This paper presents a horizontally two-dimensional theory based on a variable-coefficient Kadomtsev–Petviashvili equation, which is developed to investigate oceanic internal solitary waves propagating over variable bathymetry, for general background density stratification and current shear. To illustrate the theory, a typical monthly averaged density stratification is used for the propagation of an internal solitary wave over either a submarine canyon or a submarine plateau. The evolution is essentially determined by two components, nonlinear effects in the main propagation direction and the diffraction modulation effects in the transverse direction. When the initial solitary wave is located in a narrow area, the consequent spreading effects are dominant, resulting in a wave field largely manifested by a significant diminution of the leading waves, together with some trailing shelves of the opposite polarity. On the other hand, if the initial solitary wave is uniform in the transverse direction, then the evolution is more complicated, though it can be explained by an asymptotic theory for a slowly varying solitary wave combined with the generation of trailing shelves needed to satisfy conservation of mass. This theory is used to demonstrate that it is the transverse dependence of the nonlinear coefficient in the Kadomtsev–Petviashvili equation rather than the coefficient of the linear transverse diffraction term that determines how the wave field evolves. The Massachusetts Institute of Technology (MIT) general circulation model is used to provide a comparison with the variable-coefficient Kadomtsev–Petviashvili model, and good qualitative and quantitative agreements are found.

In the coastal ocean, large amplitude, horizontally propagating internal wave trains are commonly observed. These are long nonlinear waves and can be modelled by equations of the Korteweg-de Vries type. Typically they occur in regions of variable bottom topography when the variable-coefficient Korteweg-de Vries equation is an appropriate model. Of special interest is the situation when the coefficient of the quadratic nonlinear term changes sign at a certain critical point. This case has been widely studied for a solitary wave, which is extinguished at the critical point and replaced by a train of solitary waves of the opposite polarity to the incident wave, riding on a pedestal of the original polarity. Here we examine the same situation for an undular bore, represented by a modulated periodic wave train. Numerical simulations and some asymptotic analysis based on Whitham modulation equations show that the leading solitary waves in the undular bore are destroyed and replaced by a developing rarefaction wave supporting emerging solitary waves of the opposite polarity. In contrast the rear of the undular bore emerges with the same shape, but with reduced wave amplitudes, a shorter overall length scale and moves more slowly. 1 Variable-coefficient Korteweg-de Vries equation Large amplitude internal wave trains are commonly observed in the coastal ocean, see the reviews

A study of the propagation of a mode-2 internal solitary wave over a slope-shelf topography is presented. The methodology is based on a variable-coefficient Korteweg–de Vries (vKdV) equation, using both analysis and numerical simulations, and simulations using the MIT general circulation model (MITgcm). Two configurations are considered. One is a mode-2 internal solitary wave propagating up the slope, from one three-layer system to another three-layer system. Depending on the height of the shelf, which determines the variation of the nonlinear coefficient of the vKdV equation, this can be classified into two cases. First, the case of a polarity change, in which the coefficient of the quadratic nonlinear term changes sign at a certain critical point on the slope, and second, the case with no such polarity change. In both these cases there is a small transfer of energy from the mode-2 wave to mode-1 waves. The other configuration is when the lower layer in the three-layer system goes to zero at a transition point on the slope, and beyond that point, there is a two-layer fluid system. A mode-2 internal solitary wave propagating up the slope cannot exist past this transition point. Instead it is extinguished and replaced by a mode-1 bore and trailing wave packet which moves onto the shelf.

Based on a variable-coefficient Kadomtsev–Petviashvili (KP) equation, the topographic effect on the wave interactions between two oblique internal solitary waves is investigated. In the absence of rotation and background shear, the model set-up featuring idealised shoaling topography and continuous stratification is motivated by the large expanse of continental shelf in the South China Sea. When the bottom is flat, the evolution of an initial wave consisting of two branches of internal solitary waves can be categorised into six patterns depending on the respective amplitudes and the oblique angles measured counterclockwise from the transverse axis. Using theoretical multi-soliton solutions of the constant-coefficient KP equation, we select three observed patterns and examine each of them in detail both analytically and numerically. The effect of shoaling topography leads to a complicated structure of the leading waves and the emergence of two types of trailing wave trains. Further, the case when the along-crest width is short compared with the transverse domain of interest is examined and it is found that although the topographic effect can still modulate the wave field, the spreading effect in the transverse direction is dominant.

Tides are the dominant hydrodynamic processes in most continental shelf seas and have been proven to have a significant impact on both marine ecosystem dynamics and biogeochemical cycles. In situ and satellite observations have suggested that the spring-neap tide results in fluctuations of chlorophyll-a concentrations (Chl-a) with a fortnightly period in some shelf waters. However, a large number of missing values and low observation frequency in satellite-observed Chl-a have been recognized as the major obstacle to investigating the regional pattern showing where and to what extent of the effects of spring-neap tide on Chl-a and the seasonal variations in the effects within a relatively large region. Taking Himawari-8 as an example, a simple algorithm appropriate for geostationary satellites was proposed in this study with the purpose of obtaining a tide-related daily climatological Chl-a dataset (TDCD) and to quantitatively estimate the effects of the spring-neap tide on Chl-a variations. Based on the Chl-a time series from TDCD, significant fortnightly signals of Chl-a fluctuations and high contribution together with high explanations of the fortnightly fluctuations for Chl-a variations were found in some specific inshore waters, especially in the East China Sea, Bay of Bengal, South China Sea, and northern Australian waters. The spring-neap tide was found able to induce the spatio-temporal fortnightly fluctuations of Chl-a with an annual amplitude of 12–33% of the mean in these inshore areas. Significant seasonal variations in the fortnightly fluctuation of Chl-a were observed in the temperate continental shelf regions, while levels remained relatively stable in the tropical waters. Further analysis implied that the spatio-temporal fortnightly fluctuations of Chl-a were closely associated with the tidal current differences between the spring and neap tides. Seasonal variations in the tidal current differences were found to be a key driving factor for seasonal fluctuations of the spring-neap tidal effects on Chl-a in the temperate continental shelf regions. This study provides a better understanding of tide-related marine ecosystem dynamics and biogeochemical cycles and is helpful in improving physical–biogeochemical models.