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Nonlinear and random effects are pervasive in the atmospheric, oceanic, and climate sciences. This article gives a unified treatment of such effects from the point of view of the theory of dynamical systems and their bifurcations. Energy balance models are used to illustrate multiple equilibria, while multidecadal oscillations in the thermohaline circulation illustrate the transition from steady states to periodic behavior. Random effects are introduced in the setting of random dynamical systems, which permit a unified treatment of both nonlinearity and stochasticity. This treatment is applied to a stochastically perturbed version of the classical Lorenz convection model.

We provide an overview of numerical bifurcation methods and how they can help one understand the variability of large-scale oceanic and atmospheric flows, as well of the climate system as a whole. As a particular example, we consider the problem of how low-frequency variability of the wind-driven ocean circulation arises due to internal dynamics. The methods are illustrated with the help of a hierarchy of models of so-called double-gyre, wind-driven flows.

This chapter describes the general circulation models (GCMs) and their role in the climate modeling hierarchy. By considering a few climate problems on different time scales, it is illustrated that the general theory can be combined with powerful numerical tools to study successive bifurcations across the hierarchy of climate models, all the way from 0-D global or box models to 2- and 3-D models: atmospheric, oceanic, and coupled. The tools of analytical and numerical bifurcation theory can be applied only to 0-D thermohaline circulation (THC) models or 0- and 1-D climate models. Each bifurcation is associated with a specific linear instability of a relatively simple climate state—oscillatory in the case of Hopf bifurcations and purely exponential in the case of saddle-node or pitchfork bifurcations—where nonlinear saturation leads to more complicated climate behavior. GCMs—atmospheric, oceanic, and coupled—provide climate problem solutions that have the greatest spatiotemporal detail and the greatest degree of realism. Statistical tools, such as singular-spectrum analysis and other advanced spectral methods, have to be applied in parallel to the GCM simulations and to the relevant data sets.

Publisher Summary This chapter provides a review of current operational practice and of advanced data assimilation techniques in meteorology. Numerical models can be used to assimilate meteorological and oceanographic data, creating a dynamically consistent, complete and accurate “movie” of the two geofluids, atmosphere, and ocean in motion. The ocean's strong stratification helps determine the most energetic scales and processes for the global ocean circulation. Active research on data assimilation is burgeoning rapidly in both meteorology and oceanography. Operational NWP requirements have produced a mature data-assimilation technology in meteorology, from which climatic research has benefitted as well. Ocean is characterized by transient, energetic motions with a broad spectrum in frequency and wave number. A steady component of the circulation may not even exist, and be only a model resulting from the analysis of data sets sparse in space and time, like hydrographic data sets, for which steadiness is assumed a priori. Thus, in oceanic data-assimilation problems, the choice of a model and related data assimilation scheme and the definition of success of the assimilation process itself depend crucially on the scientific issue of interest as the starting point.

The central theme of this review article is the role that dynamics plays in estimating the state of the atmosphere and of the ocean from incomplete and noisy data. Objective analysis and inverse methods represent an attempt at relying mostly on the data and minimizing the role of dynamics in the estimation. Four-dimensional data assimilation tries to balance properly the roles of dynamical and observational information. Sequential estimation is presented as the proper framework for understanding this balance, and the Kalman filter as the ideal, optimal procedure for data assimilation. The optimal filter computes forecast error covariances of a given atmospheric or oceanic model exactly, and hence data assimilation should be closely connected with predictability studies. This connection is described, and consequences drawn for currently active areas of the atmospheric and oceanic sciences, namely, mesoscale meteorology, medium and long-range forecasting, and upper-ocean dynamics.