the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Study of the tidal dynamics of the Korea Strait using the extended Taylor method

### Guohong Fang

### Zexun Wei

### Xinmei Cui

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 of 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 four connected uniform-depth
rectangular areas and establish a theoretical model for the tides in the KS
and JS using the extended Taylor method. The model-produced *K*_{1} and
*M*_{2} 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 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 of less than 180^{∘}, causing the formation of
amphidromic points in the KS. The above phase-lag increase depends on the
angular velocity of the wave and becomes smaller as the angular velocity
decreases. This dependence explains why the *K*_{1} amphidromic point is
located farther away from the connecting cross section in comparison to the
*M*_{2} amphidromic point.

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The Korea Strait (KS, also called the Tsushima Strait) connects the East China Sea (ECS) to the southwest and the Japan Sea (the JS, also called the East Sea, or the Sea of Japan) to the northeast. It is the main route linking the JS to the ECS and the Yellow Sea and is thus an important passage for navigation. The strait is located on the continental shelf, and it has a length of approximately 350 km, a width of 250 km and an average water depth of approximately 100 m. The JS, which is adjacent to the KS, is a deep basin that has an average depth of approximately 2000 m and a depth of more than 3000 m at its deepest part. A steep continental slope separates the KS and the JS, and it presents abrupt depth and width changes (Fig. 1). Such topographic characteristics create the unique tidal waves that occur in the KS.

Ogura (1933) 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 of the tidal waves in the KS. Figure 2 displays the
distributions of the *K*_{1} and *M*_{2} 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) *K*_{1} and *M*_{2} amphidromic
points in the strait that appear in the northeast part of the KS close to
the southern coast of the Korean Peninsula and (2) the *M*_{2} amphidromic
point appears further northeast and closer to the shelf break relative to
the *K*_{1} 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
*K*_{1} and *M*_{2} tides in the KS–JS basin using the extended Taylor
method. The model idealizes the KS–JS basin into three connected
uniform-depth rectangular areas, with the effects of bottom friction and
Coriolis force included in the governing equations and with the observed
tides specified as open-boundary conditions. The extended Taylor method
enables us to obtain semi-analytical solutions consisting of a series of
Kelvin waves and Poincaré modes.

The Taylor problem is a classic tidal dynamic problem (Hendershott and Speranza, 1971). Taylor (1922) first presented a theoretical 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 conditions (Fang et al., 1991) to study tides in multiple rectangular basins (Jung et al., 2005; Roos and Schuttelaars, 2011; Roos et al., 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; Fang and Wang, 1966; Hendershott and Speranza, 1971; Webb, 1976; Fang et al., 1991; Carbajal, 1997; Jung et al., 2005; Roos and Schuttelaars, 2011; Roos et al., 2011; Wu et al., 2018).

## 2.1 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 *J* rectangular areas with length *L*_{j}, width *W*_{j} and
uniform depth *h*_{j} for the *j*th rectangular area (denoted as
Area*j*, *j*=1, …, *J*). For convenience, the shape
of the study 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.

Consider a tidal wave of angular velocity *σ* and typical elevation
amplitude *H*. We assume $H/h\ll \mathrm{1}$, and the conservation of
momentum and mass leads to the following depth-averaged linear shallow water
equations on the *f* plane:

