A numerical model, ISWFoam, for simulating internal solitary waves
(ISWs) in continuously stratified, incompressible, viscous fluids is
developed based on a fully three-dimensional (3D) Navier–Stokes equation
using the open-source code OpenFOAM^{®}. This model combines the density
transport equation with the Reynolds-averaged Navier–Stokes equation with
the Coriolis force, and the model discrete equation adopts the finite-volume
method. The

Internal solitary waves (ISWs) are commonly observed in oceans, particularly in continental shelf regions, due to strong tidal current flows over large topographic features (Huthnance, 1981), such as in the northern South China Sea (Alford et al., 2010, 2015; Cai et al., 2012). ISWs play an important role in both conveying nutrients from the deep ocean to shallower layers and promoting biological growth (Sandstrom et al., 1984). Additionally, ISWs are a potential threat to the ocean structures of resource exploration, exploitation and submarine navigation vehicles (Alford et al., 2010; Osborne and Burch, 1980). A considerable number of studies, which include field measurements, remote sensing, experiments, theoretical analysis and numerical simulations, have been carried out due to the significance of ISWs (Vlasenko et al., 2005; Apel et al., 2006; Alford et al., 2011; Guo et al., 2014).

For numerically simulated ISWs, many models have been adopted, including the
Euler equation, the inviscid or viscid incompressible Boussinesq model, the
hydrostatic model, the non-hydrostatic model, and the VOF-based two-phase
flow model. Among these models, the representative hydrostatic models
include the Naval Research Laboratory Ocean Nowcast/Forecast System (ONFS)
(Ko et al., 2008), the Regional Hallberg Isopycnal Tide Mode (RHIMT)
(Hallberg and Rhines, 1996; Hallberg, 1997) and the Ostrovsky–Hunter model.
The representative non-hydrostatic models include the Bergen Ocean Model
(BOM), the nonhydrostatic Regional Ocean Modeling System model (ROMS), the
Stanford Unstructured Nonhydrostatic Terrain-following Adaptive
Navier–Stokes Simulator (SUNTANS) and the Massachusetts Institute of
Technology general circulation model (MITgcm). For example, Zhang et al. (2012) established a variable-water-depth internal-wave numerical model in a continuously stratified fluid system based on the Euler equation. Xu and
Stastna (2020) used the viscid incompressible Boussinesq model to study
cross-boundary-layer transport (Boegman and Stastna, 2019) by the fissioning
process of shoaling ISWs. Lamb (1994) established a non-hydrostatic model,
using a second-order projection method developed by Bell and Marcus (1992),
which is used for internal-wave research including boundary-layer
instability (Aghsaee et al., 2012), reflection (Lamb and Nguyen, 2009
), and the
interaction of the tides with the topography (Lamb, 2007; Aghsaee et al., 2010). Diamessis (2005) developed a spectral multidomain penalty method
model and correctly reproduced the characteristic vorticity and internal-wave structure. Subich et al. (2013) developed a spectral collocation method
for the solution of the Navier–Stokes equations under the Boussinesq
approximation and simulated the internal wave in continuously stratified
fluid. Smedstad et al. (2003) employed the ONFS model to establish a global
ocean real-time forecasting system with an operational eddy resolution of

In summary, for continuously stratified fluids in complex ocean
environments, numerical simulation has become a leading method for ISW
investigations. However, there are presently few versatile numerical models
with shared code that can accurately simulate the ISW flow around complex
topography and submarine navigation vehicles in continuously stratified
fluids. Therefore, the main objective of this paper is to develop a solver,
referred to as ISWFoam with a modified ^{®} library.

Notably, the open-source field operation and manipulation code
OpenFOAM, as an object-oriented C

The outline of the paper is described as follows. First, in Sect. 2, the governing equations for a continuously stratified fluid are presented, and discrete forms of these equations are derived. Then, grid independence tests of the developed ISWFoam model are described in Sect. 3. Subsequently, in Sect. 4, a series of test cases are presented to verify the model. Simulation examples at the field scale are shown in Sect. 5. Finally, the conclusions are drawn in Sect. 6.

