lagrangian methods for water transport processes in a long-narrow bay- xiangshan bay, china

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2014,26(4):558-567 DOI: 10.1016/S1001-6058(14)60063-9

Lagrangian methods for water transport processes in a long-narrow bay− Xiangshan Bay, China* LIANG Shu-xiu (梁书秀), HAN Song-lin (韩松林), SUN Zhao-chen (孙昭晨) State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China, E-mail: [email protected] HU Zhan-ming (胡展铭) National Marine Environmental Monitoring Center, Dalian 116023, China (Received July 10, 2013, Revised December 25, 2013) Abstract: A better understanding of water transport processes is highly desirable for the exploitation of the ocean resources and the protection of the ocean ecological system. In this paper, the Lagrangian methods are used to study the water transport processes in Xiangshan Bay in China, a typical semi-closed and narrow-shaped bay with complex coastline and topography. A high-resolution 3-D hydrodynamic model is developed and verified, and the results from the model agree well with the field data. Based on the hydrodynamic model, the Lagrangian residual current is computed by using the particle tracking method. A concept based on the dynamical systems theory, the Lagrangian coherent structures (LCSs), is introduced to uncover the underlying structures which act as the transport barriers in the flow. The finite-time Lyapunov exponent (FTLE) fields are computed from the hydrodynamic model results to extract the LCSs. The results indicate that the LCSs act as the internal structures of the Lagrangian residual current and the Lagrangian residual current displays the residual current speed and direction of different water regimes separated by the LCSs. The water masses with different transport characteristics can be identified and their exchange ability with other water masses can be estimated by combining the Lagrangian particle tracking with the LCSs methods. The comprehensive applications of these Lagrangian methods reveal the underlying structures and the inhomogeneous characteristics of the water transport in Xiangshan Bay. Key words: water transport processes, circulation, Lagrangian coherent structures (LCSs), Xiangshan Bay

Introduction In recent years, with the rapid development of the

harbor industry, the fish culture and the social econo- my in coastal areas, the coastal water pollution becomes a very serious issue due to the excess and un- reasonable exploitations. The overabundance of pollution matters in the water, such as the organic matter,

* Project supported by the National Natural Science Foun- dation of China (Grant No. 51279028), the Foundation for In- novative Research Groups of the National Natural Science Foundation of China (Grant No. 51221961) and the Public Welfare Projects of China’s Oceanic Administration (Grant Nos. 200805086, 201105009). Biography: LIANG Shu-xiu (1972-), Female, Ph. D., Associate professor Corresponding author: SUN Zhao-chen, E-mail: [email protected]

the DIP, the DIN and the COD, etc., may cause eutrophication and even red tide. The pollution in the water could be diluted to a low concentration or transported to open seas by the seawater movement. However, the pollution may accumulate or fluctuate in the bay because of complex topography, weak current conditions or other environment factors. Therefore, it is necessa- ry to study the water transport characteristics in order to better control the water quality.

The Lagrangian and Eulerian methods are widely applied to analyze the water transport problems, e.g., the Eulerian and Lagrangian residual current, the Lagrangian particle tracking, the dye tracer study, the drift experiment, etc.. The Eulerian residual current could be obtained by averaging the velocity at a fixed position for one or several tidal cycles or by using other filtering methods. It was widely used to analyze the observed tidal currents and numerical simulation results[1,2]. However, Longuet-Higgins[3] pointed out that the mass transport velocity is not solely controlled