where *x* and *y* are coordinates in the longitudinal (along-channel) and
transverse (cross-channel) directions; *t* represents time; ${\stackrel{\mathrm{\u0303}}{u}}_{j}$ and
${\stackrel{\mathrm{\u0303}}{\mathit{\upsilon}}}_{j}$ represent the depth-averaged flow velocity
components in the *x* and *y* directions, respectively, with the subscript
*j* indicating the area number; ${\stackrel{\mathrm{\u0303}}{\mathit{\zeta}}}_{j}$ represents the free
surface elevation above the mean level; *γ*_{j} represents the
frictional coefficient, which is taken as a constant for each tidal
constituent in each area; *g*=9.8 ms^{−2} represents the
acceleration due to gravity; and *f*_{j} represents the Coriolis parameter,
which is also taken as a constant based on the average of the concerned
area. The equations in Eq. (1) for each *j* are two-dimensional linearized shallow
water equations on an *f* plane with momentum advection neglected. For any
*j*, 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) and Rienecker and Teubner (1980). When a monochromatic
wave is considered, $\left({\stackrel{\mathrm{\u0303}}{\mathit{\zeta}}}_{j},{\stackrel{\mathrm{\u0303}}{u}}_{j},{\stackrel{\mathrm{\u0303}}{\mathit{\upsilon}}}_{j}\right)$ can be
expressed as follows:

where Re stands for the real part of the complex quantity that follows;
$\left({\mathit{\zeta}}_{j},{u}_{j},{\mathit{\upsilon}}_{j}\right)$ are
referred to as complex amplitudes of $\left({\stackrel{\mathrm{\u0303}}{\mathit{\zeta}}}_{j},{\stackrel{\mathrm{\u0303}}{u}}_{j},{\stackrel{\mathrm{\u0303}}{\mathit{\upsilon}}}_{j}\right)$,
respectively; $i\equiv \sqrt{-\mathrm{1}}$ is the imaginary unit; and
*σ* is the angular velocity of the wave. For this wave, Eq. (1) can be
reduced as follows:

in which

Provided that the *j*th rectangular area, denoted as Area*j*, has a
width of *W*_{j}, has a length of *L*_{j} and ranges from *x*=*l*_{j} to
$x={l}_{j+\mathrm{1}}$ (${l}_{j+\mathrm{1}}={l}_{j}+{L}_{j}$) in the *x* direction and from
$y={w}_{j,\mathrm{1}}$ to $y={w}_{j,\mathrm{2}}$ (${w}_{j,\mathrm{2}}={w}_{j,\mathrm{1}}+{W}_{j}$) in
the *y* direction, the boundary conditions along the sidewalls within
$x\in [{l}_{j},{l}_{j+\mathrm{1}}]$ are taken as follows:

Along the cross sections, such as *x*=*l*_{j}, 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 *x* direction occurs on the
cross section;

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

if the cross section is a connecting boundary of the areas *j* and *j*+1,
with each having a different uniform depth of *h*_{j} and *h*_{j+1}.

Equation (9) is matching conditions accounting for sea level continuity and
volume transport continuity. The individual Eqs. (6) to (9), or their
combination, may be used as boundary conditions for the cross sections. The
relationship between *u*_{j} and *ζ*_{j} 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.

## 2.2 General solution

For the *j*th rectangular area, that is, for $x\mathit{\u03f5}[{l}_{j},\phantom{\rule{0.125em}{0ex}}{l}_{j+\mathrm{1}}]$ and
$y\mathit{\u03f5}[{w}_{j,\mathrm{1}},\phantom{\rule{0.125em}{0ex}}{w}_{j,\mathrm{2}}]$, the governing equations in Eq. (3) only have the
following four forms satisfying the sidewall boundary condition of Eq. (5)
(see for example Fang et al., 1991):

and

where *α*_{j}, *β*_{j}, *r*_{j,n} and *s*_{j,n}
are equal to the following:

