Lagrangian methods for water transport processes in a long-narrow bay-Xiangshan Bay, China*

LIANG Shu-xiu (梁書秀), HAN Song-lin (韓松林), SUN Zhao-chen (孫昭晨)

State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China, E-mail: sxliang@dlut.edu.cn

HU Zhan-ming (胡展銘)

National Marine Environmental Monitoring Center, Dalian 116023, China

Lagrangian methods for water transport processes in a long-narrow bay-Xiangshan Bay, China*

LIANG Shu-xiu (梁書秀), HAN Song-lin (韓松林), SUN Zhao-chen (孫昭晨)

State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China, E-mail: sxliang@dlut.edu.cn

HU Zhan-ming (胡展銘)

National Marine Environmental Monitoring Center, Dalian 116023, China

(Received July 10, 2013, Revised December 25, 2013)

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.

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 economy in coastal areas, the coastal water pollution becomes a very serious issue due to the excess and unreasonable exploitations. The overabundance of pollution matters in the water, such as the organic matter, 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 necessary 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 controlledby 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 trajectory 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 the2northeast. The water area of Xiangshan Bay is 563 km, 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.

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 tourism. 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 channels (I), the mouth of Xiangshan Bay (II), the middle and head of the bay (III), (IV).

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 continuity, the temperature, the salinity, and the density equations. In the horizontal direction, a non-overlapping 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 variable bottom topography. The equations are solved numerically by the flux calculation over an arbitrarilysized triangular mesh using the finite-volume approach. The finite-volume approach is better suited to guarantee 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

whereu,vandωare the components of the velocity in the horizontal(x,y)and vertical(σ)directions, respectively,C is the constant parameter,?ζis the area of the individual tracer control element. Then the thermal vertical eddy diffusion coefficient, AH=PrAM,Pris the turbulence Prandtl number.

The equations of the MY-2.5 turbulence closure model are given by

where q2is the turbulent kinetic energy,lis the turbulent macroscale.gis the gravitational acceleration,andis a wall proximity function. The von Karman constant k =0.4.E1,E2,E3are the close constants,,represent the horizontal diffusion of the turbulent kinetic energy and the macroscale,,are the vertical eddy viscosity coefficient and the thermal vertical eddy diffusion coefficient,is the vertical eddy diffusion coefficient of the turbulent kinetic energy. Turbulent close parameters are given by

2.2 Lagrangian residual current

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

where xTand xoare the end and start positions of the particles over the periodTandT is equal to 25 h in this study. With the integrated periodT 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,

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

2.3 Lagrangian coherent structures

where the second equality comes from taking the Taylor series expansion of the flow about point x. Since δx(t)is infinitesimal, the term ο[δx(t)2] can be considered negligible. Therefore, the magnitude of the distance is given by (using the standard vector L2-norm)

When the convention thatM?denotes the adjoint (transpose) of M are used, the symmetric matrix Δ is written as

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,which results in a vertical resolution of between 0.3 m to 5 m in most areas. The boundary conditions are extracted 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.

4. Results

4.1 The Lagrangian residual current in Xiangshan Bay

The hydrodynamic model is baroclinic, with con-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-layered 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 reflect the Lagrangian characteristics of the flow fields in an interval time t-. The meaningful LCS can be obtained with t-t0close to a typical time scale of the flow[9]. The choice of t-t0is 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. Sot-t0could be chosen as one or several tidal current periods. A series of experiments wheret-t0=6 h, 12 h, 24 h, 48 h, respectively, are performed to estimate the optimal t-t0. The results show that the internal structures are already clear and typical fort-t0=12 h. For a shorter interval time (t-t0=6h), 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,t-t0=12 h is taken for further analyses.

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 relationships 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 Foduchannel 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 governs 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.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.

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 contrast, 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 hydrodynamic 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.

This research demonstrates that the LCSs from the dynamical system theory is a useful tool to understand the water transport and mixing processes. The comprehensive applications of the LCSs and other Lagrangian methods provide a rich body of knowledge of the water transport characteristics in the bay, which can serve as a scientific basis for the control and protection of water quality.

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10.1016/S1001-6058(14)60063-9

* Project supported by the National Natural Science Foundation of China (Grant No. 51279028), the Foundation for Innovative 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

SUN Zhao-chen,

E-mail: sunzc@dlut.edu.cn