# A Computationally Efficient Shallow Water Model for Mixed Cohesive and Non-Cohesive Sediment Transport in the Yangtze Estuary

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{9}t·yr

^{−1}during the period 1953–2002 to less than 1.0 × 10

^{9}t·yr

^{−1}in the past decade [1]. Not surprisingly, bed erosion and reduction in suspended sediment concentration (SSC) have been observed in various regions of the Yangtze Estuary [1,2,3,4]. However, it has been suggested that there might be a morphological lag response to riverine sediment supply changes of about 10–30 years in seaward regions [5,6]. In this regard, it is important to conduct high-resolution 10–30 years prediction of the response of the hydro-sediment-morphodynamic system in the Yangtze Estuary. However, high-resolution numerical prediction of field scale numerical cases is computationally demanding. The idea of morphological accelerating factor (MF), which updates the bed level at each hydrodynamic time step by increasing sediment erosion and deposition fluxes (thus resultant bed level changes) using a constant MF, has found wide applications in long-term morphodynamic predictions [7,8,9,10,11,12,13,14,15,16,17]. One basic assumption of the MF approach is that the bed level changes in one tidal cycle are small so that “nothing irreversible happens within an ebb and flood phase, even when all changes are multiplied by the factor” [7,8]. Numerically, using MF = 10, Hu et al. [9] projected evolution of the Jiuduansha Shoal for 20 years. Using MF = 5, Kuang et al. [10] forecasted the 20-year evolution of Nanhui Tidal Flat after the closure of the Three Gorges Dam. Using MF ranging from 11 to 60, Luan et al. [11] conducted 20-year forecast modeling of the Yangtze Estuary with decreasing river inputs and relative sea-level rise. Using MF ranging from 100 to 400, Guo et al. [12,13] and Zhou et al. [17] investigated the role of river flow, tidal asymmetry and salinity in the long-term (600–4000 years) evolution trends of idealized estuaries. These studies greatly improved our understanding of the long-term evolution trends of the Yangtze Estuary. Nevertheless, it is still interesting to conduct intermediate-term (e.g., 10–30 years) simulations without resorting to the MF method. Such interests are motivated by both the great progress of efficient numerical schemes as well as the improvement of computing hardware. There has been great improvement in both the numerical accuracy and numerical efficiency. For example, for the regional oceanic modeling system (ROMS), Shchepetkin and McWilliams [18] proposed a new family of time-stepping algorithms that combine forward–backward feedback with the best known synchronous algorithms, allowing an increased time step due to the enhanced internal stability without sacrificing its accuracy. In addition, for finite-volume shallow water hydro-sediment-morphodynamic models, the numerical accuracy of such model builds on the use of Riemann solvers to capture shock waves and contact discontinuities [19,20,21]. The high computational cost of such models has been largely reduced by the recently proposed hybrid Local Time Step/Global Maximum Time Step (LTS/GMaTS) method [22,23]. Specifically, high computational efficiency is achieved by implementing the LTS to solve equations governing sediment-laden flows (i.e., the hydro-sediment part), and implementing the GMaTS to solve equations governing bed materials (i.e., the morphodynamic part). The simulation of a 2-year topographic evolution in the Taipingkou Channel of the Yangtze River shows that the calculation cost can be reduced by up to 90%. However, such models have not been applied to estuarine regions as they were developed for non-cohesive sediments. In this paper, the computationally efficient shallow water model [23] is improved by considering mixed cohesive and non-cohesive sediment transport, and the effects of salinity, sediment concentration and sediment diameter on the flocs settling velocity. The improved model was calibrated against field data (i.e., water level, tidal current velocity and SSC) in the Yangtze Estuary and Hangzhou Bay. Comparative numerical studies of key factors are conducted, including the bed resistance, erosion and deposition parameters and the initial bed sediment composition, as well as the Deepwater Navigation Channel (DNC). Finally, the computational efficiency of the improved model in the Yangtze Estuary is evaluated. This paper is structured as follows. In Section 2, the mathematical formulations are described. In Section 3, model setup and validations are introduced, Section 4 analyses the computational efficiency of the improved model, and Section 5 presents the conclusion.

