A Numerical Simulation of Internal Wave Propagation on a Continental Slope and Its Influence on Sediment Transport

Abstract

Significant current velocity near the sea bottom can be induced by internal waves, even for water a few hundred meters in depth. In this study, a nonhydrostatic ocean model was applied to simulate the generation and propagation of internal waves on the continental slope of the northern SCS. Based on the analyses of the vertical profiles of the currents, the propagation of internal waves along the continental slope can be categorized into six modes. The bed shear stress and the bedload transport were calculated to analyze the general characteristics of sediment transport along the continental slope of the northern SCS. Generally, there was no sediment transport on the sea bottom induced by the internal waves when the water depth was deeper than 650 m or shallower than 80 m. The downslope sediment transport dominated the slope at a water depth range of 200~650 m, while the upslope sediment transport dominated the slope at a water depth range of 80~200 m. The predicted directions of the bedload transport are coincident with the field observations of sand wave migration on the continental slope, which further confirms that the main cause of the generation and formation of sand waves on the continental slope of the northern SCS is the strong bottom current induced by the shoaling process of internal waves.

1. Introduction

Internal waves occur in density-stratified fluids because of gravitational restoring forces acting on vertically displaced fluid. They are among the most important physical phenomena in stratified waters [1] and can generate a high shearing flow velocity due to the large wave amplitude at the internal interface. This can threaten the safe operation of deep-water engineering projects, such as risers and platforms [2]. In recent years, the influence of internal waves on the seabed has attracted more attention from researchers. On the continental slope of the northern South China Sea (SCS), there are many offshore oil and gas exploration engineering projects, and a large number of sand waves on the sea bottom with water depths of 100~400 m have been observed. Meanwhile, strong currents near the sea bottom have also been observed in these areas [3]. Zhang et al. [4] believed that strong bottom currents are induced by internal waves, which is also the main cause of the generation and formation of sand waves. Therefore, the study of internal wave propagation on continental slopes and its effect on sediment transport is of importance for the design and maintenance of offshore oil and gas engineering projects.

In the past, many studies have focused on the generation mechanism of internal waves. According to their formation and propagation characteristics, internal waves have been mainly categorized as [5]: internal tides, internal solitary waves, and inertial internal waves. The continental slope of the northern SCS is a high-incidence area of internal waves, which are generated from tidal forcing across the Luzon Ridge on the eastern side of the SCS. Then, the internal waves propagate west across the deep basin, with amplitudes regularly exceeding 100 m, and dissipate extremely large amounts of energy via turbulent interaction with the continental slope [6,7,8]. The bottom current velocity induced by the internal waves can exceed 1.5 m/s on the continental slope of the northern SCS, while the bottom current induced by the barotropic tide is no more than 0.15 m/s if the astronomical tide theory is applied [9]. The strong bottom current induced by the internal waves is likely the main cause of the formation and migration of submarine sand waves on the continental slope of the northern SCS [10]. However, there is still a lack of direct evidence for the formation and migration of sand waves induced by internal waves.

The propagation process of internal waves from deep water to the continental slope and then to the continental shelf is called the shoaling process of internal waves [11,12]. In this shoaling process, the wavelength, amplitude, and period of internal waves and the bottom current change with the water depth. The main influence on sediment transport during the shoaling process is polarity conversion, i.e., the internal waves of depression are transforming into internal waves of elevation when propagating into shallow water. Liu et al. [13] observed the polarity conversion by using synthetic aperture radar. Orr and Mignerey [14] found that polarity conversion mainly occurred on slopes with depths from 264 m to 110 m through analyses of measuring data. Xu et al. [15] found that the amplitude and width of internal waves of elevation are smaller than those of internal waves of depression. According to a large number of measured and simulated results, the direction of the bottom current induced by internal waves of depression is opposite to its propagation direction, while for the internal waves of elevation, the bottom current has the same direction as the internal wave propagation [16]. Faraci et al. [17] found that the current-to-wave velocity ratio and the bed roughness have significant influence on the velocity profile during orthogonal wave–current interactions in coastal zones.