We present an ISW numerical model by solving the motion of a
three-dimensional, viscous, incompressible fluid with the Boussinesq
approximation and rigid lid hypothesis. The governing equations of the model
are

To close the above equations, the turbulence model needs to be employed. The
two-equation

Default values for

Considering the variable density field during the solution process, it is
necessary to consider the change in the density field in the turbulence
model. Therefore, we modify the turbulence model to consider the change in
density, and finally a modified

The governing equations are numerically discretised using the finite-volume
method based on the C

The momentum equation in ISWFoam is solved by constructing a predicted
velocity field and then using the pressure implicit with splitting of
operators (PISO) algorithm (Issa, 1986) to modify it.

First, only the temporal, convection and diffusion terms appear in the
discrete version of the equation momentum, and the other terms are ignored.
After this operation, we obtain an explicit expression for the predicted
velocity field

The solution process requires the velocity on the surface

ISW generation methods mainly include the gravity collapse mechanism, double push-pedal method (Fu et al., 2008), velocity-inlet method (Gao et al., 2012), mass source method (Wang et al., 2018), initialisation method, and methods addressing the interaction between tidal current and topography. For example, Hsieh et al. (2014) investigated the flow evolution of a depression ISW generated by the gravity collapse mechanism. Cheng et al. (2020) studied the interaction between ISWs and a cylinder using the gravity collapse mechanism. The initialisation method involves solving the internal solitary wave theory at the initial moment, such as the Korteweg–de Vries (KdV) equation (Grimshaw et al., 2010), the modified KdV (mKdV) equation, the extended KdV (eKdV) equation, the forced KdV equation, the Ostrovsky equation (Li and Farmer, 2011), the Miyata–Choi–Camassa (MCC) model (Miyata, 1985, 1988; Choi and Camassa, 1999) and the Dubreil–Jacotin–Long (DJL) equation (Long, 1953; Turkington, 1991; Brown and Christie, 1998; Dunphy et al., 2011), to obtain the wave surface and velocity field. The method of an interacting between tidal current and terrain that stimulates ISWs is adopted by many scholars, such as Farmer and Smith (1980), Lamb et al. (1994), and Shaw et al. (2009).

In this paper, the method of initialising the field is selected to generate
the ISWs. To increase the application range of the ISWFoam model, two
initialisation methods are provided, including solving the weakly nonlinear
models of the eKdV equation (Helfrich and Melville, 2006) and the fully
nonlinear models of the DJL equation for continuously stratified fluids
(Turkington, 1991; Dunphy et al., 2011). The Dubreil–Jacotin–Long (DJL)
equation is expressed as

By solving the above DJL equation we can obtain

Another theory of ISWFoam model wave generation involves the weakly
nonlinear models of the eKdV equation. Using the first-order stream function
for the DJL equation, we can obtain the well-known KdV equation and further
obtain the eKdV equation. For the specific derivation, please refer to the
paper by Lamb and Yan (1996). The eKdV equation (Helfrich and Melville, 2006) is

The vertical profile of the initial density is given by a hyperbolic tangent
function profile (Aghsaee et al., 2010):

To compare the DJL equation and the eKdV equation, we set up a numerical
simulation, which includes a tank that is 15 m long, 1 m wide and has a
water depth of 0.5 m. The depths of the upper (

Comparison chart of the horizontal velocity component field: DJL equation (left) and eKdV equation (right).

Figure 1 shows the comparison of the horizontal velocity component field when the DJL equation and the eKdV equation are used to generate ISWs. At the initial moment, the ISW generated by the eKdV equation is not as smooth as the ISW generated by the DJL equation, and the horizontal velocity at the interface area is discontinuous as shown in Fig. 1a and b. With the propagation of ISWs, the ISWs generated by the DJL equation are always smooth at the interface area, and the velocity field is always continuous as shown in Fig. 1a, c, e and g. Correspondingly, the ISW generated by the eKdV equation gradually produces a gradient in the vertical direction of the horizontal velocity in the interface area; thus, the interface area becomes smooth, and the velocity becomes continuous. Fig. 1d shows this evolution process, which is basically completed in 5 s as shown in Fig. 1f. At 50 s, the difference between the horizontal velocity fields of the two equations is very small as shown in Fig. 1g and h.

