A Comprehensive Lagrangian Transport Study in A Long-Narrow Bay, Xiangshan Bay, China

Study on the transport and mixing in coastal waters is of great concern to the ocean resources exploitation and ecological system protection. Lagrangian methods are direct and effective of researching mass transport. Two Lagrangian tools were adopted and combined to describe water transport in a long-narrow bay, Xiangshan Bay, China. Based on the fields of tidal velocity simulated from the 3-D hydrodynamic model, Lagrangian Coherent Structures (LCSs) and synoptic Lagrangian maps (SLMs) were calculated in the study area. Through comparison of the results, the features and relation of the two tools were discussed. The results show that the LCSs act as the separatrix of the water regions with different transport characteristics and can identify the water areas with different transport time scales. The comprehensive application of the Lagrangian tools is helpful to obtain more insight into the water transport process.


Introduction
Salinity, sediment, phytoplankton and all types of pollutants are advected by water flow in coastal waters, estuary, etc. Compared to the Eulerian method, the Lagrangian method, which can directly depict the trajectories and fates of mass in water, has been widely applied in recent years. Due to the different perspectives, various Lagrangian tools and indices have been proposed to describe mass transport.
Residual current plays an important role in the mass transport of coastal areas. Many researchers have studied residual current based on Lagrangian particle tracking, which includes numerical simulation (Liang et al., 2014;Muller et al., 2009) and drifter trajectories tracking (Hancke et al., 2014;Zhou et al., 2010). Charria et al. (2013) calculated the surface circulation in Biscay Bay based on Lagrangian drifter trajectories from 1992 to 2009. Muller et al. (2010) estimated the Lagrangian residual circulation (LRC) in Iroise Sea by the numerical method. In his study, the particle trajectories were calculated using HF radar data and the Regional Scales model. In recent years, Lagrangian Coherent Structures (LCSs) have been introduced to study ocean circulation. Shadden et al. (2005) defined LCSs as the ridge lines in spatial fields of the Finite-Time Lyapunov Exponent (FTLE). Huhn et al. (2012) studied the horizontal Lagrangian surface transport based on LCS and showed that LCSs can be interpreted as a imprints of the circulation pattern and reveal in detail the separation of the time-depend flow. Besides, the transport time scales in coastal areas is one of the important parameters to quantify the time that water remains inside a concerned area. A number of concepts, such as age (Bolin and Rodhe, 1973), half-life time (Luff and Pohlmann, 1995), residence time (Takeoka, 1984), were used as indicators to assess the time scales of mass transport. Poulain and Hariri (2013) estimated the transit and residence times in the Adriatic Sea from drifter data and numerical simulation. Moreover, the concept of synoptic Lagrangian maps (SLMs), as proposed by Lipphardt et al. (2006), was used to represent each trajectory by its origin/fate and residence time.
The Lagrangian method is a powerful tool for analyzing and understanding mass transport in ocean. However, it is difficult to interpret the characteristics of water transport and circulation within the whole study area from particle trajectories without any indices. For example, Marinone et al. (2011) used four indices to quantify the seasonal circulation in the Gulf of California. Both LRC and LCSs integrate information based on the particles' trajectories, but they ignore the water transport process. The advantages and relation between different Lagrangian tools is a subject worthy of investigation. In this paper, LCSs and SLMs were applied to study the Lagrangian transport in a long, narrow bay, Xiangshan Bay (XSB) in China. It is 62.8 km long from the mouth of the bay to the head and varies in width from 20 km near the mouth to a minimum of 3 km in the middle of the bay. We were motivated to understand the conditions and limitations of the two Lagrangian tools, as well as the relation between them for choosing adequate tools to estimate water transport in the bay. This paper has been organized in the following manner: after this introduction, Section 2 describes the study area and the hydrodynamic model used in the study; In Section 3, the study methodology is introduced; In Section 4, the results and relation between the different tools are discussed; Finally, in Section 5, the conclusions are presented.

Study area and data
2.1 Study area XSB is located along the central coast of Zhejiang Province, China (121°25′E−122°00′E, 29°23′N−29°49′N) (as shown in Fig. 1). It is a strong-tide bay and with an irregular shallow semidiurnal tide. The water from an outer sea flows into XSB from Niubishan Channel at the southwest and Fodu Channel at the northeast. There are three branching bays in XSB. They are Xihu Bay, Huangdun bay and Tie bay. The water area of XSB is 563 km 2 , and the averaged depth is about 10 m, and the maximum depth is 70 m. In the past three decades, the mariculture and harbour industries in the coastal areas have rapidly developed, and a large amount of agricultural waste, industrial sewage and pollutants are discharged into XSB. Eutrophication has be-come a serious problem in the bay, and red tides have occasionally occurred. Understanding the mass transport characteristics can provide scientific guidance for water environmental protection.

