A linear stochastic dynamical model of ENSO. Part I: Model development

Singular vector analysis and Floquet analysis are carried our on a lineariz ed variant of the Zebiak-Cane atmosphere-ocean model of El Nino-Southern Os cillation (ENSO), hereinafter called the nominal model. The Floquet analysi s shows that the system has a single unstable mode. This mode has a shape a nd frequency similar to ENSO and is well described by delayed oscillator ph ysics. Singular vector analysis shows two interesting features. (i) For any starting month and time period of optimization the singular vector is shap ed like one of two nearly orthogonal patterns. These two patterns correspon d approximately to the real and imaginary parts of the adjoint of the ENSO mode for the time-invariant basic-state version of the system that was calc ulated in previous work. (ii) Contour plots of the singular values as a fun ction of starting month and period of optimization show a ridge along end t imes around December. This result along with a study of the time evolution of the associated singular vectors shows that the growth of the singular ve ctors has a strong tendency to peak in the boreal winter. For the case of a stochastically perturbed ENSO model, this result indicates that the annual cycle in the basic state of the ocean is sufficient to produce strong phas e locking of ENSO to the annual cycle; it is not necessary to invoke either nonlinearity or an annual cycle in the structure of the noise. The structures of the ENSO mode, of the optimal vectors, and of the phase l ocking to the annual cycle are robust to a wide range of values for the fol lowing parameters: the coupling strength, the ocean mechanical damping, and the reflection efficiency of Rossby waves that are incident on the western boundary. Four variant models were formed from the nominal coupled model b y changing the aforementioned parameters in such a way as to (i) make the m odel linearly stable and (ii) affect the ratio of optimal transient growth to the amplitude of the Brst Floquet multiplier (i.e., the decay rate of th e ENSO mode). Each of these four models is linearly stable to perturbations but is shown to support realistic ENSO variability via transient growth fo r plausible values of stochastic forcing. For values of these parameters th at are supported by observations and theory, these results show the coupled system to be linearly stable and that ENSO is the result of transient grow th. Supporting evidence is found in a companion paper.