Numerical models are one of the most important theoretical tools in atmospheric research, and the development of numerical techniques specifically designed to model the atmosphere has been an important discipline for many years. In recent years, stochastic numerical models have been introduced in order to investigate more fully Hasselmann's suggestion that the effect of rapidly varying ''weather'' noise on more slowly varying ''climate'' could be treated as stochastic forcing. In this article an accurate method of integrating stochastic climate models is introduced and compared with some other commonly used techniques. It is shown that particular care must be used when the size of rapid variations in the ''weather'' depends upon the ''climate.'' How the implementation of stochasticity in a numerical model can affect the detection of multiple dynamical regimes in model output is discussed. To illustrate the usefulness of the numerical schemes, three stochastic models of El Niño having different assumptions about the random forcing are generated. Each of these models reproduces by construction the observed mean and covariance structure of tropical Indo-Pacific sea surface temperature. It is shown that the skew and kurtosis of an observed time series representing El Niño is well within the distributions of these statistics expected from finite sampling. The observed trend, however, is unlikely to be explained by sampling. As always, more investigation of this issue is required.

Stochastic descriptions of multiscale interactions are more and more frequently found in numerical models of weather and climate. These descriptions are often made in terms of differential equations with random forcing components. In this article, we review the basic properties of stochastic differential equations driven by classical Gaussian white noise and compare with systems described by stable Lévy processes. We also discuss aspects of numerically generating these processes.

We present and study the stability, convergence, and order of convergence of a numerical scheme used in geophysics, namely, the stochastic version of a deterministic "implicit leapfrog" scheme which has been developed for the approximation of the so-called barotropic vorticity model. Two other schemes which might be useful in the context of geophysical applications are also introduced and discussed.

The aim of this article is to establish the existence and uniqueness of stochastic solutions of the two-dimensional equations of the ocean and atmosphere. White noise is additive, and the solutions are strong in the probabilistic sense. Finally, from the point of view of partial differential equations, they are of the type z-weak, that is, bounded in L ∞ (L 2 ) together with their derivative in z.

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