## A discussion on turbulent and undular bores using the models from the shallow water equations and the dispersive system.

##### Abstract

The bore is a wave phenomenon that occurs in channels and rivers of shallow water. Referred to as a discharge wave, a bore is generated by a sudden increase of water flow. Bores appear in certain rivers as the tide pushes water into the river mouth. Simply described a bore is a transition between two uniform flows of different flow depth. The point of transition is referred to as the bore front. The transition between the two states of flow is often marked by violent turbulence. But there are also bores in which no turbulence is observed. Such bores are called weak bores, as in these cases, the difference in height between the two flow depths are small. The weak bore displays a unique character not present in the turbulent bore. It carries a train of undulating waves just behind the front. For this reason it is often called the undular bore.
This thesis aims to give an introductory summary on the theory of the bore phenomenon and lay out a map for further study. In this way it ought to serve as a good introduction for those not familiar with the subject and a good repetition for those who are.
In the following investigation two models of the bore will be presented in which water is treated as an ideal fluid. The first model is reached by assuming hydrostatic pressure. This gives the shallow water equations. These equations model the bore as a travelling discontinuity separating two uniform flow depths. The water flow is steady, conserving mass and momentum, but the water loses energy as it passes through the bore front. This energy loss was first pointed out by Rayleigh in and is commonly referred to as the classical energy loss. The energy loss is a trait that coincides well with turbulent bores and the shallow water equations model these bores quite well.
The shallow water equations do not model the undular bores as they would not sustain the undulating waves behind the bore front. A second model is based on a dispersive system. This is an extension of the shallow water equations where, effectively, the treatment of pressure is refined. It leads to various Boussinesq systems and in a specialized case, of all fluid moving in one direction, it leads to the well know KdV equation. Modelling a bore-like initial value by a dispersive system brings out the undulations behind the bore front, however it does not capture the full nature of the undular bore itself.
In this thesis we support the conclusion that adding frictional effects of the boundary will improve the dispersive model of the undular bore. But the arguments leading to this conclusion will be debated.