Estimate of energy loss from internal solitary waves breaking on slopes

Internal solitary waves (ISW) emerge in the ocean and seas in different forms and break on the shelf zones in a variety of ways. Their breaking on slopes can produce intensive mixing that produces such process as biological productivity and sediment transport. Mechanisms of ISW of depression interaction with the slopes related to breaking and changing polarity as they shoal. We assume that parameters that described the process of interaction of ISW in a two-layer fluid with the idealised 5 shelf-slope are: the non-dimensional wave amplitude α (wave amplitude normalized on the upper layer thickness), the ratio of the height of the bottom layer on the shelf to the incident wave amplitude β and angle γ. Based on three-dimensional αβγ classification diagram with four types of interaction with the slopes it was discussed: (1) ISW propagates over slope without changing polarity and wave breaking; (2) ISW changes polarity over slope without breaking;(3) ISW breaks over slope without changing polarity; (4) ISW both breaks and change polarity over the slope. Relations between the parameters 10 α,β,γ for each regime were obtained using the empirical condition for wave breaking and weakly nonlinear theory for the criterion of changing the polarity of the wave. In the present paper the α,β,γ diagram was validated for idealised real scale topography configurations. Results of the numerical experiments that were carried out in the present paper and results of field and laboratory experiments from other papers are in good agreement with proposed classification and estimations. Based on 85 numerical experiments ISWs energy loss during interaction with slope topography with 0.5 < γ < 90 was estimated. “Hot 15 spots” zones of high levels of energy loss were shown for idealized configuration that mimics continental shelf at Lufeng Region SCS.

waves, internal waves break in the coastal zone of the ocean. Such waves are an important components of mixing and energy dissipation (Liu (1998), Davis (2020)). Depending on the characteristics of the traveling ISW, ocean stratification and characteristics of the continental slope wave on the shelf could be shoaling in different forms. ISW often reach extreme amplitudes, for example in the South China Sea (SCS) (Klymak (2006)). They are well-known contributors to material transport, nutrient supply (Wang (2007)). ISWs in SCS are generated by the interaction of internal tides and the system of two quasi-parallel 5 ridges in the Luzon Strait (Alford (2015)). Such waves could propagate at vast distances toward Dongsha Atoll mainly in the form of mode-1 internal solitary waves where they shoal and dissipate at the shallow continental shelf (Liu (1998)).
During wave shoaling two mechanisms were expected to be essential: (i) wave breaking resulting in mixing and (ii) changing of the polarity of the initial wave of depression over the slope into a wave of elevation on the shallow zone. The conversion of the internal wave of depression into elevation waves in a two-layer stratification is expected if the thickness of the mixed 10 layer is greater than one-half the total water depth (Helfrich (1986), Cheng (2011)). Different types of the breaking locations are commonly distinguished by slope inclination, water column stratification, and wave characteristics (Vlasenko (2002), Aghsaee (2010), Boegman (2005)). A simple three-dimensional αβγ classification diagram was proposed by (Terletska (2020)) to distinguish different regimes of transformation of ISW over the shelf-slope topography. It is based on three parameters: the slope angle γ, the non-dimensional wave amplitude α (wave amplitude normalized on the upper layer thickness), and the 15 blocking parameter β that is the ratio of the height of the bottom layer on the shelf to the incident wave amplitude.
The present study focused on ISWs transformation over idealised shelf-slope topography with a two-layer stratification.
The objectives are to: (1) compare αβγ classification with the results of numerical modelling, laboratory studies , and field observations, (2) find high energy dissipation zones of ISWs passed over the shelf-slope topography based on regimes classification that includes such parameters as wave amplitude, stratification and slope, (3) for αβγ classification carry out numerical 20 modeling for idealized configuration that mimics continental shelf at Lufeng Region SCS, (4) determine energy loss of the transformation of ISWs with the shelf-slope topography.
Information about polarity change criteria and criteria of breaking in αβγ classification of regimes of ISWs transformation over shelf topography is presented in the corresponding Section. The overview of field and laboratory measurement data and their comparison with numerical modelling data are in Section "Data and methods". Dissipation of energy of ISWs breaking 25 over shelf topography discussed in Section "Estimate of energy loss in internal waves breaking on slopes". The results are summarized in the Conclusions.