and

in which ${k}_{j}=\mathit{\sigma}/{c}_{j}$ is the wave number, with
${c}_{j}=\sqrt{g{h}_{j}}$ being the wave speed of the Kelvin wave in
the absence of friction. The parameters *s*_{j,n} in Eq. (17) are of
fundamental importance in determining the characteristic of the Poincaré
modes. If $\mathrm{Re}({\mathit{\beta}}_{j}^{\mathrm{2}}-{\mathit{\alpha}}_{j}^{\mathrm{2}}{)}^{\mathrm{1}/\mathrm{2}}<\mathit{\pi}/{W}_{j}$, all Poincaré modes are bound in the
vicinity of the open, connecting or closed cross sections (see Fang and
Wang, 1966; Hendershott and Speranza, 1971, for an absence of friction),
while if $\mathrm{Re}({\mathit{\beta}}_{j}^{\mathrm{2}}-{\mathit{\alpha}}_{j}^{\mathrm{2}}{)}^{\mathrm{1}/\mathrm{2}}>n\mathit{\pi}/{W}_{j}$, the *n*th and lower-order Poincaré modes
are propagating waves. In the present study, the inequality
$\mathrm{Re}({\mathit{\beta}}_{j}^{\mathrm{2}}-{\mathit{\alpha}}_{j}^{\mathrm{2}}{)}^{\mathrm{1}/\mathrm{2}}<\mathit{\pi}/{W}_{j}$
holds for all rectangular areas shown in Fig. 3, so that all Poincaré
modes in the present study appear in a bound form. The parameter *s*_{j,n}
has two complex values for each *n*, and here we choose the one that has a
positive real part. To satisfy the equations in Eq. (3),
(${A}_{j,n},{B}_{j,n},{C}_{j,n},{D}_{j,n}$) and
(${A}_{j,n}^{\prime},{B}_{j,n}^{\prime},{C}_{j,n}^{\prime},{D}_{j,n}^{\prime}$) should be as
follows:

and

Equations (10) and (11) represent Kelvin waves propagating in the
−*x* and *x* directions, respectively; and Eqs. (12) and (13)
represent two families of Poincaré modes bound along the cross sections
*x*=*l*_{j} and *l*_{j+1} in the *j*th rectangular area,
respectively. The coefficients (${a}_{j},{b}_{j},{\mathit{\kappa}}_{j,n},{\mathit{\lambda}}_{j,n}$) determine amplitudes and phase lags of Kelvin waves and
Poincaré modes. These coefficients must be chosen to satisfy the
boundary conditions, preferably using the collocation approach.

## 2.3 Defant's collocation approach

The collocation approach was first proposed by Defant in 1925 (see Defant,
1961) and is convenient in determining the coefficients
(${a}_{j},{b}_{j},{\mathit{\kappa}}_{j,n},{\mathit{\lambda}}_{j,n}$). In the simplest case,
that is, if the model domain contains only a single rectangular area, then
*J*=1 and the index *j* has only one value: *j*=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 *N*_{1}th order so that the
number of undetermined coefficients for two families of Poincaré modes
is 2*N*_{1} and the total number of undetermined coefficients (plus
those for a pair of Kelvin waves) is thus 2*N*_{1}+2. To determine
these unknowns, we take equally spaced *N*_{1}+1 dots, which are called
collocation points, located at
$y={w}_{\mathrm{1},\mathrm{1}}+\frac{{W}_{\mathrm{1}}}{\mathrm{2}({N}_{\mathrm{1}}+\mathrm{1})}$, ${w}_{\mathrm{1},\mathrm{1}}+\frac{\mathrm{3}{W}_{\mathrm{1}}}{\mathrm{2}({N}_{\mathrm{1}}+\mathrm{1})}$, …, ${w}_{\mathrm{1},\mathrm{1}}+\frac{\left(\mathrm{2}{N}_{\mathrm{1}}+\mathrm{1}\right){W}_{\mathrm{1}}}{\mathrm{2}({N}_{\mathrm{1}}+\mathrm{1})}$ on both
cross sections *x*=*l*_{1} and *l*_{2}. At these points, one of the boundary
conditions given by Eqs. (6) to (8) should be satisfied, which yields
2*N*_{1}+2 equations. By solving this system of equations, we can
obtain 2*N*_{1}+2 coefficients (${a}_{\mathrm{1}},{b}_{\mathrm{1}},{\mathit{\kappa}}_{\mathrm{1},n},{\mathit{\lambda}}_{\mathrm{1},n}$). Because the high-order Poincaré modes, which
have great values of *n* and *s*_{1,n} 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 $\mathrm{\Delta}y={W}_{\mathrm{1}}/({N}_{\mathrm{1}}+\mathrm{1})$.

