Although running a 2D slice model is much more economical than running a 3D model from the perspective of computational resources, the 2D model cannot correctly reproduce the ISW field in the real ocean (see Appendix A). Therefore, we implement a realistic 3D non-hydrostatic primitive equation ocean solver (MIT general circulation model, MITgcm; Marshall et al., 1997) in the spherical coordinate to reproduce the ISW features in the NSCS. The model domain (115.8–123.8∘ E, 17.8–22.3∘ N; see blue box in Fig. 1a) includes the main generation site of ISWs (i.e. Luzon Strait) and the mooring station DS on the continental slope. Bathymetry data are derived from the global gridded bathymetry dataset GEBCO (https://www.gebco.net/data_and_products/gridded_bathymetry_data, last access: 17 May 2023). To keep consistency with the instrumental deploying period, we start the model at 00:00 UTC 5 August 2014 and it lasts 15 days. Previous statistical analyses, based on synthetic aperture radar (SAR) images in the NSCS from 1995 to 2001, also indicated that ISW occurrence frequencies were relatively high in August (Zheng et al., 2007). The initial model temperature and salinity profiles (see black and blue lines in Fig. 1b) are derived from the WOA18 climatology dataset (World Ocean Atlas 2018, https://www.ncei.noaa.gov/access/world-ocean-atlas-2018/, last access: 17 May 2023) by spatially averaging the monthly output in August, resulting in horizontally uniform conditions. Density and buoyancy frequency profiles are shown as black lines in Fig. 1c and d.

To ensure ISWs can be physically derived and consider the computational efficiency, the horizontal cell (Δx) is set as 500 m in both zonal and meridional directions. In order to satisfy the high-mode vertical resolution requirements, 90 vertical layers are spaced in accordance with the hyperbolic tangent function (Stewart et al., 2017), namely ranging from 5 m near the surface to 120 m near the sea bed (in the deep water). We impose a time step of Δt=10 s to satisfy the Courant–Friedrichs–Lewy (CFL) conditions in both the horizontal and vertical. The Coriolis parameter is varying with latitudes in the entire model domain. To determine whether the non-hydrostatic mode is necessary, we also run a hydrostatic model (not shown). It notes that fake internal solitary-like wave trains, also called spurious non-hydrostatic processes (Álvarez et al., 2019), are clearly visible at first glance, suggesting that the hydrostatic mode is inappropriate for a high-resolution model of ISWs. We therefore configure the model in non-hydrostatic mode.

The control run (Exp. 1, 500m_8HARs) is driven by eight main tidal constituents (M2, S2, N2, K2, K1, O1, P1 and Q1) on the four open boundaries with values that originated from the Oregon State University TOPEX/Poseidon Solution (TPXO8-atlas data) with 1/30∘ resolution (Egbert and Erofeeva, 2002). A 25 km wide sponge layer is imposed on each lateral boundary to absorb internal wave energy and avoid wave reflection back to the inner region. Quasi-steady conditions occur after 3 d (see Appendix B), so the model results are analysed over the remaining 12 d (8–20 August). The sampling rate of model outputs is at a 1 h interval for the entire model domain in the control run (500m_8HARs) and single-point outputs with a higher sampling rate of 1 min at the selected station DS for recording the local ISW properties, and thereby comparing to the in situ observations. Constant horizontal and vertical eddy viscosity and diffusivity coefficients are imposed as Ah=0.5 m2 s-1, Av=5×10-3 m2 s-1, Kh=0.5 m2 s-1 and Kv=5×10-3 m2 s-1 to eliminate grid-scale instability (Legg and Huijts, 2006). The bottom stresses are parameterized using a quadratic law with a bottom drag coefficient of Cd=2.5×10-3.

Remote sensing imagery contains lots of detailed information of ISW properties, including wave crest lines and their arrival time, which was commonly applied in the NSCS (Liu and Hsu, 2004; Zheng et al., 2007). Here we download two MODIS true-colour pictures with a horizonal resolution of 250 m at 05:15 UTC on 14 August and at 02:50 UTC on 15 August 2014, respectively. In addition, we compute the horizontal gradients of sea surface height (∇η, in the unit of cm km-1), which detects the variations in surface roughness caused by the ISW-induced convergent and divergent currents, thereby producing analogous images to the satellite images. Note that the model is hourly sampled, so we select the closest snapshots of ∇η at 05:00 UTC on 14 August and at 03:00 UTC on 15 August 2014 to compare with MODIS images.

