Hydrodynamics of the Qiantang Tidal Bore and Its Responses to Embankment, Morphology, and River Discharge

Abstract

The Qiantang tidal bore is globally renowned for its spectacular landscape and its strong impacts on the Qiantang riverbed erosion/deposition process. However, due to the extremely high spatial gradient of its water level along the propagation direction around the bore front, as well as the very swift movement of this front, the numerical reproduction of the bore formation and propagation has been a challenge for several decades. Here, using GPU acceleration and Local Time Stepping (LTS), we present a high-resolution simulation of the Qiantang tidal bore formation and propagation, achieving a 10 m resolution across 1169 km², which captures the bore dynamics and the full formation-to-decay processes while simulating 2-day tidal bore phenomena in 1.2 h. The formation mechanisms of three typical tidal bores (cross-shape bore, thread-shape bore, and returned bore) are revealed numerically. The cross-shape bore appears first and is generated by flow division around the mid-channel bars; further upstream, the thread-shape bore is formed due to the increasingly narrow river along a straight reach; and its reflection results in the returned bores at the YC bending reach. This study also highlights how variations in flow discharge affect the tidal bore. When the discharge increases to the annual mean discharge, the intensity of the tidal bore is increased, whereas extremely high flood peak discharge inhibits bore propagation. This study provides a scientific basis for conserving tidal bore landscapes and offers decision support for sustainable estuarine governance.

1. Introduction

A tidal bore, which is essentially a shallow-water shock wave with significant nonlinearity and discontinuity, is characterized by a steep-fronted surge during tidal propagation in estuaries. Worldwide, tidal bores have been reported in many estuaries, such as the Qiantang Estuary in China, the Seine and Garonne Rivers in France, the Daly Estuary in Australia, the Rokan River Estuary in Indonesia, and the Amazon River in Brazil [1,2,3,4,5]. On the one hand, as a great spectacle of nature, the tidal bore should be protected. The strong mixture of estuarine water with environmental matters like sediments, nutrients, oceanic algae, etc., induced by tidal bore forces is also of crucial importance for aquatic and benthic organisms. On the other hand, tidal bores have the potential to cause catastrophic damage to coastal engineering works (e.g., coastal levees and hydraulic structures) and endanger human life; thus, they should be managed to a controllable extent. In this regard, it is of significance to balance this contradiction in practice for which the understanding of tidal bore hydrodynamics deserves prior study.

At the microscale, studies are mainly focused on the velocity structures and turbulent features of tidal bores through laboratory experiments and small-scale delicate numerical modeling. For example, Yeh and Mok [6] compared turbulence differences between bores and hydraulic jumps, proposing fractal vortex scales during bore breaking. Using three-dimensional large-eddy simulations, Lubin et al. [7] reproduced vortex-tube structures of the bore breaking process, highlighting the important role of vortex stretching on the transport of turbulent kinetic energy. Koch and Chanson [8] demonstrated the anisotropic turbulent energy distribution of a bore front based on flume testing and modeling, using Reynolds-averaged Navier–Stokes equations. Bonneton et al. [9] established dynamic balance equations for turbulent energy production–dissipation relating to the tidal bores of the Garonne River, complementing the turbulence closure theories for shallow-water shocks. Qin et al. [10] numerically revealed the vertical velocity stratification of the undular and breaking bores, as well as their differences in turbulent energy distributions, using three-dimensional large-eddy simulations. While advancing the understanding of turbulent microscale mechanisms, these models’ high-resolution demands limit simulation scales, making them difficult to apply to practical estuarine scenarios. Conversely, many macroscale investigations are concerned about the relations of tidal bore evolution with the estuarine geometry and other estuarine dynamic processes. Based on field observations in the Mont-Saint-Michel Bay, Chanson [11] revealed the enhancement of estuarine mixing using tidal bore processes. Savenije et al. [12] established an analytical relationship between tide deformation and estuarine convergence rate for the Amazon River, proposing a dimensionless index for quantifying tidal bore intensity. By deploying a depth-integrated model, Furuyama and Chanson [13] investigated the front shape of tidal bores and their associated turbulence dissipation. Based on the Serre–Green–Naghdi equations, Filippini et al. [14] suggested that the critical condition for tidal bore formation depends on the relative importance of the distortion and dissipation of tidal waves. Using the TELEMAC 2D model, Roy-Biswas and Sen [15] found that in the Hooghly estuary (India), rapid channel narrowing could lead to the formation of tidal bores when the tidal range exceeds 4.8 m. Macroscale studies require balancing efficiency and precision in large-scale estuarine modeling, where 2D models serve as the preferred solution due to their theoretical suitability and computational efficiency. On one hand, core tidal bore mechanisms are primarily influenced by horizontal nonlinear advection and bed friction, with vertical structure contributions being limited. This enables 2D models to effectively characterize macroscale bore dynamics; On the other hand, high grid resolution demands for capturing bore dynamics significantly increase computational loads. By employing vertical averaging simplifications, 2D models can reduce computational complexity while maintaining accuracy, thus meeting the need to balance precision and efficiency in macroscale tidal bore evolution research.

The present study focuses on the tidal bore in the Qiantang Estuary (QE), China (see Figure 1). According to hydrodynamic controls, the estuary can be divided into three reaches [16,17]. The upper reach, which spans 87 km from the Fuchun Power Station to Zakou (ZK), is characterized by river discharge, and its bathymetry is effectively stable. The middle reach extends 108 km from ZK to Ganpu (GP) and is influenced by a combination of river flow and tidal currents. The lower reach, which is 85 km in length, that runs downstream from GP is also known as Hangzhou Bay (HZB) and is dominated by the tidal force [17]. As tidal flows from the East China Sea propagate upstream through HZB, the funnel-shaped shorelines concentrate tidal energy, thus enlarging the tidal range to an annual mean of 5.62 m and a maximum of 9 m at GP, respectively. Meanwhile, the giant underwater sandbar upstream of Zhapu (ZP) promotes tidal deformation and likely contributes to tidal bore development in the upstream GP reach. The tidal bore is first formed about 5 km upstream of GP, before being strengthened in the reach between Cao’e River Mouth and Yanguan (YG) and finally dissipating near Fuyang. The bore influences are extended over 120 km in the Qiantang River (QR) [18] (see details in Section 3.1).

