Contents
1 ENSO 7
1.1 An overview on ENSO process . . . 7
1.2 The mean state of the tropical Pacific region . . . 8
1.3 El Niño event . . . 12
1.4 La Niña event . . . 13
1.5 ENSO characteristics . . . 16
2 From a deterministic system to a generalized Fokker-Planck equation 19 2.1 The projective method . . . 19
2.2 The Zwanzig projection . . . 20
2.3 Correlation function and booster response to perturbations . . . 22
2.4 The generalized Fokker-Planck equation . . . 24
3 The projective method applied to the LOM model 27 3.1 LOM and ROM models . . . 27
3.2 The reference model . . . 28
3.3 The projective approach applied to the ROM system . . . 29
3.4 Equivalent forms of the Fokker-Planck equation . . . 31
3.5 Time evolution of the first and second moment of the distribution . . . . 32
3.6 The ansatz . . . 35
4 Numerical methods 37 4.1 Integrators for parabolic partial differential equation . . . 37
4.1.1 Notation and the von Neumann Linear Stability Analisys . . . 37
4.1.2 The ADI method . . . 39
4.2 Integrator for stochastic differential equations (SDE) . . . 41
4.2.1 The Heun scheme . . . 43
4.2.2 The issue of finite sampling in proximity of the barriers . . . 43
5 The stationary solution and checks on the ansatz 47
5.1 Discretization of the FPE and the integrator setup . . . 47
5.2 The stationary solution . . . 51
5.3 Time evolution of the moments . . . 53
5.4 Checks on the anstaz . . . 58
6 The mean first passage time and average duration of strong El Niño events 63 6.1 SDE integrator setup . . . 64 6.2 The mean first passage time and average duration of strong El Niño events 66
