Application of eddy viscosity closure models for the M2 tide and tidal currents in the Yellow Sea and the East China Sea

Continental Shelf Research 19 (1999) 445 — 475

Application of eddy viscosity closure models for the M tide and tidal currents in the Yellow  Sea and the East China Sea Jong Chan Lee, Kyung Tae Jung* Korea Ocean Research and Development Institute, Ansan P.O. Box 29, Seoul 425-600, South Korea Received 8 August 1997; accepted 13 January 1998

Abstract A three-dimensional mode-splitting, p-coordinate barotropic finite-difference model, with subgrid scale diffusion represented using a range of eddy viscosity closure models, is used to examine M tidal elevation and currents in the Yellow Sea and the East China Sea. Four eddy  viscosity formulations are considered: q-ql turbulence energy model (Blumberg and Mellor, 1987), Prandtl mixing length model, Davies and Furnes’ (1980) simple flow-related model with mixing length which includes the bottom boundary layer thickness, and a time and space invariant eddy viscosity with 650 cm/s. The bottom friction at the sea bed is given in a quadratic form using a constant bottom friction coefficient, c and near-bottom velocities. D A series of M tide model runs were carried out and optimal values of c were determined  D through the comparison with tidal elevation amplitudes and phases at 203 stations. From these comparisons it is shown that the M tidal charts computed with a range of eddy viscosity  formulations are in good agreement with each other when optimal values of c are chosen; D comparing with M tidal current amplitudes and phases at 15 stations, it is shown that tidal  current distributions and its profiles are in reasonably good agreement with winter-time observations in the central part of the Yellow Sea; relatively poor results are obtained near the Chinese coast where non-tidal effects such as abrupt changes in tidal current phase in the vertical due to large freshwater discharge are pronounced. It is noted that the bottom friction coefficient has a major influence on tidal elevation and tidal currents and optimal values of bottom friction coefficient are closely related to the near-bottom eddy viscosity. The considered eddy viscosity closure models appear to work well for tidal problem when the bottom friction parameter is optimized. Results indicate that for a barotropic tide the Prandtl mixing length model which can account of the boundary layer thickness could be an useful alternative to a highly complex q-ql model.  1999 Elsevier Science Ltd. All rights reserved.

*Corresponding author. 0278—4343/99/$ — See front matter  1999 Elsevier Science Ltd. All rights reserved. PII: S 02 7 8— 4 34 3 ( 98 ) 0 00 8 7— 9

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1. Introduction The Yellow Sea and the East China Sea (YSECS) are marginal seas of the northwestern Pacific Ocean, surrounded by the Korean Peninsula, the Chinese coasts and the chain of Ryukyu islands (Fig. 1a). Most of the seas are shallow except a steep continental slope adjacent to the Okinawa Trough in the southern part of the East China Sea (ECS). The region is most remarkable for the complexity of tidal pattern with large tidal range appearing along the west coast of Korea. Since Ogura (1933) presented tidal charts over Asian marginal seas based on a large number of tidal measurements, numerical simulations of tides in YSECS have been conducted by many Chinese, Japanese and Korean scientists. Choi and Fang (1993) reviewed previous works on tidal modelling of YSECS performed over last two decades including general description of tides based on empirical tidal charts. Most of the applied models are two-dimensional and highly successful. For many purpose, however, it is indeed necessary to compute tidal currents in three dimensions. There have been limited efforts of developing three-dimensional tidal models in YSECS. Choi (1984, 1990) reproduced M tide and tidal currents in YSECS  using a spectral model with the expansion of cosine functions. The equations of motion used were basically linear except for the quadratic bottom friction terms. Reasonable agreement between current meter data and model results was obtained using a constant eddy viscosity with the bottom friction coefficient which was close to that used in his two-dimensional modelling (Choi, 1980). Three-dimensional tidal modelling in YSECS is at its early stage in that the applied models are linear and that these models are based on the traditional method of parameterizing subgrid scale turbulence through a constant or flow-related eddy viscosity. As an effort of attaining information on the vertical structure of tidal currents in YSECS, a three-dimensional nonlinear barotropic tidal model is presented in this paper. The model uses the finite difference method in three-dimensions with a pcoordinate in the vertical direction, and spherical coordinates for the horizontal planes. The entire area of YSECS bounded by 117°E—130°E in longitude and 24°N—41°N in latitude is modelled with grid resolution of 1/6° in longitude and 1/8° in latitude. The vertical variation is represented by twelve layers with variable grid size. Numerical techniques such as mode-splitting, the alternating direction implicit method for the external mode and the implicit method for the internal mode are implemented. As an open boundary condition the radiation boundary condition developed by Flather (1976) and subsequently used by Davies and Furnes (1980) and Davies (1986) is used. The bottom friction at the sea bed is represented in a quadratic form using a constant bottom friction coefficient and near-bottom velocities although, strictly speaking, the coefficient of bottom friction depends upon bed roughness and a reference height (i.e. the height of bottom grid above the bed). Given a turbulence closure, the bottom friction coefficient is the only free parameter that can be used for model calibration (excluding the horizontal eddy diffusion terms). Davies and Xing (1995) and Davies and Gerritsen (1994) showed in Irish Sea calculations that agreement

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between observed and calculated water levels and current profiles depends upon both eddy viscosity closure and the bottom friction coefficient. For the vertical eddy viscosity which is critical in three-dimensional simulation, four formulations are considered: a turbulence energy closure model based on prognostic equations for both turbulence energy and mixing length (or turbulence macroscale), so-called two-equation q-ql model (Blumberg and Mellor, 1987), Prandtl mixing length model including the effect of boundary layer thickness, a simple flow-related eddy viscosity model (Davies and Furnes, 1980), and a specified vertical eddy viscosity with a constant value. In this paper emphasis is given on the influence of the bottom friction coefficient upon M tidal elevation and tidal currents in YSECS using a range of closure models  described above. The bottom friction coefficient is regarded as a part of eddy viscosity closure and, through the comparison with observed tidal elevation, the optimal values of the bottom friction coefficient is determined for each eddy viscosity formulation. A total of 203 M tidal elevation harmonics and a total of 29 M tidal current   harmonics at 15 stations are used for comparison. In addition, we try to understand the spatial variations of the vertical eddy viscosity induced by M tide (e.g. tide induced mean background turbulence) in YSECS.

2. A three-dimensional numerical model 2.1. Hydrodynamic equations The model assumes hydrostatic balance and homogeneous fluid. For spherical coordinates in horizontal directions and a normalized r-coordinate in the vertical direction, the nonlinear three-dimensional hydrodynamic equations become






u g 1 * K *u *u v *u u *u *f *u + # # # !fv"! # #F H R cos *j H *p H *p *t R cos *j R * H *p (1)






*v u *v v *v u *v g *f 1 * K *v + # # # #fu"! # #F ( *t R cos *j R * H *p R * H *p H *p





u(p)"!