Figure 2 shows the comparison of the vertical velocity component field when the DJL equation and the eKdV equation are used to generate ISWs. Since the theoretical solution of the eKdV equation only obtain the average horizontal velocity of the upper and lower layers of the fluid, there is no vertical velocity at the initial moment, as shown in Fig. 2b. With the propagation of ISWs, the vertical velocity field will gradually be generated and finally stabilised, and the stable time occurs at 5 s as shown in Fig. 2b, d, f and h. At 50 s, the difference between the vertical velocity fields of the two equations is very small as shown in Fig. 2g and h.

Comparison chart of the vertical velocity component field: DJL equation (left) and eKdV equation (right).

The ISW propagates for 10 m, and the amplitudes of the ISWs generated by the DJL equation and the eKdV equation are reduced by 9.88 % and 17.96 %, respectively, as shown in Fig. 3. Overall, the reduction in energy leads to the attenuation of the amplitude of the ISW, which in turn reduces the wave speed. Except for the difference in initial fields, the grid sizes, time step, turbulence model and other features are the same. Therefore, the initial stage of ISWs generated by the eKdV equation leads to excessive energy loss compared with those generated by the DJL equation. From the above analysis of the velocity field, we know that the method of initialising the field with the eKdV equation requires a period of movement before the jump of the velocity field develops into a field with continuous changes in velocity. In addition, the DJL equation, as a fully nonlinear model, can better reflect its superiority for internal waves with strong nonlinearity. Therefore, the wave generation of the subsequent numerical cases in this paper adopts the method of initialising the field with the DJL equation.

Time series of the interface displacement. The probe was 10 m away from the initial ISW.

These grid independence tests were performed in the horizontal and vertical
directions by applying meshes of different sizes. The sizes of the mesh
determined in this paper are calculated based on the amplitude of the ISW
and a characteristic length determined through the integration of the wave
profile (Michallet and Ivey, 1999):

To determine the appropriate mesh size, the propagation of ISWs on flat
bottoms is calculated, and the numerical results are compared with the DJL
theoretical solution. We set up a numerical simulation, which includes a
tank that is 50 m long, 0.5 m wide and has a water depth of 0.5 m. The
depths of the upper (

First, we analyse the grid independence in the horizontal direction, with a
constant cell height of

Grid independence in the horizontal direction at probe P1:

Second, we analyse the grid independence in the vertical direction, with a
constant cell width of

Grid independence analysis in the vertical direction at probe P1:

To verify the numerical model of the ISWs, the propagation of ISWs on a flat bottom section, submerged triangular ridge and slopes is calculated, and the numerical results are compared with the corresponding experimental results. To verify the correctness of Coriolis code implantation and reflect the role of local mesh refinement, the propagation of ISWs on a flat bottom section of actual ocean scale and a Gaussian ridge is calculated.

Schematic diagram of probe position (P1–P5) (Hsieh et al., 2014).

Density contours at different moments.

In this section, ISWFoam is verified by employing ISWs propagating on a flat
bottom section with Case Flat_4 in the continuously
stratified laboratory experiment described in Hsieh et al. (2014). The
physical dimensions and ultrasonic probe locations in the experiments of
Hsieh et al. (2014), as shown in Fig. 6, are adopted to establish the
numerical computation domain. We set up a numerical tank of the experiment
of Hsieh and co-authors, which includes a tank that is 15 m long, 0.5 m wide and has a stable water depth of 0.5 m. The fluid densities of the upper
(

Figure 7 shows the density contours at three different times from Case Flat_4 in the laboratory experiment of Hsieh and coworkers, showing the stable evolution of an ISW. The results also show the realistic evolution of the thickening of the pycnocline after ISW propagation because of convection and diffusion. At the same time, the propagation of the ISW is stable and unbroken.

To further verify the model, the waveform is compared between the numerical simulations and the experimental measurements, and the measurement point selection is the same as the experimental setting, as shown in Fig. 6. Figure 8 shows the comparison results between the waveform simulated by ISWFoam and the experimental results at probes P1–P5. Figure 8 shows that the results of the numerical simulations agree with the experimental results (red circle). Notably, the laboratory wave height at the probe P1 measurement point is greater than the numerical simulation results, and the wave surface of the laboratory wave is not smooth, which is caused by the wave generation method using the gravity collapse mechanism in the laboratory. In general, the model developed in this paper can simulate the generation and evolution of ISWs in continuously stratified fluids.

Comparison of the waveform between the experimental results and numerical simulation results at probes P1–P5.

