Internal Undular Bores in the Coastal Ocean

In the coastal ocean, large amplitude, horizontally propagating internal wave trains are commonly observed. These are long nonlinear waves and are often modelled by equations of the Korteweg-de Vries type, such as the variable-coefficient Korteweg-de Vries equation when the background topography varies as the waves propagate shoreward. Most emphasis has been placed on the solitary wave solutions of these model equations, whereas in reality, wave trains are more usually observed. In this review article we examine the undular bore asymptotic representation of wave trains in the framework of the variable-coefficient Korteweg-de Vries equation, placing a special emphasis on the front of the undular bore which can be represented by a simplified model as a solitary wave train. We consider applications for both propagation shorewards whenw nonlinearity increases, and for cases when the wave train passes through a critical point of polarity change, when the nonlinear coefficient in the Korteweg-de Vries equation changes sign.