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by the mean velocity, but also determined by the Stokes drift velocity. The sum of the Eulerian mean velocity and the Stokes drift velocity is called the first-order approximation of the Lagrangian residual velocity[4,5], which was successfully applied in weakly nonlinear systems[6]. In a strongly nonlinear tidal system, the Lagrangian residual current can be calculated as the net displacement over one or more tidal cycles by the Lagrangian particle tracking method. The traje- ctory analysis was also proved to be a valuable tool for determining the origin as well as the fate of specific water masses[7]. As an incorporation of the Lagrangian method, the Lagrangian coherent structures (LCSs) can be used to study some underlying flow structures that are not evident from the Eulerian field because of the chaotic horizontal transport present in many oceanic flows[8,9]. The LCSs of the flow field are defined as the attracting or repelling material curves in the finite-time Lyapunov exponent (FTLE) fields. This method was applied in several transport problems. For example, Olascoaga et al.[10] used the LCSs to study the spreading of plankton blooms on the West Florida Shelf. Lekien et al.[11] applied this method to predict the transport of contaminants and pollutions. It was also been used to study the causes of the low quality coastal water near the shoreline[12].

The purpose of the present study is to study the water transport processes in a complex topography bay, Xiangshan Bay in China by using the Lagrangian methods, and to provide some food for thought for the choice of adequate tools in estimating the water transport. In this study, a 3-D high resolution hydrodynamic model is implemented to simulate the water movement in Xiangshan Bay. The Lagrangian residual current and the LCSs are computed based on the Lagrangian particle tracking method. Furthermore, particle tracking experiments were conducted to reveal the effect of the residual current and the LCSs on the water mass transport and the water exchange. 1. Study area

Xiangshan Bay is a semi-closed narrow-shaped bay with complex topography, located in the middle coast of Zhejiang Province between 121o25′E- 122o00′E and 29o23′N-29o49′N, as shown in Fig.1(a). The bay is 62.8 km long from the bay mouth to the bay head, and its width varies from 20 km near the mouth to a minimum of 3 km in the middle of the bay. Liuheng Island locates at the mouth of the embayment which divides the entrance into Niubishan channel in the southeast and Fodu channel in the northeast. The water area of Xiangshan Bay is 563 km2, and the averaged depth is about 10 m, and the maximum depth is 70 m. Figure 1(b) is the contour map of the water depth D in Xiangshan Bay.

Fig.1 Location of the study area and measurement stations

Xiangshan Bay, the largest aquaculture base of Zhejiang Province, enjoys an excellent aquaculture condition. Since 1990, the marine cage culture saw a great development and became the main aquaculture mode in the bay. In 2005, the fish cages in the bay were increased to 65 000. The main farmed species are the yellow croaker, the weever, and the sciaenops ocellatus. Recent years witnessed a rapid development of the harbor construction and the harbor industry, such as the port logistics, the processing centers, the Wushashan and Guohua power station. Along with the rapid development of the cage culture and the harbor industry in the bay, a large amount of industrial and agricultural wastewater has been discharged into the bay. The eutrophication has become a serious problem and the red tide happens occasionally, as a serious constraint to the development of aquaculture and tou- rism. From previous studies[13], it is known that the water exchange capacity gradually drops from the mouth to the head of the bay. The period of the 90% water exchange is about 80 d near the bay head areas. A better understanding of the water transport is therefore of great importance in Xiangshan Bay. For the convenience of research, Xiangshan Bay is subdivided into four regions according to the hydrological and water quality investigations, as shown in Fig.1(a). These regions are the intersection area of two channe- ls (I), the mouth of Xiangshan Bay (II), the middle and head of the bay (III), (IV).

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2. Methods 2.1 3-D hydrodynamic model

A 3-D finite volume coastal ocean model (FVCOM, 2.7.1) is selected for the Xiangshan Bay application, which was developed originally by Chen et al.[14], based on the unstructured-grid, the finite-volume, and the 3-D primitive equations. The model governing equations consist of the momentum, the conti- nuity, the temperature, the salinity, and the density equations. In the horizontal direction, a non-overla- pping unstructured triangular grid is used to resolve the dynamics in the regions with complex shorelines and the sigma-coordinate transformation is used in the vertical direction to accommodate the irregular varia- ble bottom topography. The equations are solved numerically by the flux calculation over an arbitrarily- sized triangular mesh using the finite-volume approach. The finite-volume approach is better suited to gua- rantee the mass conservation in both the individual control element and the entire computation domain. The shape of Xiangshan Bay is complex, with numerous islands in it. Therefore, the FVCOM model is one of the best choices.

