Observations demonstrate evidence of internal solitary waves (ISWs) in coastal oceans and seas . It is generally accepted that one of the main causes of ISWs is the barotropic tide interacting with the bottom topography .

Generated by tides, ISWs of depression (where the upper-layer thickness is usually much less than the depth of the ocean) are the most energetic and can propagate thousands of kilometers from their origin . As a result, ISWs transport energy far from the location of their generation. Like surface waves, internal waves break at the coastal zone of the ocean. Such waves are an important component of mixing and energy dissipation in the ocean . The breaking of ISWs upon sloping boundaries in the coastal region also plays an important role in diapycnal mixing , biological enhancement , and resuspension of bottom deposits .

The interaction behavior of ISWs depends on the steepness of the topography and the characteristics of the solitary waves . If the slopes are smooth, much of the energy scatters upslope onto the continental shelf where it will dissipate; however, if the slopes are steeper, energy will reflect and return to the deep ocean . It is important to understand the mechanisms of the transformation of ISWs at continental slopes and identify “hot spots” of wave energy dissipation. Two shoaling mechanisms can be important: (1) the conversion of ISWs of depression into elevation waves in a two-layer stratification when the thickness of the upper mixed layer is greater than one-half of the total water depth ; and (ii) ISW breaking on the slope that occurs when fluid velocities in the wave exceed the wave phase speed, which leads to the overturning of the rear face of the wave, shear instability, and intensive mixing. Different types of breaking are commonly distinguished by slope inclination, water column stratification, and wave characteristics. Assuming analogy with surface waves, the breaking regimes of ISWs in a two-layer fluid over a slope have been classified into surging, plunging, collapsing, and fission. In these studies, the classification of breaking is based on the Iribarren number, which is a ratio of the slope to the square root of the wave steepness (amplitude divided by the wavelength). This criterion was modified by for collapsing and plunging breakers using a new wave Reynolds number that takes nonlinear wave steepening into account. A simple three-dimensional α β γ classification diagram was proposed by to distinguish different regimes of ISW interactions with the shelf-slope topography. The classification is based on three parameters: the slope angle γ; the nondimensional wave amplitude α (wave amplitude normalized on the upper-layer thickness); and the blocking parameter β, which is the ratio of the height of the bottom layer of the shelf to the incident wave amplitude.

ISWs breaking over slopes have been observed in many coastal locations worldwide . Observational studies have also shown that the amplitudes of depression ISWs in the South China Sea (SCS) could reach extreme values of over 200 m . Based on the analysis of satellite images , it has been found that most internal waves in the northeastern South China Sea are generated in the Luzon Strait. Further, solitary waves propagate westward and then diffract around the Dongsha Atoll. In the shallow-water regions of the northern SCS, changes in water depth may cause polarity conversion, leading to the transformation of depression ISWs into elevation ISWs . showed that the kinetic energy of ISWs decreased three times after changing polarity, while showed that the seasonal variations in stratification caused these seasonal variations in polarity. The present study is focused on ISW transformation over an idealized shelf-slope topography with a two-layer stratification. The objectives of this work are (1) to compare α β γ classification with the results of numerical modeling, laboratory studies, and field observations; (2) to identify high-energy-dissipation zones of ISWs that pass over the shelf-slope topography using the α, β, γ classification; (3) to apply the α, β, γ classification to numerical modeling data that mimic ISW transformation over a continental shelf in the Lufeng region (SCS); and (4) to determine energy loss as a result of the transformation of ISWs over the shelf-slope topography. Information about polarity change criteria and breaking criteria in the α β γ classification of ISW transformation regimes over shelf topography is presented in Sect. 2. An overview of field and laboratory measurement data and their comparison with the numerical modeling data are given in Sect. 3. The energy dissipation of ISWs breaking over shelf topography is considered in Sect. 4. Finally, the results are summarized in the Sect. 5.

A two-layer approximation is a simple model of stably stratified oceans and lakes. In this model, we approximated stratification using two continuous layers of depths h1 (upper layer) and h2 (lower layer) with a relatively thin pycnocline. When h1>h2, ISWs propagate in the form of elevation ISWs, whereas if h1<h2, they propagate in the form of waves of depression. In this study, we consider ISWs of depression (with amplitude ain) propagating over an idealized shelf slope with a slope γ and a minimum depth of the lower layer over the shelf of h2+. The idealized shelf-slope topography is shown in Fig. b, and the idealized configuration that mimics the continental shelf in the Lufeng region (Fig. a) in the SCS is shown in Fig. c.

It was assumed that ISW transformation over a slope is controlled by stratification, slope inclination, and amplitudes (wavelength) of the incident wave . Two possibilities that could occur with the wave during shoaling were determined: (i) ISW breaking, which was associated with gravitational instability due to the wave overturning and shear instability, and (ii) changing ISW polarity on the slope.