The tidal bore evolutions in the QE have received much attention in recent decades. The mathematical modeling of tidal bores has been actively advocated for and developed in order to achieve a more comprehensive recognition of the bore evolution processes in the QE, as well as to ensure that this achievement can be made at a relatively low cost. The work of Pan et al. [19] marks a milestone in this area. Based on structured quadrilateral grids, Pan et al. [20] were the first to develop a two-dimensional (2D) Godunov-type shallow-water model to capture the tidal bore features in the QE. Later, Pan and Xu [20] extended this model to unstructured triangular grids, establishing Kinetic Flux Vector Splitting (KFVS)-type shallow-water models, which were more adapted to the complex geometry of the QE for bore replication. By using the FVCOM model, Xie et al. [21] investigated the vertical velocity profile during bore processes, while Wang and Pan [22] examined typhoon effects on tidal bore evolutions. By developing a 2D Boussinesq-type model, Huang et al. [23] successfully reproduced the generation, development, and decay of QE tidal bores. However, it should be noted that most previous models are based on CPU simulations. Some challenges remain for efficient and accurate modeling and physics analysis, as follows: (1) To accurately capture the tidal bore, a grid resolution scaled to several meters is required, which results in an extremely expensive computation process for the overall QE model; thus, it is critically important to solve this computational overhead and to ensure a balance with accuracy. (2) Though the varied features of tidal bores in the QE have been reported previously, a delicate reproduction of the whole tidal bore landscapes (e.g., in terms of the fine resolution of tidal levels) and their formation mechanisms are still lacking in systematic studies.

Therefore, in this study, we aim to solve the above two challenges by numerically modeling the tidal bore evolution of the QE through a two-dimensional depth-averaged hydrodynamic model within a triangular finite volume framework [24,25,26]. For the first challenge, the model attempts to couple Graphics Processing Unit (GPU) acceleration with the LTS method for computation acceleration. The LTS method was first advocated by Osher and Sanders [27] in a one-dimensional conservation law solution in order to release the limit of the global minimum time step computed by the Courant–Friendrichs–Lewy (CFL) condition for explicit discretization. Using this method, each computational grid can be updated according to its own stability, which may significantly speed up computation, especially for spatially varied mesh sizes. Later, Sanders [28] combined the LTS method with the unstructured grid Godunov-type finite volume method, marking the first application of the LTS method in a two-dimensional shallow-water model. Recently, Hu et al. [25,26] enhanced the stability of this method, achieving a promising computational efficiency in both shallow-water and morphological modeling. The combination of GPU and LTS techniques has been verified in computational fluid dynamics simulations. Specifically, Dazzi et al. [29] verified its efficacy in dam-break flow modeling; Hu et al. [30] validated its application for tidal dynamics in the Yangtze River; Zhao et al. [31] extended this to hydro-sediment–morphodynamic modeling; Liu et al. [32,33] optimized the Local Time Stepping (LTS) scheme using the Single-Layer Interface Prediction–Correction scheme, significantly enhancing computational efficiency. The former achieved a 37% acceleration ratio in storm surge modeling by dynamically coupling coarse and fine time discretization while strictly maintaining mass conservation. The latter proposed a dynamic grading strategy based on this framework, achieving a 2.615-fold speedup ratio, offering a promising approach for cross-scale hydrodynamic modeling in complex coastal and marine engineering problems. For the second challenge, numerical experiments will be conducted to reproduce the tidal bore landscapes with a fine resolution, especially for tidal levels; we will systematically investigate the relationship between bore variations and estuarine geometry and river runoff in the QE. The following points are expected to be addressed: (1) assessing the model’s ability to replicate the tidal bore evolution processes of the QE, (2) clarifying the characteristics and mechanisms of the tidal bore landscapes for different geometries, and (3) evaluating the impacts of river runoff variations on bore evolution. The present work might provide a scientific tool for conserving tidal bore landscapes and offers decision support for sustainable estuarine governance.

2. Mathematical Model

2.1. Governing Equations

The model established in this study is described using two-dimensional shallow-water governing equations with the following vector form:

$${\displaystyle \frac{\partial \mathrm{U}}{\partial t}} + {\displaystyle \frac{\partial \mathrm{F}}{\partial x}} + {\displaystyle \frac{\partial \mathrm{G}}{\partial y}} ={ S}_b + S_f + S_c$$

(1)

$$\mathrm{U}=\left[\begin{array}{c}h\\ hu\\ hv\end{array}\right],\mathrm{F}=\left[\begin{array}{c}hu\\ h{u}^2+\mathrm{g}{h}^2/2\\ huv\end{array}\right], \mathrm{G}= \left[\begin{array}{c}hu\\ huv\\ h{v}^2+g{h}^2/2\end{array}\right]$$

(2a–c)

$$S_b=\left[\begin{array}{c}0\\ \mathrm{g}h{S}_{\mathit{bx}}\\ \mathrm{g}h{S}_{\mathit{by}}\end{array}\right],S_f=\left[\begin{array}{c}0\\ -\mathrm{g}h{S}_{fx}\\ -\mathrm{g}h{S}_{fy}\end{array}\right],S_c=\left[\begin{array}{c}0\\ fhv\\ -fhu\end{array}\right]$$

(2d–e)

where $U$ is the conserved variable; $\mathrm{F}$ and $\mathrm{G}$ are the convective flux vectors in the $x$ and $y$ directions, respectively; $S_b$, $S_f$, and $S_c$ are the bottom-slope source term vectors, the drag source term vectors, and Koch’s force source term vectors, respectively; $t$ is the time; $x$ and $y$ are the spatial horizontal coordinates of the Cartesian coordinate system; $h$ is the water depth; $u$ and $v$ are the depth-integrated mean velocity components in the $x$ and $y$ directions, respectively; $S_{bx} = -\partial z_b/\partial x$ and $S_{by} = -\partial z_b/\partial y$ are the depth-integrated mean velocity components in the $x$ and y directions, respectively; $S_{fx}$ and $S_{fx}$ are the resistance slopes in the $x$ and $y$ directions, respectively; $f=2{\omega }_0\sin \phi$ is the Koch force parameter; ${\omega }_0=7.27 \times {10}^{-5} \mathrm{rad}/\mathrm{s}$ is the rotation speed of the Earth; and $\phi$ is the latitude.

2.2. Empirical Relations

The friction slope is estimated using the Manning formula, as follows:

$$S_{fx}={\displaystyle \frac{n^{2}u\sqrt{u^2 + v^2}}{h^{4/3}}},S_{fy}={\displaystyle \frac{n^{2}v\sqrt{u^2 + v^2}}{h^{4/3}}}$$

(3)

where $n$ denotes Manning’s roughness coefficient.

2.3. Numerical Methods

2.3.1. Finite Volume Discretization on Unstructured Triangular Meshes

As a typical shock wave phenomenon in shallow-water flow, a tidal bore is characterized by a rapid rise in water levels, drastic velocity changes, and the rapid migration of wet–dry boundaries during its propagation [34]. This places high demands on the discontinuity-capturing ability of numerical models. To address this, the model employs the HLLC algorithm to reconstruct the Riemann states on both sides of the interface, preserving the physical discontinuity of the tidal bore front in flux calculations and accurately capturing the shock wave. Meanwhile, to simulate the dynamic wetting–drying processes during tidal bore propagation, this study uses the slope flux method to handle the bed slope source term, combined with water level gradient reconstruction techniques to dynamically mark the wet–dry cells.

