 , 1 , 1, , 1, ). Because the high-order Poincaré modes, which 25 have great values of n and 1, in Eqs. (12) and (13), decay from the boundary very quickly, it is generally necessary to retain only a few lower-order terms. In the above single-rectangle case, the spacing of collocation points is equal to ∆y = 1 /( 1 + 1).
For > 1, that is, the model contains multiple rectangular areas connected one by one, we can treat the approach in the https://doi.org/10.5194/os-2020-86 Preprint. Discussion started: 28 September 2020 c Author(s) 2020. CC BY 4.0 License.
following way. First, we may choose a common divisor of 1 , 2 , … , as a common spacing, which is denoted by ∆y, for all areas. For the th rectangle ( Fig. 3), we may select the collocation points at = ,1 + on the cross-sections = and = +1 , where ,2 = ,1 + . The number of collocation points on each cross-section in this area is equal to /∆ . Thus the number of undetermined coefficients for the Poincaré modes is selected to be = ( /∆ ) − 1. Accordingly, there will be in total ∑ (2 + 2) =1 collocation points in J areas. Note that on the cross-5 section connecting Areaj and Area(j+1), the collocation points that belong to Areaj and those that belong to Area(j+1) are located at the same positions. For the points located on the open or closed boundaries, Eqs. (6) to (8) are applicable, while for the points located on the cross-sections connecting two areas, Eq. (9) should be applied. From these ∑ (2 + 2) =1 equations, we can obtain ∑ (2 + 2) =1 coefficients ( , , , , , ), in which = 1, 2, … , and = 1, 2, … .

Tidal dynamics of the Korea Strait 10
As noted by Odamaki (1989b), the co-oscillating tides are dominant in the JS, which is mainly induced by inputs at the opening of the KS rather than those through the TGS and SYS. Furthermore, our study focuses on the KS, in which influences of the tide-generating force and the inputs from the TGS and SYS are negligible. Therefore, we idealize the KS-JS basin as a semienclosed basin with a sole opening connected to the ECS and study the co-oscillating tides generated by the tidal waves from the ECS through the opening. 15

Model configuration and parameters for the Korea Strait and Japan Sea
To establish an idealized analytical model for the KS-JS basin, we use three rectangular areas as shown in Fig. 4 to represent the study region. The first rectangle, designated as Area1, represents the KS, which is our area of focus. According to the shape of its coastline, we use two rectangles designated as Area2 and Area3 to represent the JS. We place the x-axis parallel to but 200 km away from the southeast sidewall of the KS (that is, 1,1 in Fig. 3 is equal to 200 km), and the y-axis is in the direction 20 perpendicular to the x-axis through the opening of the KS (Fig. 4). The selected depths are the mean depths calculated based on the topographic dataset ETOPO1. The K1 and M2 angular frequencies are equal to 7.2867 × 10 −5 s −1 and 1.4052 × 10 −4 s −1 , respectively. The details of the model parameters can be found in Table 1. https://doi.org/10.5194/os-2020-86 Preprint. Discussion started: 28 September 2020 c Author(s) 2020. CC BY 4.0 License.  Table 2.
Based on the depths listed in Table 1, the wavelengths of the K1 Kelvin waves in these three areas are 2686 km, 12189 km, 5 and 11398 km, respectively, and those of the M2 Kelvin waves are 1393 km, 6321 km, and 5911 km, respectively. Because the widths of the areas are all smaller than half the corresponding Kelvin wavelengths, the inequality Re( 2 − 2 ) < / as stated in the subsection 2.2 is satisfied (see also Godin, 1965;Fang and Wang, 1966;Wu et al., 2018), Thus the Poincaré modes can only exist in a bound form.
10 In addition to the parameters listed in Table 1, we need to estimate the parameters M 2 and K 1 as defined by Eq. (4).
where is the drag coefficient and 2 is the tidal current amplitude of M2, = / 2 , with representing the tidal current amplitude of the constituent i (here, we designate i=1 for M2 and i=2, 3, … for any constituents other than M2).
According to Fang (1987) and Inoue and Garrett (2007), the linearized frictional coefficient for the non-dominant constituent i is approximately equal to the following: 5 Inserting Eqs. (26) and (27) (26) and (27). Then, after inserting these values into Eq. (4), we 10 obtain rough estimates of 2 and 1 for the KS (Area1), which are approximately 0.05 and 0.09, respectively. Since the JS is much deeper and has much weaker tidal currents than the KS, we simply let 1 = 2 = 0 for both Area2 and Area3.
For the collocation approach, we take 10 km as the spacing between collocation points. Thus in this model,