N



*f 1 *Hu *Hv cos

# # dp *t R cos *j *

\ *f 1 *(HuN ) *(HvN cos ) # # "0 *t R cos *j *



(2) (3) (4)

In Eqs. (1)—(4), t is time, j and are East longitude and North latitude, p"(z!f)/(h#f) is a normalized vertical coordinate (sigma-coordinate), z is a vertical coordinate measured vertically upwards, f is the elevation of sea surface, h is water depth below the undisturbed surface, H is total water depth ("h#f), (u, v, u) are velocity components in (j, , p) directions, R is radius of the Earth, f is the Coriolis

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parameter ("2) sin ), ) is the speed of Earth’s rotation, g is the acceleration due to gravity, o is the density of sea water, (F , F ) are components of horizontal diffusion H ( terms in (j, ) directions, (qU, qU) are the components of wind stress, (q@ , q@ ) are the H ( H ( components of bottom stress, and overbar denotes depth averaged quantity (for example, uN " u dp, vN " v dp). \ \ Neglecting high order derivative terms and using a constant horizontal eddy coefficient A , horizontal momentum diffusion terms take a simple form + * * A (F , F )+ + # (u, v) (5) H ( R cos *j *   where is the mean latitude of the study area. 






2.2. Boundary conditions At land boundaries the component of depth mean currents normal to the boundary is set to zero, and along the open boundaries of the model a radiation condition developed by Flather (1976) is employed. Imposing a zero stress condition at the sea surface (p"0) gives











o *u *v "(qU,qU)"(0, 0) K ,K + + H ( H *p *p  Allowing the slip at the sea bottom (p"!1) gives

(6)

o *u *v K ,K "(q@ ,q@ ) (7) + + H ( *p *p \ H Relating the bottom stresses, through a quadratic friction law, to the bottom current one has (q@ ,q@ )"oc (u#v (u , v ) (8) D @ @ @ @ H ( where c is the coefficient of bottom friction, (u ,v ) are the horizontal components of D @ @ near-bottom velocities (velocities in the first gridpoint above the bottom). Strictly speaking, the coefficient of bottom friction depends upon the horizontally varying reference height (it is usually taken as half gridsize at the sea bed) and the roughness length z which is closely related to the bed types and forms. Though use of a variable  bottom friction coefficient is more physically attractive, we consider a constant bottom friction coefficient over the whole region because correct prescription of z is  difficult and, according to Aldridge and Davies (1993), its role in coarse grid modeling might be less important. In a mode-splitted tidal model, calculation of bottom stress can be a delicate problem because the magnitude and direction of bottom currents can change rapidly. In this study, the bottom stresses for the external mode are computed according to Martin and Delhez (1994). q "o c "u "(uL #uN ) @ D @ @

(9)

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where "u " is the total velocity at the near-bottom layer, uL is deviation of the bottom @ @ velocity from the depth averaged one at the recent internal mode time step, uN (depth averaged velocity) varies from one external mode time step to the other external mode time step. 2.3. Vertical eddy viscosity formulations Three-dimensional hydrodynamic models allow the computation of vertical shear of current but determination of the vertical eddy viscosity K representing the + turbulent momentum flux in the vertical still poses a major problem. For the physically realistic parameterization of K , a number of turbulence closure models + have been developed (Blumberg and Mellor, 1987; ASCE, 1988). Four vertical eddy viscosity formulations are considered: (a) two-equation turbulence closure model (TCM1) (b) Prandtl mixing length model derived from the q-ql equation (TCM2), (c) flow-related eddy viscosity model (EVM1) and (d) a specified constant eddy viscosity model (EVM2). 2.3.1. Two-equation turbulence closure model (TCM1) The first turbulence model considered is Blumberg and Mellor’s (1987) q-ql model, classified as a second-order turbulence closure model, in which the turbulent kinetic energy (q/2) and length scale (l) are determined by transport equations with source and sink terms. Details of q-ql model including the boundary conditions and the form of the wall proximity function are described by Blumberg and Mellor (1987) and Galperin et al. (1988). The vertical eddy viscosity in q-ql model is calculated as K "l q S + +

(10)

where S is a stability function. For the present study dealing with homogeneous + flows, it is fixed to 0.3920 ("16.6\) according to Blumberg et al. (1992). 2.3.2. Prandtl mixing length model derived from the q2-q2l equation (TCM2) The vertical eddy viscosity in traditional Prandtl’s mixing length model is calculated as





l K " + H

*u  *v  # *p *p

(11)

The above equation can be derived from the q equation also under the assumption of local equilibrium of production and dissipation of q. The length scale l in TCM2 is given by l"l ) MIN(C ,1.0) N *

(12)

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with l "0.3"p"(1!"p")H"l H (13) N N *u  *v  0.1i # (14) C " * pH *p *p Q where i is Von Karman constant ("0.4), and p ("10\ s\) is a frequency Q characteristic of tidal flow. The length scale l can be derived from the ql equation under the assumption of N local equilibrium and has a parabolic form with respect to water depth. Other types of the length scale l in the q!ql model for homogeneous flows can be referred to Lee N and Jung (1997). The second term modifies the length scale to incorporate the bottom boundary layer thickness. This model has been used previously to simulate winddriven currents in YSECS by Lee (1995), and detailed derivation and application of TCM2 can be referred to Lee and Jung (1996).





2.3.3. Flow-related eddy viscosity model (EVM1) Based on Irish Sea observations, Bowden et al. (1959) presented the eddy viscosity related to the depth-averaged velocity, that is K "a"uN "l (15) + where a is a constant (of order 2.5;10\), and l is the length scale taken as the water depth H. Davies and Furnes (1980) suggested an alternative formulation of l which is limited in deep water by the bottom boundary layer thickness. Davies and Aldridge (1993) defined the boundary layer thickness as d"c u /p (16) @ *@ Q where c is a constant of order 0.3 (c was set to 0.4 in this study) and u is a bottom @ @ *@ frictional velocity. In this study, l is set to MIN(H, d). 2.3.4. A specified constant eddy viscosity model (EVM2) The simplest way of representing the internal transfer of momentum is using a constant eddy viscosity throughout the depth. Choi (1984, 1990) reproduced tidal elevation and currents in YSESC with reasonable accuracy using this formulation. A constant eddy viscosity of 650 cm/s was also used in meteorologically-induced circulation in YSECS by Choi and Suh (1992). Correspondingly, the same value of eddy viscosity is used in calculations with EVM2.

3. Application to the Yellow Sea and the East China Sea 3.1. Numerical aspects The study area includes the whole YSECS bounded by longitudes 117°E and 130°E, and by latitudes 24°N and 41°N. The depth of deep region ('30 m) is obtained from

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DBDB5 set, while the depth of shallow region is constructed using bathymetric charts published by Hydrographic Office of Korea. Fig. 1a shows the bottom topography of YSECS where the locations of open boundaries are marked by cross symbols. The dotted lines represent isobaths between 200 and 500 m with a contour interval of

Fig. 1. (a) Bathymetric map of the Yellow Sea and the East China Sea. Open boundaries are marked by cross symbols. (b) Locations where tidal harmonics are available for comparison. Open circles denote locations for tidal elevation data and cross symbols denote locations for current data.