## 2. Mathematical Formulations

#### 2.1. Governing Equations and Empirical Relations

^{−2}is the gravitational acceleration; ${c}_{k}$ is the depth-averaged volumetric sediment concentrations of the k-th sediment class with a mean diameter of ${d}_{k}$, where the subscript $k$ = $1,2,3,\dots {N}_{sps}$, ${N}_{sps}$ is the number of sediment size class; ${C}_{sa}$ = depth-averaged salinity concentration; ${z}_{b}$ = bed elevation; ${S}_{fx}$ and ${S}_{fy}$ are the friction slopes in the $x$ and $y$ directions, which are determined by Equation (5) following [23]; ${S}_{bx}=$ $-\partial {z}_{b}/\partial x$ and ${S}_{by}=$$-\partial {z}_{b}/\partial y$ are the bed slopes in the $x$ and $y$ directions; f is the Coriolis force coefficient; ${E}_{T}=$ $\sum {E}_{k}$ and ${D}_{T}=$$\sum {D}_{k}$ are the total sediment erosion and deposition fluxes, respectively; ${E}_{k}$ and ${D}_{k}$ are the size-specific sediment erosion and deposition fluxes, which are determined by Equations (6) and (7). For cohesive sediment (${d}_{k}\le 0.03$ mm), the erosion and deposition fluxes are calculated applying the Partheniades–Krone formulations [25], whereas the erosion and deposition fluxes of non-cohesive sediment (${d}_{k}>0.03$ mm) follow the approach of Hu et al. [23]; $p$ = bed sediment porosity, which is set empirically as $p$ = 0.42; ${f}_{a,k}$ and ${f}_{s,k}$ are the sediment fractions within the bed active layer and at the interface between the active layer and those below the active layer (see details in [26]), respectively; $\eta $ = ${z}_{b}-\delta $ is the bottom elevation (i.e., the interface) of the active layer and $\delta $ = thickness of the bed active layer.

^{−6}m

^{2}·s

^{−1}is the kinematic viscosity of water; ${\alpha}_{k}$ is the ratio of near-bed to depth-averaged concentration, which is here set as ${\alpha}_{k}$ = 1; ${c}_{e,k}$ is the sediment transport capacity of the k-th size class of sediment determined using the formula of Zhang et al. [31], see Equation (12); $\tau =\rho {u}_{*}{}^{2}$ is the bed shear stress; ${u}_{*}$ is bed shear velocity with ${u}_{*}^{2}=gh\sqrt{{S}_{fx}^{2}+{S}_{fy}^{2}}$; ${\tau}_{d}$ is the critical bed shear stress for deposition (N·m

^{−2}); ${\tau}_{e}$ is the critical bed shear stress for erosion (N·m

^{−2}); M is the erosion coefficient (kg·m

^{−2}·s

^{−1}).

^{−3}is the density of sediment, ${\rho}_{w}$ = 1000 kg·m

^{−3}is the density of fresh water; ${c}_{T}$ = $\sum {c}_{k}$ is the total sediment concentration; ${c}_{p}$ is a threshold sediment concentration above which the settling velocity of the flocs decreases with sediment concentration, and ${c}_{p}$ = 1.5 kg·m

^{−3}is used; ${c}_{sap}$ and ${c}_{sa,\mathrm{min}}$ are threshold salinities: ${c}_{sap}$ = 28 ppt and ${c}_{sa,\mathrm{min}}$ = 2.8 ppt [28,29]; the reference diameter ${d}_{k}$ has a range from 0.011 to 0.022 mm, ${d}_{r}$ = 0.0215 mm is used. The above governing equations and empirical relations differ from Hu et al. [23] in the following aspects. Firstly, the Coriolis force is considered because it plays a crucial role in the dynamics of fluid and the morphological evolution of large-scale water body. Secondly, salinity movement is considered because it may cause fine-grained suspended sediments to flocculate. Thirdly, both cohesive and non-cohesive sediment transport are considered. These improvements (e.g., consideration of the Coriolis force, salinity effects, cohesive sediment transport, etc.), as compared to the previous model by Hu et al. [23], enable the present model the potential to be applied in estuarine regions, where these effects are important. See Section 3 for numerical calibration.

