A Numerical Study of Stochastic El Niño Southern Oscillations Using Wiener Chaos Expansion and Monte Carlo Methods

Stochastic climate models are mathematical representations of the Earth's climate system that integrate stochastic components to simulate the inherent uncertainties and variability observed in the climate. These models are formed to capture the complex interactions between various elements of the climate system, such as the atmosphere, oceans, land surface, and ice. However, due to the high-dimensionality and randomness, simulation of stochastic climate models given by stochastic partial differential equations (SPDEs) often requires costly expensive computational resources. Therefore, it is important to develop efficient and effective techniques. In this paper, we explore the application of Wiener chaos expansion (WCE) and Monte Carlo (MC) methods for simulating stochastic El Ni & ntilde;o Southern Oscillations (ENSO) that is modeled by coupled atmosphere, ocean and sea surface temperature (SST) mechanism in the equatorial Pacific. Initially, we first apply the WCE-based method on the simple ocean model driven by oceanic Kelvin and Rossby waves forced with white noise as a test bed problem, and show that the first few WCE-modes are able to closely approximate the theoretical variance values obtained by using the method of characteristics. Our results depict that statistical moments, (i.e., the mean and variance) of the solutions obtained from the WCE method provide remarkably accurate results with a reasonable convergence rate and error range. In light of the results of the test problem, we then employ a high-dimensional coupled linear stochastic ENSO model and show that Monte Carlo (MC) simulations with a large number of ensembles can converge to the results with few WCE modes. We also show that the WCE-based approach requires less computation time with a reasonable convergence rate. Along with the comparison of computational cost, this combination of WCE with MC methods is particularly practical when dealing with problems or complex high-dimensional stochastic models, where analytical or exact solutions are not easily available, as similar to stochastic ENSO models.