| Two main areas of theoretical research in geophysical fluid dynamics—linearized instability and nonlinear equilibration—are addressed in this thesis. First, instability of equatorial Kelvin wave in shear flow is investigated. For a simple model eigenproblem it is shown both analytically and numerically that the eigenvalue has exponentially small positive imaginary part. It is then illustrated via numerical experiments that growth rate of the equatorial Kelvin wave in the system of shallow water equations on the equatorial β-plane exhibits the same type of behavior. Next, Hermite-Pade approximants are used to calculate growth rates of the model eigenproblem from the divergent perturbation series. In the second part of the thesis the nonlinear development of baroclinic instability in the two-layer quaisgeostrophic forced/dissipative model is studied numerically in strongly supercritical regime. It is found that in large domains the zonal extent of a localized initial disturbance will spread with time, but the wavy state left behind the leading edge will not be uniform, but will have modulated structure. For a zonally confined domain it is shown that for moderate supercriticality the flow will equilibrate to a stable uniform wave train, while for strong supercriticality the wave train is unstable. |
