Journal of Oceanography, Vol. 54, pp. 651 to 668. 1998

Three-Dimensional Structure of Tidal Current in the East
China Sea and the Yellow Sea
XINYU GUO1 and TETSUO YANAGI2
1Institute for Global Change Research,

SEAVANS North 7F, 1-2-1 Shibaura, Minato, Tokyo 105-6791, Japan
2Research Institute for Applied Mechanics, Kyushu University,

6-1 Kasuga-kohen, Kasuga, Fukuoka 816-8580, Japan(Received 17 October 1997; in revised form 15 July 1998; accepted 22 July 1998)

A three-dimensional tidal current model is developed and applied to the East China Sea(ECS), the Yellow Sea and the Bohai Sea. The model well reproduces the major four tides,
namely M2, S2 , K1 and O1 tides, and their currents. The horizontal distributions of the
major four tidal currents are the same as those calculated by the horizontal twodimensional models. With its high resolutions in the horizontal (12.5 km) and the vertical(20 layers), the model is used to investigate the vertical distributions of tidal current. Fourvertical eddy viscosity models are used in the numerical experiments. As the tidal current
becomes strong, its vertical shear becomes large and its vertical profile becomes sensitive
to the vertical eddy viscosity. As a conclusion, the HU (a) model (Davies et al., 1997), which
relates the vertical eddy viscosity to the water depth and depth mean velocity, gives theclosest results to the observed data. The reproduction of the amphidromic point of M2 tidein Liaodong Bay is discussed and it is concluded that it depends on the bottom friction
stress. The model reproduces a unique vertical profile of tidal current in the Yellow Sea,
which is also found in the observed data. The reason for the reproduction of such a uniqueprofile in the model is investigated.

(An, 1977; Choi, 1980, 1984, 1989; Yanagi and Inoue,
1994; Zhao et al., 1994; Ye and Mei, 1995), which reproduces the tide in the domain from the harmonic constants
along the open boundary, according to the physical rule.
Some available harmonic constants along the coast of thedomain are used to verify the model’s results. Therefore,
this method is more logical and has recently become popular.
Table 1, which summarizes the main characteristics ofthe tidal models in the ECS, the Yellow Sea and the Bohai
Sea based on the initial value method, shows that most tidemodels are horizontally two-dimensional ones with a horizontal resolution of about 25 km. All of the two-dimensionalmodels include the nonlinear advective terms, while thethree-dimensional models (Choi, 1984, 1989) ignore them.
The horizontal eddy viscosity is said to have little influenceon the tide because it acts as a scale-selective filter, damping
the shorter waves more heavily than the long waves (Davies
et al., 1997). Including it or not in the tide model should thus
not be a serious problem. The effect of the tide generating
potential on the M2 tide in this region is said to be less than
3% (An, 1977) and is usually ignored. The effect of theearth’s curvature on the tide in this region is also said to besmall enough to be ignored in the numerical model (Yanagi
and Inoue, 1994).

1. Introduction
As one of the most important physical processes in the
East China Sea (ECS) and the Yellow Sea, the tide and tidal
current there have been investigated by field observation
data (Ogura, 1933; Nishida, 1980; Larsen et al., 1985; Fang,1986), satellite altimetric data (Yanagi et al., 1997) andnumerical models (An, 1977; Shen, 1980; Choi, 1980, 1984,
1989; Ding, 1984; Yanagi and Inoue, 1994; Zhao et al., 1994;
Ye and Mei, 1995). It may be stated that the general information about the tidal dynamics in this region is now
available. However, since we have few observed tidal current data, and most numerical models are horizontally twodimensional ones, the three-dimensional structure of tidal
current in this region is little known and must be investigatedmore thoroughly.
Among the numerical models of the tide and tidal
current in the ECS, some are based on the boundary value
method (Shen, 1980; Ding, 1984), which calculates the tidein the domain based on the harmonic constants along thecoast and ignores the nonlinear effects. Usually there are notenough tide gauges to provide the harmonic constants along
the coast of the model domain, and so the use and precisionof the boundary value method are limited.
The other models are based on the initial value method651
Copyright  The Oceanographic Society of Japan.

Keywords:⋅ Tidal current,⋅ numerical model,⋅ East China Sea,
⋅ Yellow Sea,
⋅ high resolution,⋅ vertical profile.

Table 1. Summary of the tidal models in the East China Sea, the Yellow Sea and the Bohai Sea, in which Y = Yellow Sea; B = Bohai
Sea; E = East China Sea; Four = M2, S2, K1 and O1 tides; ADV = advective terms; HEV = horizontal eddy viscosity (cm2s–1 );GF = tide-generating force; Cb = sea bed drag coefficient; Lat = latitude; Long = longitude; 2-D = horizontally two-dimensionalmodel; 3-D = three-dimensional model.

