Internal solitary waves (ISWs) emerge in the ocean and seas in various forms and break on the shelf zones in a variety of ways. This results in intensive mixing that affects processes such as biological productivity and sediment transport. As ISWs of depression propagate in a two-layer ocean, from the deep part onto a shelf, two mechanisms are significant: (1) the breaking of internal waves over bottom topography when fluid velocities exceed the wave phase speed that causes overturning of the rear face of the wave, and (2) the changing of polarity at the turning point where the depths of the upper and lower layers are equal. We assume that the parameters that describe the process of the interaction of ISWs in a two-layer fluid with an idealized shelf-slope topography are (1) the nondimensional wave amplitude, normalized on the upper-layer thickness; (2) the ratio of the height of the bottom layer on the shelf to the incident wave amplitude; and (3) the angle of the bottom inclination. Based on a proposed three-dimensional classification diagram, four types of wave propagation over the slopes are distinguished: the ISW propagates over the slope without changing polarity and wave breaking, the ISW changes polarity over the slope without wave breaking, the ISW breaks over the slope without changing polarity, and the ISW both breaks and changes polarity over the slope. The energy loss during ISW transformation over slopes with various angles was estimated using the results of 85 numerical experiments. “Hot spots” of high levels of energy loss were highlighted for an idealized bottom configuration that mimics the continental shelf in the Lufeng region in the South China Sea.

Observations demonstrate evidence of internal solitary waves (ISWs) in coastal oceans and seas

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

The interaction behavior of ISWs depends on the steepness of the topography and the characteristics of the solitary waves

ISWs breaking over slopes have been observed in many coastal locations worldwide

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

Sketch of the transformation of a depression ISW over a shelf slope. Panel

It was assumed that ISW transformation over a slope is controlled by stratification, slope inclination, and amplitudes (wavelength) of the incident wave

Three parameters,

the slope inclination

the blocking parameter

the nonlinearity parameter, which is the ratio of the incident wave amplitude to the depth of upper layer, calculated as

Internal waves in the framework of weakly nonlinear theory change their polarity at the point where the upper and lower layers are equal

For the breaking point, the criterion was taken from that proposed by

For each slope angle

We can also obtain the value of

Thus, four different scenarios of ISW interaction with the shelf-slope topography in a two-layer approximation can be realized: (1) a non-breaking regime without changing polarity, (2) a non-breaking regime with changing polarity, (3) a breaking regime without changing polarity, and (4) a breaking regime with changing polarity.

Analyzing Eqs. (

Three-dimensional

Parameters

To compare the

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

In numerical experiments, we vary the wave amplitudes (

ISW parameters from numerical experiments for the idealized Lufeng region in the SCS.

Density profile from measurements from the SCS (May)

The numerical simulations were carried out using a free-surface nonhydrostatic numerical model

The evolution of an ISW with

The model was initialized using the iterative solution of the Dubreil-Jacotin–Long (DJL)

In Fig.

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

ISW parameters from laboratory-scale numerical experiments

The characteristics of the incoming and reflected wave were recorded in the cross sections

Energy loss from breaking waves was estimated following

The relative estimation of the energy loss (

We can compare the energy dissipation for a real-scale experiment with a laboratory-scale experiment with similar values of

Zone map for internal waves' transformation over the South China Sea shelf with an initial amplitude of 50 m and a depth of the mixed layer of 50 m.

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

A three-dimensional

The output files for all of the numerical experiments reported in the paper are available from the corresponding author upon reasonable request.

KT and VM conceived the idea for the study; KT also carried out the numerical simulations, contributed to the design of figures, and participated in writing the paper. VM contributed to writing the paper and the interpretation of the results.

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

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

This article is part of the special issue “Nonlinear internal waves”. It is not associated with a conference.

This paper was edited by Zhenhua Xu and reviewed by two anonymous referees.