## Climate modeling the problem of chaos and instability

The ocean-atmosphere-cryosphere-biosphere system is a complex system in the sense that it is made of different components that may interact with each other on a very wide range of time-scales (Mitchell 1976). These interactions are generally nonlinear, that is the response is not proportionate to the amplitude of the excitation. A physical system with at least three components interacting nonlinearly with each other may be chaotic (e.g. Hilborn 2001, p. 134) (Figure 4.1). In other words,

Figure 4.1 Lorenz's model is a classic example of a chaotic system with a strange attractor (Lorenz 1963). It is made of three nonlinear equations describing the trajectory of a particle along three coordinates {x,y,z} given two parameters (s and r). The figure represents the projection of a trajectory projected on the {x,y} plane. The particle leaves its initial condition and is attracted to a certain region of the space called the attractor. Depending on the chosen parameters, the attractor may be a point, a well-defined geometric form such as a circle or an ellipse, or a "strange attractor". "The strange attractor" featured here characterizes a chaotic behavior. This means that the particle circulates around a mean path (the butterfly shape) but it never passes twice through the same point in the three-dimensional space. The shape of the attractor is a function of the parameters and can be statistically defined after a long integration time. By contrast, individual trajectories cannot be predicted because any small error on initial conditions grows exponentially with time. This is the reason why climate (the attractor) may be predicted, but weather (the location of a particle at a moment in time) cannot beyond a certain time horizon.

its evolution cannot be predicted accurately beyond a certain time horizon because any error on the initial conditions grows exponentially with time. The atmosphere is chaotic. This is the reason why we cannot forecast weather much beyond about 6 days. Yet, we can predict global warming. Indeed, conservation of energy, heat, and momentum makes it possible to predict the general evolution of a chaotic system in statistical terms. This statistical description of weather is nothing but the definition of climate.

A special problem about climate is that it features exchanges of energy between components characterized by radically different temporal and spatial scales. For example, a cloud has a typical spatial scale of the order of 1 km and a life cycle of a few hours to one or two days long. Cloud formation and the resulting precipitation may influence and be influenced by the topography of continental ice sheets that evolve over millennia. This is an example of an interaction between a component with a very short time-scale and one with a very long one. These exchanges may be extraordinarily complex and this is the reason why climate models are uncertain.

Nonlinear systems may also exhibit instability. A steady state is said to be unstable when a small perturbation is irremediably amplified by feedbacks internal to the system until it reaches a new mode of behavior. There may be several stable steady states and the one chosen by the system depends on its history. A typical example is glaciated versus nonglaciated Earth. Both states may be stable (Budyko 1969; Calov et al. 2005) and the one chosen depends on whether the preceding history is favorable or not to a glacial inception.

Finally the actual climate system has no strict steady state because oceanic and atmospheric currents vary constantly. This is why theoreticians prefer to use the notion of "attractor". The attractor is, loosely speaking, a closed trajectory that the climate system follows more or less closely (Figure 4.1).

Figure 4.2 Time- and space-scales covered by numerical models used for weather and climate prediction. Conceptual models have only a few degrees of freedom and are designed to formulate and test hypotheses in a very well-defined framework. Three examples are given here: (1) turbulence in the boundary layer; (2) stability of the ocean circulation; and (3) ice-sheet response to astronomical forcing. Comprehensive climate models (also called "general circulation models") include the largest number of degrees of freedom and are suitable to study climate dynamics on time-scales of a few decades to a few centuries. Earth models of intermediate complexity (EMICs) usually cover longer time-scales. Mesoscale and regional models simulate weather and climate over a limited domain of the globe. A mesoscale model typically covers a domain the size of the UK, and a regional model may cover Europe or the USA.

Global Continents/ oceans

Region

200 km 100 km

20 km 10 km

100 m

Comprehensive models

Regional models

EMICs

Mesoscale models

Inductive models

Year Decade Century Millennium Milankovitch

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