The calculation of the bottom friction stress may be themost important problem in the tide model. The quadratic
friction rule is often used; that is, the bottom friction stressis calculated from the velocity at a single height (for example,
1 m above the sea bottom) and a constant, i.e. the bed dragcoefficient. In the two-dimensional model, the velocity at asingle height has to be replaced by the depth mean velocity,while in the three-dimensional model this velocity is represented by the velocity of the layer nearest to the sea bottomor the depth mean velocity (Davies et al., 1997). A widelyused value for the bed drag coefficient is 0.0026. In the
numerical models for the tide and tidal current in the ECS,
the Yellow Sea and the Bohai Sea, 0.0026 or a value close
to it were used by An (1977), Choi (1980, 1984, 1989) andYanagi and Inoue (1994). However, Chinese scientists prefera small value for this coefficient. Zhao et al. (1994) assumed
that the bed drag coefficient has a horizontal variation, using
values of 0.001 west to the line from (40°N, 124.25°E) to(25°N, 120.75°E), 0.0035 in the Korea/Tsushima Strait and0.0016 in the remaining region; while Ye and Mei (1995)
used different bed drag coefficients to simulate different
tides (0.0017 for M2 tide, 0.0051 for S2 tide, 0.0055 for K1
and O1 tides). Using such small values for M2 tide, the model
can well reproduce the amphidromic point in Liaodong Bay,
while the models using 0.0026 cannot reproduce thisamphidromic point.
Few three-dimensional tide models have been reported
for the ECS, the Yellow Sea and the Bohai Sea. Choi (1984)
developed a linear three-dimensional model with three layers
in the vertical. However, compared to his horizontal twodimensional model with the same horizontal resolution andtopography data (Choi, 1980), the tide reproduced by thethree-dimensional model is not very good, especially in the
Bohai Sea. To solve this problem, Choi (1989) improved themodel’s horizontal resolution and ran the same model again.
Though the results are better than before, the tidal amplitudein the Bohai Sea is still underestimated and the amphidromic
point in Liaodong Bay is not reproduced. And, since his

652

X. Guo and T. Yanagi

model has only three layers in the vertical, the verticalvariation of tidal current cannot be resolved very clearly.
Summarizing the above tide models of the ECS, theYellow Sea and the Bohai Sea, it can be said that there is still
no complete three-dimensional tide model that can clarify
the three-dimensional structure of the tidal current in thisregion. Here, we present a high resolution (12.5 km × 12.5
km × 20 layers) three-dimensional tide model to investigate
this problem. The open boundary is located out of the
Ryukyu Islands in order to include the shelf edge in themodel. Four major tides, namely M2, S2, K1 and O1 tides, arechosen to be reproduced. After reproducing the four tides
and tidal currents, more attention is paid to the vertical
distribution of tidal current and the influence of the bottom
friction stress and interior friction on the tidal current.
2. Numerical Model
Because the effect of the earth’s curvature on the tidal
phenomena in the ECS and the Yellow Sea is small enoughto be ignored in a numerical model (Yanagi and Inoue,
1994), we formulate our problem in the Cartesian coordinate
system. Assuming constant density, the tide and tidal current are controlled by the following equations, in which thex axis is eastward, y axis northward, and z axis upward fromthe mean sea surface.
 ∂ 2u ∂ 2u ∂u r∂η ∂  ∂ u + u ⋅ ∇u − fv = −g
+  Av  + Ah  2 + 2  , (1)∂x ∂z  ∂z 
∂t
∂y 
 ∂x
 ∂ 2v ∂ 2v ∂v r∂η ∂  ∂ v + u ⋅ ∇v − fu = −g
+  Av  + Ah  2 + 2  , ( 2 )∂y ∂z  ∂z 
∂t
∂y 
 ∂x

∂η ∂ 0∂ 0+udz +vdz = 0.
∂ t ∂ x ∫− h∂ y ∫− h( 3)Here u and v are the eastward and northward velocity,respectively; η the sea surface elevation from the mean sea
surface; f = f0 + βy ( f0 = 7.73 × 10–5 s–1 at 32°N, β = 1.94 ×
10–13 s–1cm–1) the Coriolis parameter; g (=980 cm s–2) thegravitational acceleration; Ah (=107 cm2s–1, Yanagi andInoue, 1994) the horizontal eddy viscosity; h the water depth.
The vertical eddy viscosity Av represents the interior
friction of tidal current and may have a large effect on thecalculated tidal current near the sea bottom (Davies et al.,1997). Many models of the vertical eddy viscosity have beenused in the tidal calculations, from a simple constant modelto the turbulence energy model (Xing and Davies, 1996). In
this paper, we use four models described below to determine
the vertical eddy viscosity. By comparing the results of thesemodels, we want to find the best one suitable for the tidal
calculation in the ECS, the Yellow Sea and the Bohai Sea.
The first model is to set the vertical eddy viscosity as aconstant (50 cm2/s).
The second model is the mixing length model (Fangand Ichiye, 1983) and expressed as:22 ∂u  ∂v 
Av = l 2   +   .
 ∂z 
 ∂z 

( 4)

The mixing length l is defined as:z+h l ( z ) = κ ( z + h ) 1 −,1( + s )h ( 5)

where κ (=0.4) is the Von Kármán’s constant, s (=1.2) aparameter expressing the roughness of the sea surface (Fangand Ichiye, 1983).
The third and fourth models relate Av to the water depth
and the vertical mean velocity (Davies et al., 1997), as:Av = 2. 5 × 10 −3 h