The Smagorinsky eddy parameterization method and the MY-2.5 model are used to determine the horizontal and vertical diffusions for the momentum, the temperature and the salinity. The horizontal eddy viscosity coefficient is calculated by

2 22

= 0.5 + 0.5 + +mu v u vA Cx x y y

ζΩ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

(1)

where u , v and ω are the components of the velocity in the horizontal ( , )x y and vertical ( )σ dire-

ctions, respectively, C is the constant parameter, ζΩ is the area of the individual tracer control element. Then the thermal vertical eddy diffusion coefficient,

=H MA PrA , Pr is the turbulence Prandtl number. The equations of the MY-2.5 turbulence closure

model are given by

2 2 2 2 2

+ + + = +qKq D q uD q vD q qt x y D

ωσ σ σ

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

2 2

0

2 2+ +M

HK u v g KD

ρσ σ ρ σ

∂ ∂ ∂ − ∂ ∂ ∂

3

1

2+ q

Dq FB l

(2)

2 2 2 2 2

+ + + = +qKq lD q luD q lvD q l q lt x y D

ωσ σ σ

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

2 2

1 1 30

+ +MH

K u v gE l E lE KD

ρσ σ ρ σ

∂ ∂ ∂ − ∂ ∂ ∂

3

1

+ lDq W FB

(3)

where 2q is the turbulent kinetic energy, l is the tur-

bulent macroscale. g is the gravitational acceleration, 1 1 1= ( ) + ( )L z H zη− − −− − and 2 2

2= 1+ /( )W E l kL is a wall proximity function. The von Karman constant

=k 0.4. 1E , 2E , 3E are the close constants, qF , lF

represent the horizontal diffusion of the turbulent kinetic energy and the macroscale, MK , HK are the

vertical eddy viscosity coefficient and the thermal vertical eddy diffusion coefficient, qK is the vertical

eddy diffusion coefficient of the turbulent kinetic energy. Turbulent close parameters are given by

=M MK lqS , H HK lqS= , 0.2qK lq= (4)

where MS , HS are defined as the stability functions,

which are functions of the gradient Richardson number. The detailed governing equations and the finite volume discrete method could be found in Chen et al.[14]. 2.2 Lagrangian residual current

The Lagrangian residual current LU is defined as the net displacement vector of the water column over a period T and is calculated by the Lagrangian particle tracking method. It is expressed as

= T oL T

−x xU (5)

where Tx and ox are the end and start positions of

the particles over the period T and T is equal to 25 h in this study. With the integrated period T equal to 25 h, it includes approximately two tidal circles and the averaged short-term characteristics of the residual current can be depicted. The particles in the water are tracked by solving the equation,

d= [ ( ), ]

dt t

tx v x (6)

It can be solved by using the fourth order Runge-Kutta scheme, as derived from solving the discrete integral

( ) = ( ) + [ ( ), ]dn

t

nt

t t t τ τ∫x x v x (7)

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2.3 Lagrangian coherent structures In order to exactly determine the LCSs, the sta-

ndard method[8,11] is used to compute the FTLE fields from the current data. Consider an arbitrary point x in the study area at time 0t and a point close to x ,

which is written as 0= + ( )y x x tδ and assume that

0( )x tδ is infinitesimal. The flow map is denoted by

0

ttφ . After transported by the flow for a time interval

0t t− , 0( )t

tx xφ→ . Then the distance between the

two points becomes,

0

0 0

2

0 0

( )( ) = ( ) ( ) = ( ) + ( )

d

d

ttt t

t tt y x t x tx

x xx

δφ

δ φ φ δ ο −

(8) where the second equality comes from taking the Taylor series expansion of the flow about point x .

