www.elsevier.com/locate/jmarsys

Journal of Marine Systems 44 (2004) 141–151

Tidal-induced Lagrangian and Eulerian mean

circulation in the Bohai Sea

Hao Weia,*, Dagmar Hainbucherb, Thomas Pohlmannb,Shizuo Fenga, Juergen Suendermannb

a Institute of Physical Oceanography, Ocean University of China, 5 Yushan Road, 266003, Qingdao, PR Chinab Institute of Oceanography, University of Hamburg, Troplowitzstrasse 7, 22529, Hamburg, Germany

Received 13 February 2001; accepted 5 September 2003

Abstract

Long-term transport processes in coastal seas with time scales from weeks to seasons time scale are controlled by residual

circulation. In the Bohai Sea, an ultrashallow shelf sea of China, tidal residual is almost the dominant factor to the circulation

due to slight stratification and weak wind in summer. In order to establish an adequate hydrodynamic base to the ecosystem

dynamics of the Bohai Sea, the differences of tide-induced Lagrangian and Eulerian mean circulation are discussed and

calculated in this contribution. The Stokes drift is analyzed theoretically. According to Longuet-Higgins [Deep-Sea Res. 16

(1969) 431], the Lagrangian flow is the sum of the Eulerian flow and the Stokes drift that is induced by the mean kinetic energy

and coastal nonlinear interaction. Stokes drift is large in the coastal sea and in areas where the vorticity and/or divergence are

large. Vertical mass transports by Stokes drift are also the result of nonlinear interaction of current, water level and topography.

Hamburg Shelf Ocean Model (HAMSOM) is applied in the Bohai Sea to simulate the tides and tidal currents. The tide-induced

Lagrangian mean circulation and the Eulerian one are calculated at the same time. In the area where the Stokes drift is in the

same direction as the Eulerian residual, the Lagrangian one is stronger than the Eulerian one. Where they are pointing in

opposite directions, the Lagrangian one is small, like in the southwest of the Bohai Bay, Liaodong Bay and Bohai Strait. The

Lagrangian residual current flows into the Bohai Bay along its southern bank causing deposition of Huanghe River sediments.

This is in agreement with observations.

D 2003 Elsevier B.V. All rights reserved.

Keywords: Tidal induced Lagrangian and Eulerian mean circulation; Hydrodynamics model; The Bohai Sea

1. Introduction

In a coastal sea such as the Bohai Sea (Bohai), the

dominant observable motions are tidal oscillations.

The M2 tide is the principal tidal constituent. There

are two amphidromic points off Qinhuangdao and the

0924-7963/$ - see front matter D 2003 Elsevier B.V. All rights reserved.

doi:10.1016/j.jmarsys.2003.09.007

* Corresponding author.

E-mail address: weihao@lib.ouc.edu.cn (H. Wei).

Huanghe River (Yellow River) estuary. The maximum

tidal range is larger than 4 m and the maximum

current is 2.0 m/s. The K1 tide has an amphidromic

point at the southern part of the Bohai Strait (Editorial

Board Marine Atlas, 1994). For long-term transport

processes, with time scales of weeks to seasons, the

tidal residual currents, i.e. time mean circulation, are

important (Delhez, 1996). The intrinsic time scales of

the ecosystem dynamics are in the same range as the

H. Wei et al. / Journal of Marine Systems 44 (2004) 141–151142

residual circulation. In winter, the surface current in

the Bohai Sea is controlled by strong northern winds,

but the tidal residual still has an important effect due

to the nonlinear interaction of wind and tide. In

summer, some parts of the Bohai Sea are stratified,

but the horizontal gradients are small and the density

circulation is weak. So the tide-induced mean circu-

lation is even more important for the system.

The time mean circulation in a coastal sea can be

obtained by averaging the current-meter data or aver-

aging the simulation results of the instantaneous flow

at a fixed point for several tidal periods, thus the

Eulerian residual (utE ) is obtained. Another way is

dividing the net displacement of a water parcel in an

averaged time period by the time it travels, resulting in

the Lagrangian mean velocity. Longuet-Higgins

(1969) has pointed out that it is more reasonable to

use the Lagrangian mean velocity than the Eulerian

mean velocity to determine the origin of a water mass

in a time-varying flow. In the case of the Bohai Sea,

the Lagrangian mean velocity can be taken as the

Lagrangian residual (utL). The lowest order approxi-

mation of the Lagrangian residual is mass transport

velocity (Feng and Cheng, 1987) which is the sum of

the Eulerian residual and the Stokes drift (utS) that is

induced by the nonlinear interactions (Longuet-Hig-

gins, 1969; Zimmerman, 1979; Cheng and Xi, 1986;

Feng et al., 1986). utL has been proven to be a solenoid

vector satisfying the conservative condition at a ma-

terial surface (Feng and Cheng,1987; Feng, 1990). It

has been shown that the Eulerian residual is not

conserved while the Lagrangian residual is conserved

at a section of the estuary (Ianniello, 1977). So the

mass transport velocity, the lowest order approxima-

tion of the Lagrangian residual, should be appropriate

for describing the lowest order mean circulation in

coastal seas or tidal estuaries.