( u 2 + v 2 )Φ ( z )(6)where Φ(z) is a function of the depth. In the third model, it
is set to a constant (=1) and this model is referred to model
HU (a). In the fourth model, it is set to:if − h0 ≤ z
1Φ( z ) =  z + h − ( z + h0 )µ 0
if − h ≤ z < −h0 .
h − h0( 7)

Referring to Davies et al. (1997), h0 is set to 0.9 h, µ0
is set to 0.2. This model will be called model HU (b) in thispaper.
The stress at the sea surface is assumed to be zero andthe stress at the sea bottom is calculated by:Av∂ (u, v ) 1 x y
2
2=
τ b , τ b = C100 (u100 , v100 ) u100
+ v100
∂z
ρ

(

)

( 8)

where u100 and v100 are the velocity 1 m above the sea bottom
in the eastward and northward direction, respectively. C100
is the bed drag coefficient.
As pointed out by Davies et al. (1997), the bottom
friction stress in the three-dimensional model may also be
calculated from the depth mean velocity, similarly to that in
the horizontally two-dimensional model, as:Av∂ (u, v ) 1 x y=
τ b , τ b = Cb (u , v ) u 2 + v 2
∂z
ρ

(

)

(9)

where u and v are the depth mean velocity in the eastward
and northward direction, respectively. Cb is a coefficientsimilar to C100.
Due to the vertical resolution of the three-dimensionalmodel, the velocity 1 m above the sea bottom is difficult to
calculate in the model, and the velocity of the layer nearest
to the sea bottom (hereafter referred to the bottom layer) isoften used to replace it. However, no matter what coordinate
system is used (z or σ coordinate) in the numerical model,
this approximation means that the height of u100 and v100
above the sea bottom varies with the water depth. At someshallow places, the height of u100 and v100 may be less than
1 m, while at some deep places it may be larger than 1 m. In
fact, the error of the topography data is usually even larger
than 1 m. Therefore, the calculation of u100 and v100 in a threedimensional model is very difficult.
The topography of the model’s domain is shown in Fig.
1, in which the depth data are read from the chart. Depths
greater than 1000 m are set to 1000 m. This approximation
should have little influence on the propagation of tidal wave.
The grid size is 12.5 km × 12.5 km and the water column isdivided into 20 vertical layers.
The positions of the tide gauges and the corresponding
grids are plotted in the same figure. The harmonic constantsfor the major four tides can be found in Choi (1980) and will
be used to check our model’s results. Due to the limited
horizontal resolution, the positions of the corresponding
grids of tide gauges are somewhat different from their actual
positions. Apart from these tide gauges, some tidal current
observation stations are also shown in Fig. 1. Tidal current
data for these can be found in Larsen (1985) and Choi (1984,
1985, 1989). Table 2 summarizes these data, including thestation position, observation period, depth and the analyzed
harmonic constant of tidal current. It should be noted that
due to the horizontal and vertical resolution limitation, there
is a possibility that the depth of the corresponding grid is alittle different from the recorded station depth.
The above equations are solved by the finite difference
method (Guo and Yanagi, 1994). The open boundary is
Tidal Current in the East China Sea

653

654
X. Guo and T. Yanagi

Table 2. Observed tidal current data in Larsen et al. (1985), Choi (1984, 1985, 1989), in which the amplitude of tidal current (H) is in
cm/s and the local phase lag κ in degree and referred to longitude of current meter mooring stations.

Fig. 1. The topography of the East China Sea, the Yellow Sea and the Bohai Sea. The positions of the open boundary, tidal gauges andtheir corresponding grids, and the observed tidal current data are shown by the black circles, black triangles, plus symbol and blackstars, respectively.

located at the Taiwan Strait, offshore of Ryukyu Islands andthe Korea/Tsushima Strait (Fig. 1), along which the known
tidal harmonic constants are given. The necessary harmonic
constants along the open boundary are read from the co-tidal
and co-range charts presented by Nishida (1980). The major
four tides of M2, S2 , K1 and O1 are selected for reproduction.
The calculations for these tides last for five periods and theharmonic analysis is done in the last period.
3. Result
To clarify the response of the model to the parameters,
we designed 12 case runs for each tide, as shown in Table 3.

At first, the bed drag coefficient is changed from 0.0030 to
0.0010 to find the most suitable value (case 1–case 5). Thendifferent vertical eddy viscosity models are used (case 6–
case 12). The bottom friction stress is calculated from thedepth mean velocity (case 1–case 8) and the velocity of thebottom layer (case 9–case 12), respectively.
The calculated results are compared with the observed
data in three ways. The first way is to compare the calculated
harmonic constants of tidal elevation with the observed ones
at 55 tide gauge stations (Choi, 1980). A parameter Hs,
defined in Davies et al. (1997), is used in the comparison,

Tidal Current in the East China Sea

655

Table 3. Calculated cases for M2, S 2, K1 and O1 tides.