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Fig. 1. (Continued.)

100 m, and the solid lines represent isobaths shallower than 100 m with a contour interval of 20 m and isobaths deeper than 1,000 m with an interval of 1,000 m. The continental slope is located on dotted lines connecting Taiwan and Kyushu island. The maximum model depth is greater than 6,000 m on the edge of the northwestern Pacific Ocean.

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453

A standard finite difference method based on the Arakawa-C grid system was used with an alternating direction implicit method for the external mode and an implicit method for the internal mode. The horizontal grid size is 1/6° in longitude and 1/8° in latitude, and the vertical grid sizes in p-coordinate are +0.05, 0.05, 0.075, 0.075, 0.10, 0.15, 0.15, 0.10, 0.075, 0.075, 0.05, 0.05. The external and internal time steps were 186.31 sec and 931.55 sec, respectively. The detailed description of the numerical scheme can be found in Lee (1995). The coefficient of horizontal diffusion is set to a constant value of 250 m/s. The model was forced with the same M tidal input  which was identical to that used by Lee and Jung (1996a). The solutions are generated from a state of zero elevation and no motion. Time integration is carried out for 20 M tidal cycles. To calculate amplitudes and phase lags for tidal elevation and current,  model results at the last cycle stored in an interval of 0.5 lunar hour were analyzed by the Fourier method. In this study the coefficient of bottom friction is assumed to be time and space invariant. Strictly speaking, the coefficient of bottom friction depends upon the horizontally varying reference height (usually taken as half gridsize at the sea bed in Davies and Gerritsen (1994)) and the roughness length z which is closely related to  the bed types and forms (Soulsby, 1983). Davies and Gerritsen (1994) used the roughness height derived from the Chezy coefficient used in the depth-averaged calculation. Though use of a variable bottom friction coefficient is more physically attractive, we consider, in this first stage application of 3D model to YSECS, a constant bottom friction coefficient, which is equivalent to assuming that H/z  remains constant throughout the domain. Correct prescription of z is in fact very  difficult and, according to Aldridge and Davies (1993), its role in coarse grid modeling might be less important. 3.2. Results for water levels 3.2.1. Error statistics of M2 harmonics As addressed by Prandle (1982), a few of model runs were sufficient to find that the bottom friction coefficient has a major influence on tidal elevation and tidal currents along with the eddy viscosity closure. Initially we therefore carry out a series of runs to determine optimal values of the bottom friction coefficient for each eddy viscosity formulation through the comparison of computed and observed M tidal elevation  harmonics. Values of c were varied from 0.0025 to 0.01 with an interval of 0.00125. D Error statistics (mean and root-mean-square errors) are given in Table 1a for the amplitude and in Table 1b for the phase lag. Errors are defined by computed minus observed ones. A total of 203 points (marked by open circles in Fig. 1b) was used for the comparison. From the statistics, it is clear that turbulence models considered overpredict or underpredict the tidal amplitude with different c values. This is related to the D decrease or increase in the rate of bottom frictional dissipation. For a given value of c , TCM1 and TCM2 give lower dissipation of tidal energy than EVM1 and EVM2. It D is noted that phase lags are much less sensitive to the change in the bottom friction coefficient and the eddy viscosity formulation although all of the turbulence models

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Table 1a Error(computed minus observed) statistics of M tidal amplitude (unit: cm)  TCM1 mean rmse c "0.0025 D c "0.0038 D c "0.0050 D c "0.0063 D c "0.0075 D c "0.0088 D c "0.0100 D

19.57 10.75 5.47 1.33 !1.61 !4.17 !6.13

23.90 16.53 13.94 13.55 14.27 15.46 16.67

TCM2 mean rmse 19.66 10.88 5.62 1.50 !1.43 !3.98 !5.93

24.38 17.12 14.52 14.05 14.65 15.74 16.86

EVM1 mean rmse 10.62 2.09 !2.87 !6.69 !9.34 !11.61 !13.32

16.62 14.04 15.46 17.72 19.69 21.55 23.02

EVM2 mean rmse 8.97 !0.41 !6.03 !10.43 !13.56 !16.28 !18.34

15.71 14.18 16.81 20.05 22.73 25.21 27.17

Table 1b Error (computed minus observed) statistics of M tidal phase (referred to 135°E, unit: deg.) 

c "0.0025 D c "0.0038 D c "0.0050 D c "0.0063 D c "0.0075 D c "0.0088 D c "0.0100 D

TCM1 mean rmse

TCM2 mean rmse

EVM1 mean rmse

EVM2 mean rmse

1.77 2.08 2.34 2.59 2.81 3.03 3.22

1.91 2.25 2.53 2.81 3.05 3.28 3.49

1.16 1.53 1.87 2.24 2.56 2.90 3.19

1.61 2.09 2.48 2.88 3.23 3.62 4.00

15.39 15.05 14.97 14.98 15.01 15.07 15.12

15.39 15.03 14.94 14.93 14.96 15.01 15.06

15.67 15.67 15.73 15.80 15.85 15.90 15.94

15.85 16.33 16.78 17.22 17.60 18.03 18.54

show tendency to overpredict the phase lags. The optimal values of c (in a sense of D minimum rms error in tidal elevation amplitude) with TCM1 and TCM2 lie between 0.0063 and 0.0075, EVM1 between 0.0038 and 0.0050 and EVM2 about 0.0038. Among turbulence models EVM2 has the optimal value of c most close to that D frequently used in depth-averaged 2D tidal models. From the rms errors, we see that, if optimal values of c are chosen, the computed D and observed values of tidal amplitudes and phases agree each other within about $15 cm and $15°, respectively, independently of the eddy viscosity formulations. The difference in results with TCM1 and TCM2 is marginal. The behavior of EVM1 and EVM2 appears to be similar in terms of error statistics although EVM2 gives slightly larger dissipation than EVM1. Fig. 2 shows a scatter plot of amplitudes and phase lags obtained using TCM2 with c "0.0063 which is close to the optimal value. D It is clearly shown that there is no apparent bias in amplitudes and phase lags. 3.2.2. Co-phase and co-amplitude charts Fig. 3a and b is the M tidal chart obtained using TCM1 with c "0.0063 and  D EVM2 with c "0.0038 which are close to the optimal value. We can see that the D overall feature of the M tidal charts computed with two different eddy viscosity  formulations are not significantly different each other. The model results are in good agreements with the existing tidal charts derived either empirically (Nishida, 1980;

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Fig. 2. Comparison between observed and calculated values. Amplitude and phase lags of M tide are  computed using Prandtl mixing length model (TCM2) with c "0.0063. D