From the viewpoint of physics, it is easy to

understand that the Lagrangian residual rather than

the Eulerian residual can embody the coastal mean

circulation. But in practice, this theory is not used

widely now. It is taken for granted that the difference

between the two kinds of residual is very small and

can be neglected. This is not true in some real coastal

sea situations. Delhez (1996) has calculated the Stokes

drift on the Northwestern European Continental Shelf

and compared the Lagrangian and Eulerian transports

of passive tracers. He found significant differences.

In this paper, a numerical method for calculating

the two kinds of tidal residuals with an existing

hydrodynamic model (e.g. POM, HAMSOM) is in-

troduced and the two residuals in the Bohai Sea are

compared.

2. Circulation studies in the Bohai Sea

Basing on ‘‘Chinese National Comprehensive

Oceanographic Survey (1959–1960)’’, Guan (1994)

analyzed the data of current, sea temperature and

salinity, and then suggested the following scheme of

the mean circulation in the Bohai Sea. The Yellow Sea

Warm Current Extension enters the Bohai Sea through

the deep trench of the Bohai Strait and moves west-

ward into the central part of the Bohai Sea until it

meets the coast where it splits into two branches. One

branch moves toward the Liaodong Bay to form a

clockwise gyre, and the other moves toward the Bohai

Bay to form a counterclockwise gyre.

There were several numerical studies on the tidal-

induced Eulerian mean and Lagrangian mean circula-

tion in the Bohai Sea with different models. The

circulation patterns derived were quite different. With

regard to the mean Eulerian circulation, several authors

concluded by using Leendertse’s model (2- or 3-di-

mensional) that there are many small-scale local gyres

and an obvious clockwise gyre in the Liaodong Bay

and Laizhou Bay, while a counterclockwise gyre is

located in the Bohai Bay (Dou et al., 1981; Yu and

Zhang, 1987; Sun et al., 1990). Huang et al. (1999)

using HAMSOM got a pair of significant eddies near

the headland of Liaodong Peninsula. As for the mean

Lagrangian circulation, Huang and Wang (1988) de-

rived a counterclockwise gyre in the Bohai Bay and

Liaodong Bay with a particle tracing scheme. Sun et al.

(1989) calculated it by using the Stokes formula and

got almost the same result as Huang and Wang (1988)

except that the gyre in the western part of the Laizhou

Bay points in the opposite direction. Feng et al. studied

the circulation in the Bohai Sea using the weakly

nonlinear theory in coastal seas. They pointed out that

there exist a counterclockwise circulation in the central

part of the Bohai Sea and two clockwise gyres, one in

the Bohai Bay and one in the northeast corner of the

Liaodong Bay (Feng, 1990; Zheng, 1992; Wang et al.,

1993). In most of the above-mentioned papers, the

H. Wei et al. / Journal of Marine Systems 44 (2004) 141–151 143

obtained circulation near the Bohai Strait is ‘‘inflow in

the north and outflow at the southern part of the Bohai

Strait.’’ The magnitude of the tide-induced mean cir-

culation in the Bohai Sea is less than 5 cm/s except in

the area of the Laotieshan Channel where it exhibits

15–25 cm/s.

3. Schemes for Eulerian and Lagrangian mean

circulation

3.1. Schemes based on HAMSOM

Hydrodynamic equations, computing domain, nu-

merical scheme, spatial and temporal discretisation are

the same as in Hainbucher et al. (this volume). The

open boundary is chosen on the 122.5jE line from

Changshan Island to Jiming Island (Fig. 1). The

harmonic constants of the M2, S2, N2, K1 and O1

tides obtained from the ocean stations on these two

islands are interpolated at the open boundary. The

instantaneous water level and current speed (f, ut) canbe calculated. Based on this model, a scheme was

developed to calculate the Eulerian and Lagrangian

tidal-induced mean circulation at the same time.

Fig. 1. Schematic grid for the

The definition of the time-mean operator is as

follows:

hi ¼ 1

nT

Z t0þnT

t0

dt; ð1Þ

where T is the tidal period and n is the number of tidal

cycles.

The Eulerian time-mean circulation can be calcu-

lated from

utE ¼ uth i ð2Þ

Stokes drift can also be derived from the instanta-

neous current by using the following formula:

utS ¼Z

utdt � r ut� �

: ð3Þ

Then the Lagrangian residual is

utL ¼ utE þ utS: ð4Þ

Eq. (3) is Longuet-Higgins’ original formula for the

Stokes drift that is applicable to all kinds of oscilla-

tions, whereas Eq. (4) is the Stokes formula.

computational domain.