The through-water-column mooring station DS (magenta star in Fig. 1a) is located at 117∘44.7′ E, 20∘44.2′ N in the vicinity of the Dongsha Atoll, which was deployed at a water depth of ∼ 1250 m from 1 August to 28 September 2014. Three acoustic Doppler current profilers (ADCPs) measured currents ranging from a depth of 1180 m to the sea surface every 2 min with 16 m vertical bins in upper 900 and 8 m vertical bins below 900 m. The mooring was configured by temperature sensors, a conductivity–temperature–depth (CTD) sensor and conductivity–temperature (CT) sensors at different water depths. The temperature sensors were at 10, 30, 50, 90, 130, 150, 170, 250, 350, 500, 600, 700, 800, 950, 1050 and 1220 m, respectively; the CTD sensor was at 1100 m; and the CT sensors were at 20, 40, 70, 110, 150, 200, 300, 450, 550, 650, 750, 850, 1000 and 1200 m, respectively. Temporal sampling rates were 10 s for the temperature and CTD sensors and 15 s for the CT sensors, respectively. The instruments carried by the moorings generally functioned well, but CT sensors stopped working after 6 September 2014 due to the lack of power. Besides, Xu et al. (2020) indicated that an anti-cyclonic eddy dominated the region of the mooring starting in mid-September 2014, which significantly affected the local wave properties at the DS station. To avoid the impacts of background currents, we selected 15 ISWs during the spring tidal period from 00:00 UTC on 8 August to 00:00 UTC on 15 August as criteria to quantitatively evaluate the performance of sensitivity numerical experiments.

The 3D control run only runs for 15 d, which is too short to do the harmonic analysis. To validate the model accuracy in simulating the barotropic currents of eight key tidal constituents, we rerun a 3D model (Exp. 2, 500m_8HARs_BT) with the same configurations as 500m_8HARs, but we extend the duration time to 100 d and turn off the iteration of temperature and salinity to focus on the barotropic tide regimes.

As M2, S2, K1 and O1 barotropic tides are dominant in the NSCS (Ramp et al., 2004; Farmer et al., 2009), here we calculate the amplitude (U) and phase (ϕ) of the zonal velocity (Ubt) by doing the harmonic analysis over the last 90 d in Exp. 2 (500m_8HARs_BT) and compare them with the TPXO8-atlas dataset. A root-mean-square error (RMSE), referring to Cummins and Oey (1997), is computed to evaluate the model performance in the barotropic regime, which is given by 1RMSEh=12Um2+Uo2-UmUocos⁡(ϕm-ϕo), in which subscript h represents four different harmonics; U and ϕ are amplitude and phase of zonal barotropic currents with the subscripts m for model and o for observation (TPXO8-atlas). We therefore obtain the horizontal distributions of RMSE for four tidal constituents (see Fig. 2a–d). In most model domains, RMSE is less than 0.02 m s-1 but slightly more in the shallow water (e.g. Luzon Strait and the continental shelf), which is still less than 0.2 m s-1. It may be because the bathymetry derived from the GEBCO dataset and resolutions in our model differ from those in the TPXO8-atlas, thereby resulting in the discrepancy.

In addition to comparison between this model and the global tide model, we extract the DS station outputs with a high sampling rate for comparison with the in situ observations. To avoid the effects of massive high-frequency motions (i.e. environmental noises) in the observational time series on the barotropic regime, we first do the harmonic analysis for zonal barotropic velocities from 5 August to 19 September, then extract the amplitude and phase of eight key tidal constituents, and restructure the time series (see red line in Fig. 2e). In terms of the model results, we obtain the time series at station DS in the same way (see black line in Fig. 2e). It is worth mentioning that the discrepancy between the eight-harmonic restructured time series and the raw data in the model is small, since the experiment is basically driven by the eight tidal constituents and does not include any affects from the background environment. By comparing the two time series, the model reliability is validated all through the spring and neap tides. Overall, the model presents nice performance in the barotropic regime.