In this section, the validation of the numerical model is conducted through
an ISW propagating over a submerged triangular ridge with the continuously
stratified experiments described in Hsieh et al. (2015). The laboratory tank
is 12 m long and has a stable water depth of 0.5 m, with which the fluid
system has a finite thickness of the pycnocline. The specific experimental
parameters used for validation of ISWFoam include the various depths of the
upper (

Schematic illustration of the laboratory set-up and the locations of the probes (Hsieh et al., 2015).

The numerical tank is designed to reproduce the experiment described in Fig. 9. The unstructured grid and local mesh refinement based on the finite-volume method are used to construct the computational domain and discretise
the governing equations. The grid is uniform in the

Schematic of the mesh.

Comparison of the waveform between the experimental results in Hsieh et al. (2015) and numerical simulation results at probes P1–P5.

Figure 11 shows the comparison results between the waveform calculated by ISWFoam and the experimental results at probes P1–P5. In each subplot, the results of the numerical simulations (blue line) are found to be in good agreement with the experimental results (red circle). From Fig. 11a, the numerical simulation result of the probe P1 measurement after 20 s is different from the experimental results, which is caused by the different ISW generation methods. For the experimental results, the first large leading ISW is formed via the gravity collapse mechanism, which is trailed by a train of small-amplitude mode-one waves that is generated due to shear instabilities. However, the initialisation method used to generate an initial ISW for the numerical simulation in this paper is more stable than the gravity collapse mechanism, so the rear part of the ISW is relatively flat compared to the experimental results for probe P1. In Fig. 11, the waveform of the ISW gradually evolves towards a flat waveform due to the interaction between the ISW and the ridge. In general, the model developed in this paper can simulate the interaction between ISWs and structures.

To verify the ability and accuracy of simulating the ISW breaking of the
numerical model, two continuously stratified laboratory experiments (12 and
15) of Michallet and Ivey (1999) are chosen for the simulation in this
section. The experimental set-up is represented schematically in Fig. 12. We
set up a numerical tank of the experiment of Michallet and Ivey (1999),
which includes a tank that is

Schematic diagram of the laboratory set-up.“

Time series of the interface displacement. The probe was 99.8 cm away from the start of the slope.

The sponge layer on the left side, whose length is double the wave characteristic length, is checked to properly dissipate the reflected wave. Slip boundary conditions are applied to the bottom and both sides, while slip boundary conditions are assigned to the top boundaries. The boundary conditions related to the density field are no-flux boundary conditions.

The vertical profile of the initial density is given by a hyperbolic tangent
function profile

The first case of model verification is experiment 12 of Michallet and Ivey (1999) in this section. The layer thickness ratio

Comparison of the velocity field between the experimental observation results in Michallet and Ivey (1999) (left) and numerical simulation results (right).

Figure 14 shows a comparison of ISWFoam results and the experimental observations of the velocity field associated with the ISW run-up process along the slope. The model effectively reproduces laboratory tests, such as the intensity and direction of the velocity field, the location of the vortices and the occurrence of boundary-layer separation beneath the ISW. Therefore, the model developed in this paper can reflect the ISW breaking phenomenon during the propagation of ISWs along the slope.

Another laboratory experiment that more clearly shows the ISW breaking
phenomenon from Experiment 15 of Michallet and Ivey (1999) is used to
verify the numerical model presented in this paper, and the corresponding
numerical case is set corresponding to it. The layer thickness ratio

Figure 15 shows the results of the numerical simulations of the ISWs propagating along the slope and wave breaking using ISWFoam. As the ISW propagates to the slope, according to the conservation of mass, the upper fluid forward and the downward velocity of the lower fluid increasing along the slope results in the formation of a thin boundary layer, as shown in Fig. 15a, b and c. At the same time, the amplitude of the ISW increases, and the rear of the ISW gradually becomes very steep but does not overturn. With the development of the ISW, the rear waveform of the ISW cannot maintain its stability and overturns forward, resulting in wave breaking, as shown in Fig. 15d. After wave breaking occurs, the denser lower layer flow accelerates into the less dense upper layer flow, forming a mixture region, as shown in Fig. 15e. After the lower layer flow is drawn downward from beneath the ISW, a mixing region comprised of vortices is pushed upwards along the slope while the leading waveform is reflected, as shown in Fig. 15f, g and h. Figure 15i and j show the falling process of ISWs. From the perspective of the entire process of wave breaking, the steepening of the rear waveform in this case is the main reason for wave breaking.

