Penland, C., 1996: A stochastic model of IndoPacific sea surface temperature anomalies. Physica D, 98, 534-558. Reprinted in Nonlinear Phenomena in Ocean Dynamics, D. D. Holm, R. C. Malone, and L. G. Margolin (Eds.), Elsevier, 534-558.
It is often desirable to represent a rapidly varying physical process as stochastic forcing of some slower dynamical system. A review of this approximation is presented. The white-noise limit of rapidly varying processes reduces the dynamical description of the affected slower systems to stochastic differential equations, the properties of which are summarized.
As an explanation of stochastic differential equations, we review the evidence that IndoPacific sea surface temperature anomalies (SSTAs) can be represented as a stable linear process driven by spatially coherent stochastic forcing. An "inverse-modeling" approach is used. That is, the relevant parameters of the best-fit stable linear process are obtained from observations and, given these parameters, the assumptions of stability and linearity are subsequently tested. An "optimal initial structure" for growth, predicted by the model, is found to occur approximately seven months before most of the major warm and cold (El Niño/La Niña) events in the data record. The optimal structure preferentially occurs in the boreal spring, with the mature phase of the extreme event occurring in the subsequent late fall/winter.
The best model to fit the observations, including the dependence of the dynamical system on the annual cycle, is driven by stochastic forcing with periodic statistics. These statistics are inferred using a time-dependent fluctuation-dissipation relation. The variation of the stochastic forcing with the annual cycle is not the same as that of the optimal initial structure. Yet, when a linear numerical model is driven with the derived stochastic forcing, the optimal structure, including its variation with the annual cycle, is recovered. This implies that deterministic dynamics play a role in setting up the optimal structure and, therefore, a mature phase of El Niño may be predicted before the optimal structure actually appear.