For *J*>1, that is, when the model contains multiple rectangular areas
connected one by one, we can treat the approach in the following way. First,
we may choose a common divisor of *W*_{1}, *W*_{2}, …, *W*_{J} as a common spacing, which is denoted by
Δ*y*, for all areas. For the *j*th rectangle (Fig. 3), we may
select the collocation points at $y={w}_{j,\mathrm{1}}+\frac{\mathrm{\Delta}y}{\mathrm{2}}$,
${w}_{j,\mathrm{1}}+\frac{\mathrm{3}\mathrm{\Delta}y}{\mathrm{2}}$, …, ${w}_{j,\mathrm{2}}-\frac{\mathrm{\Delta}y}{\mathrm{2}}$ on the cross sections *x*=*l*_{j} and
$x={l}_{j+\mathrm{1}}$, where ${w}_{j,\mathrm{2}}={w}_{j,\mathrm{1}}+{W}_{j}$. The number of
collocation points on each cross section in this area is equal to
${W}_{j}/\mathrm{\Delta}y$. Thus the number of undetermined coefficients for the
Poincaré modes is selected to be ${N}_{j}=({W}_{j}/\mathrm{\Delta}y)-\mathrm{1}$.
Accordingly, there will be in total ${\sum}_{j=\mathrm{1}}^{J}(\mathrm{2}{N}_{j}+\mathrm{2})$ collocation points in *J* areas. Note that on the
cross section connecting Area*j* and Area(*j*+1), the collocation points that
belong to Area*j* 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
${\sum}_{j=\mathrm{1}}^{J}(\mathrm{2}{N}_{j}+\mathrm{2})$ equations, we can obtain
${\sum}_{j=\mathrm{1}}^{J}(\mathrm{2}{N}_{j}+\mathrm{2})$ coefficients
(${a}_{j},{b}_{j},{\mathit{\kappa}}_{j,n},{\mathit{\lambda}}_{j,n}$), in which *j*=1,
2, …, *J* and *n*=1, 2, …, *N*_{j}.

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 Tsugaru Strait (TGS) and Soya Strait (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 semi-enclosed 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.

## 3.1 Model configuration and parameters for the Korea Strait and Japan Sea

To establish an idealized analytical model for the KS–JS basin, we use four
rectangular areas as shown in Fig. 4 to represent the study region. The
first rectangle, designated as Area1, represents the KS, which is our focus
area. According to the shape of its coastline, we use three 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
*w*_{1,1} in Fig. 3 is equal to 200 km), and the *y* axis is in the direction
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 *K*_{1} and *M*_{2} angular velocities are equal to
$\mathrm{7.2867}\times {\mathrm{10}}^{-\mathrm{5}}$ and
$\mathrm{1.4052}\times {\mathrm{10}}^{-\mathrm{4}}$ s^{−1},
respectively. The details of the model parameters can be found in Table 1.

Based on the depths listed in Table 1, the wavelengths of the *K*_{1} Kelvin
waves in these four areas are 2686, 12 189, 11 398 and 2561 km,
respectively, and those of the *M*_{2} Kelvin waves are 1393, 6321,
5911 and 1328 km, respectively. Because the widths of the areas are all
smaller than half the corresponding Kelvin wavelengths, the inequality
$\mathrm{Re}({\mathit{\beta}}_{j}^{\mathrm{2}}-{\mathit{\alpha}}_{j}^{\mathrm{2}})<\mathit{\pi}/{W}_{j}$ as
stated in the Sect. 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.