Data
The velocity data sets for calculating the Lagrangian transport in XSB were obtained from the 3-D finite volume coastal ocean model (FVCOM, 2.7.1), which was developed originally by Chen et al. (2003), based on the unstructured-grid 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 model also has been converted to a GPU based parallelized version by Zhao et al. (2017) to improve computing efficiency and applied in Ningbo coastal water. Therefore, FVCOM is suitable for the hydrodynamic simulation in XSB.
The hydrodynamic model in XSB is implemented by two nested grids with different resolutions. The larger grid covers almost the entire Zhejiang coastal areas and there are 165450 triangle meshes in the computation area (Han, 2015), and the second gird covers XSB as its center, where the high resolution grid is used. 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 open sea boundary is driven by the main tidal constituents-M 2 in the study area. The meteorological process and river discharged are not considered in the model. The model runs with cold start as the initial condition and until it reaching steady-state. The hydrodynamic model developed in XSB has been verified by the observed data and applied to study the pollutant transport and water quality (Han et al., 2013;Liang et al., 2015). In this study, we concentrate on the horizontal surface transport in XSB and the application of the two Lagrangain tools, so horizontal velocities in the surface layer are considered. Fig. 2 shows the surface current distribution at the maximum flood time and ebb time. During the flood time, the water from outer sea flows into XSB through Niubishan channel. In XSB the direction of the current is almost parallel to the coast. It is noteworthy that the current reaching the maximum flood tide in Fodu channel is two hours earlier than that in Niubishan channel. When the current in XSB reaches the maximum flood, the current in Fodu channel is the turn of tide (Wu et al., 2015). During the ebb tide, the water flow back to outer sea along the same pathway. Water flows into the bay from multiple waterways, and the tidal flow is very complicated near the mouth of the bay.

Lagrangian Coherent Structures
LCSs are introduced by Haller and Yuan (2000) for the boundaries of coherent structures in two-dimensional turbulence and applied in aperiodic flows (Shadden et al., 2005). It 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. The LCSs are defined as the material lines in the FTLE field which separate the waters with the different transport characteristics. The FTLE field is computed from the discrete velocity data set of the hydrodynamic model by advecting a grid of artificial tracers for a finite time. Consider an arbitrary point in the study area at time and a point close to , which is written as and assume is infinitesimal. The flow map is denoted by . After transported by the flow for a time interval , ,. Then the distance between the two points becomes, x where the second equality comes from taking the Taylor series expansion of the flow about point . Since is infinitesimal, the term can be considered negligible. Therefore, the magnitude of the distance is given by (using the standard vector L2-norm) ∆ where the convention that M* denotes the adjoint (transpose) of M are used. And the symmetric matrix is Then the FTLE value at the time at the spatial posi- x τ tion with a finite integration time is given by the formula, where is the advection time and is the largest eigenvalue of the Cauchy-Green deformation tensor . The particles were tracked by the Lagrangian particle tracking (Eq. (5)) and solved by the fourth order Runge-Kutta scheme. If , the system is unstable, which means that the neighbouring points will separate from each other no matter how close they are initially. The FTLE values represents the maximum stretching rate for infinitesimally close particles over the time interval.
where v is the velocity from the hydraulic model.

Synoptic Lagrangian maps
To analyze and display the enormous amount of trajectory information, SLMs was produced by representing each trajectory according to its origin/fate and residence time (Lipphardt et al., 2006). SLMs are constructed by colour coding each particle's initial position to represent information about its residence time and/or its origin/fate.
In this study, the main information of SLMs contains the residence time of each water particle and its fate. The calculation procedure is as follows. First particles are initialized in the study area at each mapping grid cell and then integrated forward in time using Eq. (5). Each particle is tracked until it leaves the study area through the open boundary or reaches the maximum computing time. The boundaries are divided into land boundaries and ocean boundaries, which are divided into several sections. Finally, the time of each particle staying in the study area is its residence time. The boundary of each particle reaching is its fate. HAN Song-lin et al. China Ocean Eng., 2020, Vol. 34, No. 4, P. 581-588 583 4 Results and discussion