Fang, 1986; Odamaki, 1989) or from a simple data-assimilative 2D model (Lee and Jung, 1996b) showing clearly formation of three amphidromic points in the Yellow Sea (YS) and one deamphidromic point appearing in Liatung Bay. Although model results are not presented here, we could see that the amphidrome in YS moves eastwards as the value of c reduces. D Closer examination reveals that there are some differences in model results along most of the Chinese coast and the southern coast of Korea and at ECS. Discrepancy at the offshore is noticeable in the spatial distribution of co-amplitude lines of 60 cm (in southern part of ECS) and 80 cm (around the Cheju island and the Chinese side of ECS). Furthermore, we note that both models clearly overpredict tidal elevations around Cheju island. The amplitudes of observed M tide in the northern and  southern parts of Cheju island are 71.3 and 77.0 cm, respectively (see Choi (1980)). The same tendency appeared in previous 2D calculation (Lee and Jung, 1996b). Adjustment of the quadratic bottom friction coefficient did not improve the result unless a data assimilation technique was introduced. 3.3. Results for tidal currents 3.3.1. Comparison of M2 tidal currents at selected points Although it is important to check whether the model can reproduce the general feature of the M tide, the major concern in using a 3D tidal model is its ability of  reproducing current profiles. The M tidal current harmonics obtained using TCM1  and TCM2 with c "0.0063 and EVM1 and EVM2 with c "0.0038 are compared D D with two tidal current data set (attempt to obtain optimal value of c for tidal currents D is not made because of the lack of data). The first data set (C1—C9) came from a joint China—U.S.A. study on the effluent from the Changjiang Estuary and its effect on ECS

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with moorings during summer season in 1980 and 1981. A detailed description of data could be referred to Larsen et al. (1985). It is noted that many of records in this set are far shorter than 30 days which is conventionally required for an accurate tidal analysis: the data lengths of C1—C5 vary from 8 to 12 days, and the data length of C7,

Fig. 3. M tidal chart computed using (a) q-ql model (TCM1) with c "0.0063, (b) a constant eddy  D viscosity of 650 cm/s (EVM2) with c "0.0038. Degrees refer to 135°E. D

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Fig. 3. (Continued.)

C8 and C6 are 20, 23 and 31days, respectively. The water depth of each station was not reported in Larsen et al. (1985). It was possible to find the water depths of C3 and C8 from Sternberg et al. (1983), and C6 and C7 from Beardsley et al. (1983); C3 47.3 m, C6 188 m, C7 49 m, and C8 39.3 m, respectively. Estimation of water depth made on a bathymetric chart in terms of longitudes and latitudes gives 39 m for C4 and 58 m

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for C5. It was hardly possible to estimate the water depths of C1, C2 and C9 due to excessive complexity of bottom topography. The second data set (K1—K6) obtained using burst sampling current Meters during winter season (January 1986—April 1986) came from a joint current monitoring experiment in the eastern YS by Florida State University and Sung Kyun Kwan University, Korea. In this set near-bottom currents were all obtained at 1 m above the sea bed. The water depths are: K1 75 m, K2 53 m, K3 87 m, K4, 64 m, K5 95.5 m and K6 93.8 m. A detailed description of moorings and collected data was given in report by Harkema and Hsueh (1987). Table 2 summarizes observed and computed amplitudes and phases of u and v components at the 15 stations (marked by cross symbol in Fig. 1b). The comparison of results shows that there is considerable discrepancy between the observed and the computed harmonics, particularly at C1, C2, C4 and C8 (more precisely amplitudes at C2 and phase lags at C1, C4 and C8). Phase difference between the observations and Table 2a Observed and computed amplitudes (cm/s) and phase lags (deg., referred to 135°E) of the u component of M tidal current  Rig

C1 C2 C3

C4

C5

C6 C7

C8 C9 K1 K2 K3 K4 K5 K6

depth (m)

observed u u ? E

TCM1 u u ? E

TCM2 u u ? E

u ?

4 13 4 13 2 25 38 44 5 20 35 5 50 60 177 23 32 45 37 46 38 74 52 41 86 63 70 95 48

84.0 36.5 121.8 81.1 47.0 40.1 34.8 30.0 55.0 47.3 39.1 39.7 24.0 19.1 26.7 42.2 35.5 31.8 20.1 12.4 18.6 11.8 29.5 15.2 15.5 20.2 5.0 8.4 13.4

83.6 40.8 122.7 68.9 50.4 43.2 34.9 27.8 58.6 47.6 36.2 44.9 32.0 22.9 19.4 46.8 39.3 25.9 25.9 26.6 15.5 12.6 24.4 14.3 11.3 17.7 8.8 5.0 15.8

85.0 41.3 124.9 76.9 51.5 43.4 35.5 28.1 58.8 47.9 36.7 45.1 32.3 22.6 19.5 46.6 39.6 25.8 25.7 26.7 15.6 12.2 24.7 13.9 11.5 18.0 9.4 5.5 16.0

80.8 53.1 120.5 80.0 47.0 42.3 35.5 32.7 56.3 47.4 39.9 43.1 32.6 29.2 19.9 46.5 39.5 33.5 33.0 34.1 15.8 13.6 31.6 14.0 13.0 22.5 8.8 7.3 15.5

120 80 126 161 110 101 92 89 145 69 105 77 54 85 12 45 22 33 19 69 17 25 14 23 22 358 19 341 124

113 111 152 151 97 94 92 90 77 73 72 57 52 50 0 41 37 34 78 97 28 23 16 27 2 358 30 359 129

113 111 150 149 98 94 92 91 77 74 72 57 52 50 0 41 37 35 78 98 27 23 17 28 5 0 31 4 127

EVM1 u E 112 111 149 149 94 93 92 91 75 73 72 54 51 51 0 40 37 36 79 98 25 24 21 24 9 6 27 11 130

EVM2 u u ? E 71.0 57.5 97.0 78.6 43.6 40.1 35.2 33.2 54.5 46.6 40.1 41.2 32.7 29.9 19.2 45.9 39.3 33.6 33.6 35.5 15.5 14.2 31.7 14.4 13.1 22.8 7.7 6.1 14.7

114 113 155 155 95 94 93 92 74 72 71 54 51 51 1 39 37 36 79 100 25 24 22 22 12 7 25 10 131

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Table 2b Observed and computed amplitudes (cm/s) and phase lags (deg., referred to 135°E) of the v component of M tidal current  Rig

C1 C2 C3

C4

C5

C6 C7

C8 C9 K1 K2 K3 K4 K5 K6

depth (m)

observed v v ? E

v ?

v E

v ?

v E

v ?