#### 2.2. Finite Volume Discretization Using the Hybrid LTS/GMaTS Method

^{−6}m, $\Delta {t}_{ami}$ is set to the maximum value of those cells with water depth larger than 10

^{−6}m. Based on Equation (13), the globally minimum time step is computed as follows

## 3. Quantitative Accuracy

#### 3.1. Study Area and Numerical Setting

_{50}as an example) in part of the Yangtze Estuary, which is obtained by interpolation using the available measured data (see Figure 4b). For other regions, the spatial distribution of surficial bed sediment composition refers to Yang et al. [39]. There are three open boundaries, including the boundary of the Yangtze River at the Sanjiangying, the boundary of Hangzhou bay at the Qiantang River Bridge and a seaward boundary. The open sea boundary was driven by nine tidal constituents (i.e., M2, S2, N2, K2, K1, O1, P1, Q1 and M4), which are obtained from TPXO 7.2; SSC at the seaward boundary was set to zero since the open boundaries are far away from the Yangtze Estuary and mostly deeper than 100 m; saline concentration at the seaward boundary is interpolated from the HYCOM Global Reanalysis salinity data (http://apdrc.soest.hawaii.edu/data/data.php (accessed on 6 July 2020)). For the landward boundary in the Yangtze River, the time-series of SSC and water discharge measured at Datong (e.g., Figure 5a,b) are used, whereas the sediment composition uses the averaged value over 2004–2010 (Figure 5c). At the Qiantang River bridge, constant water discharge of 1000 m

^{3}·s

^{−1}and sediment concentration of 0.07 kg·m

^{−3}are prescribed. Saline concentrations at the two landward boundaries are zero. The spatial distributions of initial salinity in the whole region are interpolated from the HYCOM Global Reanalysis salinity data. The code of present model is written using Intel

^{(R)}FORTRAN. The model has realized parallel computing by using open multiprocessing (OPENMP), whereas the graphic processing unit (GPU)-acceleration and message passing interface (MPI) parallel computing is in the testing stage. Two parameters are introduced to quantify the performance of the model [40,41], including the RMSE (Equation (21)), and the SS (Equation (22)).

#### 3.2. Model Performance for Tidal Hydrodynamics

^{−1}and 24°4′, respectively, which are consistently smaller than those for the relation n = 0.013 + 0.012/h and n = 0.016. Moreover, the values of SS for 0.01+0.01/h show that it is better than another two Manning roughness coefficients while simulating the hydrodynamics (i.e., SS of 0.812–0.992, 0.862–0.965 and 0.837–0.940 for water level, tidal current velocity and direction, respectively). Therefore, the Manning roughness coefficient n = 0.01 + 0.01/h is used in the following. The comparison between simulated and observed water level as shown in Figure 6, indicating that the simulated results are in good agreement with the measured data. Figure 7 presents the comparison between simulated and observed tidal velocity and direction, for which there are some deviations between the simulated tidal velocity and measured data, especially at NCH6. The reason for this discrepancy may be that the locations of these three stations are easily disturbed by the incident flow and reflected flow from the coasts and channel. The averaged values of SS for tidal velocity and direction are 0.901 and 0.889 (n = 0.01 + 0.01/h), respectively, which are greater than 0.65. Overall, the above validation generally shows the present model’s good capability in the reproduction of tidal hydrodynamics.

#### 3.3. Model Performance for SSC and Sensitivity Analysis

^{−2}to 3.5 N·m

^{−2}; the critical shear stress for deposition is usually linked to the critical shear stress for erosion using empirical relations (e.g., ${\tau}_{d}=0.69{\tau}_{e}$ in Zhang and Xie [27]; ${\tau}_{d}=0.5{\tau}_{e}$ in Zhu et al., [49]; and ${\tau}_{d}=4\times {\tau}_{e}/9$ in Cao and Wang [50]); the erosion coefficient can vary in the range of 10

^{−6}–10

^{−3}kg·m

^{−2}·s

^{−1}. Here, three values were selected for ${\tau}_{e}$: 0.1 N·m

^{−2}, 0.4 N·m

^{−2}and 0.8 N·m

^{−2}; three values are selected for M: 10 × 10

^{−5}kg·m

^{−2}·s

^{−1}, 3.0 × 10

^{−5}kg·m

^{−2}·s

^{−1}and 5.0 × 10

^{−5}kg·m

^{−2}·s

^{−1}; and the critical shear stress for deposition is set as ${\tau}_{d}=4\times {\tau}_{e}/9$ following Cao and Wang [50]. This results in a total of nine cases. Time variations of the SSC during July 2016 (including a spring tide and a neap tide) are simulated using these nine numerical cases.