Since 0( )x tδ is infinitesimal, the term 2

0[ ( ) ]x tο δ

can be considered negligible. Therefore, the magnitu- de of the distance is given by (using the standard vector L2-norm)

0 0

0 0

d ( ) d ( )( ) = ( ), ( ) =

d d

t tt tx x

x t x t x tx x

φ φδ δ δ

0 0( ), ( )x t x tδ δ∆ (9)

When the convention that M ∗ denotes the adjoint (transpose) of M are used, the symmetric matrix ∆ is written as

0 0

*d ( ) d ( )=

d d

t tt tx x

x xφ φ

∆ (10)

The FTLE fields 0

0( )t t

t xσ − are computed by

0

0 max0

1( ) = ln ( )t t

t xt t

σ λ− ∆−

(11)

where max ( )λ ∆ is the maximum eigenvalue of the finite-time version of the Cauchy-Green deformation tensor ∆ . ∆ is computed from the flow map of the artificial particles which are placed in the study area initially. They are tracked using Eq.(7) to compute the positions in time and space. Eq.(11) represents the FTLE at the point x at time 0t with a finite integra-

tion time 0t t− . If 0

0

t ttσ − > 0, the system is unstable,

which means that the neighboring points will separate from each other no matter how close they are initially.

The FTLE fields represent the maximum stretching rate for infinitesimally close particles over the time interval 0t t− . When the FTLE fields are computed

forward in time 0( 0)t t− > , the material curves in the

FTLE fields represent the repelling LCSs, and when

0t t− < 0, they represent the attracting LCSs. In this

study, the attracting LCSs are not considered. The high FTLE values in the FTLE fields are as-

sociated with the LCSs. For most flows, the FTLE value varies over space and time. So the LCSs are also time dependent and oscillate with the flows. The finite integration time 0t t− is the key to obtain meaningful

FTLE fields. For a shorter time interval 0t t− , the

particles may have not been separated from each other and the meaningful FTLE field is not fully established. However, if a large 0t t− is chosen, the particles

would be involved in many different parts of the flow. The choice of the criterion was discussed by Huhn et al.[9] and Branicki et al.[15], and it is shown that the value should be chosen according to specific cases.

Fig.2 The nested grid of the study area 3. Model configuration and verification

The model is implemented by two nested grids with different resolutions. The larger grid covers almost the entire Zhejiang coastal areas and there are 165 450 triangle meshes in the computation area (Fig.2). The largest element scales 3 500 m. The model boundaries are determined by the observed water elevation data in 2009 from Ocean Environment Monitoring Centre of Ningbo (OEMCN). The detailed information could be found in Xiong et al.[16].

The study area covers Xiangshan Bay as its center, where the high resolution grid is used. It includes Xiangshan Bay, Fodu channel and Niubishan Channel surrounded by white lines and coastlines as shown in Fig.2. There are 18 645 elements and 9 958 nodes in the study area. The element scales vary from 80 m for complex coastlines and islands to 400 m for offshore areas. Eleven sigma layers are adopted vertically,

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which results in a vertical resolution of between 0.3 m to 5 m in most areas. The boundary conditions are ex- tracted from a large model. The meteorological parameters (the wind components, the air temperature at 2 m, the shortwave radiation, the cloudiness etc.) are obtained from the analysis of the National Center for Environmental Prediction (NCEP), with a bilinear interpolation in space and a linear interpolation in time. The net heat flux at the air-sea interface can be calculated with these parameters.

