The data used in this study are the LLC4320 simulation outputs from the Massachusetts Institute of Technology general circulation model (MITgcm), which solves the Navier-Stokes equations under the hydrostatic and Boussinesq approximations. The model is forced by the complete luni-solar tidal potential including 16 principal tidal constituents (such as M₂, S₂, K₁, and O₁) and by the 6-hourly atmospheric reanalysis dataset provided by the European Centre for Medium Range Weather Forecasts (ECMWF) (Marshall et al., 1997; Arbic et al., 2018). An Arakawa-C grid mesh is used in the simulation with a horizontal resolution of 1/48° (approximately 2 km in the SCS). In the vertical, the model has 90 z-coordinate levels, with layer thickness increasing from 1 m at the surface to around 480 m near the seabed. The hourly output of three-dimensional velocity, potential temperature, salinity, and sea surface height from 13 September 2011 to 15 November 2012 is used, with specific focus on two key topographic regions, the LS and the Nansha Islands (Fig. 1). The Luzon Strait domain is defined to encompass the Hengchun Ridge, the Lanyu Ridge, and the inter-ridge trough, which collectively constitute the double-ridge where intense internal tides are generated (Buijsman et al., 2014; Tian et al., 2025; Wang et al., 2016). The Nansha Islands domain is selected to encompass the dominant topographic features (islands and reefs), which are located in the propagation paths of internal tides coming from the LS (Tian et al., 2025; Xu et al., 2016). The interaction of the incoming internal tides with the rough islands and reefs promotes energy transfer to higher modes and hence enhances local dissipation (Vic et al., 2019).

The subtidal fields (u‾ and ρ‾) are obtained by applying a 3 d moving average to the total fields, and the perturbation fields (u′, ρ′) are obtained by subtracting the subtidal fields from the total ones. The barotropic velocity ubt′=(ubt′,vbt′) is defined as the depth-averaged velocity, and the baroclinic velocity is calculated as ubc′=u′-ubt′. A bandpass filter is then applied to extract the diurnal (D1, 0.90–1.10 cpd) and semidiurnal (D2, 1.87–2.10 cpd) tidal constituents. When diagnosing the internal tidal energy budgets and q* for individual vertical modes, a vertical modal decomposition method is employed. The modal structure function ϕn(z) can be obtained by solving the Sturm–Liouville equation (Gill, 1982): 1d2ϕn(z)dz2+N2(z)cn2ϕn(z)=0, where boundary conditions are given as ϕn(0)=ϕn(-H)=0, and H is the water depth. Here, N(z) is the buoyancy frequency, cn is the eigenspeed, and n is the mode number. Due to the orthogonality of the modes, the baroclinic velocity ubc′ and the pressure perturbation p′ can be expressed as ubc′(z,t)=∑n=1NUn(t)φn(z), p′(z,t)=∑n=1NPn(t)φn(z), where φn(z)=dϕn(z)/dz is the vertical structure for horizontal velocity and pressure, Un(t) and Pn(t) are the coefficients of individual modes, and N=5 is the total number of modes used in this study. Taking January and August 2012 as representative cases, in the LS and the Nansha Islands, modes 1–5 account for 85 % and 89 % of the total baroclinic kinetic energy, 87 % and 99 % of the diurnal baroclinic kinetic energy, and more than 98 % of the semidiurnal baroclinic kinetic energy, respectively. Modes 1–5 capture the dominant part of total baroclinic kinetic energy, while the remaining fraction (mode 6 and above) does not exceed 15 % of the total baroclinic kinetic energy, although these unretained higher modes may still contribute less to local dissipation of internal tides over rough topography (Vic et al., 2019).