4 13 4 13 2 25 38 44 5 20 35 5 50 60 177 23 32 45 37 46 38 74 52 41 86 63 70 95 48

83.4 46.1 34.3 20.2 48.0 40.6 35.6 33.7 58.7 50.8 40.1 43.0 34.6 26.9 16.8 40.7 33.8 30.0 29.5 23.3 32.8 22.4 26.0 26.1 17.0 25.4 40.0 26.0 43.8

57.4 29.2 97.0 54.2 55.3 46.2 36.9 29.5 61.2 49.3 37.4 57.3 40.2 28.7 15.9 42.4 35.0 23.2 28.2 25.9 27.9 17.9 25.1 23.2 15.8 24.6 37.9 24.2 44.7

343 342 335 335 346 341 339 338 339 335 333 306 299 298 231 296 292 290 328 335 159 140 80 80 70 58 32 26 12

58.0 29.0 100.3 55.6 56.1 46.3 37.5 29.7 61.4 49.6 37.9 57.0 40.6 28.4 16.2 42.2 35.3 22.9 27.9 25.7 27.7 18.4 24.7 23.4 15.9 24.6 37.8 24.5 44.7

343 342 333 332 347 342 339 338 339 335 333 307 299 297 230 296 292 289 328 334 160 140 79 80 69 57 31 25 12

55.8 37.5 97.8 56.9 52.0 46.1 38.3 35.3 59.3 49.5 41.6 55.8 41.8 37.5 16.6 42.7 35.7 30.3 36.7 33.8 28.2 22.6 31.2 23.7 18.7 30.9 37.8 30.2 44.3

115 325 336 312 340 331 320 316 56 333 7 326 300 325 232 307 282 293 287 312 159 149 98 75 72 53 31 2 11

TCM1

TCM2

EVM1 v E 341 341 332 332 342 340 338 338 336 334 333 304 299 299 233 295 292 291 328 334 156 144 82 80 72 60 31 28 12

EVM2 v ?

v E

44.5 37.1 75.4 61.9 47.6 43.0 37.1 34.9 56.9 48.2 41.3 54.7 41.7 37.7 15.8 42.5 35.7 30.5 36.5 33.8 28.0 23.0 32.0 24.3 20.5 31.6 37.8 31.1 43.8

338 339 339 339 341 340 339 339 335 333 332 303 299 299 234 294 291 291 329 335 152 144 83 79 75 61 32 29 12

the model results through the depth is most noticeable; there are more than 200° changes in observed phase lags, while in the computed results steady leads mostly less than 10° (except K1 with 20° difference) are found. The rms errors of amplitudes and phase lags lie about 16 cm/s and 40°, respectively; their difference obtained with variation in turbulence models are marginal (the worst results are obtained with EVM2 with 16.43 cm/s in amplitude and 41.69° in phase lags, while the best results are obtained with TCM1 with 15.99 cm/s in amplitude and 40.07° in phase lags). Calculation with TCM1, omitting the four stations, gives rms errors of about 6.51 cm/s and 18.75°, respectively, while calculation with EVM2 gives rms errors of about 6.12 cm/s and 19.56°, respectively. In what follows we examine the comparative results in more detail. At stations C1 and C2 located close to the mouth of Changjiang (or Yangtze) river, the water depth is relatively shallow and the tidal speed is fastest among the

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observations. The u component the direction of which is approximately equal to the cross-shore direction is larger than the v component the direction of which is approximately equal to alongshore direction. The models reproduced the u component fairly accurately but very poorly the v component. Observations show unusually large difference in phase lags between the surface and the bottom layers, while calculations show little differences. This may be due to the limitation on grid resolution leading to poor representation of water depth, or errors in the observational values. It is more likely that tidal regime near C1 and C2 is strongly affected by the discharge from Changjiang river which may yield strong residual flows and stratification. Intensification of the cross-shore component of surface current of C2 reflects the direct influence of the Changjiang river. It is known that the discharges of which are on average 45;10 m/s during late June through early August (and about 9;10 m/s in winter), forming a stratified system of water masses over several hundred kilometers (Beardsley et al., 1983). Effects of stratification on tidal currents were studied by many researchers. Maas and van Haren (1987) could compare the M  tidal current profiles observed in the central North Sea for the stratified and unstratified periods and explained theoretically that the reduced eddy viscosity at the pycnocline produces phase jump. Davies (1993) showed in a series of numerical experiments that the mid-water jump in tidal amplitude and phase were dependent upon the intensity of the pycnocline. The importance of sufficient resolution near the pycnocline was also addressed. Visser et al. (1994) reported that in coastal sea regions under freshwater influence a three-layer structure can be formed and then reduced eddy viscosity within the pycnocline may decouple the upper and lower layer motions, giving the currents 180° out of phase. At C3 tidal harmonics at 4 layers are available. Comparison shows that all turbulence closure models reproduce the current satisfactorily although slightly better results are obtained with EVM1 and EVM2 than with TCM1 and TCM2, especially for the u and v velocity difference in the vertical (D and D ). Calculations with TCM1 S T and TCM2 overpredict tidal current at the surface and underpredict tidal current at the bottom so that current shears in the vertical obtained using TCM1 (D "23 cm/s, S D "26 cm/s) and TCM2 (D "23 cm/s, D "26 cm/s) are larger than those comT S T puted with EVM1 (D "15 cm/s, D "17 cm/s) and EVM2 (D "10 cm/s, S T S D "14 cm/s) and than observations (D "17 cm/s, D "14 cm/s). Despite of the T S T success in reproducing the current speed, the phase leads in the vertical (i.e. differences in phase lag between the upper and the lower layers) are obviously underestimated. TCM1 and TCM2 produces about one third of the observed phase differences in the vertical, and EVM1 and EVM2 produce values 50% more smaller than TCM1 and TCM2. This may be due to poor resolution near the sea bed. Close inspection of model results also shows that the influence of varying eddy viscosity formulation is confined to the near-bottom layer. From the comparison at C4, it is also seen that the computed tidal current amplitudes are in good agreement with the observed ones. TCM1, TCM2 and EVM1 slightly overestimate the surface current, but EVM2 underpredicts the surface current velocity. The velocity differences in the vertical computed with TCM1 (D "22 cm/s, D "24 cm/s) and TCM2 (D "22 cm/s, D "24 cm/s) are slightly S T S T