Apart from the model validation in barotropic tides, we then look over the control run (500m_8HARs) in baroclinic (ISWs) regime by comparing the model results with MODIS images. Figure 3a and b both show two successive ISWs (labelled as IW1 and IW2) in the deep basin with a distance of ∼ 120 km. The lengths, curvatures and locations of IW1 and IW2 in the simulation are consistent with those in the MODIS image. However, two other ISWs occurring over the continental slope and shelf are captured in the numerical simulations, but they are not observed on 14 August in the MODIS–Aqua image due to cloud cover (Fig. 3b). Conversely, the cloud disappeared on 15 August, so the MODIS–Terra sensor gives a clear seascape painting of ISWs both in the shallow water (i.e. IW2) and deep water (i.e. IW3 and IW4). Note that IW2 in Fig. 3b and d is the same ISW, which propagates ∼ 250 km within 19 h and 35 min. All ISWs (IW2, IW3 and IW4) in Fig. 3c and d occur at the fairly close locations with analogous wave properties. From the perspective of crest-line lengths, the numerical model shows good agreement with the MODIS images, namely 131 km versus 133 km for IW2 in Fig. 3a and b, 187 km versus 198 km for IW3, and 74 km versus 69 km for IW4 in Fig. 3c and d. Note that the sea surface gradients (∇η) larger than 2 cm km-1 along the crest lines are extracted and defined as the crest-line lengths of ISWs. Besides, in the water depth shallower than 500 m, the modelled IW2 exhibits an ISW train with trailing waves, which is also shown in the MODIS image. As the model neglects wind above the sea surface and other marine dynamical processes, there are still some subtle nuances of wave characteristics between them. Overall, this model nicely demonstrates spatial distributions of ISWs in the NSCS, based on the comparison with remote sensing imagery.

To further evaluate the model performance in reproducing ISWs, we introduce the in situ observations. The vertical structure and timing of the wave arrivals, after crossing the deep basin, can be seen in details using daily plots (Fig. 4a–g) of the temperature isotherms and baroclinic (ISW-induced) velocities from 8 to 14 August at mooring DS. For clarity, only the results in upper 900 m are shown in Fig. 4, including the main wave-induced temperature fluctuations. On the basis of space–time diagram of isopycnal displacements along the main propagation paths of ISWs (not shown), the averaged nonlinear internal wave speeds are ∼ 3.0 m s-1 from the Luzon Strait to the deep basin, so it takes roughly 1.5 d for ISWs to propagate from the generation site to the targeted station (DS). We move the arrival time (i.e. 00:00 UTC 8 August –00:00 UTC 14 August) of ISWs 1.5 d forward at the station DS, so the related barotropic tides gradually increase during the spring tidal cycle at the Luzon Strait (i.e. 12:00 UTC 6 August–12:00 UTC 12 August). It explains why ISWs were relatively weak and linear from 8 to 10 August (Fig. 4a–c), but they became significant and nonlinear from 11 to 14 August (Fig. 4d–g). A single ISW was captured around 12:00 UTC from 11 to 14 August, which arrived at the location at approximately the same time every day (termed as type-A ISWs by Ramp et al., 2004). Meanwhile, a wave train, consisting of two dominant solitons and some small trailing waves, arrived at the station an hour later each day, showing the same wave characteristics as type-B ISWs in Ramp et al. (2004).

In terms of the model, we also use the daily plots (Fig. 4h–n) at station DS with 1 min sampling rate to show its similarity to the in situ observations. An increasing trend of wave amplitude and nonlinearity is obvious from 8 August to 14 August in the model results, suggesting precise depictions of barotropic tides and ISWs' characteristics. Specifically, both type-A (single solitons) and type-B ISWs (wave trains) are displayed with analogous arrival time, wave-induced (baroclinic) velocity (colour shades in Fig. 4) and wave amplitude (contours in Fig. 4) compared to those in the observations. It's worth noting that even the linear internal tides and/or hydraulic jumps around 12:00 UTC from 8 to 10 August are reproduced. Although the model omits some small wave signals (see blue arrows in Fig. 4e) in the observations, which might be induced by non-tidal processes such as background currents, the model still shows a good performance in the ISW reproduction.

