| Coupled ocean-atmosphere models are complex due to the large number of modes. To understand the predictability and dynamics of such large systems, lower dimensional projection on a subset of variables can be used to gain insight about the full model. Using certain simplifications, explicit analysis can be performed to explain the behavior of the reduced dynamics, where the neglected degrees of freedom are represented stochastically. Several techniques can be utilized to reduce the complexity of the models; in particular, the stochastic mode-reduction approach has been considered. Furthermore, due to the oscillatory behavior of the systems, mechanisms for the oscillations in predictability has been investigated and the approach of the non-equilibrium behavior to the equilibrium state is studied.;In the second half of this thesis, we apply the stochastic mode-reduction strategy to a particular class of prototype coupled ocean-atmosphere models, where self-interactions of the slow variables are given by a rotationally invariant gradient system. The problem addresses the interaction of coherent structures with noise, where the diffusion/drift term in the reduced system contains information about the full dynamics of the system. The stochastic mode-reduction strategy is utilized to derive stochastic reduced models, which gives a simple description of the phenomena that occurs from breaking the original rotational symmetry. The direction of the symmetry breaking can be predicted a priori without any information about the statistical behavior of the fast modes. Furthermore, we show a connection of the full and the mode-reduced system using the notion of predictability from the first part of thesis.;In the first half of this thesis, we examine loss of predictability in two-dimensional stochastic systems that have oscillatory behavior. We show that the information provided by an initial distribution about the state of the system decays to zero non-uniformly in time. In particular, the oscillatory behavior of the systems is responsible for the non-uniformities in predictability. Furthermore, the system as a whole will loose information, but on a subset of variables information can be gained. This return of information will lead to the notion of "return of skill". Marginal distributions will be used to study this increase in information. |
