Recent estimates of climate evolution over the coming century still differ by several degrees. This uncertainty motivates the work presented here. There are two basic approaches to apprehend the complexity of climate change: deterministically nonlinear and stochastically linear, i.e., the Lorenz and the Hasselmann approach. The grand unification of these two approaches relies on the theory of random dynamical systems. We apply this theory to study the random attractors of nonlinear, stochastically perturbed climate models. Doing so allows one to examine the interaction of internal climate variability with the forcing, whether natural or anthropogenic, and to take into account the climate system's non-equilibrium behavior in determining climate sensitivity. This non-equilibrium behavior is due to a combination of nonlinear and random effects. We give here a unified treatment of such effects from the point of view of the theory of dynamical systems and of their bifurcations. Energy balance models are used to illustrate multiple equilibria, while multi-decadal 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. The combined treatment of nonlinear and random effects is applied to a stochastically perturbed version of the classical Lorenz convection model. Climate sensitivity is then defined mathematically as the derivative of an appropriate functional or other function of the system’s state with respect to the bifurcation parameter. This definition is illustrated by using numerical results for a model of the El Niño–Southern Oscillation. The concept of a hierarchy of models is the thread that runs across this chapter, and the robustness of elementary bifurcations across such a hierarchy is emphasized.
Oceanic variability on interannual, interdecadal, and longer timescales plays a key role in climate variability and climate change. Paleoclimatic records suggest major changes in the location and rate of deepwater formation in the Atlantic and Southern oceans on timescales from millennia to millions of years. Instrumental records of increasing duration and spatial coverage document substantial variability in the path and intensity of ocean surface currents on timescales of months to decades. We review recent theoretical and numerical results that help explain the physical processes governing the large-scale ocean circulation and its intrinsic variability. To do so, we apply systematically the methods of dynamical systems theory. The dynamical systems approach is proving successful for more and more detailed and realistic models, up to and including oceanic and coupled ocean-atmosphere general circulation models. In this approach one follows the road from simple, highly symmetric model solutions, through a “bifurcation tree,” toward the observed, complex behavior of the system under investigation. The observed variability can be shown to have its roots in simple transitions from a circulation with high symmetry in space and regularity in time to circulations with successively lower symmetry in space and less regularity in time. This road of successive bifurcations leads through multiple equilibria to oscillatory and eventually chaotic solutions. Key features of this approach are illustrated in detail for simplified models of two basic problems of the ocean circulation. First, a barotropic model is used to capture major features of the wind-driven ocean circulation and of the changes in its behavior as wind stress increases. Second, a zonally averaged model is used to show how the thermohaline ocean circulation changes as buoyancy fluxes at the surface increase. For the wind-driven circulation, multiple separation patterns of a “Gulf-Stream like” eastward jet are obtained. These multiple equilibria are followed by subannual and interannual oscillations of the jet and of the entire basin's circulation. The multiple equilibria of the thermohaline circulation include deepwater formation near the equator, near either pole or both, as well as intermediate possibilities that bear some degree of resemblance to the currently observed Atlantic overturning pattern. Some of these multiple equilibria are subject, in turn, to oscillatory instabilities with timescales of decades, centuries, and millennia. Interdecadal and centennial oscillations are the ones of greatest interest in the current debate on global warming and on the relative roles of natural and anthropogenic variability in it. They involve the physics of the truly three-dimensional coupling between the wind-driven and thermohaline circulation. To arrive at this three-dimensional picture, the bifurcation tree is sketched out for increasingly complex models for both the wind-driven and the thermohaline circulation.