Three parameters, α, β, and γ, can be important for the behavior of the incident wave on a shelf slope (Fig. b, c):

the slope inclination γ, which is measured as an angle;

Let us consider the transformation of ISWs in the case of idealized topography and stratification that approximately follow the cross section in the Lufeng region in the SCS. The position of the cross section is shown in Fig. 1a. The data indicate that internal waves from the Luzon Strait propagate westward to the Dongsha Atoll and then further to the Lufeng region and that the measured current in waves is about 1.5–2.0 m s-1. Wave amplitudes obtained using synthetic-aperture radar (SAR) images (during May after a strong thermocline developed in April) at a depth of about 300 m vary from 10 to 50 m with the depth of thermocline being about 40–65 m . For numerical modeling of the idealized case that mimics the Lufeng region computational domain with a length L=18 km, a maximal depth Hmax⁡=300 m was considered (Fig. 1a). We approximated stratification in the Lufeng region using the two-layer density profile. The densities of the layers are ρ1 and ρ2 (depths h1 and h2, and H=h1+h2), and the pycnocline layer thickness is dh: 7ρ(z)=ρ1+ρ22-ρ1-ρ22tanh⁡z-h1dh.

In numerical experiments, we vary the wave amplitudes (|ain|), |ain|=20 m, |ain|=50 m, and |ain|=100 m, and we vary the slopes (γ), γ=1∘, γ=3∘, and γ=5∘. The slope inclination γ for a smooth curvilinear slope is measured as the maximal slope value. Corresponding values of α, β, and γ are given in Table 2. Density ρ1=1021.5 and ρ2=1025.5 (kg m-3), and the pycnocline layer thickness dh=15 m. The flux of salinity through the flume boundaries is also set to zero. The density profile from measurements from the SCS (May) and the initial density profile (Eq. ) are shown in Fig. . Depths layers are h1=50 m and h2=250 m for all runs.

The numerical simulations were carried out using a free-surface nonhydrostatic numerical model . The Smagorinsky model extended for stratified fluid was used to explicitly describe the small-scale turbulent mixing and dissipation effects in the ocean-scale ISWs. In total, nine (three γ and three α) runs were carried out for all cases. The spatial resolution was 4.3×1.2×1.2 m for all cases. A bottom-following sigma coordinate vertical system was used in the present model. A quasi-two-dimensional model with a resolution of 4 nodes across a wave tank with a resolution of 4200×250 nodes was used for the present calculations. No-slip boundary conditions were applied at the bottom and two end walls. Free-slip conditions were applied at the side walls. A mode-splitting technique and the decomposition of pressure and velocity fields on the hydrostatic and nonhydrostatic components were used in the numerical method; this is described in detail in .

The model was initialized using the iterative solution of the Dubreil-Jacotin–Long (DJL) equation with the initial guess obtained from a weakly nonlinear theory. The “DJLES” spectral solver package (https://github.com/mdunphy/DJLES/, last access: 19 February 2018) in MATLAB was used. The transformation of an ISW with an initial amplitude of |ain|=50 m is shown in Fig. . The minimum depth of the lower layer over the shelf is h2+=20 m, and the slope is γ=1∘. The parameters are α=1, β=0.4, and γ=1∘ and correspond to the regime of breaking with changing polarity. The ISW propagation velocity is about 1.2 m s-1, which is typical for the Lufeng region in the SCS . Using Eqs. () and (), we could find the location on the slope where the ISW would change polarity (h2+=h1=50 m), and we found ISW breaking at the place where hbr≈40 m. It can be seen from Fig.  that ISW |ain|=50 m first changes its polarity at time t=2 h 30 min and then breaks at the slope at t=2 h 40 min.

In Fig. a, a three-dimensional diagram of regimes with the cross section αβ for γ=1∘ is shown. In Fig. b, the red line corresponds to the polarity change criterion (), and the black line corresponds to the breaking criterion (). Three experiments are also marked in panel (b): exp.1 – α=0.4, β=1, γ=1∘; exp.2 – α=1, β=0.4, γ=1∘; and exp.3 – α=2, β=0.2, γ=1∘. The first experiment, exp.1, represents cases of interaction of the ISW (|ain|=50 m) with polarity change but without wave breaking. The second experiment, exp.2, represents cases in which the ISW first changes its polarity from a depression- to an elevation-type wave and then breaks (Fig. ). The final experiment, exp.3, represents cases in which the ISW breaks on the slope before it passes the changing polarity point.

An important characteristic of the wave–slope interaction is the loss of kinetic and available potential energy during the transformation. Energy transformation due to mixing leads to the transition of energy to background potential energy and then to energy dissipation. This can be estimated based on the budget of the wave energy before and after the transformation.

A calculation of energy dissipation was carried out for two configurations: (1) a real-scale experiment for the idealized Lufeng region in the SCS and (2) a laboratory-scale experiment with a trapezoid shelf-slope configuration . The parameters of ISWs from the laboratory-scale numerical experiments by are given in Table 3.