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larger than observations (D "16 cm/s, D "19 cm/s). The difference computed with S T EVM1 (D "16 cm/s, D "18 cm/s) is almost same to that of observations, but S T EVM2 (D "14 cm/s, D "16 cm/s) slightly underestimates it. It is interesting to note S T that there is large discrepancy in the observed and computed phase lags of the surface and bottom layer currents, which is believed to be related to stratification effects. It is however unusual that the phase differences in the vertical are larger than C2 and C3 which are closer to Changjiang river than C4. The rapid variation of phase lags shown in observations, giving a three-layered structure, is very unlikely for the barotropic tide. We note that Larsen et al. (1985) exclude in their report the diurnal harmonics at the near-surface measurements of C4 because they were unreasonably large. Errors in measurements may be an possible answer. Considering that the similar tendency is found at C5, there is however strong possibility that effects of oceanic current (Kuroshio) may be added to the freshwater influence. At C5 the model results are reasonably comparable with observations although turbulence models all overpredict the current throughout the depth. The velocity differences in the vertical computed with TCM1(D "22 cm/s, D "28 cm/s) and S T TCM2 (D "23 cm/s, D "29 cm/s) are, as in C3 and C4, larger than observations S T (D "21 cm/s, D "16 cm/s) and those computed with EVM1(D "14 cm/s, S T S D "18 cm/s) and EVM2 (D "11 cm/s, D "17 cm/s). The vertical shear of v curT S T rents computed with EVM1 are in better agreement than current magnitude, which may be influenced by local errors in water depth, although for u comparison is not as good. Like C4 there is a unusual change in phase lags through the depth, giving significant difference between the observations and the model results. Once again, this may be due to freshwater influence. Excellent agreement in phase is found at 50 m depth. Station C6 is located near the continental shelf edge and only the tidal harmonics near the bottom are available. Both the amplitude and the phase lag of v component are reproduced with excellent accuracy, while in the case of u component the models underpredict the current by about 7 cm/s with errors in phase lags of about 12°. At station C7 tidal harmonics at 3 layers are available. We can see that the upper two layers correspond to mid-depth and turbulence models all reproduce the current very satisfactorily. Like C3, slightly better results are obtained with EVM1 and EVM2 rather than with TCM1 and TCM2. Again, TCM1 (D "21 cm/s, D "19 cm/s) and S T TCM2 (D "21 cm/s, D "20 cm/s) show the tendency of reproducing stronger S T vertical shear than EVM1 (D "13 cm/s, D "12 cm/s) and EVM2 (D "12 cm/s, S T S D "12 cm/s). The same is true for C3, C4 and C5. EVM1 and EVM2 overpredict the T u and v components throughout the depth, while TCM1 and TCM2 overpredict the upper two layer currents but underpredict the bottom layer current. There is some evidence in the phase lags of non-tidal effects. The locations of C8 and C9 are close to C1, C2 and C3. As expected, the model results are considerably different from the observed ones, particularly for the case of phase lags. At C8 the tidal amplitudes computed with TCM1 and TCM2 are in good agreement with observations, while EVM1 and EVM2 overpredict the tidal amplitudes (u component by about 13 cm/s; v component by about 10 cm/s). At C9 TCM1 and TCM2 reproduce the v tidal amplitudes with reasonable accuracy, however

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overpredict the u tidal amplitudes by about 14 cm/s. EVM1 and EVM2 overpredict the u and v tidal amplitudes by about 10 and 22 cm/s, respectively. For the second data set (K1—K6), comparison is possible at most at two layers. In overall sense, the computed amplitudes and phase lags are in fairly good agreement with the observed ones. Throughout the stations differences in u and v amplitudes are smaller than about 5 cm/s. Errors in phase lags are smaller than about 11° except for the bottom currents at K3 and K5. Among the stations K1 is located at the far North. The models satisfactorily reproduce the observed tidal amplitudes and phase lags although discrepancy between the observed and computed values of u component is noticeable. The observed change in the vertical in the u component phase appears erroneous. For the case of v component the models give phase difference in the vertical considerably larger than other stations. Only EVM2 produces the difference smaller than observations but still larger than other stations. The observations at K3 indicate that the current profile is more or less uniform with little phase difference in the vertical. The models however give rise to large phase differences particularly in the u component. The observations at K5 show very large phase differences in the vertical in both u and v components, while the models produce large difference in u component. These are all attributed to poor resolution near the sea bed. Without presenting detailed plots, we briefly describe the spatial variation in phase contours of u and v components. In overall sense, the v component phase contours of the surface and bottom layers vary more or less regularly and are approximately parallel with each other. The u component phase contours also exhibit regular variation in most part of YSESC except for the central region of YS where complicated patterns of the phase contours appear and consequently large phase difference occurs in the vertical. We conjecture that this phenomenon is associated with the contribution of three-dimensional Poincare waves. 3.3.2. M2 tidal current ellipse To understand the spatial variation in current field, we examine tidal current ellipses of the third layer (14% from the surface) obtained using TCM1 (Fig. 4a). It is evident that circular tidal currents with a clockwise rotation are dominant over the shelf extending from the shelf edge of ECS to the entrance of YS, while rectilinear tidal currents with an anticlockwise rotation are dominant along the west coast of Korea. These features were also found in 2D results (Lee and Jung, 1996b). In accordance with the theory developed by Prandle (1982) anticlockwise current rotation generally appears in regions with more or less equidistant distribution of co-range lines. An anticlockwise rotation can be also found at the south of Sandong Peninsula and at the upper northwestern part of the Taiwan Strait. In Pohai bay anticlockwise and clockwise rotations appear together. Surface current ellipses computed with other turbulence models show similar spatial variability; difference is graphically distinguishable only in regions where tidal currents are strong and predominantly circular. Current ellipse distributions at the bottom layer show similar features to those found for the surface current, although

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current magnitudes are significantly smaller. Therefore detailed results are omitted. We instead present as Fig. 4b the spatial distribution of semi-major axes at the bottom layer (2.5% from bottom) computed with TCM1 (solid lines) and EVM1 (dotted lines). The spatial variability of the semi-major axes is generally in good agreement with

Fig. 4. (a) Near-surface M tidal current ellipses computed using q-ql model (TCM1). Results at every  fourth grid point are displayed. Plus symbol denotes an anticlockwise rotation. (b) Contours of semi-major axis of bottom tidal current ellipses using q-ql model (TCM1, solid lines) and flow-related model (EVM1, dotted lines).

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Fig. 4. (Continued.)

Fang’s (1986) results obtained using a depth-averaged model. The semi-major axes of bottom current computed with EVM1 are significantly larger than those computed with TCM1 over the entire shelf area. This tendency has already been seen in Table 2. The ratios of the bottom layer semi-major axes to the top layer semi-major axes in

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deep water ('200 m) are close to one independently of eddy viscosity formulations, while in shallow water approximately 0.6 in TCM1 and 0.8 in EVM1. 3.3.3. Time variation of current speed and eddy viscosity For the better understanding of the performance of the turbulence models we here compare the time variation in current speed ("(u#v) and eddy viscosity. In detail, we examine at C1 (h"15 m) and C6 (h"177m) the time variation in current speeds at the surface layer (Figs. 5a and 6a) and the bottom layer (Figs. 5b and 6b), the eddy viscosity at the depth of 35% from the bottom (Figs. 5c and 6c) where, according to Bowden (1959), the eddy viscosity reaches its maximum, and the depth-mean eddy viscosity (Figs. 5d and 6d). From the results, it is evident that there are common sinusoidal variations in current speeds with M frequency associated with tidal variation. The magnitudes of  eddy viscosity also vary sinusoidally with phases slightly retarded to the current. Detailed examination of the peak locations reveals that the turbulence models give different behaviors in the phases of current speeds and eddy viscosity values. We note that the response of the turbulence models is strongly dependent upon the depth. In shallow water (C1), EVM1 (normal solid lines) and EVM2 (normal dashed lines) give smaller values of surface currents and larger values of bottom currents than those computed with TCM1 (heavy solid lines) and TCM2 (heavy dashed lines), producing vertical shear smaller than those computed with TCM1 and TCM2. The magnitudes of the surface and bottom currents and the eddy viscosity computed with TCM2 are almost identical to those computed with TCM1. Over a tidal cycle the depth-mean

Fig. 5. Time series at C1 (h"15 m) : (a) current magnitude at the surface layer, (b) current magnitude at the bottom layer, (c) eddy viscosity at a depth of 35% from the bottom, (d) depth-mean eddy viscosity; computed using TCM1 (heavy solid line), TCM2 (heavy dotted line), EVM1(normal solid line) and EVM2 (normal dotted line).