Temporal and spatial variations in the ISWs breaking calculated using ISWFoam (the black line represents the waveform).

For comparison with the flow visualisation image of the experiment, a
specified thickness of the pycnocline is presented, and the pycnocline
ranges from 1003 to 1045 kg m

Comparison of the density fields between the experimental observation results in Michallet and Ivey (1999) (left) and the numerical simulation results (right).

Figure 16 compares the ISWFoam results and the experimental results of Michallet and Ivey (1999) before, during and after ISW breaking. The results indicate that some main features of the laboratory tests are reasonably well reproduced by ISWFoam, such as the profile of ISW, the location of the wave breaking point, ISW arrival time, and spatial and temporal changes in the mixture region. Therefore, the model developed in this paper can accurately simulate the ISW breaking phenomenon during the propagation of ISWs along the slope.

The ISWFoam model developed in the present paper can be used as a tool to investigate the interaction between ISWs and complex structures and topography. In this section, two numerical examples are presented to show the capability of ISWFoam on field-scale simulation.

We designed a case of an ISW propagating over a 3D Gaussian ridge. The 3D
Gaussian ridge is obtained by rotating a 2D Gaussian ridge:

Schematic of the local refinement of the grid.

Temporal and spatial variation in the ISWs propagating over a 3D Gaussian ridge.

We set up a 3D numerical tank, which includes a tank that is 3 km long, 400 m wide (

Figure 18 shows the temporal and spatial variations in the ISWs propagating over a 3D Gaussian ridge. The ISW reaches the Gaussian ridge, causing the wave surface in front of the ridge to decrease, and the wave surface behind the ridge to climb up the ridge, as shown in Fig. 18a. Due to being obstructed by the Gaussian ridge, flow around a ridge and wave surface uplift are generated on both sides of the Gaussian ridge (perpendicular to the direction of wave propagation), as shown in Fig. 18b. As the ISW propagated over the Gaussian ridge, the wave surface climbed along the ridge, and at the same time, low velocity was generated behind the ridge, as shown in Fig. 18c. Since the top of the ridge is in the pycnocline, there will be a low-velocity area behind the ridge for a period of time after the ISW passes, as shown in Fig. 18d. In general, the ISWFoam model with unstructured grids and local mesh refinement can simulate the interaction between ISWs and complex structures and topography at the field scale.

The propagation of ISWs to the shore is bound to be affected by the
continental shelf, and shallow water evolution phenomena such as nonlinear
evolution, breaking phenomena and waveform inversion occur on the
undulating continental shelf. For simplicity, this section simplifies the
continental shelf into a hyperbolic tangent terrain, and the terrain profile
formula is as follows (Lamb, 2002):

Schematic of the local refinement of the grid.

Velocity field diagram of ISW propagation (the black solid line is the iso-density contours,

We set up a 3D numerical tank, which includes a tank that is 7.5 km long,
200 m wide (

Velocity field diagram of ISW propagation (the black solid line is the iso-density contours,

Vorticity field (areas marked A–E represent transverse sections).

Figure 20 shows the waveform and velocity field when the ISW passes through the hyperbolic tangent terrain. From Fig. 20, it can be seen that the ISW breaks and has a significant waveform inversion when propagating from deep water of 100 m to shallow water of 40 m. As the ISWs propagate to the continental shelf, the water depth gradually becomes shallower, and the thickness of the lower fluid gradually decreases as shown in Fig. 20a. Due to the presence of the continental slope, the nonlinearity of the ISWs becomes stronger, and the trough velocity of the ISWs is significantly lower, which causes the waveform at the rear of the ISW to become steep, as shown in Fig. 20b. At the same time, the front waveform of the ISW gradually becomes flat and parallel to the shelf topography. As the waveform at the rear of the ISW becomes steeper and loses balance, the waveform at the rear of the ISW rolls forward, leading to the occurrence of ISW breaking phenomena, as shown in Fig. 20c. It is worth noting that the ISW breaking occurs at the rear of the ISW, while the front waveform does not break but transforms into another form of wave (referred to as the head wave) and continues to propagate steadily along the continental shelf. The breaking of ISWs causes severe disturbances in the water and excites a series of secondary waves at the tail of the head wave, represented by elevation ISWs (as shown in Fig. 20d, e), and then elevation ISWs propagate forward steadily in shallow water (as shown in Fig. 21g, h, i). The shelf slope of the case in this section is the same as the shelf slope of s8_c1c case studied by Lamb and Xiao (2014), both of which are 0.1. The research results of Lamb and Xiao (2014) show that waveform inversion of a depression ISW will occur at this shelf slope, and a series of elevation ISWs will be generated and propagate stably in shallow water. The simulated results have good agreement with that of Lamb and Xiao (2014).