Internal waves may result in significant sediment transport on the seabed. Olsthoorn and Stastna [18] derived a criterion for the resuspension of suspended sediment over a linearly sloping bottom. A parameterization for resuspension induced by internal solitary wave (ISW) was also proposed as a function of the maximum vertical velocity by Aghsaee and Boegman [19]. Tian et al. [20] found that a considerable amount of sediment was resuspended downslope due to the vertical ejection of flow by the breaking of internal solitary waves. Jia et al. [21] found that there was a good correlation between the timing of the ISW train and the suspended sediment concentration at water depths of 956~1545 m over continental slopes in the northern SCS. Afzal et al. [22] investigated the effect of streaming on the boundary layer flow for different wave current conditions and bottom roughness. Marino et al. [23] showed an increase in the turbulent ejections and sweeps due to the superposition of the shoaling waves. As for the bedload transport, la Forgia et al. [24] investigated the effects of ISWs breaking on an inclined surface, calculated the bed shear stress, and estimated the local flux of sediments entrained from the bed. Until now, most of the studies on the shoaling process of internal waves have been focused on the suspended sediment. To date, investigations on bedload sediment transport and seabed change have been rare. Meanwhile, the existing studies have mainly been based on metocean data and small-scale studies. Simulations of internal waves and their influence on the seabed in the real conditions of the northern SCS are still required.

In this study, a nonhydrostatic model was used to simulate the propagation of internal waves on the continental slope of the northern SCS and their influence on sediment transport. This paper is divided into six sections: Section 1 introduces the research background and significance of the work. Section 2 describes the governing equations and the establishment of the model. Section 3 presents the validations of the numerical model. In Section 4, the current profiles and the bottom currents induced by the internal waves from deep water to shallow water are analyzed. In Section 5, the bed shear stress and the bedload transport are calculated to indicate the characteristics of sediment transport on the continental slope of the SCS. Then, several concluding remarks are offered in the final section.

2. Numerical Model of Internal Waves

The Stanford Unstructured Nonhydrostatic Terrain-following Adaptive Navier-Stokes Simulator (SUNTANS) was employed to simulate the generation and propagation of internal waves in the SCS [25]. It is a nonhydrostatic coastal ocean numerical model based on the finite volume method and uses unstructured grids to adaptively simulate multiscale physical motions in the coastal ocean.

2.1. Governing Equations

In a rotating coordinate system, the SUNTANS model is governed by three-dimensional Navier-Stokes equations under the Boussinesq approximation [25]:

$$\frac{\partial u}{\partial t}+\nabla \cdot \left(uu\right)-fv+bw=-\frac{1}{{\rho }_0}\frac{\partial p}{\partial x}+{\nabla }_H\cdot \left({\nu }_H{\nabla }_{H}u\right)+\frac{\partial }{\partial z}\left({\nu }_V\frac{\partial u}{\partial z}\right)$$

(1)

$$\frac{\partial v}{\partial t}+\nabla \cdot \left(uv\right)+fu=-\frac{1}{{\rho }_0}\frac{\partial p}{\partial y}+{\nabla }_H\cdot \left({\nu }_H{\nabla }_{H}v\right)+\frac{\partial }{\partial z}\left({\nu }_V\frac{\partial v}{\partial z}\right)$$

(2)

$$\frac{\partial w}{\partial t}+\nabla \cdot \left(uw\right)-bu=-\frac{1}{{\rho }_0}\frac{\partial p}{\partial z}+{\nabla }_H\cdot \left({\nu }_H{\nabla }_{H}w\right)+\frac{\partial }{\partial z}\left({\nu }_V\frac{\partial w}{\partial z}\right)-\frac{g}{{\rho }_0}\left({\rho }_0+{\rho }^{\prime }\right)$$

(3)

$$\nabla \cdot u=0$$

(4)

where t is the time; u, v, and w are the components of the velocity vector $u$ in the x, y, and z directions, respectively; ${\rho }_0$ and ${\rho }^{\prime }$ are the constant reference density and density disturbance values, respectively; and ${\rho }_0+\rho$ is the total density. The components of the Coriolis force caused by the Earth’s rotation are $f=2\omega \sin \varphi$ and $b=2\omega \cos \varphi$, where $\varphi$ is the latitude, and $\omega$ is the angular velocity of the Earth’s rotation. The horizontal and vertical eddy viscosity coefficients are ${\nu }_H$ and ${\nu }_V$, and the horizontal gradient operator is given by

$${\nabla }_H=e_x\frac{\partial }{\partial x}+e_y\frac{\partial }{\partial y}$$

(5)

The pressure p is given by

$$p=p_s+p_h+q$$

(6)

where $p_s$, $p_h$, and q are the surface pressure, hydrostatic pressure, and nonhydrostatic pressure, respectively.