To evaluate the generation and dissipation of internal tidal energy, the depth-integrated baroclinic energy equation is used here. After tidal-period averaging and neglecting the tendency and advection terms, the energy equation reduces to (Niwa and Hibiya, 2004): 2DISbc=〈Conv〉-∇h⋅F, where 〈⋅〉 denotes the tidal-period mean. Conv is the depth-integrated barotropic-to-baroclinic energy conversion, indicating the local internal tide generation, which can be calculated as: 3Conv=∫-Hηρ′gwbt′dz, where η is the sea level displacement, and wbt′ is the vertical velocity induced by the barotropic tides. This vertical velocity wbt′ is given by wbt′=ubt′σ∂D∂x+∂η∂x+vbt′σ∂D∂y+∂η∂y+(σ+1)∂η∂t (Mellor, 2004), where D is the total water depth, D=H+η, and σ=(z-η)/D. A positive Conv denotes conversion from barotropic to baroclinic energy, that is, local internal tide generation, whereas a negative Conv denotes the reverse (baroclinic-to-barotropic) conversion. F is the depth-integrated baroclinic energy flux, and ∇h⋅F is the horizontal divergence of F. A positive ∇h⋅F indicates outward propagation of internal tidal energy from the local region; a negative ∇h⋅F indicates convergence of energy flux from the far field. F is given as: 4F=∫-Hηubc′p′dz, where p′ is calculated as p′(z,t)=-1H∫-Hη∫zηgρ′(z^,t)dz^dz+∫zηgρ′(z^,t)dz^. DIS_bc is the depth-integrated local dissipated internal tidal energy, inferred as the residual of the simplified depth-integrated baroclinic energy equation (Eq. 2) rather than a diagnosed dissipation from the model. A positive DIS_bc denotes net dissipation of baroclinic energy inferred from the energy budget; a negative DIS_bc denotes a net gain of baroclinic energy. In some studies, the dissipation can also be diagnosed from the simulation using the velocity gradients and the horizontal and vertical eddy viscosity coefficients (Nikurashin and Legg, 2011). However, in the LLC4320 simulation, horizontal eddy viscosity is determined internally by a parameterization scheme and is not provided as an output; therefore, we cannot exactly diagnose dissipation from the model output. There exist both connections and differences between the inferred dissipation and the diagnosed dissipation: the former not only includes the latter (i.e., explicit numerical dissipation), but also covers other physical processes such as bottom friction dissipation. Fortunately, their quantitative discrepancy is not critical. For example, assuming a constant horizontal eddy viscosity, Fan et al. (2024) compared the diagnosed dissipation from the LLC4320 simulation with the inferred dissipation from the energy budget equation. They found that these two dissipation fields exhibit comparable magnitudes and broadly consistent spatial patterns across the northern SCS, despite localized discrepancies attributed to the constant horizontal eddy viscosity assumption.

The local internal tidal dissipation ratio, q*, is defined as the ratio of the area-integrated DIS_bc to Conv, i.e., q*=∫sds(DISbc)/∫sds(Conv). Here, the spatial integration domain s corresponds to the red-boxed regions shown in Fig. 1. Based on the above calculations, the internal tidal energy budgets and q* for different tidal components and different vertical modes can be estimated. In this way, q* is not necessarily restricted between 0 and 1. q*>1 occurs when the area-integrated divergence of the baroclinic energy flux is negative, indicating a convergence of far-field internal tidal energy. In this case, the inferred local dissipation exceeds the barotropic-to-baroclinic energy conversion. For individual vertical modes, q*<0 occurs when the area-integrated divergence of the baroclinic energy flux of this mode exceeds its local barotropic-to-baroclinic energy conversion, yielding a negative inferred dissipation. The divergence of the baroclinic energy flux is furnished by both the local conversion and the net energy transfer from other vertical modes, and may therefore exceed the local barotropic-to-baroclinic energy conversion. Incomplete months (September 2011 and November 2012) are excluded when computing the monthly mean values of q*. January and August 2012 are selected as winter and summer months in the simulated year, respectively.

