In this chapter we review stochastic modeling methods in climate science. First we provide a conceptual framework for stochastic modeling of deterministic dynamical systems based on the Mori-Zwanzig formalism. The Mori-Zwanzig equations contain a Markov term, a memory term and a term suggestive of stochastic noise. Within this framework we express standard model reduction methods such as averaging and homogenization which eliminate the memory term. We further discuss ways to deal with the memory term and how the type of noise depends on the underlying deterministic chaotic system. Second, we review current approaches in stochastic data-driven models.We discuss how the drift and diffusion coefficients of models in the form of stochastic differential equations can be estimated from observational data. We pay attention to situations where the data stems from multi-scale systems, a relevant topic in the context of data from the climate system. Furthermore, we discuss the use of discrete stochastic processes (Markov chains) for example, stochastic subgrid-scale modeling and other topics in climate science.
The climate system is characterized by the mutual interaction of complex systems each involving entangled processes running on spatial scales from millimeters to thousands of kilometers, and temporal scales from seconds to millennia. Given current computer power it is impossible to capture the whole range of spatial and temporal scales and this will also not be possible in the foreseeable future. Depending on the question we pose to the climate system – be it forecasting regimes in the atmosphere or simulating past ice ages – we have to make a decision as to what components to include in the analysis and as to what scales to resolve. A corollary of this decision is that each numerical scheme inevitably fails to resolve so-called unresolved scales or subgrid-scales. However, typically one is only interested at the slow processes active on large spatial scales. For example, for weather forecasts it is sufficient to resolve large-scale high and low pressure fields rather than smallscale fast oscillations of the stratification surfaces, whereas for climate predictions with a coupled ocean-atmosphere model we may want to distill the slow dynamics of the ocean ignoring weather systems interacting with the ocean on fast time-scales of days.