The density field is determined by

$$\rho =\rho (p,s,T)$$

(7)

where s is the salinity, and T is the temperature, which are solved by the following convection–diffusion equations:

$$\frac{\partial s}{\partial t}+\nabla \cdot \left(us\right)={\nabla }_H\cdot \left({\gamma }_H{\nabla }_{H}s\right)+\frac{\partial }{\partial z}\left({\gamma }_V\frac{\partial s}{\partial z}\right)$$

(8)

$$\frac{\partial T}{\partial t}+\nabla \cdot \left(uT\right)={\nabla }_H\cdot \left({\kappa }_H{\nabla }_{H}T\right)+\frac{\partial }{\partial z}\left({\kappa }_V\frac{\partial T}{\partial z}\right)$$

(9)

where ${\gamma }_H$ and ${\gamma }_v$ are the mass diffusion coefficients in the horizontal and vertical directions, respectively, and ${\kappa }_H$ and ${\kappa }_v$ are the horizontal and vertical thermal diffusivity, respectively. The variation in the water density is governed by:

$$\frac{\partial \rho }{\partial t}+\nabla \cdot \left(u\rho \right)={\nabla }_H\cdot \left({\gamma }_H{\nabla }_H\rho \right)+\frac{\partial }{\partial z}\left({\gamma }_V\frac{\partial \rho }{\partial z}\right)$$

(10)

2.2. Turbulence Model

2.2.1. Vertical Eddy Viscosity Coefficient

In this model, the Mellor Yamada 2.5-order turbulence model [26] is used to solve the vertical eddy viscosity coefficient and the eddy diffusion coefficient by

$$\begin{array}{c}\frac{\partial q^2}{\partial t}+u\cdot \nabla q^2=\frac{\partial }{\partial z}\left(ql{S}_q\frac{\partial q^2}{\partial z}\right)+2{\nu }_V\left[{\left(\frac{\partial u}{\partial z}\right)}^2+{\left(\frac{\partial v}{\partial z}\right)}^2\right]\\ +\frac{2g}{{\rho }_0}{\kappa }_V\frac{\partial \rho }{\partial z}-\frac{2q^3}{B_{1}l}+F_q\end{array}$$

(11)

$$\begin{array}{c}\frac{\partial q^{2}l}{\partial t}+u\cdot \nabla \left(q^{2}l\right)=\frac{\partial }{\partial z}\left(ql{S}_q\frac{\partial q^{2}l}{\partial z}\right)+l{E}_1{\nu }_V\left[{\left(\frac{\partial u}{\partial z}\right)}^2+{\left(\frac{\partial v}{\partial z}\right)}^2\right]\\ +\frac{l{E}_{1}g}{{\rho }_0}{\kappa }_V\frac{\partial \rho }{\partial z}-\frac{q^3}{B_1}\tilde{W}+F_l\end{array}$$

(12)

where $q^2/2$ is the turbulent kinetic energy, l is the characteristic length of turbulence, $F_q$ and $F_l$ are the horizontal diffusion terms, and $\tilde{W}$ is the wall approximation function, which is given by

$$\tilde{W}=1+E_2{\left(\frac{l}{\kappa L}\right)}^2$$

(13)

$$\frac{1}{L}=\frac{1}{\left(h-z\right)}+\frac{1}{\left(H+z\right)}$$

(14)

where h and H are the water level and static water depth, respectively, and $\kappa =0.41$ is the von Karman constant. The vertical eddy viscosity coefficient ${\nu }_V$ and the vertical eddy diffusion coefficient ${\kappa }_V$ are given by

$${\nu }_V=lq{S}_M$$

(15)

$${\kappa }_V=lq{S}_H$$

(16)

where $S_M$ and $S_H$ are the stability functions and can be obtained from Galperin et al. [27]. B₁, E₁, and E₂ in the above equations are constants, taken as 16.6, 1.80, and 1.33, respectively.