In addition to the parameters listed in Table 1, we need to estimate the
parameters ${\mathit{\mu}}_{{M}_{\mathrm{2}}}$ and ${\mathit{\mu}}_{{K}_{\mathrm{1}}}$ as defined by Eq. (4). Since *M*_{2} has the
largest tidal current in the KS (Teague et al., 2001), and we assume that
the tidal currents are rectilinear, the linearized frictional coefficient
for *M*_{2} is approximately equal to the following, after Pingree and
Griffiths (1981), Fang (1987) and Inoue and Garrett (2007),

where *C*_{D} is the drag coefficient and ${U}_{{M}_{\mathrm{2}}}$ is the tidal current
amplitude of *M*_{2}, ${\mathit{\u03f5}}_{i}={U}_{i}/{U}_{{M}_{\mathrm{2}}}$, with *U*_{i}
representing the tidal current amplitude of the constituent *i* (here, we
designate *i*=1 for *M*_{2} and *i*=2, 3, …for any constituents
other than *M*_{2}). 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:

Inserting Eqs. (26) and (27) into Eq. (4), we can obtain the parameter *μ*. Teague et al. (2001) provided tidal current harmonic constants at 10
mooring stations along two cross sections in the KS. The averaged values of
the major semi-axes of the tidal current ellipses at these stations are
0.154, 0.119, 0.101 and 0.062 m s^{−1} for *M*_{2}, *K*_{1}, *O*_{1} and *S*_{2},
respectively. Here, we use these values and *C*_{D}≈0.0026 to
estimate the parameters in Eqs. (26) and (27). Then, after inserting these
values into Eq. (4), we obtain rough estimates of ${\mathit{\mu}}_{{M}_{\mathrm{2}}}$ and ${\mathit{\mu}}_{{K}_{\mathrm{1}}}$ 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 ${\mathit{\mu}}_{{K}_{\mathrm{1}}}={\mathit{\mu}}_{{M}_{\mathrm{2}}}=\mathrm{0}$ for both Area2 and
Area3.

For the collocation approach, we take 10 km as the spacing between collocation points. Thus in this model, a total of 198 collocation points are used to establish 256 equations, and the parameters of 3 pairs of Kelvin waves and 125 pairs of Poincaré modes can be obtained. Along the open boundary of the KS, the open-boundary condition Eq. (8) is employed, with the value of $\widehat{\mathit{\zeta}}$ equal to the observed harmonic constants from the global tide model DTU10 (Cheng and Anderson, 2011). Along the cross sections connecting Area1 with Area2 and Area2 with Area3, the matching conditions Eq. (9) are applied. Along the solid cross sections, condition Eq. (6) is used.

## 3.2 Model results and validation

The obtained analytical solutions of the *K*_{1} and *M*_{2} tides using the
extended Taylor method are shown in Fig. 5a and b, respectively. The maximum
amplitude of the *K*_{1} 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.

The maximum amplitude of the *M*_{2} 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
most parts of the JS vary from 150 to 210^{∘}. A
degenerated amphidromic point appears near the entrance of the Tartar
Strait. 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.

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 *K*_{1} 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 *M*_{2} tide,
the greatest phase-lag errors are approximately 64^{∘} near the
entrance of the Tartar Strait due to the existence of a degenerated
amphidromic point near this area (Fig. 2b).

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 *K*_{1} and *M*_{2} are 0.014 and 0.032 m,
respectively; and those of the phase lags are 7.0 and
5.2^{∘}, respectively.

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.

## 3.3 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 Fig. 7a–b for
*K*_{1} and in Fig. 8a–b for *M*_{2}.

For the *K*_{1} tide in the KS, the superposition of the incident
(northeastward) and the reflected (southwestward) Kelvin waves 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 superposed Kelvin waves, the
latter can basically represent the total tidal pattern, including the
counter-clockwise amphidromic system.