To quantitatively identify the model accuracy, we select 15 ISWs (marked as red arrows in the left column of Fig. 4), extract their wave properties (i.e. arrival time, maximum wave-induced velocity, propagation direction and maximum mode-1 wave amplitude), and compare them between in situ observations and numerical simulations. We obtain wave propagation direction by computing the angle of baroclinic zonal and meridional components in the layer with maximum velocity. The maximum mode-1 wave amplitude (A1) is extracted from the mooring data and model outputs by least-squares fitting density perturbation profiles ρ′(z) to normalized modal structure function Wn(z), following the similar procedures to those described by Buijsman et al. (2010a) and Rayson et al. (2012). Although the mode-1 wave amplitude can also be extracted by least squares fitting the horizontal baroclinic velocity, Rayson et al. (2019) suggested that the method in velocity field was fuzzy with unidirectional internal waves. The modal structure function can be resolved by a shear-free Taylor–Goldstein equation with the background stratification N2(z), which is given by 2d2Wn(z)dz2+N2zcnWnz=0, with the boundary conditions Wn0=Wn-H=0. Subscript n represents the mode number and cn is the phase speed of the linear internal waves in nth mode. The buoyancy perturbation b(z), depending on density perturbation ρ′(z), is written as 3bz=-gρ′zρ0, in which ρ0 is the reference density. Following the internal wave polarization relationships (Gerkema and Zimmerman, 2008), we fit the wave amplitudes (An) in different vertical modes to bz in both in situ observations and numerical simulations via 4bz=∑n=15AnN2zWn(z). Here, we select the first five vertical modes (n=1–5) to do the least-squares fitting and mainly discuss the mode-1 wave amplitude (A1) due to its significant dominance (Fig. 4).

According to the above approaches, we extract the four wave properties for 15 ISWs and plot Fig. 5, in which observation and model results are shown in red and green, respectively. First, we list the arrival time of ISWs on the two sides of Fig. 5. The bias between observation and model is always smaller than 1.5 h, and the root mean square deviation (RMSD) is 0.71 h, indicating accurate depiction of ISW arrival time in the control run (500m_8HARs). Second, the maximum baroclinic velocity (Fig. 5a) and the averaged values (0.98 and 1.18 m s-1, respectively) are shown in the solid lines. It is suggested that the model underestimates the baroclinic velocity due to neglect of some background non-tidal signals, thereby introducing a RMSD of 0.41 m s-1. Third, the averaged propagation directions of ISWs are ∼ 285 and ∼ 291∘, respectively (the angle measured anticlockwise from north) in the model results and observational data with a RMSD of 8.35∘. It is worth mentioning that the type-A ISWs mainly propagate westward, while the type-B ISWs propagate north-westward in both observation and model, verifying the model's reliability to some extent. Finally, the averaged maximum mode-1 wave amplitude (∼ 108 m) in the model is close to that (∼ 99 m) in the observation. Nonetheless, the RMSD of mode-1 wave amplitude is 37.27 m. Overall, the control run can basically reproduce various wave properties of ISWs observed in the vicinity of the Dongsha Atoll in the NSCS.

Various 3D models with different resolutions were implemented to simulate ISWs in the NSCS in previous studies (e.g. Vlasenko et al., 2010; Zhang et al., 2011; Lai et al., 2019). However, which resolution is adequate to satisfy the ISW prediction precision and save computational resources to the utmost in the meantime has yet to be discussed. Here, we run two sensitivity experiments (Exps. 250m_8HARs and 1000m_8HARs) with horizonal resolutions of 250 and 1000 m, to respectively compare the model performance in different aspects with the control run (resolution of 500 m).