^{−2}and M = 3.0 × 10

^{−5}kg·m

^{−2}·s

^{−1}results in good agreements at five stations (NG3, CS9S, CS6S, NCH1, NCH9; see Figure 3b for the specific positions of these stations). However, the results of CS3S, CSWS and NCH4 stations during spring tide from this case deviates more from the measured data, as compared to another five stations. This indicates that the specification of ${\tau}_{e}$ and M should be site-dependent. This is understandable because the values of ${\tau}_{e}$ and M are both functions of the bed density, porosity, composition, consolidation and evolution of the sediment under the complex and mixed effects of the physical and biological interaction process. These characteristics vary significantly in space and in time. As a compromise, these parameters (i.e., ${\tau}_{e}$ = 0.4 N·m

^{−2}, ${\tau}_{d}$ = 0.18 N·m

^{−2}and erosion coefficient M = 3.0 × 10

^{−5}kg·m

^{−2}·s

^{−1}), which give the satisfactory agreements at most stations, are used in the following.

## 4. Computational Efficiency

^{−3}m·s

^{−1}and 1°56′, respectively, which are significantly less than the RMSE in Table 1.

## 5. Conclusions

^{−3}m·s

^{−1}(for tidal current velocity) and 1°56′ (for tidal current direction). This advantage, along with its well-demonstrated quantitative accuracy of its capability to deal with mixed cohesive and non-cohesive sediments, provide a basis for long-term morphodynamic modeling in estuarine regions.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Yang, S.; Milliman, J.D.; Li, P.; Xu, K. 50,000 dams later: Erosion of the Yangtze River and its delta. Glob. Planet. Chang.
**2011**, 75, 14–20. [Google Scholar] [CrossRef] - Li, P.; Yang, S.; Milliman, J.D.; Xu, K.; Qin, W.; Wu, C.; Chen, Y.; Shi, B. Spatial, temporal, and human-induced variations in suspended sediment concentration in the surface waters of the Yangtze estuary and adjacent coastal areas. Estuaries Coasts.
**2012**, 35, 1316–1327. [Google Scholar] [CrossRef][Green Version] - Yang, S.; Milliman, J.D.; Xu, K.; Deng, B.; Zhang, X.; Luo, X. Downstream sedimentary and geomorphic impacts of the Three Gorges Dam on the Yangtze River. Earth Sci. Rev.
**2014**, 138, 469–486. [Google Scholar] [CrossRef] - Luan, H.; Ding, P.; Wang, Z.; Ge, J.; Yang, S. Decadal morphological evolution of the Yangtze Estuary in response to river input changes and estuarine engineering projects. Geomorphology
**2016**, 265, 12–23. [Google Scholar] [CrossRef] - Zhao, J.; Guo, L.; He, Q.; Wang, Z.; van Maren, D.S.; Wang, X. An analysis on half century morphological changes in the Changjiang Estuary: Spatial variability under natural processes and human intervention. J. Mar. Syst.
**2018**, 181, 25–36. [Google Scholar] [CrossRef][Green Version] - Zhu, C.; Guo, L.; van Maren, D.S.; Tian, B.; Wang, X.; He, Q.; Wang, Z. Decadal morphological evolution of the mouth zone of the Yangtze Estuary in response to human interventions. Earth Surf. Landf. Process.
**2018**, 44, 2319–2332. [Google Scholar] [CrossRef] - Roelvink, J.A. Coastal morphodynamic evolution techniques. Coast. Eng.
**2006**, 53, 277–287. [Google Scholar] [CrossRef] - Roelvink, J.A.; Reniers, A.J.H.M. A Guide to Coastal Morphology Modeling. Advances in Coastal and Ocean Engineering; World Scientific Publishing Company: Singapore, 2011; Volume 12. [Google Scholar]
- Hu, K.; Ding, P.; Wang, Z.; Yang, S. A 2D/3D hydrodynamic and sediment transport model for the Yangtze Estuary, China. J. Mar. Syst.
