Conceptual climate models plays an important role in the study and understanding of the Earth climate system. They are mostly used to get insight on the processes or mechanism of the climate system. Therefore a rigorous study of a conceptual model developed to study a certain phenomena is crucial. An example of conceptual models of climate is the PP04 model for climate change, in particular the behaviour of the ice-ages . This models the transition from a glacial to an inter-glacial state through a sudden release of oceanic Carbon Dioxide into the atmosphere. This process can be cast in terms of a Filippov dynamical system, with a discontinuous change in its dynamics. The work of this thesis is to perform a careful analysis of the PP04 model using the techniques from the theory of non-smooth dynamical systems. In particular, an analysis of this model is performed for cases of no forcing, periodic forcing and quasi-periodic forcing with both two and three modes. Earlier approaches to studying the PP04 model have used the theory of smooth dynamical systems. These approaches have limited the findings of the dynamics and bifurcations observed in this model. Therefore a thorough study of the model in a non-smooth dynamical systems framework is needed. This is the motivation for the research work in this thesis.
The thesis begins with a review on the development of simple conceptual models of climate and how dynamical systems theory was introduced in the construction of such models in showing response of ice sheets to astronomical forcing. A review on the literature of dynamical systems (both smooth and non-smooth systems) and in particular Filippov systems follows. The analyses of the PP04 model focuses on the presence of the discontinuity boundary due to sudden release of Carbon Dioxide. The results reveals that, the PP04 model has a rich and novel, dynamical structure. The PP04 model with no forcing has periodic solutions that are destroyedat discontinuity induced bifurcations. In addition, the periodically forced PP04model, exhibits synchronised periodic solutions with subtle regions of existence whichdepend on the amplitude and frequency of the forcing. The orbits can be created or destroyed in both smooth and discontinuity induced bifurcations, in particular a grazing bifurcation. The changes in orbital stability at the bifurcations show behaviour similar to that observed at the Mid-Pleistocene Transition. The quasi-periodically forced PP04 model exhibits quasi-periodic time solutions that resemble those observed in paleoclimate records. The solutions are observed to depend on the phase of the forcing with possibility of chaotic response when the amplitude of the forcing is increased. The study of orbits and the transitions between them is done for all three types of forcing considered and comparisons made with actual climate dynamics. We conclude by stating further work to be done.
|Date of Award||16 Sep 2020|
|Sponsors||Botswana International University of Science and Technology|
|Supervisor||Paul Milewski (Supervisor) & Chris Budd (Supervisor)|