Stochastic Climate Models Version 1.0 (Preliminary Draft, Figures to come later)

The field of climate physics is devoted to the study of the climate system as a dynamical system , that is, to the study of its spatial and temporal variability. By climate system, we refer to that tightly-coupled collection of subsystems composed of the atmosphere, the ocean, the cryosphere (icy regions), the lithosphere (land), and the biosphere (life). This system is not closed: external forcings are of crucial importance to its dynamics. In particular, solar radiation is the primary source of energy that drives climatic variability. Current physical theory implies that the sun’s radiative dynamics are independent of the climate states of its satellites, so the sun itself is not a part of the climate system but enters the problem merely as an external forcing. The climate system displays variability over a broad range of temporal and spatial scales, from dust devils to hurricanes to ice ages. As a general rule, larger spatial scale processes tend to be associated with longer time scales. Figure 1 presents a qualitative illustration of the relative amounts of variability in the climate system over a broad range of time scales (14 decades! ). Prominent in this figure are the sharp peaks associated with daily and annual variability, the broad peak between 3 and 7 days associated with synoptic-scale weather systems, and peaks at 20 ka, 40 ka, and 100 ka associated with glaciation cycles (a fairly strong peak at 4 years associated with the El Niño-Southern Oscillation (ENSO) phenomenon seems to be missing). This paper is concerned primarily with variability on time scales ranging from a day to 100 ka. Over this range, as a gross generalisation, the relative variance tends to increase with timescale. This increase in relative variance with period is not monotonic, however, and the power spectrum is characterised by a number of peaks and troughs, some of which have been discussed above. While climate variability does tend to increase with period, it also tends to be concentrated in relatively narrow bands. This suggests several natural separations of temporal scales. Of these, perhaps the most important is that between weatherandclimate. In technical parlance, weather refers to variability in the climate system at time scales less than about 14 days, while climate refers to variability at scales greater than about 14 days. As Figure 1 illustrates, the relative variance of the climate system has a rather shallow, broad valley at timescales on the order of a few weeks, so the ambiguity in the precice specification of the two scales is not problematic. However, while the time scales can be separated conceptually, they arenot separated dynamically. The equations of motion of the components of the climate system are strongly nonlinear, and interactions between different space and time scales (both within the same and between different subsystems) are observed to play a prominent role in the dynamics. In particular, the atmosphere and ocean are turbulent, a situation with results in the existence of upper time bounds on the predictability of instantaneous states of either system [1]. It is impractical to model exactly the entire range of climatic variability simultaneously, either numerically or analytically, so simplifying assumptions have to be made to r nder the problem tractable. These assumptions are guided by physical intuition, but the ultimate judge of their appropriateness is the descriptive or predictive power of the resulting approximate theories. When modelling higher-frequencycomponents of variability (also referred to as fastvariables), the low-frequencycomponents of variability ( slowvariables) are usually simply taken to be constant. For example, one expects intuitively that a model designed to forecast the weather three days in advance need not take into account the changes in global ice-cover associated with glaciation cycles (although one might need a different model during an ice age than one uses now). Note that the timescale separation is not wholly unambiguous: such a model may require a description of the changes in polar ice-cover associated with the annual cycle. The situation is somewhat different when modelling the lowerfrequency variability in the climate system. The integrated, aggregate effect of the fast variables can conceivably have a strong impact on the slower components, but because of the broad range of scales involved this cannot be modelled exactly. The analogous problem occurs in theories of turbulence, in which case the dynamical effects of the fast variables on the slow are typically modelled as (usually nonlinear) functions of these slow variables in so-calledclosure schemes . This approach has been quite successful at addressing certain problems which arise in turbulence and climate theory (thus its longevity), but it has its limitations. Another approach to modelling the effect of the fast variables on the slow is to treat the former not as deterministic variables,but asstochasticvariables, so that the slow variables evolve following dynamical equations with stochastic forcing. The chaotic components of the system often have well-defined statistical properties and these can be built into approximate stochastic representations of the high-frequency variability. The resulting models