The governing equations are discretized and solved using the numerical method described by Hu et al. [24,25,26]. Within the finite volume method framework, the equations are discretized onto unstructured triangular meshes and are updated using a modified LTS method to enhance computational speed. The discretization for cell ($i$) can be written as follows:

$$U_i^{**} = U_i^{*} - [{\displaystyle \frac{{∆t}_{LTS\text{-}i}}{A_i}}\sum _{j=1}^3{\mathrm{E}}_{nij}∆L_{ij} +{ ∆t}_{LTS\text{-}i}(S_{bi } + S_{fi} + S_{ci}\left)\right]$$

(4)

where $U_i^{*}$ and $U_i^{**}$ represent two consecutive time layers between $t$ and $t + \Delta T$; ${∆t}_{LTS\text{-}i}$ is the hierarchical time step of the $i$th cell; $∆T$ is the global time step; Ai is the area of the $i$th cell; ${\mathrm{E}}_{nij} = ({{\mathrm{F}}_{nx} + {\mathrm{G}}_{ny})}_{ij}$ is the convective flux of the $j$th face of the $i$th cell; $n_{ij}{{ = (n}_x + n_y)}_{ij}$ is the outer normal vector of the $j$th face of the $i$th cell; Δ$L_{ij}$ is the $j$th face length of the cell ($i$); and $S_{bi}$, $S_{fi}$, and $S_{ci}$ are the bottom-slope source term, the resistance term, and Koch’s force term of the ith cell, respectively. The convective flux is calculated using the HLLC Riemann operator [35]. The bottom slope source term is calculated using the slope flux method [36]. The resistance slope is calculated using the splitting point-implicit method [37]. $S_{ci}$ is calculated using empirical equations.

2.3.2. GPU Parallel Computing Architecture

Capturing the discontinuous features of the tidal bore front requires high-resolution mesh accuracy. However, traditional CPU serial computing faces limitations due to computational power bottlenecks, making it challenging to meet the simulation demands for tidal bore dynamics on large-scale grids. To tackle this issue, this study introduces GPU-accelerated parallel computing technology, enhancing computational efficiency through a parallelization algorithm based on the CUDA Fortran architecture (developed by NVIDIA Corporation, Santa Clara, CA, USA/The Portland Group (PGI), Lake Oswego, OR, USA). [30]. Specifically, the GPU’s multi-level parallel strategy and hierarchical memory system significantly optimize the execution efficiency of computational tasks (see Figure 2). First, the computational tasks are decomposed into multiple kernel functions, with each function being assigned to the GPU and being made responsible for an independent layered computational structure. Different kernel functions use distinct thread block matrix layouts, which improve task partitioning and scheduling. Additionally, the hierarchical design of the GPU memory system, especially the optimized configuration of shared memory and registers, reduces global memory access latency, further enhancing the efficiency of large-scale computations. In conclusion, the implementation of GPU parallel computing technology overcomes the computational-scale limitations of traditional methods in simulating tidal bore dynamics.

2.3.3. The Local Time Step Estimations

Most depth-integrated numerical models operate using a global time step, where every computational grid utilizes the same time increment. This time step is determined according to the minimum value that meets the CFL stability condition across all grids. However, in the QE, grid sizes vary significantly, and the tidal bore flow velocities can change rapidly. As a result, the overall time step size is often constrained by the local fine grids or high local velocities, which decreases the model’s computational efficiency. To address this issue, this study employs the LTS technique for numerical simulations of the tidal bore.

By adjusting the grading index in the LTS method, two tidal bore events were simulated within a 24 h period to analyze how the LTS method optimizes model efficiency. The simulation results indicate that the LTS method significantly enhances the computational efficiency of tidal bore simulations by modifying the grading index for time steps in different regions. Without this method, the model required 13.86 h to complete the 24 h tidal bore simulation. However, as the maximum grading number of time steps (m_max) increased from 1 to 6, the computational time gradually decreased. When m_max was set to 1, the duration was reduced to 4.51 h, representing a 67.4% reduction compared to the unoptimized scenario. With m_max set to 2, the time further decreased to 2.39 h, achieving a cumulative reduction of 82.8%. When m_max was increased to 6, the time dropped to just 0.6 h, resulting in a total efficiency improvement of 95.7% (see Figure 3).

The implementation of the LTS technique consists of three steps, as follows:

(1)

Compute the local time step $∆t_i$ for each cell, using the following equation:

$$t_i = \left\{\begin{array}{cc}Cr\underset{j = 1,2,3}{\min }({\displaystyle \frac{R_{ij}}{U_{ij} + \sqrt{\mathrm{g}{h}_i}}})& h_i > h_{cr}\\ \underset{i = 1, Nc}{\max }∆t_i& h_i < h_{cr}\\ \mathrm{Arbitrary} \mathrm{value}& h_i = 0\end{array}\right.$$

(5)

where $R_{ij}$ is the distance from the centroid of cell (i) to its jth face; $U_{ij}$ is the depth-averaged velocity normal to the jth face of cell (i); $h_i$ is the water depth of cell (i); $h_{cr}$ is a threshold water depth, which is specified as ${10}^{-6}$ m; $N_c$ is the total number of cells; and the Courant number Cr = 0.6.

(2)

Compute the global minimum time step $∆t_{\min }$ and the potential grade exponent $m_i^p$ for each cell (i), using the following equations:

$$∆t_{\min } = \underset{i=1, \mathrm{Nc}}{\min }∆t_i$$

(6)

$$m_i^p=\min \left\{\mathrm{int}\left[{\displaystyle \frac{\log \left(∆t_i/∆t_{\min }\right)}{\log 2}}\right],m_{\mathrm{user}}\right\}$$

(7)

(3)

Compute the actual grades for cells and faces, determine the LTS time step ${∆t}_{LTS\text{-}i}$ for each cell, and compute the global maximum time step $∆T$, using the following equations:

$$m_{fj} = \min (m_{jL}^p,{ m}_{jR}^p)$$

(8a)

$$m_i=\min (m_i^p,{ m}_{i1}^p,{ m}_{i2}^p,{ m}_{i3}^p)$$

(8b)

$${∆t}_{LTS\text{-}i}=2^{m_i}∆t_{\min }$$

(8c)

$$m_{\max }=\underset{i=1, \mathit{Nc}}{\min }(m_i)$$

(8d)

$$∆T=2^{m_{\max }}∆t_{\min }$$

(8e)

where $m_{fj}$ is the flux update grade for jth face; $m_i$ is the flux update grade for cell (i); $m_{jL}^p$ and $m_{jR}^p$ are the potential grades of the left and right cells adjacent to jth face; $m_{i1}^p$, $m_{i3}^p$, and $m_{i3}^p$ are the potential grades of the three neighboring cells around cell (i) (Figure 4); and $m_{\max }$ is the maximum actual grade among all cells. During computation, ${∆t}_{LTS\text{-}i}$ represents the local time step for cell (i), while the global maximum time step $∆T$ defines a complete update cycle for all variables.