τb

Case

1
2
3
4
5
6
7
8
9
10
11
12

Av

Sea bed drag coefficientEq. (9)Eq. (9)Eq. (9)Eq. (9)Eq. (9)Eq. (9)Eq. (9)Eq. (9)Eq. (8)Eq. (8)Eq. (8)Eq. (8)

M2

S2

K1

O1

0.00300.00250.00200.00150.00100.00150.00150.00150.00150.00150.00150.00150.00300.00250.00200.00150.00100.00300.00300.00300.00300.00300.00300.00300.00300.00250.00200.00150.00100.00300.00300.00300.00300.00300.00300.00300.00300.00250.00200.00150.00100.00300.00300.00300.00300.00300.00300.003050 cm 2 /s
50 cm 2 /s
50 cm 2 /s
50 cm 2 /s
50 cm 2 /s
Mixing length
HU (a)HU (b)
50 cm 2 /s
Mixing length
HU (a)HU (b)

Table 4. Comparison between the observed and calculated M2, S2, K1 and O1 tides and tidal currents, in which Hs defined by Eq. (10);H = water elevation; U = u-component of tidal current; V = v-component of tidal current; A = Amphidromic points of M2 and S2
tides in Liaodong Bay.

M2

Case

1
2
3
4
5
6
7
8
9
10
11
12

S2

K1

Hs-H

Hs -U

Hs -V

A

Hs-H

Hs-U

Hs -V

A

Hs -H

Hs-U

Hs-V

Hs -H

Hs -U

Hs -V

65.2
63.4
62.0
61.4
63.0
89.6
61.8
65.5
70.9
68.6
62.6
61.5

24.9
22.9
19.2
19.2
19.2
20.3
19.2
13.3
22.0
13.7
21.4
17.3

30.4
28.9
27.2
23.1
21.7
20.0
20.8
18.8
24.0
19.3
23.0
21.4

No
No
No
Yes
Yes
No
Yes
No
Yes
No
Yes
Yes

23.2
24.0
25.3
27.2
30.3
30.6
23.1
23.8
28.7
24.8
25.6
24.4

9.2
9.6
10.2
11.0
12.1
10.1
9.1
7.4
11.3
8.1
10.7
9.3

10.5
10.4
12.1
10.8
11.5
9.5
9.6
8.9
10.8
9.8
10.7
10.2

Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes

13.6
14.2
14.9
16.0
17.8
16.4
12.7
13.2
14.2
13.5
13.9
13.7

5.6
5.8
6.0
6.3
6.8
5.2
5.7
4.7
6.4
5.7
6.3
5.8

5.6
5.7
5.8
5.9
6.2
6.5
6.0
5.6
6.4
6.2
6.3
6.1

7.8
8.2
8.7
9.4
10.4
7.8
7.7
7.0
9.3
7.6
9.0
8.3

4.6
4.7
4.8
5.0
5.1
4.6
4.6
4.1
5.0
4.8
5.0
4.7

4.1
4.3
4.5
4.8
5.1
4.0
3.9
3.6
4.6
4.3
4.4
4.3

Hs =

1 N
∑
N i =1( HCi2 + HSi2 )(10 )HC = Aocos(αo) – Accos(αc)
HS = Aosin(αo) – Acsin(αc)
where, Ao and αo are the observed tidal amplitude and phaseat the tide gauge station; Ac and αc the calculated ones at thestations for comparison. Values of Hs for the 12 cases areshown in Table 4.
The second way is the comparison of the calculated
tidal current with the observed data summarized in Table 2.

656

O1

X. Guo and T. Yanagi

Similarly, the Hs of the u and v components of the tidal
current are calculated and also shown in Table 4.
The third way is the comparison of the co-tidal and corange charts based on the observed data and calculated
results (Fig. 2). Since some cases cannot reproduce theamphidromic point of M2 tide in Liaodong Bay, the appearance or not of this amphidromic point is presented inTable 4 too.
3.1 Comparison of the model results and observed data
3.1.1 M2 tide
Since many papers have suggested the existance of theamphidromic point in Liaodong bay (Ogura, 1933; Nishida,1980; Fang, 1986) and some models have failed to reproduceit (Choi, 1980, 1984, 1989; Yanagi and Inoue, 1994), whether
or not it is reproduced will be an important factor in choosing
the model parameters. With the decrease of the bed dragcoefficient (0.0030 to 0.0015), the calculated M2 tidal elevation along the coast increases and Hs-H, the difference
between the simulated and observed data, becomes small, as
shown in Table 4. As the bed drag coefficient is equal to0.0015, the amphidromic point in Liaodong Bay begins toappear. If the bed drag coefficient decreases further, Hs-H
becomes large again, but the amphidromic point in Liaodong
bay can be reproduced clearly. Therefore, the bed dragcoefficient for M2 tide simulation is decided as 0.0015.
Using the same bed drag coefficient (0.0015), theconstant (case 4) and HU (a) vertical eddy viscosity models
can reproduce the amphidromic point in Liaodong Bay
while the mixing length and HU (b) models cannot. Furthermore the Hs-H of the former two cases are smaller thanthe latter two cases. However, as for the reproduction of thetidal current, the mixing length and HU (b) models are better
than or at least as good as the constant and HU (a) models.
Therefore the decrease of the vertical eddy viscosity near thesea bottom in the mixing length and HU (b) models seems
to be good for the reproduction of tidal current, especiallythat near the sea bottom, but not good for the tidal elevation.
Instead of Eq. (9), Eq. (8) is used to calculate the bottom
friction stress in cases 9–12. Obviously, the influence of thebottom friction stress on the reproduced vertical profile ofthe tidal current becomes more important since the velocityof the bottom layer is used to calculate the bottom friction
stress directly. From Table 4, we see that cases 9 and 11 giveworse results than cases 4 and 7, while cases 10 and 12 give
better results than cases 6 and 8. The reason for this difference is thought to be that the models with the decreasing
eddy viscosity near the sea bottom better reproduce the tidal
current near the sea bottom than the vertically constant eddy
viscosity models.
3.1.2 S2 tide
The simulation of the S2 tide shows a different responseto the change of the bed drag coefficient than M2 tidesimulation. The decrease of the bed drag coefficient from0.0030 causes the model results to deviate further from theobserved results. The amphidromic point in Liaodong Bay
can be reproduced in all cases. As the bed drag coefficientincreases above 0.0030 (not shown here), the amphidromic
point becomes unclear and Hs for the tidal currents also
increases. Therefore the bed drag coefficient for S2 tidesimulation was decided to be 0.0030.
As the vertical eddy viscosity model and the bottom
friction stress calculation are changed, the S2 tide simulation
is not so sensitive as the M2 tide simulation. However, themixing length model gives worse result than the othermodels.
3.1.3 K1 and O1 tides
K1 and O1 tide simulations show the same response to