**2009**, 77, 114–136. [Google Scholar] [CrossRef] - Kuang, C.; Liu, X.; Gu, J.; Guo, Y.; Huang, S.; Liu, S.; Yu, W.; Huang, J.; Sun, B. Numerical prediction of medium-term tidal flat evolution in the Yangtze Estuary: Impacts of the three gorges project. Cont. Shelf Res.
**2013**, 52, 12–26. [Google Scholar] [CrossRef] - Luan, H.; Ding, P.; Wang, Z.; Ge, J. Process-based morphodynamic modeling of the Yangtze Estuary at a decadal timescale: Controls on estuarine evolution and future trends. Geomorphology
**2017**, 290, 347–364. [Google Scholar] [CrossRef] - Guo, L.; van der Wegen, M.; Roelvink, D.J.A.; He, Q. The role of river discharge and tidal asymmetry on 1D estuarine morphodynamics. J. Geophys. Res. Earth Surf.
**2014**, 119, 2315–2334. [Google Scholar] [CrossRef][Green Version] - Guo, L.; van der Wegen, M.; Roelvink, D.J.A.; Wang, Z.; He, Q. Long-term, process-based morphodynamic modeling of a fluvio-deltaic system. Part I: The role of river discharge. Cont. Shelf Res.
**2015**, 109, 95–111. [Google Scholar] [CrossRef] - Lesser, G.R.; Roelvink, J.A.; Van Kester, J.A.T.M.; Stelling, G.S. Development and validation of a three-dimensional morphological model. Coast. Eng.
**2004**, 51, 883–915. [Google Scholar] [CrossRef] - Van der Wegen, M.; Roelvink, J.A. Long-term morphodynamic evolution of a tidal embayment using a two-dimensional, process-based model. J. Geophys. Res. Ocean.
**2008**, 113. [Google Scholar] [CrossRef][Green Version] - Van der Wegen, M.; Wang, Z.B.; Savenije, H.H.G.; Roelvink, J.A. Long-term morphodynamic evolution and energy dissipation in a coastal plain, tidal embayment. J. Geophys. Res. Earth Surf.
**2008**, 113. [Google Scholar] [CrossRef][Green Version] - Zhou, Z.; Chen, L.; Tao, J.; Gong, Z.; Guo, L.; van der Wegen, M.; Townend, I.; Zhang, C. The role of salinity in fluvio-deltaic morphodynamics: A long-term modelling study. Earth Surf. Process. Landf.
**2020**, 45, 590–604. [Google Scholar] [CrossRef] - Shchepetkin, A.F.; McWilliams, J.C. The regional oceanic modeling system (ROMS): A split-explicit, free-surface, topography-following-coordinate oceanic model. Ocean. Model.
**2005**, 9, 347–404. [Google Scholar] [CrossRef] - Toro, E.F. Shock-Capturing Methods for Free-Surface Shallow Flows; John Wiley & Sons Ltd.: Chichester, UK, 2001. [Google Scholar]
- Cao, Z.; Pender, G.; Wallis, S.; Carling, P. Computational dam-break hydraulics over erodible sediment bed. J. Hydraul. Eng.
**2004**, 130, 689–703. [Google Scholar] [CrossRef] - Hu, P.; Cao, Z. Fully coupled mathematical modelling of turbidity currents. Adv. Water Resour.
**2009**, 32, 1–15. [Google Scholar] [CrossRef] - Hu, P.; Lei, Y.; Han, J.; Cao, Z.; Liu, H.; Yue, Z.; He, Z. An improved local-time-step for 2D shallow water modeling based on unstructured grids. J. Hydraul. Eng. ASCE
**2019**, 145, 06019017. [Google Scholar] [CrossRef] - Hu, P.; Lei, Y.; Han, J.; Cao, Z.; Liu, H.; He, Z. Computationally efficient hydro-morphodynamic modelling using a Hybrid local-time-step and the global maximum-time-step. Adv. Water Resour.
**2019**, 127, 26–38. [Google Scholar] [CrossRef] - Hu, P.; Han, J.; Li, W.; Sun, Z.; He, Z. Numerical investigation of a sandbar formation and evolution in a tide-dominated estuary using a hydro-sediment-morphodynamic model. Coastal Eng. J.
**2018**, 60, 466–483. [Google Scholar] [CrossRef] - Partheniades, E. Erosion and deposition of cohesive soils. J. Hydraulic. Div. ASCE
**1965**, 91, 105–139. [Google Scholar] [CrossRef] - Hu, P.; Cao, Z.; Pender, G.; Liu, H. Numerical modelling of riverbed grain size stratigraphic evolution. Int. J. Sediment. Res.
**2014**, 29, 329–343. [Google Scholar] [CrossRef] - Zhang, R.; Xie, J. Sedimentation Research in China: Systematic Selections; China and Power Press: Beijing, China, 1993. (In Chinese) [Google Scholar]
- Lin, Q.; Wu, W. A one-dimensional model of mixed cohesive and non-cohesive sediment transport in open channels. J. Hydraul. Res.
**2013**, 51, 506–517. [Google Scholar] [CrossRef] - Wu, W.; Qi, H.; Wang, S. Depth-averaged 2-d calculation of tidal flow, salinity and cohesive sediment transport in estuaries. Int. Sediment. Res.
**2004**, 19, 1–10. [Google Scholar] - Wu, W.; He, Z.; Lin, Q.; Sanchez, A.; Marsooli, R. Non-equilibrium sediment transport modeling-extensions and applications. Sediment. Transp. Monit. Modeling Manag.
**2013**, 21, 179–212. [Google Scholar] - Zhang, R.; Xie, J.; Wang; River., M. Sediment Dynamics; China and Power Press: Beijing, China, 1989. (In Chinese) [Google Scholar]
- Hou, J.; Liang, Q.; Simons, F.; Hinkelmann, R. A 2d well-balanced shallow flow model for unstructured grids with novel slope source term treatment. Adv. Water Resour.