2.3.4. Numerical Structure of GPU-LTS

Building upon the GPU parallel computing architecture and the LTS estimation framework, the following gives a detailed explanation of the numerical structure of the GPU-LTS implementation (Figure 5).

Based on the parallel computing workflow framework established by Hu et al. [30], the model initiates on the host by reading data, setting initial conditions, and computing basic parameters. The necessary data is then transferred to the device. The subsequent LTS estimation phase executes entirely on the device: Kernel ‘Cell_Time_Step’ computes locally allowable maximum timesteps; Reduction algorithm finds the global minimum timestep Δt_min; Kernel ‘Cell_Potential_Exponent’ computes the potential grade exponent; Kernel ‘Cell_Modified_Exponent’ modifies the potential grade exponents around dynamic/static fronts; Kernel ‘Face_Final_Exponent’ computes the grade exponent of each face; Kernel ‘Cell_Final_Exponent’ computes the actual grade exponent of cells; and Reduction Algorithm finds m_max. Parameters Δt_min and m_max are subsequently returned to the host. The computation then enters a cyclic phase, iterating S_c from 1 to $2^{m_{\max }}$. Updated boundary conditions transfer to the device at S_c = 1 when applicable. Each iteration employs adaptive kernel activation, Face_Flux computes numerical fluxes exclusively for dynamically activated interfaces, while Cell_Variable updates conserved quantities solely for timestep-eligible cells (incorporating friction estimation and Equation (4)). Sc increments post-execution until S_c > $2^{m_{\max }}$, triggering a macro-timestep update (t = t + ΔT) and restarting the LTS estimation phase to close the computational loop.

3. Study Area and Model Setup

3.1. Tidal Bores in the QE

The QE is a macro-tidal convergent estuary that is located immediately south of the Yangtze Estuary (Figure 6) [21]. Its length is 282 km, and its width narrows from 98.5 km at the mouth to less than 1 km at the landward end [38]. When tidal flow propagates upstream of GP, tidal bores form near the Jianshan (JS) reach and develop a variety of bore landscapes, with the most representative being cross-shape tidal bore, thread-shape tidal bore, and returned tidal bore [39]. In the JS reach, the riverbed is wide and shallow, with sandbars in the river channel and a divided main channel, often generating cross-shaped tidal bores (Figure 6c). The Yanguan (YG) reach is characterized by a straightened channel with a U-shaped cross-sectional profile, where thread-shaped tidal bores frequently develop [18] (Figure 6b). The Yancang (YC) reach with a bend angle of approximately 130°, where a 648 m YC spur dike is constructed across the river channel, frequently generates returned tidal bores [23] (Figure 6a).

3.2. Model Setup

The concerned tidal bore reaches from ZK-GP, where a fine grid resolution and downstream tidal boundary conditions are required. However, the local tidal level measured at the GP station may be insufficient to represent the tidal characteristics along the whole GP cross-section, which is particularly important for the accurate reproduction of bore evolution (ZK-ZP) in relation to high-resolution tidal bore modeling. For the large-scale model, the computational area covers the QE, Yangtze Estuary, and China Marginal seas (see Figure 7a); the mesh size ranges from 50 to 20,000 m. According to previous studies [39], the river boundary of the Yangtze Estuary is located at Sanjiangying cross-section, for which the observed daily discharge series of the Datong station is used. Meanwhile, the river boundary of the QE is located at ZK, where a given discharge enters the computational domain [17,40]. At the open sea boundary, a water level time series is applied based on the global ocean model TPXO 7.2, which incorporates 13 astronomical tidal constituents (M2, S2, N2, K2, K1, O1, P1, Q1, MF, MM, M4, MS4, and MN4). The 2012 bathymetry used 1:10,000 field data from the Zhejiang Institute of Hydraulics and Estuary (October 2010, 1985 National Height Datum). The 2016 bathymetry employed 5 m resolution data, which was retrieved from China electronic navigational charts (same datum). Computational meshes were generated using SMS 11.2 software, and discrete topographic data scatters were interpolated onto mesh nodes using the inverse distance weighted method.

The initial conditions assumed still water with zero velocity, as well as a water level of zero, as referenced in China’s 1985 national height datum. For the small-scale model, the computational domain is from ZK to ZP (see Figure 1). To capture the bore features as accurately as possible, the finest meshes are distributed in the JS reach. Specifically, the mesh sizes are progressively increased from 10 m at JS to 30 m at YC. Upstream of YC, the mesh sizes are further gradually increased to 100 m at ZK. Downstream of JS, the mesh sizes are increased to 200 m at ZP. In total, the number of computational meshes is 600914 for bore simulation. At the upstream river boundary, a constant discharge is imposed (see

Table 1. Numerical cases.

Cases	Bathymetry and Shoreline	Features	Status	Discharge (m3/s)	Simulation Period
R1	2012	YC spur dike	YES	954 m3/s	12–10 October 2012
Y1			NO
M1	2016	Mid-Channel Bar	YES		8–10 August 2016
M2			NO
D1	2012	YC spur dike	YES	600 m3/s	10–12 October 2012
D2			YES	15,000 m3/s

); at the downstream boundary (along the ZP–AD cross-section in Figure 1), the computed tidal levels from the large-scale model are used. Additionally, to ensure that hydrodynamic transport reaches a stable state, the initial computation time were set to 3 October 2010 and 1 August 2016, each preceded by a 7-day spin-up period. Subsequent simulations captured tidal bore dynamics from 10–12 October 2010 and 8–12 August 2016.