the change of the bed drag coefficient and their responses are
similar to S2 tide simulation. A value of 0.0030 is used inthese two tide simulations.
With the exception of the mixing length model, theother three vertical eddy viscosity models do not give very
different results. This is because these components of tidal
currents are weak and their vertical shears are small, compared to the M2 tidal currents. The reason for the poor resultof the mixing length model will be mentioned below.
3.2 Co-tidal and co-range charts
Figure 2 shows the observed co-tidal and co-range
charts of M2, S2, K1 and O1 tide (Nishida, 1980) and thecalculated values according to case 7. As for the M2 and S2
tides (Figs. 2(a) and (b)), except for the Bohai Sea, where theamplitude is underestimated, the calculated results are wellconsistent with the observed ones. Four amphidromic points
are reproduced, in which the positions of two amphidromic
points in the Yellow Sea are almost the same as the observed
ones, while some deviation exists in the positions of theother two amphidromic points in the Bohai Sea, which may
be attributed to the model’s resolution, which cannot represent
the coast and topography well.
As for the K1 and O1 tides (Figs. 2(c) and (d)), also
except for the Bohai Sea, where the amplitude is overestimated, the model reproduces the observed data well. Referring the model results of M2 and S2 tides, it can be concluded
that our model tends to underestimate the semi-diurnal tides
in the Bohai Sea but to overestimate the diurnal tides there.
The same trend also exists in the results of the horizontally
two-dimension model (Yanagi and Inoue, 1994), whose
topography data is the same as that used in the present
model. Therefore the error in reading depth data from thechart and the choice of the grids along the coast line may be
a possible reason for such under- or over-estimation.
3.3 The horizontal distribution of tidal current
Figure 3 shows the horizontal distribution of tidal
current ellipses of four major tides at the sea surface.
Basically, the distributions of M2 and K1 tidal currents are
similar to those of S2 and O1, respectively. The strongest
semi-diurnal tidal currents appear at the offshore area of
Changjiang River Mouth and the western Korean coast,
while the strongest diurnal tidal currents appear in Bohai
Sea. Note that the major and minor axes of the M2 and S2
tidal current ellipses at the offshore area of ChangjiangRiver Mouth are nearly the same, so the tidal mixing thereis therefore strong at any time. But on the continental shelf(water depth less than 200 m), the major axis is clearly
longer than the minor axis, which means that the tidal
mixing across the depth contour is stronger than that alongit.
The horizontal distributions of tidal currents presented
here are nearly the same as those calculated by the horizontal

Tidal Current in the East China Sea

657

Fig. 2. The observed (left) and calculated (right) co-tidal (solid line) and co-range (broken line) charts of the M2, S2, K1 and O1 tides.
The observed ones are after Yanagi and Inoue (1994), which are based on Nishida (1980).

658

X. Guo and T. Yanagi

Fig. 2. (continued).

Tidal Current in the East China Sea

659

Fig. 3. The horizontal distribution of the M2, S2 , K1 and O1 tidal current ellipses on the sea surface.