**2013**, 52, 107–131. [Google Scholar] [CrossRef] - Liang, Q.; Marche, F. Numerical resolution of well-balanced shallow water equations with complex source terms. Adv. Water Resour.
**2009**, 32, 873–884. [Google Scholar] [CrossRef] - Audusse, E.; Bouchut, F.; Bristeau, M.; Klein, R.; Perthame, B.T. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput.
**2004**, 25, 2050–2065. [Google Scholar] [CrossRef] - He, Z.; Wu, T.; Weng, H.; Hu, P.; Wu, G. Numerical investigation of the vegetation effects on dam-flows and bed morphological changes. Int. J. Sediment. Res.
**2017**, 32, 105–120. [Google Scholar] [CrossRef] - Hu, P.; Li, Y. Numerical modeling of the propagation and morphological changes of turbidity currents using a cost-saving strategy of solution updating. Int. J. Sediment. Res.
**2020**, 35, 587–599. [Google Scholar] [CrossRef] - Lu, C.; Jia, X.; Han, Y.; Bai, Y. Numerical simulation of sudden silting in the Yangtze Estuary deepwater channel by the wave of typhoon. Adv. Water Sci.
**2018**, 29, 696–705. (In Chinese) [Google Scholar] - Song, D.; Wang, X. Suspended sediment transport in the Deepwater Navigation Channel, Yangtze River Estuary, China, in the dry season 2009: 2. Numerical simulations. J. Geophys. Res. Ocean.
**2013**, 118, 5568–5590. [Google Scholar] [CrossRef] - Yang, H.; Yang, S.; Meng, Y.; Zhu, Q.; Wu, C.; Shi, B. Sediment distribution patterns in the Yangtze Estuary and comparison of particle size measurement methods. Shanghai Land Resour.
**2017**, 38, 75–79. (In Chinese) [Google Scholar] - Willmott, C. On the validation of models. Phys. Geogr.
**1981**, 2, 184–194. [Google Scholar] [CrossRef] - Warner, J.C.; Geyer, W.R.; Lerczak, J.A. Numerical modeling of an estuary: A comprehensive skill assessment. J. Geophys. Res. Ocean.
**2005**, 110. [Google Scholar] [CrossRef] - Allen, J.I.; Somerfield, P.J.; Gilbert, F.J. Quantifying uncertainty in high-resolution coupled hydrodynamic-ecosystem models. J. Mar. Syst.
**2007**, 64, 3–14. [Google Scholar] [CrossRef] - Dou, X.; Li, L.; Dou, G. Numerical model study on total sediment transport in the Yangtze Estuary. Hydro Sci. Eng.
**1999**, 2, 136–145. (In Chinese) [Google Scholar] - Chen, G.; Niu, Y. Marine Atlas of Bohai Sea, Yellow Sea and East China Sea; China Ocean Press: Beijing, China, 1993. (In Chinese) [Google Scholar]
- Hu, K.; Ding, P.; Zhu, S.; Cao, Z. 2-D current field numerical simulation integrating Yangtze Estuary with Hangzhou Bay. China Ocean. Eng.
**2000**, 14, 89–102. [Google Scholar] - Ge, J.; Ding, P.; Chen, C.; Hu, S.; Fu, G.; Wu, L. An integrated East China Sea-Changjiang Estuary model system with aim at resolving multi-scale regional-shelf-estuarine dynamics. Ocean. Dyn.
**2013**, 63, 881–900. [Google Scholar] [CrossRef] - Yao, Z.; He, R.; Bao, X.; Wu, D.; Song, J. M_2 tidal dynamics in bohai and yellow seas: A hybrid data assimilative modeling study. Ocean. Dyn.
**2012**, 62, 753–769. [Google Scholar] [CrossRef] - Ge, J.; Shen, F.; Guo, W.; Chen, C.; Ding, P. Estimation of critical shear stress for erosion in the Changjiang estuary: A synergy research of observation, GOCI sensing and modeling. J. Geophys. Res. Ocean.
**2015**, 120, 8439–8465. [Google Scholar] [CrossRef][Green Version] - Zhu, Q.; Prooijen, B.V.; Wang, Z.; Yang, S. Bed-level changes on intertidal wetland in response to waves and tides: A case study from the Yangtze River delta. Mar. Geol.
**2017**, 385, 160–172. [Google Scholar] [CrossRef] - Cao, Z.; Wang, Y. Numerical Modeling of Hydrodynamics and Sediment; Tianjin University Press: Tianjin, China, 1994. (In Chinese) [Google Scholar]

