Fluid-Dynamic Models of Geophysical Waves

Geophysical waves are waves that are found naturally in the Earth\u27s atmosphere and oceans. Internal waves, that is waves that act as an interface between uids of dierent density, are examples of geophysical waves. A uid system with a at bottom, at surface and internal wave is initially considered. The system has a depth-dependent current which mimics a typical ocean set-up and, as it is based on the surface of the rotating Earth, incorporates Coriolis forces. Using well established uid dynamic techniques, the total energy is calculated and used to determine the dynamics of the system using a procedure called the Hamiltonian approach. By tuning a variable several special cases, such as a current-free system, are easily recovered. The system is then considered with a non- at bottom. Approximate models, including the small amplitude, long-wave, Boussinesq, Kaup-Boussinesq, Korteweg-de Vries (KdV) and Johnson models, are then generated using perturbation expansion techniques, that is using small arbitrary parameters. Solutions are obtained that model waves that move without change of form called solitary waves. These waves can be referred to as solitons when their particle-like behaviour is considered. The Johnson model is used to model the `birth\u27 of new solitons when a single soliton hits an underwater ramp. The presented models have applications for climatologists, meteorologists, oceanographers, marine engineers, marine biologists and applied mathematicians