Path integral, Fokker-Plank equation and transition matrices in climate dynamics

The Stochastic Ordinary and Partial Differential Equations turn out to be a very important tool in the understanding and modeling the Climate System. The resolution of this kind of problems is far from trivial. The only meaningful quantities are those derived by ensemble mean over the noise. In order to address these issues, with a particular focus on ENSO for its relevance at global scale, three different methods of investigation are presented. The first method is inspired by the Statistical Mechanics and Quantum Field Theory. It is shown how it is possible to find a generating functional from this stochastic system, and how, using functional differentiation, to derive all the n-points functions of the system studied, in particular how to derive variance and correlation. Since there are problems due to boundaries conditions, typical problems related to the ocean dynamics, this methods has been used to study the Stochastic Barotropic Vorticity Equation in a channel with periodic boundaries. This equation has been studied in three different configurations: with or without the damping term and with damping and mean flow with profile depending only on the meridional coordinate. The second method is based on the resolution of the Fokker-Planck equation related to the stochastic system via eigenfunctions expansion. The ENSO model studied has been derived using a rotation in the space of variables of the Recharge Oscillator. It is suggested a new way to consider ENSO, as a system that can jump between two states, one positive and one negative, represented by the potential wells which arise by the non-linearity that damps the system. The jumps are possible thanks to the stochastic fluctuations. It is suggested a possible mechanism that could explain the asymmetry in the Sea Surface Temperature Anomalies probability density function in the ENSO zone. In particular, taking into account the MJO effect, the double well potential is modified becoming asymmetric producing an asymmetric probability density function for the anomalies. Using this model, exploiting a periodic growth rate, it is studied also the possible cause of the predictability barrier, another important feature of ENSO. The third method shows how the transition probability matrices can be used to deal climatic phenomena. In particular, these matrices have been used to define a predictability index of ENSO using their entropy. These matrices turned out to be a possible instrument to check models in respect to the observations. Not only the long-time seasonal PDF could be checked, but also the single transitions for different states in different periods.