For the *M*_{2} tide, the highest amplitude of the superposition of two
Kelvin waves is approximately 0.95 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 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
counter-clockwise amphidromic system structure (Fig. 8b). Therefore, the
*M*_{2} 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 their amplitudes quickly approach 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 analyse the characteristics of the incident (northeastward) and reflected (southwestward) Kelvin waves.

The incident and reflected *K*_{1} Kelvin waves are shown in Fig. 7c and
d, respectively. The area-mean amplitude of the incident Kelvin wave in the
KS is 0.248 m, and that of the reflected Kelvin wave is 0.190 m, which is
77 % of the incident Kelvin wave. On the connecting cross section, the
section-mean amplitude of the incident Kelvin wave is 0.243 m, and the
section-mean phase lag is 151.6^{∘}. The section-mean amplitude of
the reflected Kelvin wave is 0.194 m, which is 80 % of the incident Kelvin
wave. The section-mean phase lag is 295.8^{∘}, indicating that the
phase lag increases by 144.2^{∘} when the wave is reflected. The
amphidromic point of the superposed Kelvin wave is 137 km away from the step
and close to the northwest shore of the KS.

The incident and reflected *M*_{2} Kelvin waves are shown in Fig. 8c and
d, respectively. The area-mean amplitude of the incident Kelvin wave in the
KS is 0.471 m, and that of the reflected Kelvin wave is 0.439 m, which is
93 % of the incident Kelvin wave. This ratio is larger than the *K*_{1}
tide because the bottom friction of *M*_{2} is smaller and less energy is
lost in the propagation process. On the connecting cross section, the mean
amplitude of the incident Kelvin wave is 0.462 m, and the phase lag is
97.9^{∘}. The mean amplitude of the reflected Kelvin wave is 0.447
m, which is 97 % of the incident Kelvin wave, and the phase lag is
approximately 266.4^{∘}, with a phase-lag increase of
168.5^{∘}, which is closer to 180^{∘} as compared to the
corresponding value of the *K*_{1} tide. Accordingly, the *M*_{2}
amphidromic point of the superposed Kelvin wave shifts to approximately 21 km away from the step. A comparison between Figs. 7a and 8a shows that
the amphidromic point of *K*_{1} is located west of that of *M*_{2}. This
result reproduces well the observed phenomenon as seen from Fig. 2.

The above results indicate that the relation of the amplitudes and phase
lags of the reflected Kelvin wave with the incident wave plays a decisive
role in the tidal system in the KS, especially in the formation of
amphidromic points, for both the *K*_{1} and *M*_{2} tides.

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 steep continental slope between the KS and JS, and to the northeast of this
slope, the JS is much deeper and wider than the KS. Thus, the channel is
idealized to contain two areas, with the first one (Area1) having uniform
depth *h*_{1} and uniform width *W*_{1} and the second one (Area2) having
uniform depth *h*_{2} and uniform width *W*_{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.

If the second area is semi-infinitely long, allowing for the wave radiating out from the second area freely, then 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 (see for example Dean and Dalrymple, 1984)

where *H*_{I} is the amplitude of the incident wave and *κ*_{T} is
called the transmission coefficient, which is equal to

where

with ${c}_{j}=\sqrt{g{h}_{j}}$ representing the wave
speed in the *j*th area; *j*=1, 2. *c*_{j} is in fact proportional
to $\sqrt{{h}_{j}}$. The amplitude of the reflected wave *H*_{R} is

where *κ*_{R} is called the reflection coefficient and is equal to
the following:

If *ρ*>1, namely, if $\sqrt{{h}_{\mathrm{2}}}{W}_{\mathrm{2}}>\sqrt{{h}_{\mathrm{1}}}{W}_{\mathrm{1}}$,
then *κ*_{R}<0, Eq. (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.

