Numerical Generation of Stochastic Differential Equations in Climate Models

Abstract

This chapter summarizes numerical procedures for evaluating the solutions of classical stochastic differential equations (SDEs) in climate prediction and research. It discusses the central limit theorem, which directs the way a system with scale separation may be approximated as an SDE. It also discusses an extension of the traditional central limit theorem usually employed by geoscientists to justify the use of Gaussian distributions. Informally, the classical central limit theorem states that the sum of independently sampled quantities is approximately Gaussian. The SDEs are averaged over a large temporal interval so that the fast timescales collectively act as Gaussian stochastic forcing the slow, coarse grained system. The chapter provides a review of stochastic Taylor expansion and relates it to the development of stochastic numerical integration methods. It also provides an overview of stochastic numerical methods as used in climate research.