To develop a more robust parameterization of q*, this study not only adopts a physical-based empirical formulation but also uses the XGBoost model jointly, which can make up for the limitations of traditional physical-based empirical formula in complex dynamic environments. The model inputs include 6 physical features, namely the internal tidal shear averaged within the bottommost 500 m, stratification, barotropic tidal kinetic energy, mode-1 internal tidal energy, the proportion of mode-1 internal tidal energy in the total internal tidal energy, and the mode-1 internal tidal group velocity. The data from the LS and the Nansha Islands are sequentially combined into a training set, a validation set, and a test set with proportions of 70 %, 15 %, and 15 %, respectively. Considering the differences in the physical processes governing q* between the internal tidal source regions and sink regions, this study constructs two separate models, one for the LS and one for the Nansha Islands. The Mean Absolute Error (MAE) is adopted as the objective loss function, i.e., MAE=1n∑i=1ny^i-yi, where yi and y^i denote the actual and predicted values, respectively, and n is the number of samples. An early stopping iteration setting of 500 rounds and 800 rounds are applied for models of the LS and Nansha Islands, respectively.

The internal tidal energy budget in the LS is characterized by intense generation, strong outward radiation, and relatively low q*. The barotropic-to-baroclinic conversion in the LS ranges from 40 to 50 GW, of which 15–25 GW radiates out of the LS (Fig. 2g), yielding a q* value of approximately 0.5 (Fig. 2h). The semidiurnal wavelength better matches the alignment of the double ridges in the LS, resulting in a larger semidiurnal tide generation (20–30 GW) than diurnal tide generation (10–25 GW) (Fig. 2g). The semidiurnal dissipation (10–20 GW) and q* values (0.6–0.8) are also higher than those of the diurnal tide, whose dissipation ranges from 1 to 10 GW and q* from 0.1 to 0.6 (Fig. 2g and h). The semidiurnal conversion, dissipation, and q* are larger in summer than in winter (Fig. 2g and h). The diurnal q* reaches minima of 0.1 and 0.3 in February–March and June–July, respectively (Fig. 2h).

The summer spatial distributions demonstrate that internal tide generation and dissipation are concentrated over and around the double ridges (Fig. 2a, b, e and f). Substantial internal tidal energy propagates westward and northwestward from the western ridge into the SCS (Fig. 2c and d). The semidiurnal component plays a dominant role in the internal tide generation and dissipation (Fig. 2a, b, e and g). The q* values ranging from 0.49 to 0.57 in the LS are comparable to those in other representative internal tide generation regions; for example, You et al. (2023) estimated a value of about 0.58 for the M₂ internal tide at the Aleutian Ridge, suggesting that q* values in submarine ridges serving as internal tide source regions share a similar magnitude.

The internal tidal energy budgets and q* in the LS exhibit differences among individual tidal components and vertical modes. Both diurnal and semidiurnal internal tide generation are dominated by modes 1–2, with mode 1 accounting for roughly 50 % of the total generation and exhibiting an energy flux divergence greater than those of other modes (Fig. 3j and k). The divergence of diurnal mode-1 energy flux is even larger in magnitude than the local generation (Fig. 3j), resulting in negative q* values (Fig. 3l), implying that the internal tidal energy generated in the LS radiates outward as low-mode internal tides intensively. Tian et al. (2025) indicated that the diurnal tide has a larger decay scale than the semidiurnal tide, which favors more outward radiation of the diurnal tide as mode 1. By contrast, although the semidiurnal energy budget is also dominated by modes 1–2, the proportion of mode-1 energy flux divergence is relatively smaller, and its difference between January and August 2012 is weaker than that of the diurnal tide (Fig. 3j and k). Regarding local dissipation, q* for modes 3–5 of both the diurnal and semidiurnal components ranges from 0.8 to 2.7, significantly larger than that of modes 1–2 (<0.8) (Fig. 3l and m).

The strongly positive generation regions for modes 1–2 internal tides are concentrated at the crests of the double ridges, whereas the high-generation regions for mode 3 are distributed along the ridge flanks (Fig. 3a–c). The energy budget of the mode-1 internal tide is characterized by substantial energy propagating northwestward and westward into the SCS, as well as eastward into the Pacific Ocean (Fig. 3d). The majority of the mode-1 semidiurnal energy flux enters the SCS, with peak values reaching 25 kW m⁻¹ (Fig. 3d). Local dissipation gradually weakens and becomes spatially dispersed with increasing mode number (Fig. 3g–i). At the Mid-Atlantic Ridge and the Hawaiian Ridge, Vic et al. (2018) indicated that a dominant proportion of mode-1 internal tide generation facilitates the outward radiation of substantial energy as low-mode internal tides, yielding small values of the dissipation ratio. This result confirms the robust far-field radiation capacity of internal tides and the smaller q* values of mode 1 relative to those of higher modes in the LS.