LCSs in XSB
The calculation and results of LCSs in the study area are described in this section. The FTLE fields are obtained by calculating the stretching rate for infinitesimally close particles over a time interval. Therefore, the FTLE field is affected by the particle resolution, the integration time and the initial calculation time. In the previous study (Han, 2015), the effect of the particle resolution on the FTLE field was studied and the result shows that the particle resolution 200 m is appropriate in XSB. τ τ The finite integration time ( ) represents the Lagrangian transport time scales that the LCSs display. To obtain meaningful FTLE fields, the integration time should be chosen carefully. For a shorter integration time, the particles may not have been separated from each other and the meaningful FTLE field is not fully developed. Alternately, for a large integration, the particles would be involved in many different parts of the flow. Previous studies indicate that the value should be chosen according to each specific case (Huhn et al., 2012;Branicki and Wiggins, 2010). Therefore, a series of experiments with integration time of 0.5, 1, 4 and 6 times of the M 2 tidal component period (T is a M 2 tidal component period, which is equal to 12 h) were performed to estimate the optimal integration time, and the results are shown in Fig. 3. As shown in the figure, when equals T/2, four remarkable LCSs (L1−L4) develop in the XSB. However, the LCSs is not evolved fully because it only includes the ebb tidal process. The LCSs appear more clearly τ τ τ τ in Fig. 3b when equals to T. When equals 4T and 6T, some new LCSs appear such as L5 and L6 in Fig. 3c. Meanwhile, some noisy structures appear in the study area, and the FTLE is tending to the same value in the XSB for =6T. Overall, the value of FTLE decreases with increasing integration time. The LCSs are more identifiable and clearly for the integration time equals T and 4T. The initial release time of the particles will affect the development and distribution of the LCSs in XSB. The variation of the LCSs in one M 2 tidal component period (T) were investigated by comparing the distribution of the FTLE fields at four typical initial times (the high slack water, the maximum ebb velocity, the low slack water and the maximum flood velocity), and the results are shown in Fig.  4. The largest FTLE value in the picture represents the largest separation rate between two neighbouring particles. The LCSs are the "ridge" shaped structures connected by the larger FTLE values. In XSB, distinct LCSs mainly developed in the mouth of the bay, and only some fragmentary large values appear near the islands in the bay. There are four LCSs, L1, L2, L3 and L4, as shown in Fig. 4a. L1 always exists in Fodu Channel and divides the channel into two parts. The other LCSs (L2, L3, and L4) develop in Niubishan Channel and divide the channel into several parts. The shapes and locations of the LCSs move along with the tidal current movement. L2, L3, and L4 LCSs, evolved into different shapes in Niubishan Channel during a tidal period.

SLMs in XSB
The marine environmental capacity and pollutant trans- Fig. 3. FTLE fields for the integration time of (a) =T/2, (b) =T, (c) =4T, (d) =6T. port are closely related to water exchange in a bay. The SLMs were adopted to study the features of water exchange in XSB. They were obtained by first calculating the movement trajectory of water particles and the residence time using Lagrangian particle tracking and then drawing the fates of water particles and residence time on the contour map. SLMs express the residence time and water transport pathway in the study area.
XSB is a semi-enclosed bay and water exchanges with open seas through Fodu Channel in the northeast and Niubishan Channel in the southeast. The study area includes the narrow water body XSB, Fodu and Niubishan Channels. The open boundaries were divided into two segments to better understand the fate of the water in the bay. In addition, four typical initial release times (t 0 =0, T/4, T/2 and 3T/4) were chosen to study their effects on the water exchange. Particles are initialized in the study area with the interval of 200 m. Then they were tracked for 30 days. The fate and the residence time of each particles were recorded. The time series of the percentage of particles that remain in the bay and leave the bay from Niubishan channel and Fodu channel are shown in Fig. 5. The black lines represent the percentage of the particles that remain in XSB, and the red and blue lines represent the percentage of particles that leave from Niubishan and Fodu Channels, respectively. The lines show amplitude, periodic fluctuations, which indicate that some particles leave the study area during the ebb tide and come back during the flood tide. The red lines in Figs. 5a and 5b during the first two days, show a bigger fluctuations than that in Figs. 5c and 5d. It means that more particles are transported in and out of the study area at the open boundary of Niubishan channel. And 8.3%−9.1% of the particles released at t 0 =0 and T/4 leave Niubishan channel in 30 days, which is 1.6%−2.6% higher than those released at t 0 =T/2 and 3T/4. The blue lines in Figs. 5a and 5d during the first day, show a sharp increase to 10.5%−14.8%, which illustrate lots of particles leave the Fodu channel. There are 40.1%−41.3% percentage of particles leave Fodu channel in 30 d for the particles released at t 0 =0 and 3T/4, which is 2.3%−9.7% higher than the particles released at T/4 and T/2. Therefore, there are fewest particles staying in the bay in 30 days when the particle released at t 0 =0. It shows that the initial time has effect on the residence time and the fate of particles, especially for the first few days.
To study the water particles' residence time and transport path in the study area, the SLMs for the release time t 0 =0 are shown in Fig. 6. The contour map in Fig. 6a represents the residence time of water in the bay. In Fig. 6b, the different colours represent the transport pathways of water particles. The red colour means that the particles will remain in the study area in the calculation period. The light blue and dark blue colours mean that the particles flow out of the study area from Niubishan and Fodu Channel, respectively. The figure shows that the water residence time of most particles between the mouth of XSB and Niubishan Channel is approximately 0.4−8 days and the particles mainly leave the study area from Fodu Channel. A small amount of particles are transported to XSB fjord or island and the water residence time is significantly longer, which is more than 24 days. The particles in Fodu Channel flow out of the study area in 0.4 days. The water residence time in XSB fjord is more than 30 days.

