Stochastic perturbations of El Nino Southern Oscillations (ENSO) : a Wiener chaos approach

The phenomena of El Nino Southern Oscillations (ENSO) is modeled by coupled atmosphere-ocean mechanism together with sea surface temperature (SST) budget at the equatorial Pacific and has a significant impact on the global climate.  We consider a modeling framework that was originally developed by Majda and co-workers in (Chen et al. 2018; Thual et al. 2016), which is physically consistent and amenable to detailed analysis. The coupled model is mainly governed by the equatorial atmospheric and oceanic Kelvin and Rossby waves and it is shown that stochastic forcing gives rise to the model anomalies and unpredictable behavior. The purpose of our work is to investigate the influence of randomness on the model dynamics,  construct the appropriate model components with stochastic noise and calculate the statistical properties. We also provide analytical and numerical solutions of the model to prove the convergence of the numerical scheme developed in our work. 

We use Wiener-Chaos Expansion (WCE) to study stochastic ENSO models. The WCE method is based on reducing stochastic partial differential equations (SPDEs) into an infinite hierarchy of deterministic PDEs called propagators-Fourier modes (Lototsky and Rozovsky, 2006) and represents the stochastic solution as a spectral decomposition of deterministic components with respect to a set of random Hermite bases. We solve the WCE propagators, which are forced by a set of complete orthonormal bases,  by applying numerical integration and finite-difference methods. We compare WCE-based results with Monte Carlo simulations of SPDEs.

Our results depict that the mean and variance of the solutions obtained from the WCE method provide remarkably accurate results with a reasonable convergence rate and error range.  We first test the WCE-based method on the ocean  model with white noise and show that 10-Fourier modes are able to approach the theoretical variance values. We also show that the OU process with a specific noise strength and dissipation over a one-time period can be recovered with less than 50-Fourier modes for the ENSO model.  To illustrate the particular weight of variance, we also generate the ensembles of solutions by using different stochastic bases. We also derive the analytical formulation of propagators for the coupled model with nonlinear SST by using the properties of Wick polynomials that construct the foundation of numerical schemes.