Regimes of ISW transformation over slope-shelf topography
The two-layer approximation is a simple model of the stably stratified oceans and lakes. In this model we approximated stratification by the two continuous layers of depths h 1 (upper layer) and h 2 (lower layer) with relatively thin pycnocline. In 30 the case when h 1 > h 2 internal solitary waves propagate in the form of elevation ISW, and if h 1 < h 2 in the form of the waves of depression. In this study we will consider ISW of depression (with amplitude a in ) propagating over an idealized slope-shelf with a slope γ and the minimum depth of the lower layer over the shelf h 2+ . Idealised shelf-slope topography is shown in Fig.   1 b, and the idealised configuration that mimics continental shelf at Lufeng Region ( Fig. 1 a) SCS is shown in Fig. 1 c . It could be assumed that transformation on a slope is controlled by stratification, slope inclination and amplitudes (wavelength) of the incident wave (Terletska (2020)). They determine two possibilities that could occur with the wave during shoaling: (i) ISW breaking, that is associated with gravitational instability due to shear instability and wave overturning and (ii) 5 depression type ISW changing the polarity on the slope. There are different breakers behaviour of ISW in a two-layer fluid over slope were classified by (Aghsaee (2010), Boegman (2005)) into four types: surging, plunging, collapsing, and fission.
Classification is based on the ratio of the wave slope and the bottom slope, and was modified for collapsing and plunging breakers with the use of new wave Reynolds number that taking into account nonlinear wave steepening [Nakayama (2019)].
According the (Terletska (2020)) three parameters α β γ can be important for behaviour of the incident wave on slope-shelf ( fig. 1 b,c): 1. slope inclination γ (measured as an angle); 2. blocking parameter β is the ratio of the height of the minimum depth of the lower layer over the shelf h 2+ (Fig. 1 (1) This idea comes from numerical and laboratory experiments as the "degree of blocking" that is an important parameter that controls the loss of energy into transmitted and reflected waves passing the obstacle (Vlasenko (2002), Wessels (1996). Parameter B was modified in (Talipova (2013)) where considered ISWs (as depression and elevation type) passing over underwater step γ = 90 o . It was shown that for the case of ISW of depression for values β > 3 the ISW 10 transformation over underwater step is weak (when the dynamics of ISW could be described by weakly nonlinear theory), for 2 < β < 3 interaction is moderate (when the main mechanism for ISW breaking over bottom step produce shear instability), for 0.4 < β < 2 interaction is strong with maximal energy loss (when ISW produced the flow that results in a jets and vortices), −0.9 < β < 0.4 so-called 'transitional regime' (when the step height between strong interaction and full reflection from the step) and for β < −0.9 full reflection from the underwater step. 15 3. nonlinearity parameter that is the ratio of the incident wave amplitude to the depth of upper layer: Internal waves in a framework of weakly nonlinear theory change their polarity in the point where the upper and lower layers are equal (Grimshaw (2004)). Notice that numerical experiments using full Naiver-Stokes equations (Maderich (2010)) confirm the applicability of the Gardner equation to predict turning point h 1 = h 2 even for the wave of large amplitude. This 20 relation for turning point can be expressed through parameters and using For breaking point criteria was taken criterion from Vlasenko (2002). It was build based on the Naiver-Stokes numerical model simulations data. It was found that the ratio of the amplitude of the incident wave a in into the value of the undisturbed thickness of the lower layer in point where the wave breaking takes place h br (Fig. 1 b,c), is the function of the slope γ: For each slope angle γ the blocking parameter value β br that divide zone of non-breaking regime for β > β br and breaking regime for β < β br can be found from (3) and (4) We can also obtain value α br that divide zone 4 on breaking regime when ISW first breaks α > α br and when wave first change polarity and then breaks α < α br . It can be found from(3) using (5) that yields Thus four different scenarios of internal solitary wave interaction of with a shelf-slope topography in a two layer approximation can be realized: (1) breaking regime over the slope-shelf; (2) non-breaking regime over the slope-shelf, (3) polarity change regime and (4) regime of non-polarity change.
Analysing equations (3) (5) we conclude that parameters α, β, γ control the processes as the wave breaking as the wave 10 polarity change. A three-dimensional diagram with the dependence on parameters α, β, γ (αβγ diagram) is introduced in Fig. 2 with four zones: (zone 1) ISWs transform without changing polarity and wave breaking; (zone 2) ISWs transform with changing polarity without breaking; (zone 3) ISWs break without changing polarity; (zone 4) ISWs break with changing polarity. In the space of α, β, γ these regimes are separated by the surfaces (3)and (5).   (2020)) was shown that results of field observations (Moum (2003), Osborn (1980), Orr (2003), Nam (2010), Fu (2016)), laboratory experiments (Helfrich (1986), Cheng (2011)) and numerical experiments that simulate ISW transformation in laboratory scales (Talipova (2013), Terletska (2020)) are in good agreement with proposed classification. All data were identified as belonged to corresponding diagram domain. km, and maximal depth H max = 300 m was considered (Fig. 1 a). We approximated stratification in the Lufeng Region by the two-layer density profile. The densities of the layers are ρ 1 and ρ 2 (depths h 1 and h 2 and H = h 1 + h 2 .) and the pycnocline layer thickness dh :  The numerical simulations were carried out using the free-surface non-hydrostatic numerical model Kanarska (2003),Maderich (2012) as a nonhydrostatic extension of the Princeton Ocean Model (POM). The Smagorinsky model extended for the stratified fluid (Siegel (1994)) was used to explicitly describe the small-scale turbulent mixing and dissipation effects in the ocean scale NISW. Totaly 9 (three γ and three α ) runs with resolution of the computational domain 42002504 were carried out for  (2003)) and initial density profile (7) used for numerical calculations all cases. Spatial resolution was 4.31.21.2 m for all cases. Bottom following, sigma coordinate vertical system was used in present modeling. The quasi-two-dimensional model with a resolution of 4 nodes across the wave tank was used for present calculations. No-slip boundary conditions were applied at the bottom and two end walls. The free-slip conditions were applied at the side walls. Mode-splitting technique and decomposition of pressure and velocity fields on hydrostatic and nonhydrostatic components were used in numerical method and it is described in detail in (Maderich (2012)).

