Time scale interaction in low-order climate models

Over the last decades, the study of climate variability has attracted ample attention. The observation of structural climatic change has led to questions about the causes and the mechanisms involved. The task to understand interactions in the complex climate system is particularly di±cult because of the lack of observational data, spanning a period of time typical for natural climate variability. One way around this problem is to represent the earth s climate in a computer model, as a set of prognostic equations. A disadvantage of this approach is that, if the model under consideration is to faithfully represent the climate system, it has to be large in terms of the number of degrees of freedom. This puts it out of reach of the ordinary analysis of dynamical systems theory. Alternatively, we can impose symmetries, consider limits of physical parameters, exploit perturbation theory and use Galerkin approximation to obtain simplified models of the earth s climate. Such models should highlight some isolated aspects of climate dynamics. A feature these simplified models have in common is the presence of widely different time scales. Throughout this thesis the emphasis is on the question to what extent the slow time scales play a role in the model s dynamics. The slow time scales are related to ocean dynamics and the fast time scales to atmospheric dynamics. The atmosphere model, studied here, was introduced by Edward Lorenz (1984). In chapter 2 a derivation of this model is given and it is shown that the Lorenz-84 model describes the jet stream in the mid-latitude atmosphere, and planetary waves, which can grow if the jet stream becomes dynamically unstable. The Lorenz-84 model is coupled to two different low-order ocean models. In chapter 3, it is coupled to Stommel s two box model. Stommel s model mimics the thermohaline circulation in the North Atlantic ocean. The typical time scale of variability of this circulation is of the order of centuries. This will be the longest time scale in the coupled models. In chapter 4, the Lorenz-84 model is coupled to an ocean model formulated by Leo Maas (1994). A physical description of the coupling is given. Apart from the overturning circulation, Maas model represents a wind driven gyre. There is coupling through exchange of heat at the surface and through wind shear forcing. The latter acts on a time scale of about one year, in between the fast atmospheric time scale and the slow overturning time scale. The simplified models are sets of coupled, nonlinear, ordinary differential equations. These can be analysed with the aid of dynamical systems theory. The emphasis will be on bifurcation analysis. Also, the time scale separation leads to the presence of small parameters in the equations. The consequences for the behaviour of the coupled models are explored by means of singular perturbation theory. In both coupled models, intermittent behaviour is observed. The slow subsystem, i.e. the ocean model, repeatedly pushes the fast subsystem, i.e. the atmosphere model, through a sequence of bifurcations. Thus, the ocean model plays an active role in the coupled system. Secondly, in the Lorenz-Maas model a periodic solution is shown to exist, with a period on the slow, overturning time scale. Along this solution the behaviour of the coupled model is dictated by internal ocean dynamics. Both these phenomena occur near a critical point of the coupled system, in agreement with the general idea that in climate models the slow components can play an active role near such critical points and are passive otherwise.