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Fig. 6. Same as Fig. 5 but at C6 (h"177 m).

current speeds computed with EVM1 and EVM2 are greater than those computed with TCM1 and TCM2. Comparing the peak locations, we note that TCM1 and TCM2 give phase retardation compared with EVM1 and EVM2. Calculation using u and v tidal harmonics of TCM1 and EVM2 shown in Table 2 gives difference in times of current maximum by about 4 min. TCM1, TCM2 and EVM1 give rise to the maximum eddy viscosity values about 200—210 cm/s on average with modulation $100 cm/s. From Fig. 5d, it is noted that EVM1 gives the depth-mean eddy viscosity higher than TCM1 and TCM2 by about 170%. In deep water (C6), the turbulence models all produce almost identical results in the magnitude of surface currents, but some difference in the magnitude of bottom currents. Unlike the current speeds, the (maximum) values of eddy viscosity at the depth of 35% from the bottom are very different. TCM1 gives the eddy viscosity of about 420 cm/s with modulation of about $60 cm/s, while TCM2 gives the eddy viscosity of about 90 cm/s with modulation of about $40 cm/s. EVM1 shows very large modulation of about $100 cm/s with a mean value of about 150 cm/s. Difference in the eddy viscosity values computed with TCM1 and TCM2 is related to the length scales. The length scales in TCM1 are asymptotically given by l in Eq. (10c), while the N length scales in TCM2 and EVM1 are given by Eqs. (10b) and (12), respectively. Values of the bottom boundary layer thickness computed with EVM1 and TCM2 (d+0.4 ) +0.0038 ) (0.18),;10"44 m in EVM1, d+0.4 +0.0063(0.13),; 10"41 m in TCM2) are less than the water depth. Comparison of Figs. 5c and d with Figs. 6c and d shows that the eddy viscosity closure in terms of current speed can be a good simplification to the more sophisticated

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turbulence models if the bottom boundary thickness is greater or equivalent to the water depth. 3.3.4. Profiles of tidally averaged vertical eddy viscosity Fig. 7a and b displays the tidally averaged eddy viscosity profiles at C3 and K3 computed with TCM1 for c "0.0038, 0.0063 and 0.0088. We see that the profiles of D eddy viscosity computed with TCM1 are basically parabolic. At C3 the eddy viscosity maximum appears at the depth of about 40% from the bottom with values of 380—430 cm/s, while at K3 the eddy viscosity maximum appears at the depth of about 30% with values of 170—240 cm/s. Comparison of the eddy viscosity profiles at C3 and K3 shows that throughout the depth the eddy viscosity values increase as the bottom friction coefficient increases. This is because with increase in the bottom friction coefficient turbulence kinetic energy (q/2) is enhanced but the length scales are little changed with variation in the bottom friction (The eddy viscosity values at

Fig. 7. Profiles of tidally averaged eddy viscosity at: (a) C3 and (b) K3 computed using q-ql model (TCM1) with c "0.0038 (heavy solid line), 0.0063 (dotted line) and 0.0088 (normal solid line). Profiles of D tidally averaged eddy viscosity at: (c) C1 (h"15 m) and (d) C6 (h"177 m) using TCM1 (heavy solid line), TCM2 (heavy dotted line) and EVM1 (normal solid line).

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C3 and K3 computed with TCM2 and EVM1 also increase because values of the boundary layer thickness are proportional to the bottom friction velocity). As a consequence of the boundary condition of ql, the eddy viscosity in TCM1 falls to zero in the near-surface and bottom boundary layers.

Fig. 8. (a) Ratios of bottom boundary layer thickness to water depth in percent computed using Prandtl mixing length model (TCM2). Contours of tidally averaged depth-mean eddy viscosity computed using (b) q-ql model (TCM1), (c) Prandtl mixing length model (TCM2) and (d) flow-related model (EVM1).

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Fig. 7c and d shows the tidally averaged eddy viscosity at C1 (h"15 m) and C6 (h"177 m) obtained using TCM1 (heavy solid lines), TCM2 (dashed lines) and EVM1 (normal solid lines) with optimal bottom friction coefficients, respectively. From Fig. 7c, it is evident that the eddy viscosity values at C1 computed with TCM1 and TCM2 are almost identical each other. The water depth at C1 is less than the

Fig. 8. (Continued.)

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bottom boundary layer thickness (d+0.4;+0.0063 ) (0.25),;10"80 m in TCM2). The maximum values of eddy viscosity computed with TCM1 and TCM2 are slightly lower than that computed with EVM1. The bottom boundary layer thickness at C6 is about 40 m which is far smaller than the water depth. Fig. 7d clearly shows that the eddy viscosity computed with TCM2 has a maximum at 10% above the

Fig. 8. (Continued.)

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bottom and becomes negligibly small outside the bottom boundary layer, while profiles of eddy viscosity computed with TCM1 are almost parabolic and show a maximum at a depth of about 30% above the bottom, which is approximately equal to the top of the bottom boundary layer.

Fig. 8. (Continued.)

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3.3.5. Spatial distribution of bottom boundary layer thickness and eddy viscosity It is interesting to examine the spatial distribution of the bottom layer thickness in that its relative magnitude to water depth determines the importance of the effects of tidal currents in wind-driven simulations. Fig. 8a shows the ratios in percentage of the bottom boundary thickness to the water depth (e.g. d/h) obtained using TCM2. It is evident that the bottom boundary thickness exceeds the water depth along the coastal regions. In the central trough of the YS where the bottom boundary thicknesses are smaller than water depth and their ratios reduce markedly near the shelf edge. From the values we may conclude that tidal background turbulence must be taken into account in wind-driven simulation in YSESC to identify the existence of upwind flows in winter or warm water intrusion as a part of Kuroshio branches. Fig. 8b shows the spatial distributions of tidally averaged depth-mean eddy viscosity obtained using TCM1. We see that along the coastline the eddy viscosity values are low. It is noted that the eddy viscosity values in southern part of the shelf edge are very large even though tidal currents are very weak and the vertical shear of tidal currents is negligibly small. In the middle of YS the eddy viscosity values are also large even though the maximum currents (semi-major axis) are less than 20 cm/s. In northwestern Pacific Ocean where the depth is greater than 2000 m the eddy viscosity increases dramatically. These physically unrealistic results are due to the fact that there is no restriction on the length scale in TCM1 for the barotropic case (while the length scale in EVM1 is limited to minimum of water depth and the boundary layer thickness). The length scale in TCM1 is closely related to the water depth and a wall proximity function in ql equation (see Lee and Jung (1997) for the relation of the length scale and the wall proximity function in the q-q l model). Fig. 8c and d is the spatial distributions of tidally averaged depth-mean eddy viscosity obtained using TCM2 and EVM1, respectively. The minimum values are found along the coastline and also in deep offshore region. The maximum value of eddy viscosity computed with TCM2 is 300 cm/s except in the northern part of Taiwan, while the maximum value of eddy viscosity computed with EVM1 is greater than 650 cm/s. The spatial distributions of eddy viscosity computed with TCM2 and EVM1 are similar each other, although EVM1 produces eddy viscosity two times larger than that of TCM1. Comparing Fig. 8b—d, we see that all turbulence models produce as a common feature large values of the eddy viscosity at the western part of Cheju island and at the northern part of Taiwan. This is due to the fact that both the depth and tidal current are relatively large.

