Penland, C., 2003: A stochastic approach to nonlinear dynamics: A review. Bull. Amer. Met. Soc. (electronic supplement), 84, ES43-ES52.
The need to numerically model the interactions between geophysical processes having different timescales has led many modelers to represent rapidly varying components of the system as stochastic forcing. The methods these stochastic modelers use to do this are almost as numerous as the modelers themselves. Yet, there does exist a prescription for making the stochastic approximation in a systematic manner consistent with the multiscale dynamics.
Many of us are familiar with some form of the central limit theorem (CLT), which states how sums of weakly dependent quantities are approximately Gaussian distributed. There is another version that states the conditions under which a multiscale dynamical system may be approximated as depending on the realizations of a whitenoise process, that is, as a stochastic differential equation (SDE). This more general version is what the most commonly appropriate stochastic approximation is based on. We review this approximation as derived in the historical literature and summarize some of the more easily accessed publications. Since it is necessary to understand the basic concepts of Gaussian white noise before we can decide whether or not the CLT is applicable to a real system, we review the dynamical properties of the drunkard's walk, white noise, and the Wiener process in the next section. In all sections here ,we use the standard notation where vectors are boldfaced and matrices are bold sans serif. Other quantities are scalars unless otherwise specified. The applicability of the CLT is discussed in the section titled "The White Noise Approximation," an example of the white noise approximation is provided, and two different kinds of drunks are identified there. Much of the material in these two sections may also be found in Penland (1996); however, the importance of that material, particularly to the following sections in this review, justifies its reproduction here. The section titled "Numerical Generation of Stochastic Differential Equations" is concerned with the numerical solution to SDEs, and final remarks are provided in the last section. We alert the reader that we only consider a stochastic approach based on the classic theory of SDEs; in particular, we do not consider the important "fractional Brownian motions" or "random cascade models." This is why the title is "A stochastic approach to nonlinear dynamics" rather than "Stochastic approaches to nonlinear dynamics."