2.2.2. Horizontal Eddy Viscosity Coefficient

The Smagorinsky model was used to calculate the horizontal eddy viscosity coefficient:

$${\nu }_H=\frac{1}{2}{C}_{s}A{\left[{\left(\frac{\partial u}{\partial x}\right)}^2+\frac{1}{2}{\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)}^2+{\left(\frac{\partial v}{\partial y}\right)}^2\right]}^{1/2}$$

(17)

where $C_s$ is the Smagorinsky constant, and A is the area of the control volume.

The viscosity coefficient is a scalar element, which is defined at the center of the three-dimensional volume element. The partial derivative of the corresponding velocity in the volume element is calculated by Green’s theorem:

$$\nabla \varphi =\frac{1}{A}{\displaystyle {\int }_A\nabla \varphi \mathrm{d}A=}\frac{1}{A}{\displaystyle {\oint }_S\varphi n}\text{'}\mathrm{d}S=\frac{1}{A}{\displaystyle \sum _{m=1}^{N_s}{\varphi }_m}n\text{'}\mathrm{d}{f}_m$$

(18)

where $\varphi$ is u or v.

2.3. Model Setup

In the SUNTANS model, a three-dimensional unstructured finite volume prism element is used in a z-coordinate system. A horizontal grid is composed of Delaunay triangles. The vertical layer thickness does not change in the horizontal direction. The horizontal velocity is perpendicular to the vertical surface of the three-dimensional volume element and is defined at the center of the surface. The vertical velocity w is defined at the Voronoi points at the top and bottom surface of each volume element and perpendicular to the surface. The eddy viscosity, diffusion coefficient, and scalar variables, such as q, s, T, and the nonhydrostatic pressure, are defined at the center of the three-dimensional volume element.

To simulate the generation and propagation of internal waves in the SCS, the computation domain shown in Figure 1 was selected. The topography data were obtained from the ETOPO1 [28] database with an accuracy of 1/60°, and the maximum water depth in the model was over 6500 m. The velocity and water level at the east boundary were obtained from the OSU Tidal Inversion Software (OTIS) with an accuracy of 1/30° [29]. A 5 km sponge layer was set at the other three boundaries to avoid reflection. The model was initialized with horizontally uniform stratification of the temperature. The salinity was obtained from observations, and the initial velocity field was quiescent [30].

The total computational domain is shown in Figure 1 and covered the west edge of the Pacific Ocean and the northeast part of SCS. The internal waves in the SCS Sea are generated by the interaction of barotropic tidal waves with complex terrain in the Luzon Strait. The generation and propagation of internal waves on the continental slopes in the northern SCS can be well captured in the present model. To obtain a high resolution for the simulation of internal waves propagating on the continental slope, the layer thickness was set at a constant of 5 m near the sea surface within a 220 m water depth and then increased at a ratio of 1.08 for water depth over 220m. The horizontal grid size was approximately 4 km at the open boundaries and 1 km near the islands. As there is almost no energy dissipation when the internal waves propagate in the deep-water area, a larger horizontal mesh size of 4.0 km was sufficient. With the internal wave propagating on the continental slope, the mesh size was gradually reduced to 1.0 km. In the present study, a specific area with a uniform fine mesh of 100~500 m was set on the continental slope to investigate the flow and sediment transport characteristics of the internal waves. The refined area had an oval shape with a size of 300 km × 150 km, as shown in Figure 1. Considering the computational stability, a time step of 7.5 s was used for the grids in the refined area, while a time step of 15 s was used for the grids outside the refined area.