660

X. Guo and T. Yanagi

two-dimensional models (Choi, 1980; Zhao et al.,1994). As
the results of a three-dimensional tidal model, we want to
pay more attentions to the vertical variation of tidal current.
3.4 The vertical profiles of tidal current at some stations
Among 15 stations in Larsen et al. (1985) and Choi
(1984, 1985, 1989), four stations are chosen to represent theshallow water, the continental shelf, the shelf edge and the
Yellow Sea. The observed M2 and K1 tidal currents data andthe vertical profile of calculated M2 and K1 tidal currents byfour vertical eddy viscosity models are presented in Figs. 4
and 5, respectively.
3.4.1 M2 tidal current
a. Shallow water
Stn. M2 represents the typical shallow water tidal
current, with a depth of only 15 m and a very strong tidal
current (60–100 cm/s). The calculated current there is very
sensitive to the vertical eddy viscosity. The constant verticaleddy viscosity model (case 4) produces a large verticalcurrent shear, while the other ones cause the current to varysmoothly from the sea bottom to the sea surface, as shownin the upper panel of Fig. 4. Such a difference is explainedby the fact that the vertical eddy viscosity (50 cm2/s) used in
the constant model is so small that the bottom friction stressproduces a large vertical current shear. In the HU (a) (case7), HU (b) (case 8) and mixing length models (case 6), thecalculated vertical eddy viscosities are large. So the bottom
friction stress only produces a small current shear in thevertical.
b. Continental shelfStn. MS represents the tidal current on the continental
shelf, whose depth is about 50 m and where the tidal current
is relatively large (30–50 cm/s). Compared to Stn. M2, thecalculated tidal current is not so sensitive to the vertical eddy
viscosity. However, the constant vertical eddy viscosity
model (case 4) gives an unnatural current profile below the
mid-depth, as shown in the second panel of Fig. 4, which
may be attributed to the small vertical eddy viscosity too. Itshould be noted that such an unnatural profile can also befound at Stn. M5, Stn. M7 and Stn. CM7. At these stations,the bottom friction stress is not so large as that at Stn. M2 dueto the decrease of tidal current. And the increased depthmeans that the bottom friction stress only influences a smallpart of the water column, while at Stn. M2 it influences thewhole water column. Consequently, the large vertical current shear only appears near the sea bottom. Beyond therange influenced by the bottom friction stress, there is no
factor to produce the vertical shear. So the current tends tokeep the same magnitude in the vertical direction. On theother hand, the other three models result in a large verticaleddy viscosity, which distributes the influence of the bottom
friction stress to a large range of the water column and thecurrent varies smoothly from the sea bottom to the surface.

c. Shelf edge
At the shelf edge (Stn. SB), the calculated current is notsensitive to the vertical eddy viscosity due to its weak tidal
current, which produces a small bottom friction stress and asmall vertical eddy viscosity in the HU (a) (case 7), HU (b)(case 8) and mixing length (case 6) models. However, thecalculated current is smaller than the observed one, nomatter what the u and v components are. Two possibilities
may be considered. The first one is that the numerical model
does not represent the topography there well. The second
one is that the baroclinic tidal current accompanied by theinternal tide is probably included in the observed data, but is
not included in the numerical model. We prefer the second
explanation because of the topography there and the periodof observation (June, 1980).
d. Yellow Sea
Stn. F is located on the Yellow Sea, close to the western
Korean coast. As shown in Fig. 3, the M2 tidal current there
has a strong north-south component, which can be found in
Fig. 4 too, in which the v component is clearly larger than the
u component. The vertical profile of the v component at thisstation is very similar to those on the continental shelf, andthe constant vertical eddy viscosity model (case 4) produces
the unnatural distribution, too. However, the u component at
Stn. F shows a different profile. Its amplitude becomes largenear the sea bottom. Such a profile is also shown in theobserved data there and can be found in the calculated
results at Stn. B, Stn. D and Stn. I too. A detailed discussion
will be given below.
3.4.2 K1 tidal current
The vertical distributions of K1 tidal currents at four
stations are shown in Fig. 5. On the whole, the K1 tidal
current is not sensitive to the change of the vertical eddy
viscosity. In fact, since the K1 tidal current is very much
weaker than the M2 tidal current, the vertical eddy viscosities
calculated by the four models do not have such a great
difference as those in the M2 tide calculations. Also, as thecurrent is weak, the bottom friction stress is small too, which
in turn means a small vertical current shear. Thus, theunnatural vertical profile as seen in the M2 tidal current at
Stn. MS produced by the constant vertical eddy viscosity
model does not appear here.
In the shallow water (Stn. M2), the K1 tidal current hasa relatively large variation with the change of the verticaleddy viscosity, as shown in Fig. 5. The HU (a) model (case7) gives better results than the others. On the continental
shelf (Stn. MS), the model overestimates the observed
current somewhat. At the shelf edge (Stn. SB), the model
reproduces the observed current well. In the Yellow Sea(Stn. F), the K1 tidal currents have similar characteristics as
the M2 tidal current, that is, the v component is larger thanthe u component by one or two times and the u component
becomes large near the sea bottom. We will discuss thisissue in some detail later.

Tidal Current in the East China Sea

661

Fig. 4. The vertical profiles of the amplitudes and phases of u and v components of the M2 tidal currents at 4 stations Stn. M2, Stn. MS,
Stn. SB and Stn. F whose positions are shown in Fig. 1. The different types of line represent the results of different cases as shown
above the upper panel. The observed tidal current data are shown by the black stars.