First, the spatial distributions of ISWs are exhibited via the snapshots of sea surface height gradients (∇η) at 12:00 UTC on 12 August 2014. In the control run (500m_8HARs), three ISWs (labelled as IWB1, IWA1, and IWB2 from west to east) with distinct crest lines successively occur between 116 and 120∘ E (see Fig. 6a), in which IWB1 and IWB2 are internal wave packets with trailing waves (type-B wave) and IWA1 is a single soliton (type-A wave). As IWB1 approaches the continental slope and shelf, the leading wave front fully steepens with a narrow characteristic half-width, suggesting its strong nonlinearity. IWB2 also shows up as a wave packet with many secondary waves in the developing stage, although its nonlinearity is slightly weaker than IWB1's. Conversely, the single soliton IWA1 with relatively long crest line and broad characteristic half-width is about to pass mooring station DS (marked as green star in Fig. 6). In comparison, the Exps. 250m_8HARs and 1000m_8HARs reproduce these three waves as well, but with some subtle discrepancies between them. In Exp. 250m_8HARs, more details of wave properties are clarified (Fig. 6b). Specifically, the secondary waves of IWB1 and IWB2 are more visible than those in 500m_8HARs. However, in Exp. 1000m_8HARs, some fine structures of ISWs are not well resolved. For instance, only one secondary wave is found behind the leading wave of IWB2, and the south portion of IWA1 crest line is barely observed (Fig. 6c).

Then, we select a transect along the main propagation path of ISWs (shown in dashed line in Fig. 6a) at 12:00 UTC on 12 August 2014 to compare the vertical structure of ISWs among three experiments (see Fig. 7). In Fig. 7, blue (yellow) colour shades represent westward (eastward) baroclinic velocity and contours are temperature isotherms. Linear internal waves, such as internal wave beams near the generation site (120–121∘ E), are nicely reproduced in all numerical experiments. Nonetheless, nonlinear internal waves present different wave characteristics in different cases. In Exp. 500m_8HARs, the single soliton IWA1 and the wave packet IWB2 with a series of trailing waves are apparent in the slice, but IWB1 is not included (Fig. 7a). In Exp. 250m_8HARs, IWA1 and IWB2 occur at the same location as those in Exp. 500m_8HARs. IWA1 show similar properties in two cases, but the secondary waves of IWB2 are better described in Exp. 250m_8HARs. By comparison, IWA1 shows its weak nonlinearity with small vertical displacement and broad characteristic half-width (i.e. horizontal distance between the wave front and wave trough) in Exp. 1000m_8HARs. Besides, only one secondary wave appears in the IWB2 packet in Exp. 1000m_8HARs.

Last, a two-day time segment of observational temperature and baroclinic velocities from 18:00 UTC on 11 August to 18:00 UTC on 13 August 2014 at the station DS is extracted to demonstrate the sensitivity model capability of simulating vertical structures of ISWs over the continental slope (Fig. 8). In the control run (500m_8HARs, Fig. 8b), two wave packets and two single solitons successively arrive at the station, keeping the consistency with the observation, although their characteristic half-widths are slightly broader than those in the field measurements (Fig. 8a). Meanwhile, some small fluctuations, occurring in the observations, are not included in the control run. In Exp. 250m_8HARs (Fig. 8c), the half-widths are narrower than those in the Exp. 500m_8HARs, which agree better with the real internal wave field. Besides, more fluctuations, i.e. those small wave signals (09:00 UTC on 12 August and 09:00 UTC on 13 August) in front of the single solitons, are reproduced in this experiment. Conversely, in Exp. 1000m_8HARs, internal wave trains can still be reproduced with relatively weak nonlinearity, but the single solitons are not correct due to their tiny amplitudes and linear wave structures.