2.4. Mesh Sensitivity Analysis

As mentioned above, a refined area was specifically set on the continental slope for the accurate simulation of the flow velocity induced by internal waves. The sensitivity of the mesh density in the refined area on the numerical results of the internal wave was conducted with four mesh levels: (1) Mesh01—a mesh size of 1000 m with a total mesh number of approximately 300,000; (2) Mesh02—a mesh size of 500 m with a total mesh number of 400,000; (3) Mesh03—a mesh size of 200 m with a total mesh number of 600,000; and (4) Mesh04—a mesh size of 100 with a total mesh number of 1,000,000. As shown in Figure 2, the numerical results with different mesh levels were compared with the measured results of velocity at point A1 (see Figure 1), which was obtained from Zhang et al. [3]. Overall, the simulated results of the peak value and phase of the velocities with Mesh 03 and Mesh 04 coincide well with the measured values, with errors of less than 10%. However, there was a slight fluctuation in the velocity in the low-velocity region for Mesh04. Considering the computational efficiency and the stability of the simulation, Mesh03 was adopted in the following simulations.

3. Model Validation

The validation of the SUNTANS model was divided into two parts: one part was for the simulation of the barotropic tides, and the other part was for the simulation of internal waves. The simulated results were compared with the measured data or published results.

3.1. Barotropic Tide Simulation

For the simulation of the barotropic tides, the measured velocities at location L1 on the eastern ridge of the Luzon Strait (see Figure 1) in June 2005 were adopted [30]. The simulated depth-averaged velocity was compared with the measured results, as shown in Figure 3. The simulation results are consistent with the measured data in terms of the phase angle but slightly smaller than the measured results in terms of the peak value. This may be due to two reasons. One is the aspect of the mesh size, as it was not refined at the simulation point. On the other hand, in the field measurements, the velocity profile in the whole water depth could not be well measured due to the large water depth, which may have resulted in a difference from the simulation results. The difference between the numerical and measured results of the averaged peak values is less than 10%, and the overall correlation coefficient is R² = 0.896.

3.2. Internal Wave Simulation

Internal waves are usually characterized by the variation in the temperature in the water, and their amplitudes can be analyzed from the isotherm. Here, the simulated isotherm at location S7 (see Figure 1) is compared with the published results of Zhang et al. [30] to validate the numerical model. As shown in Figure 4, the simulated results of the time series of isotherm at S7 are well coincident with the published results in both the peak values and the phase angle during 24~30 May. Then, the amplitudes of the internal waves obtained from the isotherm were plotted against the empirical formula by Ning et al. [31], as shown in Figure 5. In the study by Ning et al. [31], the relationship between the amplitude of internal waves and the water depth was formulated based on a large amount of remote sensing data near the Dongtan Islands in the SCS. As shown in Figure 5, the numerical results of the internal wave amplitude are consistent with the empirical formula proposed by Ning et al. [31], with a correlation coefficient of R² = 0.852.

The current velocity near the bottom induced by the internal waves at location A1 during 24~30 May was further output and compared with the measured data from Zhang et al. [3], as shown in Figure 6. The simulation results are generally in good agreement with the measured results from the peak value and variation period of the velocities, although there is a slight phase shift between the two sets of data, which may have been caused by the large spatial resolution of the seabed topography (~5 km) at the assessed area. Considering that the main purpose of the present study was to investigate the direction and strength of sediment transport on the continental slope induced by internal waves, a slight phase shift in the velocities has little effect on the direction of bottom currents and the resultant sediment transport.

From the above simulations of internal waves in terms of the barotropic tide, the isotherm, the amplitude of internal waves, and the current velocity near the bottom, it is shown that the present nonhydrostatic numerical model had high accuracy and credibility for the simulation of the propagation of internal waves on the continental slope of the northern SCS. It can be applied to analyze the characteristics of the bottom currents induced by internal waves and their effects on sediment transport along the continental slope.

4. Flow Characteristics of Internal Waves on the Continental Slope

The flow characteristics due to the shoaling process of internal waves on the continental slope were studied. The variation in the isotherm, the vertical profile of the current velocity, and the near-bottom current velocities were analyzed. For the current simulation case, a total of 120 processors were used in the parallel computing. It took about 72 h to simulate the whole process of the generation and propagation of internal waves on the continental slope in the northern SCS.