H. Wei et al. / Journal of Marine Systems 44 (2004) 141–151144

3.2. Theoretical analysis to the mechanism of Stokes

drift

Supposing that there is a harmonic wave propagat-

ing only in the x-direction with phase speed c, the

northern and eastern speed components for Stokes

drift are derived from formula (3) as

uS ¼hu2 þ v2i

c

Zvdx

� �ð5Þ

vS ¼Z

vdtd

� �ð6Þ

In formulae (5) and (6), x is the vorticity and d is the

divergence of the oscillating current,

x ¼ Bv

Bx Bu

Byð7Þ

d ¼ Bu

Bxþ Bv

Byð8Þ

Formula (5) and (6) mean that the Stokes drift is

generated by the mean kinetic energy term

hu2 + v2i, the mean nonlinear interaction of the

net displacement mvdt with vorticity x and diver-

gence d. For the Bohai Sea, the average depth

is 18 m so the phase speed (c ¼ffiffiffiffiffigh

p) is less

than 14 m/s. Tidal current in this shallow sea is

quite strong (1 m/s in magnitude). So the Stokes

drift could be as large as 7 cm/s if there is no

rotation.

The transport associated with the Stokes drift in a

water column can be written as

Hus ¼Z s

b

Zudt

Bu

BxþZ

vdtBu

By

� �dz ð9Þ

For periodical functions A and B exists the

relationship

A

ZBdt

� �¼ B

ZAdt

� �ð10Þ

Applying Eq. (10) to Eq. (9) and using the

continuity equation, it can be concluded that

Hus ¼ ufh i þ B

ByHu

Zvdt

� �: ð11Þ

Vertically integrated mass flux in y-direction is

Hvs ¼ hvfi BH

Bxu

Zvdt

� �: ð12Þ

This transport is determined by the nonlinear

interaction between the current and water eleva-

tion (term hu1i), between topography, current and

net displacement (second term of rhs).

So utL is different from utE as theoretical analysis

shows. For the sake of mass conservation, Stokes drift

transport should be concluded.

4. Results and analysis

4.1. Tides and tidal currents simulation

Results of tidal waves simulated with this model

are in good agreement with the observations. Two

amphidromic points for M2 tide and one for K1 are

simulated by using this model. The average error of

the amplitude at 19 stations is 5 cm and that of the

phase is about 10j. The simulated tidal level (solid

line) of Tanggu, Qinhuangdao and Longkou which are

representative for three bays are consistent with the

monitoring data (dashed line) (Fig. 2).

Simulated tidal currents also agree with our obser-

vation during the R/V DongFangHong2 Spring Cruise

in 1999 (Fig. 3). We have chosen stations where no

stratification existed. E1 (118.5jE, 38.5jN, 22-m

depth) is a station in the Bohai Bay, where the

strongest current is about 80 cm/s and the tidal current

is alternating in east–west direction. At station E3

(119.5jE, 38.5jN, 26-m depth), a station in the

central basin, the current is quite weak ( < 46 cm/s)

and rotates clockwise. B1 (119.5jE, 37.74jN, 16-mdepth), the shallowest station located in the Laizhou

Bay, shows a little counterclockwise rotating current

with a maximum value of 66 cm/s in north–south

direction. The northeastern wind of 7 m/s blowing

Fig. 2. Tidal level (cm) of (a) Tanggu, (b) Qinghuangdao, (c) Longkou, 1–31, July 1996. Monitored data are drawn in dashed line and simulated

in solid line.

H. Wei et al. / Journal of Marine Systems 44 (2004) 141–151 145

continuously at B1 is mainly responsible for the

difference between simulated and observed data.

4.2. Comparison of Eulerian and Lagrangian time-

averaged circulation in the Bohai Sea

The differences between Eulerian and Lagrangian

time-averaged circulations are caused by the Stokes

drift; therefore, the distribution of ut are discussed

first (Fig. 4). At the surface layer, utS produces a

clockwise flow in the Liaodong Bay and the Bohai

Bay and a counterclockwise flow in the Laizhou

Bay. It flows out of the Bohai Strait. The areas of

A utSA>1 cm/s are located near the eastern bank of

the Liaodong Bay, the northern bank of the Bohai

Bay and the Bohai Strait. Topography gradients in

Fig. 3. Comparison between simulated tidal current and observations at three anchor stations in the Bohai Sea, May 1999. Thick solid line—v

component simulated, thin solid line—u component simulated. Thick dashed line—v component observed, thin dashed line—u component

observed.