Fig.3 Comparison of simulated and measured tidal elevations at

T12 station

The validity and the accuracy of the hydrodynamic model are tested by comparing the simulated results with the field data. The field data was measured during 6-20, 2009-7-10, 2009 by OEMCN (Beijing Time). The observed stations are shown in Fig.1(b), including one tidal elevation station (T12), four tidal current stations (0916-0919) and thirteen water quality stations (XS01-XS13). Besides the boundary conditions and the meteorological parameters, the main hydrodynamic parameters used for the model calibration are the bed roughness, the horizontal and vertical diffusions for temperature and salinity. The detailed drag coefficient calculating formula and the selected value could be found in Xiong et al.[16]. In the turbulence model, the horizontal Prandtl number is set as 1 and the background mixing is 10–6 m2/s. Comparisons between the simulations and the field data at some typical stations of tidal elevation η , tidal current speed U , direction D , temperature T and salinity S are shown in Figs.3-5. The simulated results indicate that Xiangshan Bay is a strong tide bay, typically of irregular semi-diurnal shallow water tides. The average tidal range is larger than 3.0 m and the maximum current speed is about 1.8 m/s. Analysis of the horizontal distribution of the tidal current fields shows that the tidal current is the rectilinear current in the Xiangshan Bay fiord and is the rotary current in the outer area (most area of the region (I)). The current in the bay is of the semi-diurnal current type. During the flooding tide, the water from outer sea flows into Xiangshan Bay through Niubishan channel and Fodu channel. And in the Xiangshan Bay fiord the direction of the current is almost parallel to the coastline. During the ebb tide, the water flows back to outer sea along the same pathway. It is worth noting that the water flows

Fig.4 Comparison of simulated and measured tidal current

speeds and directions at 0917 station back from Fodu channel 2 h earlier than the beginning of the flood current in Xiangshan Bay. Figure 5 shows the comparisons between the simulations and the mea- surements of the temperature and the salinity during April to October at XS07 station. The highest sea surface temperature is above 30oC and occurs in July- August. Overally, the results show that the model can simulate the water movement and the thermohaline structure in Xiangshan Bay adequately. 4. Results 4.1 The Lagrangian residual current in Xiangshan

Bay The hydrodynamic model is baroclinic, with con-

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Fig.5 Comparison of simulated and measured Temperature and

Salinity at XS07 station

Fig.6 Distribution of the Lagrangian residual current field (m/s)

siderations of the tide, the wind, the river runoff, and the surface heat exchange. Hence the Lagrangian residual currents computed in this study consist of the tide-induced, river-, wind-, and density-driven currents. Figs.6(a) and 6(b) show the distribution of the Lagrangian residual current field at the surface and bottom layers. In the Lagrangian residual currents

there are few eddies or circulations because of the narrow-long shape of Xiangshan Bay. The direction of the Lagrangian residual currents is mainly parallel with the shoreline in the fiord. The surface Lagrangian residual current from Fodu channel flows to the Xiangshan Bay fiord and sees an intense outgoing current in the Xiangshan Bay fiord. The outgoing current meets the current from Fodu channel at the mouth of the bay and flows to Niubishan channel from the south area of the Xiangshan Bay mouth. The Lagrangian residual speed is 0.03 to 0.18 m/s in the fiord and 0.12 m/s to 0.3 m/s in the region (I). This means that the water column net displacement at the mouth of the bay is larger than that in the fiord. In the vertical direction, the Lagrangian residual current at the bottom layer is distinctly different, especially in the fiord (Fig.6(b)). In most areas of region (I), the Lagrangian residual current speed decreases from surface to bottom. The bottom Lagrangian residual current from Fodu channel is divided into two branches. One flows to the Xiangshan Bay fiord and the other flows to Niubishan channel. In the fiord, it shows an incoming current and the current speed is about 0.03 m/s- 0.12 m/s. At both surface and bottom layers, the Lagrangian residual current speed decreases gradually from the mouth to the head of the bay. The two-laye- red circulation in the vertical direction is mainly due to the density stratification in the fiord[1]. 4.2 The Lagrangian coherent structures in Xiangshan

Bay In this section, the FTLE fields in Xiangshan Bay

are calculated by using the surface velocity simulated by the 3-D hydrodynamic model. The FTLE fields re- flect the Lagrangian characteristics of the flow fields in an interval time 0t t− . The meaningful LCS can be

obtained with 0t t− close to a typical time scale of

the flow[9]. The choice of 0t t− is based on the flow

characteristics in the study area. The tidal current in our study area is rectilinear and rotary current with a period of about 12 h. So 0t t− could be chosen as one or several tidal current periods. A series of experiments where 0 =t t− 6 h, 12 h, 24 h, 48 h, respectively,

are performed to estimate the optimal 0t t− . The results show that the internal structures are already clear and typical for 0 =t t− 12 h. For a shorter interval

time 0( = 6 h)t t− , the FTLE value is relatively small and the LCSs have not developed fully, because the particles are not separated from each other. For a longer interval time, the structures are similar but some small fragile structures grow. To display the typical structures clearly, 0 =t t− 12 h is taken for further analyses.