**Figure 2.**Sketch of the numerical structure (revised from Hu and Li [36]).

**Figure 3.**(

**a**) Model domain embedded with a set of meshes; (

**b**) part of the bed topography. Also included in Figure 3b are stations for the available measured data; (

**c**) local meshes considering the DNC engineering; (

**d**) local meshes without considering the DNC engineering.

**Figure 4.**(

**a**) Spatial distribution of bed sediment composition (using d

_{50}as an example) in part of the Yangtze Estuary, which was obtained by interpolation using the available measured data in (

**b**). (

**b**) Indications for the available measured surficial bed sediment compositions in the Yangtze Estuary (data in Area I were measured in January 2016, whereas data in Area II were measured in July 2016).

**Figure 5.**Basic details of water discharge, sediment concentration, and composition. (

**a**) Water discharge of the Yangtze Estuary during July 2016; (

**b**) the sediment discharge at Datong station during July 2016; (

**c**) the averaged riverine suspended sediment compositions at Datong in July from 2004 to 2010.

**Figure 6.**Verification of water level (dots denote the observed data; line denotes the simulation data).

**Figure 7.**Verification of depth-averaged tidal velocity and direction (dots denote the observed data; line denotes the simulation data).

**Figure 8.**Distributions of the co-amplitude (blue dashed lines; unit: meter) and co-phase (red solid lines; unit: degree) of the M2, S2, K1 and O1 tide constituents around the East China Sea, Yellow Sea and the Bohai Sea.

**Figure 9.**Comparison of simulated and measured SSC under the different critical shear stresses and erosion coefficients at 8 stations shown in Figure 3b. The graphs on the left and right sides of break-line symbol show the variation of SSC during spring tide and neap tide, respectively.

**Figure 10.**Comparison of simulated and measured SSC during spring tide under the different bed sediment composition, and subfigures (

**a**–

**h**) represent the comparison results at 8 stations for SSC shown in Figure 3b, respectively.

**Figure 11.**Comparison of simulated and measured SSC during spring tide under the influence of Deepwater Navigation Channel (DNC), and subfigures (

**a**–

**h**) represent the comparison results at 8 stations for SSC shown in Figure 3b, respectively.

**Figure 12.**(

**a**) Calculation efficiency varies with ${m}_{user}$; (

**b**) the relative difference of tide level varies with ${m}_{user}$; (

**c**) the relative difference of the tidal current velocity changes with ${m}_{user}$; (

**d**) the relative difference of the tidal current direction changes with ${m}_{user}$.

**Table 1.**Model performance (RMSE and SS) in simulating hydrodynamics when using different roughness estimations.

Stations | Hydrodynamics | n = 0.01 + 0.01/h | n = 0.013 + 0.012/h | n = 0.016 | |||
---|---|---|---|---|---|---|---|