To quantitatively evaluate the model performance of sensitivity experiments, we present the bias of five wave properties of 15 ISWs (marked as red arrows in Fig. 4) between model results and observational data in Fig. 9. The biases of arrival time are generally smaller than 1 h (see black and blue circles in Fig. 9a) for Exps. 500m_8HARs and 250m_8HARs, whose RMSDs are 0.71 and 0.67 h, respectively. In contrast, the bias for Exp. 1000m_8HARs is larger than 1 h (red circles in Fig. 9a) and its RMSD is 0.79 h. In terms of the wave-induced velocity (Fig. 9b), the RMSDs are 0.38, 0.41 and 0.48 m s-1 in Exps. 250m_8HARs, 500m_8HARs and 1000m_8HARs, respectively. The RMSDs of propagation directions are very close (∼ 8.5∘) in the three experiments (see Table 3). As for the mode-1 wave amplitudes, Exps. 250m_8HARs and 500m_8HARs overestimate the wave amplitudes in most cases (see positive biases in Fig. 9d), thereby resulting in RMSDs of 38.12 and 37.27 m, respectively. Conversely, Exp. 1000m_8HARs would underestimate the wave amplitudes of majority ISWs with dominant negative biases in Fig. 9d, resulting in a RMSD of 40.28 m (Table 3). Last but not least, Exps. 500m_8HARs and 1000m_8HARs inaccurately depict characteristic half-widths of ISWs with RMSDs of 1.07 and 2.41 km, while Exp. 250m_8HARs performs well with a RMSD of 0.64 km (Fig. 9e). The relative difference of RMSD suggests that Exp. 250m_8HARs increases to 40 % accuracy of predicting characteristic half-widths by comparing Exp. 500m_8HARs. From the perspective of computational resources, Exps. 250m_8HARs, 500m_8HARs and 1000m_8HARs spend 20.4×104 CPU hours, 4.6×104 CPU hours and 1.0×104 CPU hours, respectively.

In summary, the control run with a resolution of 500 m can basically reproduce the principal ISW field in the NSCS, while the sensitivity model with a higher resolution of 250 m would be a better solution to identify wave properties, in particular of the wave nonlinearity. Nonetheless, a 250 m resolution model spends nearly a 5-fold computational resources of a 500 m resolution model in the same model domain. Besides, the model with a lower resolution of 1000 m underestimates the nonlinearity of ISWs, thereby resulting in an inaccurate ISW field in the NSCS.

The 3D/2D models with different numbers of barotropic tidal constituents (e.g. single harmonic, four harmonics and eight harmonics) were commonly imposed to investigate the generation mechanisms of ISWs in the NSCS in previous studies (e.g. Li, 2014; Buijsman et al., 2010a; Jin et al., 2021). However, whether a single tidal constituent can satisfy the reproduction of a real ISW field and how many tidal constituents are required for a realistic ISW model are still questions. Here, we run three sensitivity experiments (Exps. 500m_1HAR, 500m_4HARs and 500m_13HARs) with different numbers of tidal harmonics to answer the questions by comparing the model performance with the control run (500m_8HARs).

We first discuss the model requirements of tidal constituents from the point of view of the ISW horizontal distributions and look back to Fig. 6. Note that time series of zonal barotropic currents at the generation site (Luzon Strait) are presented at the bottom left for each panel, where single, four and eight tidal constituent(s) are shown in green, magenta and blue, respectively. By comparing Exp. 500m_1HAR (Fig. 6d) and 500m_8HARs (Fig. 6a), we find that the single M2 tidal harmonic is not adequate to reproduce ISWs in the NSCS, so only some linear internal tides are detected on the sea surface via ∇η. In contrast, Exp. 500m_4HARs (Fig. 6e) nearly recreates the analogous scenario of ISWs to Exp. 500m_8HARs, where IWB1, IWA1 and IWB2 appear at the same locations. Nonetheless, the crest-line length (∼ 134 km) of IWB2 in Exp. 500m_4HARs is slightly shorter than that (∼ 167 km) in Exp. 500m_8HARs, and the secondary waves of IWB2 are unclear in Exp. 500m_4HARs (see Fig. 6e). ∇η in Exp. 500m_13HARs are not presented in Fig. 6, since it shows the exact same spatial patterns of ISWs as those in Exp. 500m_8HARs, suggesting the principle eight tidal constituents are fine enough to satisfy accurate reproduction of the horizontal features of ISWs in a realistic oceanic model.

