Brian D. Ewald

Accurate Integration of Stochastic Climate Models with Application to El Niño

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.

On modelling physical systems with stochastic models: diffusion versus Levy processes

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.

STOCHASTIC SOLUTIONS OF THE TWO-DIMENSIONAL PRIMITIVE EQUATIONS OF THE OCEAN AND ATMOSPHERE WITH AN ADDITIVE NOISE

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∞(L2) together with their derivative in z.

Numerical Generation of Stochastic Differential Equations in Climate Models

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.

Analysis of stochastic numerical schemes for the evolution equations of geophysics

Abstract We present and study the stability and 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.

Computing Expected Transition Events in Reducible Markov Chains

We present a closed-form, computable expression for the expected number of times any transition event occurs during the transient phase of a reducible Markov chain. Examples of events include time to absorption, number of visits to a state, traversals of a particular transition, loops from a state to itself, and arrivals to a state from a particular subset of states. We give an analogous expression for time-average events, which describe the steady-state behavior of reducible chains as well as the long-term behavior of irreducible chains.

Weak Versions of Stochastic Adams-Bashforth and Semi-implicit Leapfrog Schemes for SDEs

Abstract We consider the weak analogues of certain strong stochastic numerical schemes, namely an Adams-Bashforth scheme and a semi-implicit leapfrog scheme. We show that the weak version of the Adams-Bashforth scheme converges weakly with order 2, and the weak version of the semi-implicit leapfrog scheme converges weakly with order 1. We also note that the weak schemes are computationally simpler and easier to implement than the corresponding strong schemes, resulting in savings in both programming and computational effort.

On the Uniqueness of Invariant Measures for the Stochastic Infinite Darcy–Prandtl Number Model

The infinite Darcy–Prandtl number model is an effective reduced model for describing convection in a fluid-saturated porous medium. It is well known that the deterministic model does not possess a unique invariant measure. In this work, we study the dynamics of the infinite Darcy–Prandtl number model, under an additive stochastic forcing of its low modes. This is the so-called stochastic infinite Darcy–Prandtl number model. We prove that the stochastically forced system, does indeed possess a unique invariant measure.

The analysis of discrete transient events in Markov games

The evolution of a system from the transient phase into a steady-state or asymptotic phase is an important area of study in engineering and the mathematical sciences. While analytic methods exist for determining the steady-state behavior of a system, the transient analysis is typically more difficult. Transient analysis is often approached in either an ad hoc, case-by-case manner or is performed by simulation. In this paper we explore the transient analysis of absorbing Markov chains by counting discrete-time events. We derive a closed-form expression for the expectation of these events and give some examples. We then show how several singleagent systems may be combined into a multi-agent system where the interactions between agents can be analyzed. This affords a model for analyzing competition. For example, we can determine advantages to specific players and determine the expected number of lead changes. After developing these ideas we present simulation results to verify our methods.

SOME REMARKS ON THE NUMERICAL APPROXIMATION OF STOCHASTIC DIFFERENTIAL EQUATIONS

The aim of this article is to discuss the convergence of some numerical stochastic schemes in geophysical fluid dynamics (GFD) and to make some remarks on the numerical analysis of stochastic differential equations (SDEs).