3.5 The vertical distribution of tidal current along a section
Although we know something about the vertical distribution of tidal current at some points, we still have no idea
about the vertical distribution of tidal current over the wholedomain. Therefore, we chose a vertical section in the domain
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X. Guo and T. Yanagi

as shown in Fig. 6 to see how the tidal current varies along
this section.
The amplitudes of u and v components of M2 and K1
tidal current calculated by the HU (a) model (case 7) isshown in Fig. 7. The regions where the vertical shear of

Fig. 5. The vertical profiles of the amplitudes and phases of u and v components of the K1 tidal currents at the same 4 stations as Fig. 4.

current is large is section C–D for M2 tidal current, section
B–C for K1 tidal current. Obviously, the current varies
greatly in the vertical when the current itself is strong. So
from the horizontal distribution of tidal current ellipses (Fig.
3), we can deduce the area where the vertical shear of tidal
current is large.

4. Discussion
4.1 Reproduction of the amphidromic point in Liaodong
Bay
There are only two different points in the calculations
for the M2 and S2 tides. One is their periods and another is
Tidal Current in the East China Sea

663

Fig. 6. The position of a chosen section along which the amplitude
and phase of the M2 and K1 tidal currents are shown in Fig. 7.

the harmonic constants along the open boundary. The question is, why cannot the amphidromic point of M2 tide in
Liaodong Bay be reproduced in the large bed drag coefficient cases, while the amphidromic point of S2 tide can be
reproduced under the same bed drag coefficient? To answer
this question, we performed the following two experiments,
in which the bed drag coefficient is kept as 0.0025. The firstone was to use the S2 tidal period in the M2 tide calculation.
This calculation is the same as the S2 tide simulation using
the harmonic constants of M2 tide along the open boundary.
The result is that the amphidromic point in Liaodong Bay
cannot be reproduced. The second experiment consisted ofusing the harmonic constants of S2 tide along the open
boundary in M2 tide calculation. In this experiment, theamphidromic point in Liaodong Bay is reproduced.
So it is the harmonic constants along open boundarythat influence the appearance of the amphidromic point in
Liaodong Bay. We consider the physical explanation to bethat the large amplitude of M2 tide produces a strong tidal
current, which in turn induces a larger bottom friction stressthan the S2 tide, if the same bed drag coefficient is used. Thislarger bottom friction stress moves the amphidromic point
in Liaodong Bay from the central axis of the bay too far, thatis, into the land. When the bed drag coefficient decreases or
the tidal current becomes weak, the bottom friction stressdecreases and the amphidromic point appears in Liaodong
Bay.
In order to gain more support for this explanation, we
carried out two further experiments using the M2 tide period.

Fig. 7. The distributions of the amplitudes and phases of u and v components of M2 tidal current (a) and those of K1 tidal current (b).
664

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The first one uses the S2 tide phase data and the M2 tide
amplitude data along the open boundary. The second one
uses the S2 tide amplitude data and M2 tide phase data along
the open boundary. The bed drag coefficient is still kept as
0.0025. The calculated results show that the amphidromic
point in Liaodong Bay cannot appear in the first experiment
but can appear in the second experiment, which means thatthe phase data of M2 or S2 tide along the open boundary havelittle influence on the reproduction of the amphidromic
point in Liaodong Bay, but the amplitude data along theopen boundary, which in turn the tidal current and thebottom friction stress, influence the reproduction of thatpoint mainly.
It was also stated that the poor grid resolution and theunnatural orientation of the grid system with respect to thecoast’s shape should be responsible for the failure in reproducing the amphidromic point in Liaodong Bay (Larsen
et al., 1985). However, the improvement of the model’s
resolution from 25 km to 12.5 km and the modification of thecoastline using the fine grids did not introduce any clear
improvement of the results over the calculated result using
the coarse grids. Therefore the main factor influencing the
reproduction of the amphidromic point of M2 tide in Liaodong
Bay should be the bottom friction stress.
4.2 The unique vertical profile of the tidal current at Stn. F
Usually, the tidal current becomes weak near the sea
bottom due to the effect of bottom friction. The profiles ofu component of tidal current at the stations in the Yellow Sea

such as Stn. F are therefore unique. The question naturally
arises of whether such profiles really exist in the nature andwhy the model can produce such profiles. In the present stateof our knowledge, it is difficult to answer the first question
because the observed data are scarce in the vertical. However, we may try to answer the second question by analyzing
the calculated results in detail.
Using a linear model, the momentum equation about
the u component, neglecting the horizontal viscosity, may
be expressed as:∂u∂η ∂  ∂ u = fv − g
+
 Av
.
∂t
∂x ∂z  ∂z 

(11)

The tidal current is controlled by the Coriolis force,pressure gradient and the vertical shear stress. In otherwords, it may be explained that the tidal current is driven by
the sum of the three terms on the right, which are referred toas the driving force for convenience.
Figure 8 shows the u component of tidal current and thedriving force over one tidal period at Stn. M2 and Stn. F.
These results are obtained in case 4 for M2 tide without theadvective and horizontal viscosity terms. The periodic
variations of the current and the driving force and the 90
degree phase difference between them may be found clearlyat both stations. The difference between the two stations is
the vertical distribution of the current and the driving force.
From Eq. (11), we know that a strong tidal current must be