In the simulation, we established partitioned values for Manning’s roughness coefficient based on the range identified in prior research, tailored to distinct hydro-sedimentary characteristics and tidal bore dynamics [40], with parameters calibrated against field measurements. Specifically, the roughness coefficients were set as follows: n = 0.01 for the ZK–CQ segment, n = 0.005 for the CQ–YG segment, and n = 0.004 for the YG–ZP segment. A Courant number of 0.6 is used for the CFL condition and m = 6 is set for the LTS method.The computer is equipped with a 12th Gen Intel Core i7-12700 CPU (12 cores; Intel Corporation, Hillsboro, OR, USA/Penang, Malaysia) and an NVIDIA GeForce GTX 1650 GPU (896 CUDA cores; TSMC, Taiwan, China/Shenzhen, China). The software environment operates on Ubuntu 18.04.6 LTS (Linux kernel 4.15), utilizing the CUDA Toolkit 11.8 for parallel computing tasks alongside the NVIDIA 560.94 driver. Here, six numerical cases are conducted for different purposes (see Table 1). Case R1 considers an annual mean discharge of 954 m³/s and is the reference case for tidal flow validation and the reproduction of the tidal bore evolution. In comparison, Case Y1 removes the YC spur dike to determine how the returned tidal bores are affected. Cases M1 and M2 are compared to illustrate the impacts of mid-channel bars on cross-shaped tidal bores. The influences of river runoff are examined by comparing Case R1 with Cases D1 and D2, for which an annual mean dry season discharge of 600 m³/s and an extremely high flood peak discharge of 15,000 m³/s are used, respectively, based on historical records. Note that the 2012 bathymetric data indicated a stable southward-flowing main channel in the JS reach, without mid-channel bars. In contrast, the 2016 bathymetric data showed the formation of a mid-channel bar within the JS reach’s river channel, resulting in the main channel splitting into two branches—south and north. To investigate how the mid-channel bar affects the formation of cross-shaped tidal bores, the simulations in Cases M1 and M2 utilize the 2016 bathymetric data.

4. Reproduction of Tidal Bore Phenomena

4.1. Tide Validation and Bore Reproduction

Figure 8 and Figure 9 compare the computed tidal level and velocity (magnitude and direction) of Case R1 (model settings detailed in Table 1) with observed data from Houdou (HD) and Dingqiao (DQ) stations in the bore evolution reach during 10–12 October 2010. Good agreements are achieved between the computation and observation, suggesting the model’s ability to simulate bore-affected tidal flow processes. For a more quantitative evaluation, the widely used Correlation Coefficient (CC), Root Mean Square Error (RMSE), and the Sum of Squares (SS) are adopted to assess the model’s performance [41,42,43,44], as follows:

$$RMSE = {\left[{\displaystyle \frac{1}{N}}\sum _{i = 1}^N{(M_i - O_i)}^2\right]}^{1/2}$$

(9)

$$CC={\displaystyle \frac{1}{N}}\sum _{i=1}^N{\displaystyle \frac{{(M}_i - \overline{M})(O_i - \overline{O})}{{[\sum _{i=1}^N{(M_i - \overline{M})}^2\sum _{i=1}^N{(O_i - \overline{O})}^2]}^{1/2}}}$$

(10)

$$SS=1 - {\displaystyle \frac{\sum _{i=1}^N{{(M}_i - O_i)}^2}{\sum _{i=1}^N{(O_i - \overline{O})}^2}}$$

(11)

where $M_i$ and $O_i$ are the ith simulated and measured values, respectively; N is the total number of data; and $\overline{M}$ and $\overline{O}$ are the mean values of the simulations and measurements, respectively. The performance correlations for CC are categorized as very strong (>0.9), strong (0.7–0.9), moderate (0.7–0.5), and poor (<0.2). Meanwhile, the performance levels for SS are classified as excellent (>0.65), very good (0.65–0.5), good (0.5–0.2), and poor (<0.2).

As shown in

Table 2. Model performance (RMSE, CC, and SS) in simulating hydrodynamics.

Stations	Hydrodynamics	RMSE	CC	SS	Level
DQ	Tidal level	0.194 m	0.982	0.976	Excellent
HD		0.212 m	0.871	0.968
DQ	Tidal current velocity	0.191 m/s	0.941	0.938
HD		0.224 m/s	0.937	0.942
DQ	Tidal current direction	20°	0.908	0.896
HD		22°	0.912	0.882

, the RMSE values for tidal level, velocity, and velocity direction simulations at the DQ and HD stations are all below 0.25 m, 0.25 m/s, and 25°, respectively. Concurrently, most of the CC and SS values exceed 0.9. Water level validation during the tidal bore shows abrupt rises from 0.1 m to 3.53 m within 15 min after the bore’s arrival at 12:40 on 10 October, as well as from 0.05 to 3.41 m within 15 min after the bore’s arrival at 13:11 on 11 October at the DQ station (see Figure 9). These results indicate that the model performs well in hydrodynamic simulations at the DQ and HD stations, effectively reproducing the discontinuous features in the tidal bore propagation process, such as sudden water level changes, rapid increases in velocity, and sharp shifts in velocity and direction.

4.2. Analysis of Tidal Bore Characteristics

4.2.1. Definition and Calculation Methods of Tidal Bore Characteristic Parameters

(1)

Tidal bore height (H) and tidal bore velocity (v)

The tidal bore height is one of the indicators of the strength of the tidal bore. During tidal bore propagation, the bore front advances upstream with a defined hydraulic head (H) and propagation speed (C) (see Figure 10), demarcating the flow field into distinct pre-bore and post-bore zones. The pre-bore water depth ($h_u$) is the initial water depth before the tidal bore arrives, with a pre-bore velocity ($v_u$) opposite to that of the tidal bore front’s direction. The arrival of the bore front induces an abrupt change in water levels and velocities, resulting in significant stochastic uncertainty in single-point measurements of pre- and post-bore water levels ($h_u$, $h_d$). To address this, Lin [38] proposed the use of the time-averaged water depth during pre- and post-bore periods to determine $h_u$ and $h_d$. Meanwhile, Koch and Chanson [45] determined $h_d$ by using the average water depth of the first wave crest and the trough of the undular bore in physical model experiments.

Following the work of Lin [38], this study calculates the tidal bore height (H) as follows:

$$H = h_d - h_u$$

(12)

where $h_u$ and $h_d$ denote the pre-bore and post-bore mean water depth, respectively.

The average water depths before and after the passage of the tidal bore front are denoted as $h_u$ and $h_d$, respectively. Concurrently, the average velocities of the ebb current and flood current during the same period are calculated as the pre-bore velocity ($v_u$) and post-bore velocity ($v_d$), respectively. The maximum flood current velocity observed over the full tidal cycle is defined as the maximum tidal bore velocity (v_max) [18].

(2)

Tidal bore propagation speed (C) and Froude number (Fr)

In addition to tidal bore height and tidal bore velocity, tidal bore propagation speed (C) and Froude number (Fr) are critical parameters for analyzing the hydrodynamic characteristics of tidal bores. Tidal bore propagation speed (C) refers to the advancing speed of the bore front. By neglecting source terms in the 1D hydrodynamic continuity and momentum equations, C can be obtained as follows [38]:

$$C = v_d + {\displaystyle \frac{\sqrt{\mathrm{g}{h}_u(h_d+h_u)}}{2h_d}} = v_u+{\displaystyle \frac{\sqrt{\mathrm{g}{h}_d(h_d+h_u)}}{2h_u}}$$

(13)

where $h_u$ and $h_d$ represent the pre-bore and post-bore mean water depth; $v_u$ and $v_d$ represent the pre-bore and post-bore average velocities.

