# Publications

## A linear stochastic dynamical model of ENSO. Part I: Development

### Citation

Thomson, C.J., Battisti, D.S. 2000. A linear stochastic dynamical model of ENSO. Part I: Development. *Journal of Climate* 13:2818-83.

### Abstract

Singular vector analysis and Floquet analysis are carried out on a linearized variant of the Zebiak-Cane atmosphere-ocean model of El Ninño-Southern Oscillation (ENSO), hereinafter called the nominal model. The Floquet analysis shows that the system has a single unstable mode. This mode has a shape and frequency similar to ENSO and is well described by delayed oscillator physics. Singular vector analysis shows two interesting features. (i) For any starting month and time period of optimization the singular vector is shaped like one of two nearly orthogonal patterns. These two patterns correspond 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 calculated in previous work. (ii) Contour plots of the singular values as a function of starting month and period of optimization show a ridge along end times around December. This result along with a study of the time evolution of the associated singular vectors shows that the growth of the singular vectors 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 phase 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 locking to the annual cycle are robust to a wide range of values for the following 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 by changing the aforementioned parameters in such a way as to (i) make the model linearly stable and (ii) affect the ratio of optimal transient growth to the amplitude of the first Floquet multiplier (i.e., the decay rate of the ENSO mode). Each of these four models is linearly stable to perturbations but is shown to support realistic ENSO variability via transient growth for plausible values of stochastic forcing. For values of these parameters that are supported by observations and theory, these results show the coupled system to be linearly stable and that ENSO is the result of transient growth. Supporting evidence is found in a companion paper.