RMSE | SS | RMSE | SS | RMSE | SS | ||

SDK | water level | 0.363 m | 0.812 | 0.214 m | 0.937 | 0.297 m | 0.871 |

JGJ | 0.286 m | 0.905 | 0.372 m | 0.795 | 0.412 m | 0.782 | |

NCD | 0.157 m | 0.992 | 0.210 m | 0.942 | 0.326 m | 0.865 | |

BCZ | 0.218 m | 0.952 | 0.179 m | 0.976 | 0.268 m | 0.921 | |

YSG | 0.229 m | 0.966 | 0.279 m | 0.912 | 0.358 m | 0.836 | |

BL | 0.229 m | 0.946 | 0.258 m | 0.927 | 0.279 m | 0.891 | |

DS | 0.259 m | 0.922 | 0.288 m | 0.883 | 0.331 m | 0.857 | |

LH | 0.179 m | 0.975 | 0.351 m | 0.837 | 0.458 m | 0.753 | |

SS | 0.145 m | 0.981 | 0.196 m | 0.955 | 0.287 m | 0.889 | |

ZH | 0.218 m | 0.944 | 0.188 m | 0.968 | 0.275 m | 0.916 | |

Mean value | 0.228 m | 0.940 | 0.254 m | 0.913 | 0.329 m | 0.858 | |

CS9S | Tidal current velocity | 0.287 m·s^{−1} | 0.877 | 0.316 m·s^{−1} | 0.847 | 0.375 m·s^{−1} | 0.815 |

NGN4S | 0.186 m·s^{−1} | 0.965 | 0.198 m·s^{−1} | 0.947 | 0.227 m·s^{−1} | 0.935 | |

NCH6 | 0.305 m·s^{−1} | 0.862 | 0.315 m·s^{−1} | 0.851 | 0.371 m·s^{−1} | 0.821 | |

Mean value | 0.259 m·s^{−1} | 0.901 | 0.276 m·s^{−1} | 0.882 | 0.324 m·s^{−1} | 0.857 | |

CS9S | Tidal Current direction | 13°37′ | 0.940 | 13°58′ | 0.925 | 14°26′ | 0.907 |

NGN4S | 24°4′ | 0.837 | 24°15′ | 0.827 | 24°32′ | 0.813 | |

NCH6 | 15°20′ | 0.889 | 15°48′ | 0.882 | 16°5′ | 0.876 | |

Mean value | 17°20′ | 0.889 | 17°74′ | 0.878 | 18°36′ | 0.875 |

Method | m_{user} | Max Time Step (s) | Computational Cost (h) | Reduction in Computational Cost | Relative Difference | ||
---|---|---|---|---|---|---|---|

Water Level (m) | Tidal Current Velocity (m·s ^{−1}) | Tidal Current Direction (°) | |||||

GMiTS | - | 0.42 | 289.10 | - | - | - | - |

LTS+ GMaTS | 0 | 0.42 | 289.10 | 0 | - | - | - |

1 | 0.85 | 211.58 | 26.81% | 0.0006 | 1.2 × 10^{−4} | 0°08′ | |

2 | 1.70 | 116.03 | 59.87% | 0.0007 | 2.1 × 10^{−4} | 0°10′ | |

3 | 3.40 | 68.75 | 76.22% | 0.0008 | 2.5 × 10^{−4} | 0°18′ | |

4 | 6.81 | 46.72 | 83.84% | 0.0015 | 6.0 × 10^{−4} | 0°45′ | |

5 | 13.62 | 35.10 | 87.86% | 0.0023 | 1.1 × 10^{−3} | 1°30′ | |

6 | 27.24 | 31.88 | 88.97% | 0.0040 | 2.1 × 10^{−3} | 1°45′ | |

7 | 54.47 | 28.98 | 89.98% | 0.0070 | 3.0 × 10^{−3} | 1°56′ |

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## Share and Cite

**MDPI and ACS Style**

Hu, P.; Tao, J.; Ji, A.; Li, W.; He, Z. A Computationally Efficient Shallow Water Model for Mixed Cohesive and Non-Cohesive Sediment Transport in the Yangtze Estuary. *Water* **2021**, *13*, 1435.
https://doi.org/10.3390/w13101435

**AMA Style**

Hu P, Tao J, Ji A, Li W, He Z. A Computationally Efficient Shallow Water Model for Mixed Cohesive and Non-Cohesive Sediment Transport in the Yangtze Estuary. *Water*. 2021; 13(10):1435.
https://doi.org/10.3390/w13101435

**Chicago/Turabian Style**

Hu, Peng, Junyu Tao, Aofei Ji, Wei Li, and Zhiguo He. 2021. "A Computationally Efficient Shallow Water Model for Mixed Cohesive and Non-Cohesive Sediment Transport in the Yangtze Estuary" *Water* 13, no. 10: 1435.
https://doi.org/10.3390/w13101435