The tidal bore in the QE can be classified into two categories based on its morphology: undular tidal bores and breaking tidal bores. The tidal bore is weak, creating a series of waves that move forward in parallel, which is referred to as an undular tidal bore. However, as the intensity increases, it transforms into a forward-moving roller, which is referred to as the breaking tidal bore [39]. Prior research has utilized the Fr to categorize the morphology of tidal bores through flume experiments, field observations, and numerical simulations [23,46]. According to the critical criteria established in these studies, a tidal bore is generated as follows: when $Fr \le 1$, no tidal bore occurs; when $1 < Fr < 1.7$, it appears as an undular tidal bore; and when $Fr \ge 1.7$, it develops into a breaking tidal bore. The formula for calculating Fr is as follows:

$$F_r = {\displaystyle \frac{C - v_u}{\sqrt{{\mathrm{g}h}_u}}}$$

(14)

where $h_u$ is the pre-bore mean water depth; $v_u$ is the pre-bore average velocity; and $C$ is the tidal bore propagation speed.

4.2.2. Distribution of Tidal Bore Characteristic Parameters Along the River Reach

The previous section outlined the definitions and calculation methods for tidal bore characteristic parameters. In this section, we will analyze the distribution of tidal bore parameters along the QE to better understand the mechanisms behind bore propagation. Numerical simulations indicate that bore evolution occurs in three distinct spatial phases—generation, development, and decay. It is important to note that although tidal bores are localized phenomena, their entire evolutionary process is closely linked to tidal evolution [46]. Therefore, to fully understand the formation and propagation mechanisms of tidal bores, we need to integrate bore characteristics with the hydrodynamic evolution of tidal flow, focusing on tidal bore features across different river reaches. In the QE, the funnel-shaped geometry and sandbar topography enhance shallow-water effects on tidal progression, resulting in nonlinear tidal distortion. This distortion steepens the tidal front, ultimately leading to bore formation. Here, two widely used parameters are employed to quantify the characteristics of tidal asymmetry—the M4/M2 amplitude ratio (A_M2/A_M4) in strength and the 2M2−M4 phase difference (2φ_M2−φ_M4) in direction [47,48,49]. For the former, a larger value represents stronger overtide effects and tidal asymmetry. For the latter, values between 0 and 180° indicate a flood tidal asymmetry, while a variation in the range of 180–360° refers to an ebb tidal asymmetry [39].

(1) The tidal bore generation phase (GP-JS)

The tides at the GP section exhibit quasi-sinusoidal characteristics, with smooth tidal curves and the transition between flood and ebb tides occurring near high tidal levels (Figure 11). Under the annual mean discharge (954 m³/s), the tidal range at GP is 4.85 m, with a flood-to-ebb duration ratio of 0.83 and an A_M2/A_M4 of 0.15 (Figure 12 and Figure 13), indicating weak flood tidal asymmetry. As the tidal flow propagates toward JS, the tidal range decreases to 3.99 m, the flood-to-ebb duration ratio decreases to 0.54, and the A_M2/A_M4 increases to 0.35 (Figure 12 and Figure 13), signifying enhanced flood tidal asymmetry that triggers tidal bore generation. At this stage, H reaches 1.64 m, v_max is 1.28 m/s, C is 5.79 m/s, and the Fr is 1.81 (Figure 13), with the tidal bore exhibiting a breaking bore morphology. As shown in Figure 12b, the arrival of the bore front induces an abrupt flow field transition, with instantaneous reversal from ebb to flood flow. The flood velocity peaks at 1.28 m/s within 10 min. Subsequently, the flow velocity gradually decays, while ebb dominance is restored approximately 1.5 h after the high tidal level. This phenomenon continues to appear in the sections upstream of JS.

(2) The tidal bore development phase (JS-YC)

The combined effects of the continued channel contraction and elevation of the longitudinal sandbar topography drive tidal asymmetry to its peak intensity. At the YG section, the A_M2/A_M4 rises to 0.50, the flood-to-ebb duration ratio decreases to 0.33, and the tidal range decreases to 3.89 m (Figure 12 and Figure 13). Concurrently, C increases to 6.89 m/s, v_max increases to 1.44 m/s, Fr rises to 1.97, and H reaches 1.77 m (Figure 13), significantly enhancing the tidal bore intensity, which maintains a breaking bore morphology. Notably, a multi-peaked flood velocity phenomenon emerges at YG (Figure 11c), which is likely attributed to strong interactions between discharge and tidal flow in its straight-channel reach. Despite the decreasing tidal range from JS to YG, the flood tidal asymmetry intensifies, and tidal bore parameters such as v_max, H, and Fr reach their maximum values. This phenomenon may be attributed to the uplift of the sandbar topography in the upstream section, which increases low water levels and reduces the tidal range. Additionally, the shallow-water effect increases the strength of tidal distortion, allowing the tidal bore to continue growing in intensity even as tidal energy dissipates [46]. As the bore propagates to YC, the tidal range declines to 3.71 m and the flood-to-ebb duration ratio decreases to 0.25, though the rate of decrease slows (Figure 13b). A_M2/A_M4 remains stable, with no significant change in flood tidal asymmetry compared to YG; while H, v_max, and Fr begin to decline, C continues to increase (Figure 12 and Figure 13) and the breaking bore morphology persists.

(3) The tidal bore decay phase (YC-QB)

The tidal bore gradually weakens as it moves upstream of the YC section (YC-QB). Between YC and QB, the tidal range decreases from 3.71 m to 2.06 m. The ratio of flood duration-to-ebb duration increases from 0.25 to 0.20, while the A_M2/A_M4 decreases from 0.5 to 0.4 (Figure 12 and Figure 13), indicating a reduction in flood tidal asymmetry. This change may be attributed to the growing influence of upstream runoff, which suppresses tidal nonlinear deformation. C continues to increase to 7.68 m/s, whereas v_max decreases to 1.16 m/s. Additionally, the Fr decreases to 1.59, H drops to 1.35 m (Figure 13), and the tidal bore’s intensity diminishes, changing from a breaking bore to an undular bore.

The longitudinal variation in tidal dynamics exhibits two key characteristics. First, the tidal range decreases by 2.79 m from the GP to QB sections (Figure 13), reflecting continuous tidal energy dissipation during propagation. Second, the A_M2/A_M4 shows a slight increase in HZB. However, in the upstream river sections of GP, it demonstrates a significant trend of initially increasing and then decreasing, peaking at 0.5 in the YG-YC reach (Figure 12). Correspondingly, the tidal bore evolution follows these trends. While C increases longitudinally, v_max, H, and Fr reach their maxima at YG due to channel bed elevation and bank narrowing (Figure 13). Subsequently, the increased influence of runoff leads to greater energy dissipation, which gradually attenuates the tidal bore. This results in a bore intensity distribution that first increases and then decreases, which aligns with the tidal asymmetry variability trends.

