Analytic and numerical studies of envelope solitons in geophysical flows

While not as widely discussed as long wave soliton models of atmospheric and oceanic phenomena envelope soliton solutions to the nonlinear Schrodinger equation have been proposed to describe events such as mid latitude planetary scale dipole blocks, Rossby wave breaking in the extratropical tropopause and equatorial Rossby-gravity ocean waves. Despite these applications, most studies have been limited to the case of constant zonal shear, and little work has been done on the effects of baroclinicity or variable shear on these weakly nonlinear structures. The purpose of this thesis is twofold. Firstly, numerical methods will be developed, based on the spectral element method, for representing geophysical flows in a hierarchy of systems of increasing physical complexity, from barotropic QG flows through to layered shallow water flows capable of representing baroclinic instability and ageostrophic effects. Particular emphasis will be given to splitting methods used to solve the different barotropic and baroclinic modes, and to the use of the rigid lid constraint for removing fast barotropic gravity waves. Parallel to this, weakly nonlinear perturbation analysis will be applied to investigate the evolution and stability of envelope solitons for quasi-geostrophic flows, with particular emphasis on the effects of variable shear, stratification and topography. For shear flows undergoing topographic and baroclinic instability for which the analysis cannot be applied, the numerical models developed here will be used to investigate the evolution of these instabilities.