In contrast to the LS, the internal tidal energy budget of the Nansha Islands is characterized by low local generation, strong energy convergence, and a high value of q*. The local generation of internal tides in the Nansha Islands is weak, fluctuating between 0.6 and 1.3 GW, whereas the incoming internal tidal energy flux can reach 1.5 GW (Fig. 4g). This energy budget (dominated by far-field input) directly elevates q* values to 1.0–3.0 (Fig. 4h). Both the local generation and the energy flux of the diurnal tide dominate those of the semidiurnal tide (Fig. 4g), maintaining the q* of the diurnal tide in the range of 1.5–3.0, which is larger than the q* of the semidiurnal tide varying between 1.0 and 2.0 (Fig. 4h).

Both the generation and local dissipation of internal tides in the Nansha Islands exhibit a scattered patchy pattern, with high values concentrated around reefs (Fig. 4a, b, e and f). All the energy budget terms of the diurnal tide are larger than those of the semidiurnal tide (Fig. 4a–f), with a significant incoming diurnal energy flux of approximately 1.5 kW m⁻¹ at the southwestern and northern boundaries of the Nansha Islands (Fig. 4c). Kelly et al. (2013) estimated that at continental margins, incident mode-1 internal tidal energy is partitioned among reflection, transmission, and scattering to higher modes. Lahaye et al. (2020) demonstrated that incident low-mode internal tides can be scattered to higher modes through interactions with steep topography. Subsequently, high-mode internal tides are susceptible to breaking near their generation sites (Siyanbola et al., 2024). Modal scattering redistributes internal tidal energy rather than dissipating it; the breaking of the scattered high modes constitutes the irreversible dissipation. It is therefore reasonable that far-field internal tidal beams trigger intense topographic scattering, which supplies sufficient energy to sustain a high local dissipation ratio (q*>1) despite weak local generation in this region.

In terms of vertical modes, the internal tidal energy budget terms in the Nansha Islands are significantly smaller in magnitude than those in the LS, with strong low-mode energy convergence and high values of q*. The energy budget in the Nansha Islands is dominated by modes 1 and 2 (Fig. 5j and k). Unlike in the LS, here the local generation of mode 1 is weaker than that of mode 2 (Fig. 5j and k). This energy budget, dominated by far-field input of mode-1 energy flux, directly results in the q* values of mode 1 exceeding those of mode 2, as shown in Fig. 5l and m. The energy budget terms for modes 3–5 are all near zero with highly variable q* (Fig. 5j–m), indicating that local generation into these modes contributes little to the regional dissipation. Diurnal tides dominate in the Nansha Islands, with all energy budget terms exceeding those of the semidiurnal tide and showing pronounced differences between January and August 2012 (Fig. 5j and k). The q* values of the mode-1 diurnal tide can exceed 4 (Fig. 5l).

In the Nansha Islands, all internal tidal energy budget terms decrease with increasing mode number, except for the mode-2 energy conversion (Fig. 5a–i). The mode-3 energy budget terms are generally less than one-third of those for modes 1–2. Two dominant incoming energy fluxes exist in the Nansha Islands, originating from the northeast and the southwest, respectively. With increasing mode number, the energy flux from the southwestern shelf slope of the SCS decays more rapidly than that from the northeastern source, suggesting that the reflection from the southwestern shelf slope of the SCS may furnish local dissipation in the Nansha Islands. Siyanbola et al. (2024) pointed out that in weak generation regions, intense local dissipation depends on sufficient convergence and slope reflection of far-field low-mode internal tidal beams. This is consistent with our results that the elevated mode-1 dissipation and q* (>1) in the Nansha Islands are potentially dominated by the incoming mode-1 energy flux, particularly the reflection from the southwestern shelf slope of the SCS.