Estimate of energy loss in internal waves breaking on slopes
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 energy to background potential energy and to the energy dissipation. It can be estimated based on the budget of the wave energy before and after the transformation.
It was carried out calculating of energy dissipation for two configurations -for real scale experiment for idealised Lufeng 5 Region in SCS and laboratory scale experiment with the trapezoid shelf-slope configurations (Terletska (2020)). Parameters of ISW in numerical experiments of laboratory scale experiments (from (Terletska (2020)) are given in table 2.
The characteristics of the incoming and reflected wave were recorded in the cross-sections x r , which are located near the foot of the slope, and the wave passing on the shelf, in the cross-section x l (Fig. 1 b,c). The sum of kinetic and available potential that is the pseudoenergy energy was calculated by integration in time of the energy fluxes of these waves at the sections x l (of 10 transmitted waves) x r (of incident and reflected waves). The detailed method of estimation of available potential energy was discussed in (Talipova (2013)). The relative estimation of the energy loss δE loss is given by where P SE in -is pseudoenergy of incident wave, P SE tr and P SE ref -is pseudoenergy of transmitted and incident ones respectively. The energy loss for mixing during the interaction of the wave with the slope of δE loss (%) from the blocking 15 parameter β is shown in fig. 6 a. This field of values γβ is built by 39 numerical experiments described in table 2, 37 numerical experiments from (Talipova (2013)) for γ = 90 o and 9 experiments from the present study. δE loss was estimated for the wide the range of slopes 0.5 o < γ < 90 o and blocking parameters −2 < β < 8. ISW energy loss for limiting case of underwater step when γ = 90 o was compared with the results of laboratory experiments (Wessels (1996)) and (Chen (2007)) 6 b. It can be seen that interaction in zone 4 is the most dissipative. With this type of interaction, energy losses reach up to 55%. For slopes from 20 5 o < γ < 90 o dependence of the energy dissipation from the blocking parameter β has almost the same pick shape as in the limiting case γ = 90 o . For mild slopes γ we will expect increase of dissipation for all range of blocking parameter values β.
We can compare the energy dissipation for real scale experiment with the laboratory scale experiment with a close values of α and β for slope γ ≈ 1. Consider the cases with α = 1 and β = 0.4 for slope γ = 1 • (real scale from table 1 ) and α = 1 and  Wessels (1996) and Chen (2007)  To build a zone map for the shelf zone the direction of propagation of internal waves is necessary. It could be found using approach for estimating the geographic location of high-frequency nonlinear internal waves from (Jackson (2012)), amplitudes of the incoming internal waves, the depth of the mixed layer. Fig. 7 shows an example of a map with zones corresponding to the 5 different regimes of interaction described above. These maps were constructed for the case of internal waves with an amplitude of a in = 50 m and a mixed layer depth h 1 = 50 (Junmin (2003)). On this map the black line -is isobath 120 m (shelf), the violet line is polarity change curve h 1 = h 2 and the red area -is the zone of internal wave breaking where h 1 + h br > H.

Conclusions
A three-dimensional αβγ classification diagram with four types of interaction of ISWs with the slopes was discussed. Relations 10 between the parameters α, β, γ for each regime were obtained using the empirical condition for wave breaking and weakly nonlinear theory for the criterion of changing the polarity of the wave. The regimes are: (1) ISW propagates over slope without changing polarity and wave breaking; (2) ISW changes polarity over slope without breaking;(3) ISW breaks over slope but without changing polarity; (4) ISW both breaks and change polarity over a slope. The diagram was validated for realistic topography configurations. Numerical modeling of the idealized configuration that mimics the continental shelf at Lufeng