4. Conclusions In this paper a three-dimensional hydrodynamic model has been applied to the Yellow Sea and the East China Sea to reproduce M tide using a range of eddy  viscosity closure models which include q-ql turbulence energy model (TCM1), the Prandtl mixing length model (TCM2), a simple flow- related model (EVM1) and the constant eddy viscosity formulation (EVM2). With a quadratic bottom friction law

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applied at the sea bed influence of the bottom friction coefficient and the eddy viscosity formulations upon tidal elevations and tidal currents have been examined. We stress in this study that the bottom friction coefficient should be regarded as a part of eddy viscosity closure model and its optimal value should be chosen carefully with respect to eddy viscosity formulations. The comparison of the observed and computed tidal elevation harmonics at a total number of 203 points shows that the optimal value of bottom friction coefficient change sensitively according to the eddy viscosity formulations; TCM1 and TCM2 give the optimal values lying between 0.0063 and 0.0075, EVM1 about 0.0038—0.0050 and EVM2 about 0.0038. The turbulence models produce tidal amplitudes and phase lags coinciding with observed ones within $15 cm and $15°, respectively, when the optimal values of bottom friction coefficient are used. However, it should be noted that the optimal values might change if the near-bottom grid resolution are chosen differently because the coefficient of bottom friction is in the model a function of the reference height, i.e. the height of bottom grid above the sea bed, and bed roughness. To investigate the ability of reproducing tidal current profiles model results have been compared with a total of 29 harmonics at 15 stations. The model produces current profiles with largely varying accuracy; considerable errors are found at stations near Changjiang river particularly in the phase lags. It is noted that the phase leads in the vertical (i.e. differences in phase lag between the upper and the lower layers) are underestimated. Baroclinic modelling with enhanced resolution of the bottom boundary layer may be necessary to improve the accuracy. Close inspection of current profiles shows that the influence of varying eddy viscosity formulation is confined to the near-bottom layer. In general TCM1 and TCM2 produce relatively smaller near- bottom currents than EVM1 and EVM2, having larger optimal bottom friction coefficients. Although the spatial distributions of current computed with a range of eddy viscosity formulations show similar features, the eddy viscosity values are very different in YS. EVM1 produces the tidally averaged depth-mean eddy viscosity of about 50—550 cm/s, while TCM1 and TCM2 produce the eddy viscosity of about 50—350 cm/s. The major difference occurs in deep sea regions off the shelf edge; TCM1 gives large values of eddy viscosity being an order of water depth, while TCM2 and EVM1 give eddy viscosity values less than 50 cm/s. In previous studies on winddriven circulation in YSESC tidal effects have always been neglected. The ratio of the bottom boundary layer thickness to water depth indicates that tidal background turbulence must be taken into account in modelling of wind-driven flows in YSESC to identify the existence of upwind flows in winter or warm water intrusion as a part of Kuroshio branches. The eddy viscosity closure models appear to show similar features when the bottom friction parameter is optimized, even though the profiles and values of the vertical eddy viscosity are different from each other. Results indicate that for a barotropic tide the Prandtl mixing length model which takes account of the boundary layer thickness could be an useful alternative to a highly complex q-ql model. In this study estimating optimal values of the bottom friction coefficient for current was not possible because limited amount of current measurements are available in

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YSECS. A considerable amount of data of good quality are indeed necessary to make such estimate. As a final remark we emphasis that long-term measurements of tidal current profiles using modern instruments such as acoustic doppler current meters and a near-bottom layer current measuring system are indispensable to the threedimensional modelling in this region.

Acknowledgements This study has been partly supported by 1997—98 STAR project (BSPN 97345-001092-2) funded by Ministry of Science and Technology, Korea. The valuable comments of a referee, which helped to improve our manuscript, are greatly appreciated.

References Aldridge, J.N., Davies, A.M., 1993. A high resolution three-dimensional hydrodynamic tidal model of the Eastern Irish Sea. Journal of Physical Oceanography 23, 207—224. ASCE Task Committee on turbulence models in hydraulic computations, 1988. Turbulence modeling of surface water flow and transport: Part I-IV. Journal of Hydraulic Engineering 114, 970—1073. Beardsley, R.C., Limbeburner, R., Dunxin, Hu., Kentang, Le., Cannon, G.A., Pashinski, D.J., 1983. Structure of the Changjiang River plume in the East China Sea during June 1980. In: International Symposium on Sedimentation on the Continental Shelf, with special reference to the East China Sea, China Ocean Press, pp. 243—260. Blumberg, A.F., Mellor, G.L., 1987. A description of a three-dimensional coastal ocean circulation model. In: Heaps N.S. (ed.), Three-dimensional coastal ocean models, AGU Coastal and Estuarine Series, pp. 1—16. Blumberg, A.F., Galperin, B., O’Connor, D.J., 1992. Modelling vertical structure of open-channel flows. Journal of Hydraulic Engineer, ASCE 118, 1119—1134. Bowden, K.E., Fairbairn, L.A., Hughes, P., 1959. The distribution of shearing stresses in a tidal current. Geophysical Journal of the Royal Astronomical Society 2, 288—305. Choi, B.H., 1980. A tidal model of the Yellow Sea and the Eastern China Sea. Korea Ocean Research and Development Institute Report 80—02, 72p. Choi, B.H., 1984. A three-dimensional model of the East China Sea. In: Ichiye, T. (Ed.), Ocean Hydrodynamics of Japan and East China Sea. Elsevier Oceanography Series, vol. 39, pp. 209—224. Choi, B.H., 1990. A fine-grid three-dimensional M2 tidal model of the East China Sea. In: Davies, A.M. (Ed)., Modeling Marine System, CRC publications, pp. 167—185. Choi, B.H., Fang, G., 1993. A review of tidal models for the East China and Yellow Seas. Journal of the Korean Society of Coastal and Ocean Engineers 5, 151—171. Choi, B.H., Suh, K.S., 1992. Computation of meteorologically-induced circulation on the East China Sea using a three-dimensional numerical model. Journal of the Korean Society of Coastal and Ocean Engineers 4, 45—58. Davies, A.M., 1986. A three-dimensional model of the northwest European continental shelf with application to the M tide. Journal of Physical Oceanography 16, 797—813.  Davies, A.M., Furnes, G.K., 1980. Observed and computed M2 tidal currents in the North Sea. Journal of Physical Oceanography 10, 237—257. Davies, A.M., 1993. Numerical problems in simulating tidal flows with a frictional-velocity- dependent eddy viscosity and the influence of stratification. International Journal for numerical methods in fluids 16, 105—131.

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