4.1. Polarity Conversion of Internal Waves Due to Shoaling

Figure 7 shows the variation in the isotherm along cross-section P1 (see Figure 1) during internal wave shoaling on the continental slope at different time instants. The propagation of specific internal waves denoted with arrows from deep water to shallow water is also presented. Generally, the internal waves showed three different wave shapes. As shown in Figure 7a,b, the internal waves were propagated in the deep sea over 150 m and had a depression interface with a reverse “T” shape. The isotherm had sharp peaks with internal waves up to 80 m in amplitude. As shown was Figure 7c, when the internal waves propagated to a water depth of 100~150 m, there is an abrupt change in the water depth. The internal waves were split into a few high-frequency internal solitary waves with amplitudes of 20~30 m. At this stage, the internal waves had approximately equivalent peaks and troughs in the isotherm. When the internal waves traveled at shallow water depths of less than 100 m, they showed an elevation interface with an amplitude smaller than 20 m instead of a depression shape. The interface became smoother and wider again with a single peak, as shown in Figure 7d,e. With the decrease in the water depth, the amplitude of the internal wave decreased. The internal waves disappeared when the water depth was shallower than 30 m. The interaction of the internal waves with the seabed was similar to the interaction of the surface wave with the bed topography in the coastal environment, and the wave transformation processes occurred with refraction, diffraction, shoaling, and breaking [32]. The conversion of the depression internal waves to elevation internal waves during the shoaling process described above is called polarity conversion, which usually induces a change in the bottom current direction, which results in diverging migration directions of the sand waves [4].

4.2. Current Induced by the Internal Waves

The vertical profiles of the current induced by internal waves during their propagation on the continental slope were investigated. The water depth ranged from 1300 m to 30 m along cross-section P1, as shown in Figure 1. As shown in Figure 8a,b, for water depths ranging from 1300 m to 300 m, the velocity had a single profile of the depression type. Generally, the current velocity induced by internal waves had a shear profile. The current velocity near the sea surface had a negative speed of up to −1.5 m/s, which is the same as the direction of the propagation of the internal waves. The velocity decreased quickly with the water depth and had a zero value around a water depth of 200 m. The current velocity in the lower layer had an opposite direction to the upper layer, with nearly uniform values.

For water depths from 300 m to 150 m, i.e., as shown in Figure 8c–h, the velocity profiles showed both depression and elevation types simultaneously. However, there were also small differences among the velocity profiles. As shown in Figure 8c,d, i.e., the water depths ranged from 250~300 m, and the bottom current velocity in the positive direction was stronger than that in the negative direction, indicating that the depression type of internal waves was dominant. As shown in Figure 8e,f, i.e., water depths ranged from 250~200 m, and the bottom currents in both directions were nearly equivalent, indicating that the depression and elevation types of internal waves were equivalent. For water depths of 200~150 m, the elevation type of internal waves was dominant, with the negative bottom current stronger than the positive bottom current, as shown in Figure 8g,h.

When the internal waves were propagated in water with depths of 150~80 m, the velocity had a single profile with the elevation type of internal waves, i.e., the bottom current had negative values while the upper layer went in the opposite direction, as shown in Figure 8i,j. For very shallow water depths of 80~30 m, the velocity had a typical tidal current profile, i.e., the velocity throughout the water column had a uniform direction, and the current speed decreased with the water depth, as shown in Figure 8k,l.

Based on the above analyses, the current velocity profile during internal wave shoaling on the continental slope can be categorized into six modes: (1) a depression mode (~[1300 m, 300 m]), (2) a depression-dominant mode (~[300 m, 250 m]), (3) an equilibrium mode (~[250 m, 200 m]), (4) an elevation-dominant mode (~[200 m, 120 m]), (5) an elevation mode (~[120 m, 80 m]), and (6) a tidal mode (~[80 m, 30 m]).

Figure 9 further shows the time series of the bottom current velocity, which corresponds to the six modes of the velocity profiles in Figure 8. As shown in Figure 9a, the bottom current had a downslope direction, which corresponds to the depression mode shown in Figure 8a,b. As shown in Figure 9b, the positive current velocity had a very large value, while the negative value was very small, which corresponds to the depression-dominant mode shown in Figure 8c,d. With the internal wave propagation upslope of the continental shelf, the bottom current velocity had an equivalent mode, elevation-dominant mode, elevation mode, and tidal mode, as shown in Figure 9c,f, respectively.