H. Wei et al. / Journal of Marine Systems 44 (2004) 141–151146

these areas are largest. Vorticity and divergence are

also large due to the shape of the coastline (headland

ect.). According to formulae (11) and (12), A utSA in

these areas is larger due to the strong nonlinear

interaction of the net displacement with the vorticity

and divergence. A utSA is less than 1 cm/s in the

lowest layer.utE at the surface layer is the same as that of the

results of Huang et al. (1999). There are many pairs

of eddies near the headland and A utEA < 1 cm/s in

the central basin and the Laizhou Bay (Fig. 5). The

patterns of the Eulerian residual and the Stokes drift

are obviously different. They even run in the

opposite directions at the eastern bank of the Laiz-

hou Bay, southern bank of the Bohai Bay and the

northwest part of the Liaodong Bay. They are of the

same order and make the Lagrangian residual pat-

tern significantly different from the Eulerian one.

In the surface layer (Fig. 6), utL does not exhibit as

many eddies in the Bohai Bay and the Liaodong Bay

Fig. 4. (a) Stokes drift at the surface layer of the Bohai Sea. (b) Stokes drift at the lower layer of the Bohai Sea.

H. Wei et al. / Journal of Marine Systems 44 (2004) 141–151 147

Fig. 5. (a) Eulerian tidal residual at the surface layer of the Bohai Sea. (b) Eulerian tidal residual at the lower layer of the Bohai Sea.

H. Wei et al. / Journal of Marine Systems 44 (2004) 141–151148

Fig.6. (a) Lagrangian tidal residual at the surface layer of the Bohai Sea. (b) Lagrangian tidal residual at the lower layer of the Bohai Sea.

H. Wei et al. / Journal of Marine Systems 44 (2004) 141–151 149

H. Wei et al. / Journal of Marine Systems 44 (2004) 141–151150

as utE. The maximum Eulerian residual is about 8 cm/

s, while the maximum Lagrangian residual is about

30 cm/s. In contrast to A utEA, A utLA is larger than 1

cm/s in the central basin and the Laizhou Bay; where,

the Stokes drift and the Eulerian residual are in the

same direction. The two kinds of residuals are in

opposite directions near the southwest coast of the

Bohai Bay. utL forms a clockwise flow in the Bohai

Bay and the Liaodong Bay. The clockwise gyre in the

Bohai Bay is also given in the results of Feng (1990)

and Wang et al. (1993). The artificial jellyfish exper-

iment supported this result (Jiang et al., 1997). These

currents can transport sand discharged from the

Huanghe River into the Bohai Bay making the Bohai

Bay a sand sink. This was proven by observations in

the Sino-American joint investigations on the sedi-

ments of the Huanghe River estuary in 1985–1986

(Wright et al., 1990; Shi, 1994) and its distributions

of salinity and nutrients (Lue, 1985). In the satellite

color images (Jiang et al., this volume), the high

turbidity area can be observed near the southern bank

of the Bohai Bay.utL enters the Bohai Bay and the Liaodong Bay in

their shallow parts and leaves these bights along the

deep trench. This is consistent with observations of

the surface current as given in the Marine Atlas of the

Bohai Sea (Editorial Board, 1992). utL is weak in the

corner of the three bays. This means that there is an

unfavorable condition for water exchange in these

areas.

From the flow structure across sections of the Bohai

Strait, it can be concluded that the transport of utE and

utL are different. Transports by utE enter the Bohai Sea

by the eddy through the narrow northern part and

central part of this strait. The net flux of utE is

1.62� 10 2 Sv. Transports by uzt

L run into the Bohai

Sea through the northern deep channel with a flux of

0.16 Sv.

5. Conclusions

(1) The Eulerian and Lagrangian tidal-induced mean

circulations are significantly different in coastal

seas. This difference cannot be considered as

negligible. The latter rather than the former can

embody the total time-averaged circulation that a

water mass experience.

(2) The Lagrangian tidal residual is stronger than the

Eulerian one in most parts of the Bohai Sea and

they can point in opposite directions in some

areas. This will result in different transports, for

example, of the sand discharged from the Yellow

River into the Bohai Bay.

(3) The Lagrangian residual circulation should be

considered as the base of the ecosystem dynamics

in a coastal sea. We can deduct that it is easy to

calculate the Lagrangian residual including tide–

wind– thermohaline effects from an existing

baroclinic model by using Stokes formula.

Since the tidal excursion is about 10 km, a grid of 1/

12j in latitude and longitude may not resolve the net

displacement in one tidal period reasonably. A finer

grid is required for a better resolution of the tidal cycle.

Acknowledgements

We are indebted to the Federal Ministry of

Education and Science (BMBF), Germany, which

supported the present work under contract No.O3-

F0189A, and to National Natural Science Foundation

of China (NSFC) under contract No.49576298 and

State Oceanic Administration of China (SOA).

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