We then consider the difference of ISW vertical structures in sensitivity experiments with various tidal forcing via the selected transect and mooring station DS. In Exp. 500m_1HAR, only linear internal waves are captured from the generation site to the slope, suggesting that single M2 tidal constituent without amplification factors can only contribute to internal tides and linear internal wave beams in NSCS (see Figs. 7d and 8e). Unless the magnitudes of M2 barotropic tides are amplified, ISWs are likely to be generated (e.g. Yuan et al., 2020). In Exp. 500m_4HARs (Figs. 7e and 8f), the single soliton IWA1 is reproduced with a smaller amplitude and weaker nonlinearity than that in Exp. 500m_8HARs. Besides, the secondary waves of IWB2 are barely observed in Exp. 500m_4HARs, which are much clearer in Exp. 500m_8HARs (Figs. 7a and 8a). Figure 8a and g depict the striking similarity of wave characteristics between Exp. 500m_8HARs and Exp. 500m_13HARs.

Last, we quantitatively estimate the sensitivity model capability of reproducing ISWs, by computing the biases and RMSDs of five wave properties (see Fig. 9 and Table 3) in the cases with different tidal forcing. Since Exp. 500m_1HAR cannot predict ISWs with significant amplitudes, we exclude it in the following analysis. In terms of Exp. 500m_13HARs with 13 tidal constituents, the biases and RMSDs of five wave properties are very close to those in the control run with eight harmonics (see overlapped black circles and magenta triangles in Fig. 9 and Table 3). Conversely, Exp. 500m_4HARs shows significant difference in the biases and RMSDs of five wave properties from the control run. Specifically, in Fig. 9a, the RMSD of arrival time (0.81 h) is larger in Exp. 500m_4HARs than that in Exp. 500m_8HARs (0.71 h). In addition, Exp. 500m_4HARs underestimates averaged wave-induced velocity for about 38 % and averaged mode-1 wave amplitude for about 15 %, which result in large negative values of biases (see magenta triangles in Fig. 9b and c), corresponding to 0.58 m s-1 and 43.69 m of RMSDs, respectively. In terms of the characteristic half-widths, Exps. 500m_4HARs and 500m_13HARs with RMSDs of 1.10 and 1.01 km show analogous performance to the control run Exp. 500m_8HARs with a RMSD of 1.07 km.

In summary, the model with 8 (or 13) primary tidal constituents can accurately reproduce the real ISW field in the NSCS, while the sensitivity model with four key harmonics (M2, S2, K1 and O1) would underestimate the magnitudes of some secondary wave within a wave packet. In addition, the model only driven by M2 tide can only characterize wave properties of linear internal waves (tides) instead of ISWs.

As ISWs generate via tide-topography interaction in the stratified water, the stratification selection is crucial to directly affect the model capabilities. Here, we extract the background stratification from the in situ measurements at mooring station DS as the initial condition to run the sensitivity experiment 500m_Real_N2, and we compare the model results with the control run (500m_8HARs) with a climatological stratification from the WOA18 dataset.

In the model results, the spatial distribution of ∇η in Exp. 500m_Real_N2 shows an analogous pattern of ISWs to that in Exp. 500m_8HARs. Specifically, three ISWs (i.e. IWB1, IWA1, and IWB2) appear at the same location in the two experiments with similar horizontal wave characteristics (Fig. 6a and f). The visible difference is that the crest line length of the secondary wave of IWB2 is longer with a stronger nonlinearity in Exp. 500m_Real_N2. We then look over the difference of ISW vertical structures between two cases from the perspective of x–z plane along the transect (Fig. 7a and f) and time series at station DS (Fig. 8a and h). It is clearly shown that Exp. 500m_Real_N2 with the real stratification can better characterize the nonlinearity of the single soliton IWA1 and the secondary wave of wave train IWB2. Besides, the comparison with field measurements reveals that Exp. 500m_Real_N2 shows a better precision (13 %) in predicting the arrival time (i.e. RMSD of 0.62 h) of ISWs than the control run (i.e. RMSD of 0.71 h) with the climatological stratification. However, the RMSD of the propagation direction of ISWs is larger in the realistic-stratification case (14.74∘) than that of the control run (8.35∘). Last, Exp. 500m_Real_N2 nicely describes the characteristic half-widths of ISWs (RMSD of 0.58 km), which improves 46 % accuracy by comparing that in Exp. 500m_8HARs (RMSD of 1.07 km). To sum up, although the model with climatological stratification works well, applying the real background stratification as the model initial condition would improve the model performance in predicting some wave properties, including arrival time, wave-induced velocity, wave amplitude and characteristic half-width.