5. Sediment Transport on the Continental Slope Induced by Internal Waves

The different types of bottom current velocities along the continental slope resulted in different characteristics of sediment transport. Here, the bed shear stress and the bedload transport were calculated further to investigate the general characteristics of sediment movement on the continental slope of the northern SCS.

5.1. Bed Shear Stress

We made two assumptions in the calculation of the bed shear stress along the continental slope: (1) the sediment particle size of D₅₀ = 0.45 mm was set and unified along the continental slope of the northern SCS, which resulted in a unified critical shear stress of τ_cr = 0.24 N/m² according to the empirical formula by Soulsby and Whitehouse [33]; (2) the bottom shear stress ${\tau }_b$ was calculated by a logarithmic function:

$${\tau }_b=\frac{\rho {\kappa }^{2}u\left|u\right|}{{\ln }^2\left(z/z_0\right)}$$

(19)

where $\kappa =0.41$ is the Karman constant. In this study, the velocity at the vertical grid that was nearest to the seabed was used to compute the bed shear stress. The values of z were dependent on the thickness of the horizontal layer close to the seabed. The roughness length of the bottom z₀ (=d₅₀/12) was taken as 3.8 × 10⁻⁵ mm in this study.

As shown in Figure 10, the bottom current velocities at 16 positions along cross-section P1 (see Figure 1) from deep to shallow water were selected, and the corresponding bed shear stresses were calculated with Equation (19). As shown in Figure 9, the bottom current velocities generally had both positive and negative values, which indicates that the bed shear stress alternated between the upslope and downslope directions. Thus, in Figure 10, the maximum shear stress in the upslope direction (τ_max−) and that in the downslope direction (τ_max+) at each point are presented. The critical bed shear stresses in both directions, i.e., τ_cr = ±0.24 N/m², are also plotted in Figure 10. The downstream bed shear stress that exceeded the critical value was mainly located at points 3~10, while the upslope bed shear stress exceeding the critical value was mainly located at points 9~12. The maximum bed shear stresses at points 1, 2, 13, 14, 15, and 16 were always lower than the critical value.

Figure 11 further shows the maximum bed shear stress varying with the water depth from 1300 m to 30 m. It can be seen that in water depths of 1000 m to 150 m on the continental slope (i.e., points 3~10), the maximum bed shear stresses downslope were larger than the critical value, which was induced by the internal waves of depression. During the propagation of internal waves on the continental slope, the maximum bed shear stress in the downslope direction increased first and reached its peak value at a water depth of approximately 550 m (point 6). Then, it decreased with an increasing water depth. The maximum bed shear stress in the upslope direction had a roughly stable value ranging from 0.05~0.20 N/m² in water depths of 1300 m to 300 m. When the internal waves propagated into the shallow water within a 300 m depth, polarity conversion occurred, which enhanced the internal waves of elevation. The bed shear stress in the upslope direction increased from 0.10 to 0.6 N/m² when the water depth changed from 300 m to 150 m. The maximum values were obtained at a water depth of 150 m and then decreased quickly with a decreasing water depth due to the weakening of internal waves. The effect of internal waves on the bottom disappeared entirely at a water depth of 50~60 m.

5.2. Bedload Transport

As seen in Figure 11, the bed shear stress had both positive and negative values during the propagation of internal waves on the continental slope, which resulted in complex sediment transport on the seabed. Here, the sediment transport rate of the bedload along the continental slope is investigated further. The volumetric transport rate per unit width q_b can be calculated by [34]:

$$q_b=\Phi \sqrt{\left(s-1\right)g{D}_{50}^3}$$

(20)

where $\Phi$ is the nondimensional bedload transport rate. Here, the empirical formula by Meyer-Peter and Muller [35] was adopted:

$$\Phi =8{\left(\theta -{\theta }_{cr}\right)}^{1.5}$$

(21)

where $\theta$ and ${\theta }_{cr}$ are the Shields parameter and its critical value for sediment particle movement, respectively. The Shields parameter is calculated from the bed shear stress:

$$\theta =\frac{\tau }{\rho \left(s-1\right)g{D}_{50}}$$

(22)

The critical Shields parameter is obtained from the empirical formulae by Soulsby and Whitehouse [34].