Figure 7 shows four typical LCSs in one tidal cu-

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rrent period for the case in 8-9 July 2009. The color contours in the figures represent the FTLE values

0

0( )t t

t xσ − . The higher value (in gray color in the figures)

means higher separation rate between two neighboring particles. The LCSs are defined as the ridges of the FTLE field. As can be seen from Fig.7, the LCSs are moving separatrixes along with the tidal current movement and they evolve periodically in time as the ebb and flood tidal current oscillates. The period is about 12 h, the same as the period of the dominant tidal component in the study area. In the ebb current, a re- markable LCS develops in the regions (I) and (II) and it moves along the southwest-northeast directions (as shown in Figs.7(a) and 7(b)). This structure acts as a separatrix between the north shore water and the south shore water. It evolves into the north-south directions in Niubishan channel at a low slack water (as shown in Fig.7(c)), which divides Niubishan channel into two parts. Meanwhile another LCS from the Xiangshan Bay fiord develops in the region (II). The evolution process of the LCSs in the flood current is almost a re- verse process. In Xihu Bay, there are two material curves which divide the bay into different parts. Particles separate mainly due to the fact that one of them tran- sports towards a coastal boundary or an island. Due to complex coastlines and numerous islands in the regions (III) and (IV), the FTLE value is high in this area, but there is no clear LCS. Some noisy structures appear near Niubishan channel because the forward transported particles leave the computational domain. In the previous study[17], the LCSs were obtained based on the hydrodynamic model with a mere consi- deration of the tide. Compared with the two LCSs, the structures of the LCSs display similar patterns in both cases. However, there are some differences worth no- ticing. The effect of the multi-forcing factor is to increase the value of the FTLE and make the structures more complex. In the regions (I) and (II), the LCSs extend to the left bound of the region (II). At a low slack water, a LCS develops across the region (II) but none under the tide only condition. In addition, the LCSs in the regions (III) and (IV) are more convo- luted. This shows that the effect of the multi-forcing factors could increase the mixing and change the characteristics of the transport. 5. Discussions

In the definitions of the Lagrangian residual current and the LCSs, the former represents the mean velocity of the water particles and the latter represent the separation rates of the adjacent water particles during a given interval time. The both methods are based on the Lagrangian particle tracking method and are dependent on the initial tidal current phase. The rela- tionships between them are explored in this paper by

comparing the results of the Lagrangian residual current and the LCSs, computed at the same time. The results are shown in Figs.6(a) and 7(c) and the initial tidal current phase is at a low slack water. The outgoing current in the fiord and the water from Fodu

Fig.7 Lagrangian coherent structures in one tidal period for the

case in 8-9 July 2009

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Fig.8 Time series of particles’ distribution released at high tide (The gray color particles are initially placed in north of the LCS, the

black color particles are initially placed in south of the LCS) channel meet at the mouth area of the bay and the directions of them are opposite (see Fig.6(a)). This may cause the water particles flowing in different directions and increase the separation rate. As expected, there is a material line with a high separation rate appears at the dividing line of the two currents (see Fig.7(c)). The Lagrangian residual currents in different water regimes are separated by the LCSs with similar residual current speeds and directions. The LCS in the region (II) acts as the separatrix of the outgoing current in the fiord and the water from Fodu channel. The Lagrangian residual current in the north of the LCS flows to the fiord as an outgoing current for the south region. With the two Lagrangian methods, the water transport characteristics are revealed from different

viewpoints. The LCSs act as the internal structure of the Lagrangian residual current.