Fig. 7. (continued).
Tidal Current in the East China Sea

665

Fig. 8. The time variations in the M2 tidal cycle of the u component and the driving force in the x direction at Stn. M2 and Stn. F in
the case 4.

driven by a large driving force, while a small driving force
may only produce a weak tidal current. The vertical distributions of the driving forces and their corresponding current
in Fig. 8 explain this rule very well.
To find the reasons responsible for the large bottom
driving force at Stn. F, the time variation of the Coriolis
force, the pressure gradient, the difference between themand the vertical shear stress at Stn. M2 and Stn. F are plottedin Fig. 9. The variations of Coriolis force and pressure
gradient themselves have no special characteristic. But theirdifferences, especially the vertical distribution of their difference at the two stations, are very clear. Because the ucomponent at Stn. F is small, the vertical shear stress there
caused by the bottom friction stress is small too, and lessthan the difference of Coriolis force and pressure gradientby one order. On the other hand, the v component at Stn. F
is large, which induces a large Coriolis force in the u
direction. It may be said that the force balance at Stn. F ismainly the balance between the pressure gradient and the
Coriolis force. Therefore, the phase difference between the
Coriolis force and the pressure gradient and the decrease of
the Coriolis force near the sea bottom are the reasons
responsible to the large driving force at the sea bottom of
Stn. F, which in turn produces a strong tidal current there. It
should be noted that this unique profile is reproduced in
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X. Guo and T. Yanagi

cases 9–12 too.
The thickness of the bottom Ekman layer (BEL) relates
to this unique profile too. As the vertical eddy viscosity near
the bottom is as small as that in K1 tide simulations and cases
4 and 6 of M2 tide simulations, the BEL is thin. Then the ucomponent of tidal current becomes large near the bottom.
But in the cases 7 and 8 of M2 tide simulations, the verticaleddy viscosity near the bottom is large and makes the
thickness of BEL large too. So the unique profile cannot be
reproduced.
4.3 Summary of the four vertical eddy viscosity models
Concluding the results of the four vertical eddy viscosity
models, we know that the constant vertical eddy viscosity
model may be used in the shallow water, where the tidal
current is very strong and the current shear is large. As thewater depth increases and the tidal current become weak, theconstant vertical eddy viscosity model produces an unnatural
current profile near the sea bottom. Of course, by changing
the value of the constant, this unnatural current profile may
disappear. But it should be difficult for the constant verticaleddy viscosity model to reproduce the correct vertical profileof tidal current on the whole continental shelf well.
As for the mixing length model, it usually underestimates tidal current amplitude because it produces a large

Fig. 9. The time variations in the M2 tidal cycle of the pressure gradient, the Coriolis force, their difference and the vertical shear stressin x direction at Stn. M2 and Stn. F in the case 4.

interior friction on the continental shelf. In fact, deciding themixing length is the key point of this model. Since themixing length is basically proportional to the water depth, asmall current shear may produce a very large vertical eddy
viscosity in the deep water. Therefore, before we know more
about the mixing length, this model is not recommended for
use to simulate the tide and tidal current on the continental
shelf. At least, it is not suitable for the ECS and the Yellow
Sea.

As for the HU models, their performances are relatively good. Compared to the HU (a) model, model HU (b)produces a large vertical current shear near the sea bottom,
but cannot produce the amphidromic point of M2 tide in
Liaodong Bay. Therefore, as a conclusion, the HU (a) modelis thought to be the best vertical eddy viscosity model for
simulating the tide and tidal current in the ECS, the Yellow
Sea and the Bohai Sea.

Tidal Current in the East China Sea

667

5. Conclusion
Using a high resolution three-dimensional numericalmodel, the tides and tidal currents of M2 , S2, K1 and O1
constituents in the ECS and the Yellow Sea are reproduced
well. With 12.5 km resolution in the horizontal and 20 layers
in the vertical, the model presents the vertical distribution of
tidal currents in the ECS and the Yellow Sea for the firsttime.
Four vertical eddy viscosity models are used in the
numerical experiments. As the tidal current becomes strong,
its vertical shear becomes large and its vertical profile
becomes sensitive to the vertical eddy viscosity model. As
a conclusion, the HU (a) model, which relates the verticaleddy viscosity to the water depth and depth mean velocity,
gives the closest results to the observed data.
The reason for the reproduction of the amphidromic
point of M2 tide in Liaodong Bay has been discussed and is
attributed to the bottom friction stress calculation. The
model reproduces a unique vertical profile of tidal current in
the Yellow Sea, which is shown in the observed data too.
The reason for producing such a profile in the model has
been investigated.
Due to the limitations of the observed data, especiallythe observed tidal current data, some of the model resultscannot be verified in detail. On the other hand, it remains a
problem to be solved in the future, to include the baroclinic
tide and tidal current in the numerical model and to improve
the precision of topography data.
Acknowledgements
The authors are grateful to Mr. K. Inoue for providing
topographic data. The calculation was carried out on theworkstation of Department of Civil & Environmental Engineering, Ehime University, and the workstation of Ocean
Research Institute, University of Tokyo. The figures are
drawn by “DENNOU” graphic library. This study is a partof the MASFLEX Project sponsored by the Science andTechnology Agency of Japan.
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