Model results and validation
The obtained analytical solutions of the K1 and M2 tides using the extended Taylor method are shown in Fig.5a and 5b, 20 respectively. The maximum amplitude of the K1 tide is 0.34 m, which appears at the southwest corner of the KS. The amplitude decreases from southwest to northeast, and a counter-clockwise tidal wave system occurs in the northeast part of the KS, with amplitudes less than 0.05 m near the amphidromic point. A co-tidal line with a phase lag of 210° runs from the amphidromic point in the KS into the southwest JS.
https://doi.org/10.5194/os-2020-86 Preprint. Discussion started: 28 September 2020 c Author(s) 2020. CC BY 4.0 License.  The maximum amplitude of the M2 tide is 1.02 m, which appears at the westernmost corner of the KS. The amplitude decreases gradually from southwest to northeast along the direction of the strait, and the amphidromic point occurs at the junction of the KS and JS. The amplitudes near the amphidromic point are lower than 0.1 m, and the phase lags in the most 10 part of the JS vary from 150° to 210°. The comparison with the tidal charts based on data from DTU10 (Fig.5c, d) shows that the model-produced tidal systems agree fairly well with the observations. https://doi.org/10.5194/os-2020-86 Preprint. Discussion started: 28 September 2020 c Author(s) 2020. CC BY 4.0 License.
To quantitatively validate the model results, we first extract the data along the solid boundary of the model for comparison as shown in Fig.6. For the K1 tide, the model-produced amplitudes and phase lags along the boundary in the JS both agree well with the observed data, although small differences occur at the northern corner of the JS. For the M2 tide, the greatest phaselag errors are approximately 70° at the northernmost corner of the JS due to the existence of a degenerated amphidromic point near this area (Fig. 2b). 5 For further validation, we select 16 tide gauge stations where harmonic constants are available from the International Hydrographic Bureau (1930). The station locations are shown in Fig. 4. The result of the comparison is given in Table 2, which also shows that the model results are consistent with the data obtained from gauge observations: the RMS (root mean square) differences of amplitudes of K1 and M2 are 0.014 and 0.031 m, respectively; and those of the phase lags are 7.4° and 6.4°, respectively. 10 Although the theoretical model greatly simplifies the topography and boundary, the amplitude and phase-lag differences of these two tidal constituents are very small in the KS and its surroundings and the basic characteristics of the tidal patterns are well retained (Fig. 5). These findings show that the simplification of the model is reasonable and the extended Taylor method is appropriate for modelling the tides in the KS-JS basin. Therefore, it is meaningful to use the model results for theoretical analysis. 5

Tidal waves in the Korea Strait
To reveal the relative importance of the Kelvin waves versus Poincaré modes in the modelled Korea Strait, the superposition of Kelvin waves and that of the Poincaré modes are given in the upper panels of Fig.7 for K1 and in the upper panels of Fig.8 for M2.
For the K1 tide in the KS, the superposition of the incident (northeastward) and the reflected (southwestward) Kelvin waves 10 appears as a counter-clockwise amphidromic system, with the amphidromic point located near the middle of the strait, but closer to the southeast coast of Korea (Fig.7a). The highest amplitude of the superposed Kelvin waves is 0.3 m, and the mean difference from the observations is less than 0.03 m. The superposition of all Poincaré modes has amplitudes of approximately 0.1 m near the cross-sections on both left and right sides, and a counter-clockwise amphidromic point exists nearly at the centre of the strait (Fig. 7b). Since the amplitudes of the superposed Poincaré modes are significantly smaller than those of the 15 superposed Kelvin waves, the latter can basically represent the total tidal pattern, including the counter-clockwise amphidromic system.
For the M2 tide, the highest amplitude of the superposition of two Kelvin waves is approximately 0.96 m, which appears at the southwest corner of the strait (Fig. 8a). The amplitude decreases from southwest to northeast along the strait, and the amphidromic point appears near the cross-section connecting to the JS, where a topographic step exists. The maximum 20 deviation of the amplitudes of the superposed Kelvin waves from the observations is 0.06 m, and the structure of the superposed Kelvin waves is consistent with the observation. The amplitudes of the superposed Poincaré modes are generally less than 0.2 m on both left and right sides of the KS, and they decay rapidly towards the middle of the strait, thus forming a counterclockwise amphidromic system structure (Fig. 8b). Therefore, the M2 tide in the KS is also mainly controlled by Kelvin waves.
The above results show that the Poincaré modes only exist along the open boundary and the connecting cross-section and 25 their amplitudes quickly approach to zero away from these cross-sections. In fact, these properties of the Poincaré wave are inherent in any narrow strait. Therefore, in the following, we will focus on Kelvin waves and analyze the characteristics of the incident (northeastward) and reflected (southwestward) Kelvin waves.