The pollution in Xiangshan Bay is mainly transported out of the bay in the ebb tidal current. The circulation at the mouth of the bay is crucial for the water exchange with open seas. From the Lagrangian residual current and the LCSs in the Section 5, it can be seen that the circulation is relatively complex in the regions (I) and (II). The existence of the LCSs gove- rns the transport and it is verified numerically that the flux of the particles through these distinguished lines could be negligible[8]. Particle tracking experiments are conducted further to reveal the relationship among the Lagrangian residual current, the LCSs and the water exchange characteristics in different regimes.

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Two parts of artificial particles are assigned in the region (II), which are initially located in both sides of the LCS (see Fig.7(a)) as shown in the top left panel of Fig.8 and different colors are used to distinguish them.

Fig.9 Percent of particles left in Xiangshan Bay (The solid line

A and the dash line B represent the percentages of black particles and gray particles stayed in Xiangshan Bay when being released at high tide. The dot line C and the dash dot line D represent the percentages of two group particles stayed in Xiangshan Bay when being released during a tidal period)

The particles are released at high tide and will be

tracked for one tidal current period. The locations of the particles are given every 2 h in Fig.8. It is shown that almost no mixing occurs between the two groups of particles within 12 h. Analysis of Fig.8 shows that few particles initially in north of the LCS (gray color particles) are transported out of the region (II). In con- trast, a large number of particles initially in south of the LCS (black color particles) are flushed out. It is of a particular concern that hardly any particles are transported out from Fodu channel, even for the particles in north of the LCS. After 12 h, the gray color particles return back to the region (II) and most of them are also in north of the LCS. In order to quantify the number of the particles that flush out of the region (II) and the time it takes, the particles are tracked until the percentage of particles retained in Xiangshan Bay (all the regions except (I)) takes a relatively stable value. Figure 9 shows the time series of the percentage P of the particles remained in Xiangshan Bay. It is shown that 87% of the particles originally located in north of the LCS still stay in Xiangshan Bay, but the percentage is only 16% for the particles located in south of the LCS. The result of another couple of particles which are continuously released for 12 h is shown in Fig.9 with the line C and D. It is also shown that the percentage of the particles retained in Xiangshan Bay for the particles originally located in the north of the LCS is larger than that in the south. It reveals that the water exchange rate is much higher in the south region of the LCS. 6. Conclusions

In the present paper, a high resolution, 3-D hy-

drodynamic model is developed to simulate the water movements. Based on the hydrodynamic model, the Lagrangian residual current, the Lagrangian particle tracking method and the LCSs are used to analyze the water transport processes in Xiangshan Bay. The main conclusions are as follows:

(1) The distribution of the Lagrangian residual current shows that it is an intense outgoing current in the Xiangshan Bay fiord at the surface layer and an incoming current at the bottom layer. At both surface and bottom layers, the Lagrangian residual current speed decreases gradually from the mouth to the head of the bay. The LCSs are the moving separatrixes of different water masses and they evolve periodically with time as the ebb and flood tidal current oscillates. The period is the same as the period of the dominant tidal component in the study area.

(2) The distributions of the Lagrangian residual current and the LCSs are both dependent on the initial tidal current phase. With the two Lagrangian methods, the water transport characteristics are obtained from different points of view. The LCSs act as the transport barriers in the bay and the internal structures of the Lagrangian residual current. The Lagrangian residual current displays the residual current speed and direction of different water regimes separated by the LCSs.

(3) The water masses with different transport characteristics can be distinguished by the LCSs and their exchange ability with other water masses can be estimated by the Lagrangian particle tracking methods. It is shown that some obvious LCSs exist at the mouth area of the bay and the water exchange abilities are different among these water regimes separated by the LCSs. The water exchange rate is higher in the south of the LCSs area than in the north area.

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lagrangian methods for water transport processes in a long-narrow bay- xiangshan bay, china

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