This paper presents an analysis of a coupled ocean-atmosphere model used to study ENSO (El Nino-Southern Oscillation). Our interest here is in the growth of initial error: that is, the predictability of ENSO. The analysis proceeds by constructing a linear model that optimally fits the behavior of the original nonlinear coupled model. By construction, this approximate linear model has only a few degrees of freedom. Because the linear model is so much smaller than the original, it is possible to understand it in much finer detail, indirectly offering insight into the properties and behavior of the original model.As it turns out, even linear models with only a few degrees of freedom can have rather elaborate and surprising short-term error behavior. It has been shown that if a system is not self-adjoint that there is a possibility of error growth in a mode completely unrelated to the classic notion of a fastest growing linearly unstable mode. This holds for simple linear models as well. The work here explores error growth in simple non-self-adjoint systems, showing that the error growth is an inherent part of the system.Constructing a linear approximation to the coupled model gave us two major results: a cyclical principal oscillation pattern (POP cycle) that gives a clear ENSO cycle and error growth that is due to the non-self-adjoint nature of the system. The POP-cycle method is a generalization of the established POP method that allows the model to have an explicit seasonal cycle. The POP cycle isolates and clarifies features previously noted in the literature, such as the off-equatorial storage and westward propagation that appears 90-degrees out of phase with ENSO events. The non-self-adjoint error growth explains both the previously noted seasonality of error growth in the coupled model and the previously noted 6 month initial growth rate.The analysis done here is not dependent on the structure of the model: it could be performed on data or GCM model output. It could also be extended to characterize more/less predictable states in a nonlinear model, a characterization beyond the seasonal dependence considered here.
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