Discussion on the formation mechanism of amphidromic points
To explore the tidal dynamics of the KS-JS basin, especially the formation mechanism of amphidromic points, we consider the simplest case: a one-dimensional tidal model in channels. In the one-dimensional case, the amphidromic point is equivalent to the wave node. As previously mentioned, an important feature of the topography of the KS-JS basin is that there is a sharp continental slope between the KS and JS, and to northeast of this slope, the JS is much deeper and wider than the KS. Thus, 5 the channel is idealized to contain two areas, with the first one (Area1) having uniform depth ℎ 1 and uniform width 1 and the second one (Area2) having uniform depth ℎ 2 and uniform width 2 . Therefore, the idealized channel contains abrupt changes in depth and width at the connection of these two areas. An incident wave enters the first area and propagates toward the second area passing over the topographic step. For simplicity, we neglect friction. Dean and Dalrymple (1984) have presented a solution for a tidal waves travelling in such a channel; however, in their study, 10 the wave was allowed to radiate out from the second area freely, which implies that the second area is assumed to be semiinfinitely long. Their solution shows that a part of the wave is reflected at the connecting point and another part is transmitted into the second area. The amplitude of the transmitted wave is where is the amplitude of the incident wave and is called the transmission coefficient, which is equal to 15 where is called the reflection coefficient, and is equal to the following: If ρ > 1, namely, if �ℎ 2 2 > �ℎ 1 1 , then < 0, (32) can be rewritten in the form The above equation indicates that at the connecting point, the reflected wave changes its phase lag by 180°. Therefore, the superposition of incident and reflected waves in Area1 has the minimum amplitude at the connecting point. This theory explains how the reflected wave can be generated by abrupt increases in water depth and basin width, and why the reflected wave there has a phase lag opposite to the incident wave. https://doi.org/10.5194/os-2020-86 Preprint. Discussion started: 28 September 2020 c Author(s) 2020. CC BY 4.0 License.
The complete solution for this case is as follows (see Appendix for derivation): where 1 represents the phase lag of the incident wave at the opening of Area1; = / is the wave number, with = � ℎ representing the wave speed in Areaj, j=1, 2; and 1 = 1 1 . This solution for the K1 and M2 constituents for h1=99 m, L1=350 km, W1=230 km, h2=2039 m, and W2=700 km is plotted with the blue curves in Fig. 9. 5 However, Sect. 3.3 shows that the phase-lag changes of the reflected waves relative to the incident waves are not exactly equal to 180° but rather are smaller than 180°, and the discrepancy increases with the decreasing angular frequency. To explain this discrepancy, we improve the above theory by introducing the reflected wave in the second area. In fact, the JS is represented with a semi-closed area in the two-dimensional model (Sect. 3.1), namely, all boundaries except those connected to KS are solid ones (Fig. 4). Therefore, in the following one-dimensional model, the second area is closed at its right end so that the 10 reflection will occur at this end. In this case, the solution becomes more complicated and is dependent on the length of the second area 2 . The reflection coefficient now has the following form (see Appendix for derivation): in which is determined by the following equations: � cos δ = 1+cos 2 2 [(1+cos 2 2 ) 2 +( sin 2 2 ) 2 ] 1/2 , sin δ = sin 2 2 [(1+cos 2 2 ) 2 +( sin 2 2 ) 2 ] 1/2 , where 2 = 2 2 .
The complete solution for this case is as follows: where ε = 2 −1 . and are determined by the following relations: 20 The first terms on the rhs (right-hand side) of the two equations in Eq. (37) represent the waves propagating in the positive x direction, and the second terms are those propagating in the negative x direction. This solution for the K1 and M2 constituents for the case h1=99 m, L1=350 km, W1=230 km, h2=2039 m, L2=1150 km, and W2=700 km is plotted with the red curves in Fig.