5. Formation Mechanisms of Typical Tidal Bores

5.1. Cross-Shaped Tidal Bores in Mid-Channel Bar Reaches

When the tidal flow turns its direction from the west to the north at the Cao’E River mouth, it is divided into two streams by the underwater mid-channel bars. These two streams confluence at the upper tail of the mid-channel bars (near JS station), forming the cross-shaped tidal bore landscape (see Figure 6). The reproduction of the cross-shaped tidal bore (Case M1) is illustrated in Figure 14a,b for the contours of tidal level and velocity, respectively. The cross-shaped tidal bore consisted of the west and east bores, and the bore fronts exhibit an arc shape along the cross-section. The water level at the west bore front sharply increases from 0.5 to 1.0 m, on average, or by 1.5 m locally, implying a strong tidal bore formation. Meanwhile, the corresponding flow velocity could be as high as 2.5–4 m/s, showing its fast propagation towards the northeastern direction. However, the east bore is comparably weaker. The water level rises from 0.5 to 1.0 m with sparser contours, while the flow moves towards the northwestern direction at a speed of 1.0–2.0 m/s. To reveal the impacts of the mid-channel bar on the formation of the cross-shaped tidal bore, the results of Case M2, in which the mid-channel bar is removed, are also shown in Figure 14c,d for comparison. Under these conditions, the cross-shaped tidal bore no longer exists. Instead, a thread-shaped tidal bore is formed along the cross-section, with the water level increasing from 0.5 to 1.5 m. The flow velocity at the cross-section could reach 1.5–3.0 m/s on average, even reaching up to 3.5 m/s at a small part of the west-hand side. This implies that, in general, the single bore is stronger than the original east bore but is slightly weaker than the original west bore. Based on the above, it can be concluded that the mid-channel bar is not necessary for tidal bore development in this reach, but it is critical in relation to the formation of the cross-shaped tidal bore landscapes here.

5.2. Thread-Shaped Tidal Bores in Straight–Narrow–Shallow Reaches

When the cross-shaped tidal bore moves further upstream into the western-directed narrow channel between YG and JS, it gradually evolves into a thread-shaped tidal bore (see Figure 6). In this reach, the channel is narrowed to about 5 km wide and becomes very shallow due to the top of the giant underwater sandbar, resulting in significant tidal deformation (see Figure 12). This may contribute to the formation of the thread-shaped tidal bore. Compared with the cross-shaped tidal bore, the tidal energy is more concentrated in such narrow–shallow channels, leading to the strengthening of the tidal bore landscape. As illustrated in Figure 15 for Case R1, the water level contours sharply increase from 0.5 to 2.0 m at YG with a maximum flow velocity of up to 5 m/s. Across the YG section, the bore front nearly moves simultaneously, resulting in almost straight contours of water level paralleling with each other at the front; meanwhile, the flow is universally directed upstream (to the west) with equally distributed velocity.

5.3. Returned Tidal Bores in Sharp Turning Corners

When the tidal flows encounter the YC turning corner, the landscape of the returned bore could be developed (see Figure 6). This could be attributed to two factors—the sharp turning of the channel from the west–east to the north–south direction and the relatively long spur dike at YC. Case R1 models the co-function of these two factors in the formation of the returned tidal bore. As shown in Figure 16a, the returned tidal bore features a high tidal level near the east side of the corner and the spur dike, with a negative water surface slope referring to the west-directed tidal flows. At the bore front, the tidal level increases sharply from 3.5 to 5.0 m at the north part of the corner, while rising from 2.0 to 3.5 m in the channel center near the head of the spur dike. Due to the conjunction of the reflected wave and the west-moving tidal flows, the flow velocity near the east side of the corner and the dike is very low; however, the west-moving tidal flows maintain a fast propagation of up to 5 m/s in the front and around the corner (Figure 16b). In particular, at the near upstream of the dike head, the flow velocity increases to 7 m/s. When the YC dike is removed in Case Y1, the returned tidal bore can still be formed (Figure 16c). However, the bore moves closer to the riverbank and extends to parts of the south-directed channel upstream of YC. The water level rises from 3.0 to 4.0 m at the bore front only near the north edge of the corner, while it shows an increase from 2.0 to 3.0 m in other parts of the bore front. Without the shading effects of the spur dike, the flow velocity in the corner area is higher than that in Case Y1 (Figure 16d); however, it also releases the flow concentration at the location of the dike head, shrinking the highest flow velocity region. Thus, the turning corner plays a critical role in the formation of the returned tidal bore, while the construction of the spur dike may increase the tidal bore height and strengthen the bore landscape at YC.

6. Discussion on Tidal Bore Responses to Different Runoffs

6.1. Longitudinal Characteristics of Tidal Bores

In the above simulations, the annual mean river discharge of the QE is used for tidal bore reproduction. In nature, however, the river runoff could be changed due to seasonal and annual variations. It remains unclear how different river runoffs would affect the evolution of tidal bores. Here, the impacts of low discharge (annual mean dry season discharge) in Case D1 and extremely high discharge (extremely high flood peak discharge) in Case D2 are analyzed and compared with that of the annual mean discharge in Case R1.

The longitudinal variations in tidal bore characteristics are consistent between Case R1 and D1. From the downstream JS section to the upstream QB section, H first increases and then decreases, reaching its peak at the YG section. Concurrently, v_max and Fr exhibit identical longitudinal trends, while C progressively increases upstream (Figure 17). Although Cases R1 and D1 share similar variation patterns in tidal bore characteristics, the values of each parameter magnitude differ with increasing runoff. Specifically, H ranges from 1.38 to 1.71 m in Case R1 and from 1.35 to 1.77 m in Case D1, while Fr increases from 1.61–1.96 to 1.59–1.99, indicating enhanced bore intensity. Conversely, v_max decreases from 1.22–1.53 m/s to 1.16–1.44 m/s, and C reduces from 4.48–6.68 m/s to 4.45–6.61 m/s (Figure 17). This reduction likely stems from intensified pre-bore velocity opposing bore propagation under higher runoff conditions.

In Case D2, the extremely high flood peak discharge cause significant changes in the parameters of the tidal bore. The Fr value drops to 0.87–1.1, which leads to an overall weakening of the bore intensity along the JS-YC reach. Notably, in the upstream sections CQ and QB, Fr falls below the critical threshold of 1.0, resulting in the complete suppression of tidal bores due to the dominance of runoff, as illustrated in Figure 17. Along the JS-YC reach, H decreases to 1.28–1.44 m, v_max reduces to 1.04–1.22 m/s, and C declines to 3.27–4.79 m/s (Figure 17). Despite this, all parameters still reach their peak values at YG. Notably, the minor increases in runoff observed in Cases R1 and D1 lead to enhancements in both H and Fr values. However, the substantial increase in runoff to 15,000 m³/s in Case D2 paradoxically reduces both parameters.

