| To better understand internal waves in the atmosphere and the ocean, both of which are density-stratified fluids, we develop and study two-dimensional models of internal waves between two layers of inviscid fluids bounded by top and bottom walls.;In the first part of this dissertation, we obtain a hydrostatic model from Euler equations by making the shallow-water assumption. The system of equations describing this model is of mixed type, i.e. hyperbolic or elliptic depending on its state. We determine a criterion under which the system starting from hyperbolic initial data does not become elliptic through its evolution up to wave breaking. This is accomplished by considering simple waves on the system's phase plane. Although we do not make the Boussinesq approximation---which ignores effects of density stratification in inertia terms but retains such effects in buoyancy terms---to derive this system, our formulation lends itself to such an approximation by an adjustment of a parameter in the system.;In the second part, we derive a nonhydrostatic model by relaxing the shallow-water assumption, which introduces dispersion into the resulting system of equations, and we also make the Boussinesq approximation to simplify the system. It is known that this system is mathematically ill-posed despite the fact that physically stable internal waves have been observed. We remedy this problem by an exchange of spatial and temporal derivatives in a dispersive term, which yields a region on the phase plane in which the system is stable. To see how closely out stable model approximates the unstable system, we compare their solitary waves. With background shear, we show that solitary waves may cross the mid-level between the top and the bottom walls. Furthermore, solitary waves of elevation type are possible when the lower fluid layer is thicker than the upper fluid layer and vice versa in the presence of sufficiently strong background shear. We also examine time-dependent solutions of the stable model, including interactions of its solitary waves. Finally, we derive classical and modified Korteweg-de Vries (KdV) equations for the unstable system and the stable model. |
