Abstract

The general theory of stochastic differential equations is presented in this chapter, including the theoretical background on how measured statistics from time series can be used to develop a stochastic parameterization. The general rules of stochastic calculus, including the important and often overlooked differences between Ito and Stratonovich calculus, are mentioned, and references are provided in which more detail may be found. We discuss how Stratonovich calculus is usually appropriate for fluid systems, whereas Ito calculus is often appropriate for data assimilation. We also discuss some common numerical pitfalls awaiting the unwary modeler, and warn against unsophisticated random number generators. Finally, we offer a selection of examples showing the importance of the variability of unresolved scales in an ocean model and, by citation, a variety of methods that have been employed.

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