In this study we investigate the flow of a Boussinesq fluid contained in a rotating, differentially heated spherical shell. Previous work, on the spherical shell of Boussinesq fluid, differentially heated the shell by prescribing temperature on the inner boundary of the shell, setting the temperature deviation from the reference temperature to vary proportionally with -cos 20, from the equator to the pole. We change the model to include an energy balance equation at the earth's surface, which incorporates latitudinal solar radiation distribution and ice-albedo feedback mechanism with moving ice boundary. For the fluid velocity, on the inner boundary, two conditions are considered: stress-free and no-slip. However, the model under consideration contains only simple representations of a small number of climate variables and thus is not a climate model per se but rather a tool to aid in understanding how changes in these variables may affect our planet's climate. The solution of the model is followed as the differential heating is changed, using the pseudo arc-length continuation method, which is a reliable method that can successfully follow a solution curve even at a turning point. Our main result is in regards to hysteresis phenomenon that is associated with transition from one to multiple convective cells, in a dfferentially heated, co-rotating spherical shell. In particular, we find that hysteresis can be observed without transition from one to multiple convective cells. Another important observation is that the transition to multiple convective cells is significantly suppressed altogether, in the case of stress-free boundary conditions on the fluid velocity. Also, the results of this study will be related to our present-day climate. / UOIT

O objetivo deste trabalho é construir rigorosamente variedades de soluções definidas implicitamente por equações não-lineares em dimensão infinita. Usando um método de continuação a múltiplos parâmetros aplicado a uma projeção em dimensão finita, uma triangulação da variedade é construída e usada para construir localmente a variedade no espaço de dimensão infinita. Aplicamos este método para encontrar equilíbrio da equação de Cahn-Hilliard. Estudamos também bifurcações cúspides, com o objetivo de encontrar as condições necessárias para a existência das mesmas em qualquer dimensão finita. / The goal of this research is to rigorously compute implicitly defined manifolds of solutions of infinite dimensional nonlinear equations. Using a multi-parameter continuation method on a finite dimensional projection, a triangulation of the manifold is computed and is then used to construct local charts of the global manifold in the infinite dimensional domain of the operator. We apply this method to find the equilibria of the Cahn-Hilliard equation. We also studied cusp bifurcations, in order to find the necessary conditions for the existence of the same in any finite dimension.

O objetivo deste trabalho é construir rigorosamente variedades de soluções definidas implicitamente por equações não-lineares em dimensão infinita. Usando um método de continuação a múltiplos parâmetros aplicado a uma projeção em dimensão finita, uma triangulação da variedade é construída e usada para construir localmente a variedade no espaço de dimensão infinita. Aplicamos este método para encontrar equilíbrio da equação de Cahn-Hilliard. Estudamos também bifurcações cúspides, com o objetivo de encontrar as condições necessárias para a existência das mesmas em qualquer dimensão finita. / The goal of this research is to rigorously compute implicitly defined manifolds of solutions of infinite dimensional nonlinear equations. Using a multi-parameter continuation method on a finite dimensional projection, a triangulation of the manifold is computed and is then used to construct local charts of the global manifold in the infinite dimensional domain of the operator. We apply this method to find the equilibria of the Cahn-Hilliard equation. We also studied cusp bifurcations, in order to find the necessary conditions for the existence of the same in any finite dimension.