Vorticity field of transverse sections.

The vortex structure has an important influence on the material transport at the bottom of the shelf, so it is very necessary to study the vortex structure when the ISW breaks. Figure 22 show the vorticity field of the ISW at the breaking stage. With the occurrence of ISW breaking, a significant anticlockwise vortex structure is generated below the waveform at the rear of the ISW, as shown in Fig. 22a. With the propagation of the head wave, the vortex climbs along the shelf, the vortex continues to develop horizontally and vertically during the upward climb, and the vertical scale is about one-third of the local water depth (as shown in Fig. 22b, c). As the vortex structure continues to climb, the vorticity decays, and the vortex structure gradually disappears, as shown in Fig. 22d and e. Combined with the velocity field in Fig. 20, it can be seen that the vorticity before and after the ISW breaks is the largest, and the vortex structure is the most obvious. As the wave train of elevation ISWs propagates steadily, the vortex structure climbs up the shelf and gradually disappears.

Velocity vector field of wave train in shallow water areas.

Vorticity field of wave train in shallow water (the black solid line is the waveform).

In order to analyse the vortex of the three-dimensional structure, Fig. 23 show the vorticity field diagram on the transverse section, and the position of the transverse section corresponds to the marked section in Fig. 22. It can be seen from Fig. 23 that the vortex also evolves in the transverse section. Obviously, the bottom vortex structure generated by ISW breaking shows three-dimensional non-uniform features.

The velocity vector field of the head wave and the wave train of elevation ISWs in the shallow water area are shown in Fig. 24. The head wave generated by the breaking of the ISW loses the original wave shape of the ISW, the wave height becomes smaller, the wavelength becomes longer, and the velocity field is still in the form of upper layer forward and lower layer backward (as shown in Fig. 24a, b). In Fig. 24c and d, the velocity field and waveform of the entire wave train following the head wave are stable, the velocity field of each wave is backwards in the upper layer and forwards in the lower layer, and the wavelength gradually becomes longer as the wave train propagates. As the wave train propagates in shallow water, there is a large vorticity in the crest and trough areas of each wave, and it propagates forward steadily as the wave train propagates, as shown in Fig. 25. Generally, the waveform inversion and breaking phenomenon of ISWs is well indicated, and the propagation and evolution of the wave train generated by waveform inversion is also accurately described through ISWFoam simulation.

In this paper, a numerical model referred to as ISWFoam with a modified

ISWFoam using the finite-volume method with unstructured grids and local
mesh refinement can accurately simulate the generation and evolution of
ISWs, the ISW breaking phenomenon, waveform inversion of ISWs and the
interaction between ISWs and complex structures and topography. The method
of initialising the ISW using weakly nonlinear eKdV equation models requires
a period of movement before the jump of the velocity field develops into a
field with continuous changes in velocity. The DJL equation wave generation
method that considers the vertical velocity and the horizontal velocity
along the vertical gradient is better than the eKdV equation wave generation
method that only provides the horizontal average velocity. Using ISWFoam to
simulate an ISW with infinite wave length, the metric for the appropriate
mesh size is given as follows: the dimensions of the horizontal grid are
1

The ISWFoam code developed in this article can be downloaded for free from

All data can be accessed by contacting the corresponding author Qinghe Zhang (qhzhang@tju.edu.cn).

QZ and JL jointly developed this numerical method to calculate internal solitary waves in continuously stratified fluids. JL developed the code. TC performed the computations. QZ and JL jointly analysed the calculation results and wrote the paper together.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research has been supported by The National Key Research and Development Program of China (grant no. 2021YFB2601100) and the National Natural Science Foundation of China (grant no. 51509183).

This paper was edited by James Kelly and reviewed by two anonymous referees.