The complete solution for this case is as follows (see also Dean and Dalrymple, 1984):

where *θ*_{1} represents the phase lag of the incident wave at the
opening of Area1; ${k}_{j}=\mathit{\sigma}/{c}_{j}$ is the wave number, with
${c}_{j}=\sqrt{g{h}_{j}}$ representing the wave speed in Area*j*, *j*=1, 2; and
*χ*_{1}=*k*_{1}*L*_{1}. This solution for the *K*_{1} and *M*_{2}
constituents for *h*_{1}=99 m, *L*_{1}=350 km, *W*_{1}=230 km,
*h*_{2}=2039 m and *W*_{2}=700 km is plotted with the blue curves in Fig. 9.

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 reflection will occur at
this end. In this case, the solution becomes more complicated and is
dependent on the length of the second area *L*_{2}. The reflection
coefficient *κ*_{R} now has the following form (see Supplement for
derivation):

in which *δ* is determined by the following equations:

where *χ*_{2}=*k*_{2}*L*_{2}. Equation (36)
indicates that the length, width and depth of Area2 are also important in
determining the phase-lag increase of the reflected wave relative to the
incident wave in Area1. Comparison of Eq. (35) with Eq. (33) indicates that
the phase-lag increase is now 2*δ* instead of *π*. The
difference $\mathrm{\Delta}=\mathit{\pi}-\mathrm{2}\mathit{\delta}$ characterizes the influence of Area2 upon
the phase-lag increase at the connection of two areas. To show the influence
of the length, width and depth of Area2 on the value of Δ, we first
retain the width and depth unchanged and increase the length by 10 %; it
is shown that the value of Δ for *K*_{1} is reduced by 15 %
(reduced to 10.44^{∘} from 12.27^{∘}) and
that the value of Δ for *M*_{2} is reduced by 37 % (reduced to
2.37^{∘} from 3.78^{∘}). Next we retain the
length and depth unchanged and increase the width by 10 %; it is shown
that the value of Δ for *K*_{1} is reduced by 9 % (reduced to
11.16^{∘} from 12.27^{∘}) and that the value
of Δ for *M*_{2} is reduced by 9 % (reduced to
3.44^{∘} from 3.78^{∘}). Then we retain the
length and width unchanged and increase the depth by 10 %; it is shown
that the value of Δ for *K*_{1} increases by 1 % (increases to
12.42^{∘} from 12.27^{∘}) and that the value
of Δ for *M*_{2} increases by 9 % (increases to
4.12^{∘} from 3.78^{∘}).

The complete solution for this case is as follows:

where $\mathit{\epsilon}=\mathrm{2}{E}^{-\mathrm{1}}$. *E* and *ϕ* are determined by the
following relations:

The first terms on the 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 *K*_{1} and *M*_{2} constituents for the case *h*_{1}=99 m,
*L*_{1}=350 km, *W*_{1}=230 km, *h*_{2}=2039 m, *L*_{2}=1150 km and
*W*_{2}=700 km is plotted with the red curves in Fig. 9.

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 point by 2*δ*. 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
Δ*x* away from the connecting point with

The above relationship can also be obtained from the first equation of Eq. (37). The dependence of 2*δ* on *σ* for the case
*h*_{1}=99 m, *L*_{1}=350 km, *W*_{1}=230 km, *h*_{2}=2039 m,
*L*_{2}=1150 km and *W*_{2}=700 km is plotted in Fig. 10. This figure
shows that 2*δ*=0 when *σ*=0 and 2*δ* increases
with increasing *σ*, although it is always less than
180^{∘}. In particular, $\mathrm{2}\mathit{\delta}=\mathrm{167.7}{}^{\circ}$ when
$\mathit{\sigma}={\mathit{\sigma}}_{{K}_{\mathrm{1}}}$ and $\mathrm{2}\mathit{\delta}=\mathrm{176.2}{}^{\circ}$ when $\mathit{\sigma}={\mathit{\sigma}}_{{M}_{\mathrm{2}}}$. Based on this theory, the *M*_{2} and *K*_{1} 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-thirds of the changes may be due to the effect of the Coriolis force. The
solution of phase-lag changes at the cross section in the two-dimensional
rotating basin involves interactions among three Kelvin waves (an incident
and a reflected Kelvin wave 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 semi-infinite 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 problem, we will
presently leave it for a future study.

