PNNL

Abstract

New ideas in mathematics, even seemingly insignificant and purely impractical, can have unforeseen and important consequences that may not become apparent at the time they were suggested and occasionally not even for several decades. One of such ideas was proposed by gifted Russian mathematician, A.A. Markov (1856-1922) in the beginning of the 20th century (Basharin et al., 2004). Radically innovative for that time, it advanced substantially 19th-century works in the probability theory by generalizing various limit laws established for independent random events (e.g., the law of large numbers) to dependent ones. Our tour of such events will include stochastic radiative transfer in a cloudy atmosphere. No attempt will be made to provide a comprehensive survey of all available methods and approaches and perhaps several relevant and influential works on these topics are not included here. Instead, we will take a short detour and discuss a small number of landmark ideas and their role in solving important problems. There will be a couple of intermediate stops on the way, where we acknowledge major efforts of passionate scientists after whom several popular mathematical models are named. A substantial part of this chapter concentrates on theoretical basis of these models, illustrates their "aesthetic" mathematical value and physical significance and discusses how they can be applied to current challenges in the atmospheric and climate science.