The volumetric transport rates at different points along the continental slope were calculated using the above equations. As the propagation of internal waves on the continental slope was simulated in one week in the present study due to the significant computing cost, the mass transport per unit width on the seabed within one week was estimated based on the obtained bottom current velocity. As shown in Figure 12. Q_b+, Q_b−, and Q_bn denote the downslope, upslope, and net sediment transport, respectively. Generally, there was no sediment transport induced by internal waves on the sea bottom when the water depth was deeper than 650 m or shallower than 80 m. Downslope sediment transport dominated in the water depth range of 200~650 m, which corresponds to the depression-dominant mode of internal waves, i.e., the direction of the bottom current was opposite to the propagation of internal waves. The upslope sediment transport dominated the slope at water depths ranging from 80~200 m, which corresponds to the elevation-dominant mode of internal waves, i.e., the direction of the bottom current was the same as the propagation of internal waves. The above sediment transport characteristics are generally coincident with the field observations of sand wave movement on the continental slope of the northern SCS, i.e., the movement trend of sand waves between the deep sea and polarity conversion area was seaward, while the sand waves moved shoreward at relatively shallow water depths. In the polarity conversion region, sand wave movement shows a variety of directions on the sea bottom [4]. In the present study, the general trend of sediment transport on the continental slope was analyzed due to the relatively large mesh size in the regional ocean model, and the sediment particle size was assumed to be uniform. Moreover, the bottom current velocities induced by internal waves had different values among months according to field observations. For the accurate prediction of sand wave movement on slopes, especially polarity conversion areas, a specific sand wave model with high-resolution scanned topography data and real-time bottom currents induced by internal waves is still needed.

6. Conclusions

In this study, the nonhydrostatic ocean model SUNTANS was used to simulate the generation and propagation of internal waves on the continental slope of the northern SCS. The numerical model was validated with the measured or empirical values, including the current velocity induced by the barotropic tide in the Luzon Strait, the time-variant isotherm, the amplitude of the internal waves, and the bottom current induced by the internal waves. It is indicated that the present numerical model has high accuracy and credibility for the shoaling process of internal waves on the continental slope of the northern SCS.

According to the value and direction of the bottom current, the propagation of internal waves along the continental slope from deep to shallow water can be categorized into six modes: a depression mode (~[1300 m, 300 m]), a depression-dominant mode (~[300 m, 250 m]), an equivalent mode (~[250 m, 200 m]), an elevation-dominant mode (~[200 m, 150 m]), an elevation mode (~[150 m, 80 m]), and a tidal mode (~[80 m, 30 m]). The bottom shear stresses induced by internal waves on the continental slope of the northern SCS were calculated based on the obtained bottom current velocity. Then, the general characteristics of sediment transport on the slope were analyzed by the bedload transport. Generally, there was no sediment transport on the sea bottom induced by internal waves when the water depth was deeper than 650 m or shallower than 80 m. Downslope sediment transport dominated the slope in the water depth range of 200~650 m, which corresponds to the depression-dominant mode of internal waves. The upslope sediment transport dominated the slope at water depths ranging from 80~200 m, which corresponds to the elevation-dominant mode of internal waves, i.e., the direction of the bottom current was the same as the propagation of internal waves. The predicted directions of the bedload transport coincide well with the field observations of the migration of sand waves on the continental slope, which further confirms that the main cause of the generation and formation of sand waves on the continental slope of the northern SCS is the strong bottom current induced by the shoaling process of internal waves. Based on the present model, the long-term variation of bottom currents can be obtained for the simulation of sand wave evolution on the continental slopes. It is also noted that for the accurate prediction of sand wave movement, especially polarity conversion areas, high-resolution scanned topography data and parameters of seabed properties are required. Meanwhile, extensive measurements of the sediment concentration near the bottom and the real-time bottom current induced by internal waves are still encouraged. As for the prediction of long-term sediment transport and sand wave migration under internal waves, artificial intelligence techniques, such as machine learning, can also be applied [36].