The characteristics of the incoming and reflected wave were recorded in the cross sections xr, which are located near the foot of the slope, and the wave passing on the shelf was recorded in the cross section xl (Fig. b, c).

Energy loss from breaking waves was estimated following and from the budget of depth-integrated pseudoenergy. To find the balance of the total energy, we have calculated the total energy of the incident, reflected, and transmitted waves before the slope and on the plateau using the depth-integrated pseudoenergy flux F(x,t): 8F(x,t)=∫-H0EPSE+pUdz, where p is pressure disturbance due to a passing wave; U represents the horizontal velocities; and EPSE is the pseudoenergy density, which is the sum of the kinetic energy density Ek and the available potential density Ea (part of the potential energy available for conversion into kinetic energy). For the calculation of Ea, we used a reference density profile that was obtained by an adiabatic rearranging of the density field. Volume integration of these flows outside of the mixing zone then allows us to estimate the energy of the incoming PSEin waves, the reflected PSEref waves, and the transmitted PSEtr waves on the plateau. 9PSEin=∫xrL∫-H0EPSEdzdx=-∫t1t2F(xr,t)dt,PSEtr=∫0xl∫-H0EPSEdzdx=-∫t3t4F(xl,t)dt,PSEref=∫xlL∫-H0EPSEdzdx=∫t5t6F(xr,t)dt, where t1, t2, t3, t4, t5, and t6 are the time intervals at which incoming, reflected, and transmitted waves pass the given cross section.

The relative estimation of the energy loss (δEloss) is then given by 10δEloss=(PSEin-PSEtr-PSEref)/PSEin, where PSEin is the pseudoenergy of the incident wave, and PSEtr and PSEref​​​​​​​ are the pseudoenergy of transmitted and incident waves, respectively. The energy loss from mixing during the interaction of the wave with the slope of δEloss (%) from the blocking parameter β is shown in Fig. a. This field of values, γβ, is built by 39 numerical experiments described in Table 3, 37 numerical experiments from for γ=90∘, and 9 experiments from the present study. δEloss was estimated for a wide the range of slopes (0.5∘<γ<90∘) and blocking parameters (-2<β<8). ISW energy loss for the limiting case of an underwater step when γ=90∘ was compared with the results of laboratory experiments by and (Fig. b). It can be seen that wave transformation in Zone 4 is the most dissipative. With this type of transformation, energy losses reach up to 55 %. For slopes in the range of 5∘<γ<90∘, the dependence of the energy dissipation on the blocking parameter β has almost the same pick shape as in the limiting case γ=90∘. For mild slopes of γ, we expect an increase in dissipation for all ranges of blocking parameter values of β.

We can compare the energy dissipation for a real-scale experiment with a laboratory-scale experiment with similar values of α and β and a slope of γ≈1. Considering cases with α=1 and β=0.4 for a slope of γ=1∘ (real scale experiments in Table 2) and α=1 and β=0.3 for a slope of γ=1.5∘ (laboratory-scale experiments in Table 3) (Zone 4 – wave breaking regime with polarity change), the difference is about 5 % (δElossrs=62 % and δElossls=57 %) for strong mixing.

To build a zone map for the shelf zone, the direction of the propagation of internal waves, the amplitude of incoming waves, and the stratification should be defined. These parameters could be found using the approach for estimating the geographic location of high-frequency nonlinear internal waves from , the amplitudes of the incoming internal waves, and the depth of the mixed layer. Figure  shows an example of a map with zones corresponding to the different regimes of interaction described above. These maps were constructed for the case of internal waves with an amplitude of ain=50 m and a mixed layer depth of h1=50 . On this map, the black line​​​​​​​ is the 120 m isobath (shelf), the violet line is the polarity change curve h1=h2, and the red area is the zone of internal wave breaking (where h1+hbr>H).

A three-dimensional α β γ classification diagram describing four types of ISW interaction with the slopes is discussed. Relations between the α, β, and γ parameters for each regime were obtained using the empirical relation for wave breaking conditions and weakly nonlinear theory for the criterion of the changing the polarity of the wave. The distinguished regimes are as follows: (1) the ISW propagates over a slope without changing polarity and wave breaking, (2) the ISW changes polarity over a slope without wave breaking, (3) the ISW breaks over a slope but without changing polarity, and (4) the ISW both breaks and changes polarity over a slope. The diagram is validated for realistic topography configurations. Numerical modeling of the idealized configuration that mimics the continental shelf in the Lufeng region (SCS) is carried out. The results of numerical experiments from the present study and from other laboratory experiments are in good agreement with the proposed classification and estimations. Based on present numerical experiments, internal solitary loss of wave energy from transformation over slope topography is estimated. We concluded that the results of field, laboratory, and numerical experiments are in good agreement with the proposed classification, which can be used for the identification of “hot spots” of energy dissipation in the ocean.