In this paper, we establish a theoretical model for the KS–JS basin using
the extended Taylor method. The model idealizes the study region as three
connected flat rectangular areas, incorporates the effects of the Coriolis
force and bottom friction in the governing equations, and is forced by
observed tides at the opening of the KS. The analytical solutions of the
*K*_{1} and *M*_{2} tidal waves are obtained using Defant's collocation
approach.

The theoretical model results are consistent with the satellite altimeter
and tidal gauge observations, which indicates that the model is suitable and
correct. The model reproduces well the *K*_{1} and *M*_{2} tidal systems in
the KS. In particular, the model-produced locations of the *K*_{1} and
*M*_{2} amphidromic points are consistent with the observed ones.

The model solution provides the following insights into the tidal dynamics
in the KS. (1) The tidal system in each rectangular area can be decomposed
into two oppositely travelling Kelvin waves and two families of Poincaré
modes, with Kelvin waves dominating the tidal system due to the narrowness of
the area. (2) The incident Kelvin wave from the ECS through the opening of
the KS travels toward the JS and is reflected at the connecting
cross section between the KS and JS, where abrupt increases from the KS to
JS in water depth and basin width occur. (3) The phase lag of the reflected
wave at the connecting cross section increases by less than 180^{∘}
relative to that of the incident wave, thus enabling the formation of the
amphidromic points in the KS. (4) The phase-lag increase of the reflected
wave relative to the incident wave is dependent on the angular frequency of
the wave and becomes smaller as the angular frequency decreases. This
feature explains why the *K*_{1} amphidromic point is located farther away
from the connecting cross section in comparison to the *M*_{2} amphidromic
point. (5) The length, width and depth of the JS is also important in
determining the phase-lag increase of the reflected Kelvin wave in the KS.

The ETOPO1 data (https://doi.org/10.7289/V5C8276M, Amante and Eakins, 2009) are from the National Geophysical Center, USA (https://www.ngdc.noaa.gov/mgg/global/, last access: 14 June 2016). The DTU10 tidal data are from DTU Space, Danish National Space Center, Technical University of Denmark (ftp://ftp.space.dtu.dk/pub/DTU10/DTU10_TIDEMODEL, last access: 3 February 2012, Cheng and Andersen, 2011).

The supplement related to this article is available online at: https://doi.org/10.5194/os-17-579-2021-supplement.

GF conceived the study scope and the basic dynamics. DW performed calculation and prepared the draft. ZW and XC checked model results.

The authors declare that they have no conflict of interest.

We sincerely thank Joanne Williams for handling our paper and thank David Webb and Kyung Tae Jung for their careful reading of our paper and constructive comments and suggestions which were of great help in improving our work.

This research has been supported by the National Natural Science Foundation of China (grant nos. 41706031 and 41821004).

This paper was edited by Joanne Williams and reviewed by Kyung Tae Jung and David Webb.

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- Abstract
- Introduction
- The extended Taylor method and its application to multiple rectangular areas
- Tidal dynamics of the Korea Strait
- Discussion on the formation mechanism of amphidromic points
- Summary
- Data availability
- Author contributions
- Competing interests
- Acknowledgements
- Financial support
- Review statement
- References
- Supplement

*K*

_{1}amphidromic point is located farther away from the shelf break separating the Korea Strait and Japan Sea in comparison to the

*M*

_{2}amphidromic point.

- Abstract
- Introduction
- The extended Taylor method and its application to multiple rectangular areas
- Tidal dynamics of the Korea Strait
- Discussion on the formation mechanism of amphidromic points
- Summary
- Data availability
- Author contributions
- Competing interests
- Acknowledgements
- Financial support
- Review statement
- References
- Supplement