Relation between LCSs and SLMs
The relation between LCSs and SLMs is discussed below. The results of LCSs and SLMs were related to the initial release time and the evolution of the LCSs in a tidal period (as shown in Fig. 4) has a good consistence with the fate of the water particles as shown in Fig. 5. There were several LCSs in the open boundary of Niubishan channel during the ebb tide (Figs. 4c and 4d), which prevent the water particles at the north side of LCSs leaving this channel. Therefore, more particles released at t 0 =0 and T/4 leave Niubishan channel than those released at t 0 = T/2 and 3T/4 during the first day. During a tidal period, a remarkable LCS L1 develops in Fodu channel and is almost parallel with the coastline. It separates the channel into two parts. The particles at the south side of L1 will leave the study area and the particles at the north side of L1 will not leave the study area in T. Fig. 4d shows that more particles will be assigned at the south side of L1 initially. It can deduce that more particles will leave Fodu channel when the particles are released at t 0 =3T/4, and it is perfectly illustrated by the blue lines in Fig. 5d in the first day. Therefore, the LCSs with the integration time T could explain the effect of the water exchange and the fate of particles in T. τ The LCSs act as the separatrix of the water regions with different residence times and fates. As shown in Fig. 4c and Fig. 6, there is a good correspondence between many structures of LCSs and SLMs in XSB. New LCS L5 with the integration time ( =4T) in the mouth of the bay developed which departed the water by the line from P1 (near the west corner of Liuheng Island) to P2. L6 in Fodu channel departed the waterway to two sections. The LCSs clearly separated regions with different values of residence time and the residence time on both sides of the LCS was significantly  different. The residence time was more than 4 days on the north side of L6 and less than 0.4 days on the south side. On the left side of LCS L5, the water residence time was more than 30 days, and on the right side of L5, it was less than 20 days. The LCSs also separate water with different fates. The particles in the south side of L6 almost leave Fodu channel, and the particles in the north side of L6 leave Niubishan channel. Similarly, the particles in the north side of L5 still stay in the bay in 30 days and the particles in the south side of L6 almost leave XSB from Fodu channel and Niubishan channel.

Conclusions
In the present paper, the LCSs and SLMs were applied to study the mass transport in XSB in China. The results by different descriptive methods were analyzed and compared in detail. The main conclusions are as follows: (1) The LCSs are the moving separatrixes of different water masses and they evolve periodically with time as the ebb and flood tidal currents oscillate. The LCSs L1 (Fig.  4a), L5 and L6 (Fig. 4c) in XSB are worth being paid attention to, which divides the water into several regions with different transport characteristics in the mouth of XSB.
(2) The residence time and the fate of particles in the bay are affected by the initial time and vary in space distribution. The water residence time of most particles between the mouth of XSB and Niubishan Channel is approximately 0.4−8 days and the particles mainly leave the study area from Fodu Channel. The particles in Fodu Channel leave the study area in 0.4 days. The water residence time in XSB fjord is more than 30 days.
(3) The LCSs separate the water regions with different transport characteristics. The results indicate that the LCSs clearly depart from regions with different values of residence time and the fates. In the previous research (Liang et al., 2014), it also shows that the LCSs act as the internal structures of the Lagrangian residual current. However, it is difficult to interpret the role of the seperatix lines only by the field of LCSs. The comprehensive application of the different Lagrangian tools could discover and help to best understand the mass transport characteristics.