25
Equation (35) indicates that the amplitude of the reflected wave in the first area is equal to that of the incident wave. This result is natural because friction is not considered and no dissipation is present during wave propagation. Equation (35) also indicates that the phase lag of the reflected wave at the connecting point is greater than that of the incident wave at the same https://doi.org/10.5194/os-2020-86 Preprint. Discussion started: 28 September 2020 c Author(s) 2020. CC BY 4.0 License. point by 2 . Since the node of the superposition of the incident and reflected waves appears at the place where the phase lags of these two waves are opposite, the first node should appear at ∆ away from the connecting point with ∆ = (π − 2δ)/(2 1 ).
The above relationship can also be obtained from the first equation of Eq. (37). The dependence of 2 on σ for the case h1=99 m, L1=350 km, W1=230 km, h2=2039 m, L2=1150 km, and W2=700 km is plotted in Fig. 10. This figure shows that 5 2 = 0 when σ = 0 and 2 increases with increasing σ, although it is always less than 180°. In particular, 2 = 167.7° when = 1 and 2 = 176.2° when = 2 . Based on this theory, the M2 and K1 amphidromic points should be located at 7.4 and 45.9 km away from the connecting point, respectively. Compared with the two-dimensional model results given in Sect. 3.3, this theory roughly explains one third of the changes. The remaining two third of the changes can be attributed to the effect of Coriolis force. The solution of phase-lag changes at the cross-section in the two-dimensional rotating basin 10 involves interactions among three Kelvin waves (an incident and a reflected Kelvin waves in Area1 and a transmitted Kelvin wave in Area2) and two families of Poincaré modes at the connecting cross-section (one family in each area). Taylor (1922), Fang and Wang (1966), and Thiebaux (1988) have studied the Kelvin-wave reflection at the closed cross-section of semiinfinite rotating two-dimensional channels. In their studies, only two Kelvin waves and one family of Poincaré modes were involved. In comparison to their studies, the present problem is much more complicated. Because of the complexity of the 15 problem, we will presently leave it for a future study.

Summary
In this paper, we establish a theoretical model for the KS-JS basin using the extended Taylor  A one-dimensional model is also given in this paper to reveal the underlying basic dynamics of tides in the KS.

a. Basic Equations
We study tidal wave propagation in channels with abrupt depth/width changes. To be specific, we consider a one-dimensional problem corresponding to the model shown in Fig. 3. For simplicity, Area3 is combined into Area2, and the Coriolis force and friction are neglected, then Eqs. (10) and (11) in the Sect. 2.2 of the text can be simplified as follows: 5 where = / is the wave number, with = � ℎ representing the wave speed in Areaj, j=1, 2; = �ℎ / ; 1 is the coordinate at the opening of Area1; and 2 = 1 + 1 is the coordinate of the connecting point of Area1 and Area2, 15 where an abrupt change in depth and/or width occurs. In Eqs. (A1) to (A8), we have changed the notations ,1 , ,1 , ,2 and ,2 from Eqs. (10) and (11) to ,− , ,− , ,+ , and ,+ (j=1, 2), respectively, to indicate the directions of wave propagation.
That is, ,+ ( ) and ,+ ( ) represent the complex amplitudes of tidal level and tidal current of the tidal waves that travel in the positive x direction in Areaj, respectively; and ,− ( ) and ,− ( ) represent those travelling in the negative x direction in Areaj, respectively. 20 The open boundary condition at = 1 can be specified as follows: where and 1 represent the amplitude and phase lag of the incident wave at the opening of Area1, respectively. From Eqs.
(A9) and (A4) we obtain Therefore, where 30 The matching conditions at = 2 = 1 + 1 are as follows: and To use the relationship among tidal elevations instead of tidal currents, we multiply Eq. (A15) by 1 /ℎ 1 1 and obtain 5

b. Solution for the case with semi-infinite Area2
Here, we first investigate a simpler case that has been previously studied by Dean and Dalrymple (1984). In this case, Area2 10 is assumed to be semi-infinitely long so that the wave can propagate freely in the positive x direction without reflection, meaning that 2 = 0. Thus, the terms 2,− in Eqs. (A6), (A14) and (A16) are all equal to zero. From Eqs. (A14) and (A16) and 15 where and are called reflection and transmission coefficient respectively. These coefficients are equal to the following: and If ρ > 1, namely, if �ℎ 2 2 > �ℎ 1 1 , then < 0. It is more desired to write Eq. (A20) in the following form: Finally, we obtain the following solution: https://doi.org/10.5194/os-2020-86 Preprint. Discussion started: 28 September 2020 c Author(s) 2020. CC BY 4.0 License.
which is Eq. (34) in the text.
We have also solved the problem with the channel containing three areas corresponding to the idealized domain shown in Fig. 3. The solution is quite cumbersome and does not show significant differences from the above two-area solution (for example, it gives 2 = 167.8° when = 1 ; and 2 = 176.3° when = 2 ); therefore, the details of the solution are not given here. Author contributions. GF conceived the study scope and the basic dynamics. DW performed calculation and prepared the draft. 10 ZW and XC checked model results.
Competing interests. The authors declare that they have no conflict of interest.