A STUDY ON THE NONLOCAL SHALLOW-WATER MODEL ARISING FROM THE FULL WATER WAVES WITH THE CORIOLIS EFFECT

Abstract

The Equatorial Undercurrent is a significant feature of the geophysical waves near the equator, which is one of the key factors to explain El Niño phenomenon. However, based on β-plane approximation, the classical theory of geophysical waves ignored the vertical structure of the Equatorial Undercurrent. To obtain a better description of the equatorial waves, in this dissertation, I study the rotational-Camassa-Holm (R-CH) equation, which is a mathematical model of long-crested water waves near the equator, propagating mainly in one direction with the effect of Earth's rotation under the f-plane approximation. R-CH equation can be derived by following the formal asymptotic procedures. Such a model equation is analogous to the Camassa-Holm approximation of the two-dimensional incompressible and irrotational Euler equations and has a formal bi-Hamiltonian structure. Its solutions corresponding to physically relevant initial perturbations is more accurate on a much longer time scale. It is shown that the deviation of the free surface can be determined by the horizontal velocity at a certain depth in the second-order approximation. The effects of the Coriolis force caused by the Earth rotation and nonlocal higher nonlinearities on blow-up criteria and wave-breaking phenomena are also investigated.