Runoff variation nonlinearly regulates tidal bore intensity. When the annual mean discharge reaches 954 m³/s, key tidal bore parameters (e.g., bore height and Fr) increase systematically, enhancing bore propagation. Conversely, extreme reservoir releases suppress tidal dynamics via dominant runoff momentum, causing complete bore disappearance upstream of CQ and degrading the spectacle value of the tidal bore. This necessitates coordinated reservoir operations to optimize discharge strategy, balancing flood control and tidal bore landscapes conservation.

6.2. Velocity Distributions Across Reaches of Distinct Geometry

Following the analysis of tidal bore evolution along the river channel, the lateral distributions of maximum tidal bore and ebb velocities at representative cross-sections under Cases D1, R1, and D2 were further examined. It should be noted that the maximum flood velocity over a complete tidal cycle is defined as the maximum tidal bore velocity (v_max) in Section 4.2.1. The lateral distributions of v_max and the maximum ebb velocity are closely related to shoreline morphology and riverbed topography. The JS cross-section and its downstream reaches feature a wide channel with complex shoal-channel distributions [17]. In contrast, the upstream sections of JS exhibit a simpler shoreline morphology and riverbed topography. In the QE cross-section, flood–ebb velocities generally decrease upstream, particularly upstream of the CQ cross-section, where both maximum flood and ebb velocities approach 1 m/s, and the tidal bore enters the decay phase of propagation (Figure 11f). Consequently, the YG and CQ cross-sections were selected. Characterized by a relatively simple shoreline morphology, an uncomplicated riverbed topography, and higher flood–ebb velocities, these cross-sections effectively represent the transverse distribution characteristics of v_max and the maximum ebb velocity in the QE.

The CQ cross-section is characterized by a broad shoal on the left bank and a main channel on the right, while the YG cross-section features a central shoal flanked by minor channels on both banks (Figure 18). In Case D1, both cross-sections demonstrate consistent v_max distributions with higher values in shoals and lower values in the main channels. However, the lateral distribution of maximum ebb velocity exhibits an inverse pattern. At the YG cross-section, calculated v_max values are reported at 2.86 m/s at the main channel (YG1) compared to 3.52 m/s at the shoal (YG3), resulting in a 1.23-fold lateral difference (Figure 18a). Similarly, at the CQ cross-section, v_max decreases from 1.63 m/s at the shoal (CQ1) to 0.92 m/s in the main channel (CQ5), creating a 1.77-fold lateral difference (Figure 18b). As shown in Figure 18c,d, when the discharge increases to 954 m³/s (Case R1), the maximum ebb velocities at both cross-sections increase overall, while the v_max decreases overall. However, the lateral distribution pattern observed in Case D1 remains consistent. The v_max ratio between the main channel and the shoal remains stable at 1.23 for the YG cross-section but decreases to 1.53 at the CQ cross-section. Under the extremely high flood peak discharge, Figure 18e,f demonstrate significantly increased maximum ebb velocities alongside notable reductions in v_max. The v_max ratios rise to 1.50 at the YG cross-section and 1.89 at the CQ cross-section, indicating that enhanced river discharge suppresses tidal dynamics while preserving the topography-driven lateral velocity distributions.

Overall, the lateral distribution of v_max shows consistently lower values in the main channels and higher values in the shoals. This pattern is the opposite of the distribution characteristics observed for maximum ebb velocities. As discharge increases, there is a systematic attenuation of v_max at both the YG and CQ cross-sections, accompanied by an amplification of ebb velocities and an increase in v_max ratio between the main channels and the shoals. These phenomena may occur because during ebb tide, the main channel, which has a larger cross-sectional area, serves as the primary pathway for the ebb flow. This results in a higher maximum ebb velocity in the channels and a lower maximum ebb velocity in the shoals. As upstream discharge increases, the dominance of the ebb flow also increases, leading to an increase in ebb velocity. Since the ebb velocity opposes the velocity of the tidal bore, the increase in ebb velocity causes a decrease in v_max.

The YG reach, as the peak tidal bore intensity zone (see details in Section 4.2), exhibits significantly elevated embankment scouring risks due to hydrodynamic concentration. This necessitates prioritized construction and maintenance of protective structures in this reach. Concurrently, velocity distribution patterns across shoals and channels at the YG and CQ cross-sections require differentiated protective measures tailored to topographic features, specifically mitigating scouring in shallow shoals.

7. Conclusions

The present study numerically investigates the tidal bore evolution of the QE through two-dimensional depth-averaged hydrodynamic modeling within a finite volume framework. On the one hand, the use of unstructured triangular meshes can adequately describe the complex terrains of the QE; on the other hand, the grid nesting from the large-scale tidal flow model to the small-scale tidal bore model can provide a more accurate cross-sectional tidal flow boundary for tidal bore development, allowing for a refined resolution of 10 m that is necessary for capturing the main features of bore evolution. By integrating GPU parallel acceleration with the LTS algorithm, a balance between computational efficiency and accuracy could be achieved. The main findings include the following:

(1)

The model accurately captures abrupt water level changes, rapid velocity increases, and sudden flow direction shifts at the tidal bore front and replicates the three-phase evolution (generation, development, and decay) of tidal bores in the QE. Importantly, it achieves this simulation with a high computational efficiency, requiring only 1.2 h to simulate a 2-day tidal bore phenomenon. Despite these advantages, 2D models cannot resolve 3D turbulence in tidal bore fronts, thereby constraining precise characterization of energy dissipation mechanisms. Future studies should thus progress to 3D modeling frameworks to holistically capture multi-scale bore hydrodynamics.

(2)

The morphology, intensity, and distribution of tidal bores are influenced by natural geomorphology and artificial structures. The cross-shaped bore appears first and is generated by flow division around mid-channel bars; after its removal, the tidal bore landscape transitions to a thread-shaped tidal bore. Further upstream, the thread-shaped bore is formed due to the increasingly narrow river along a straight reach. At the YC bending reach, the reflection of the bore results in returned bores, while spur dikes further increase the tidal bore height and strengthen the bore landscape. Moreover, removing a spur dike shifts the bore closer to the riverbank.

(3)

When the discharge increases to the annual mean discharge, the intensity of the tidal bore increases while v_max is reduced. However, extremely high flood peak discharge can inhibit bore propagation by enhancing ebb dominance. In the main channels, the maximum ebb velocities are higher and v_max is lower, contrasting with shoals, where the opposite is true. These lateral differences remain consistent, even under varying discharge conditions.