Classification of climatic system models

No one doubts that the study of any physical phenomenon should be based on observational data where possible. But let us imagine that there are no such data, or not enough of them. What to do then? Of course, one can always put off the solution of the problem in hand until better times when the volume of empirical information will be sufficient and conclusions following its analysis will obtain credit. But what if we cannot afford the luxury of carrying out costly field measurements, or the time scale of phenomena to be examined is great (say, comparable with the duration of one generation or even the whole of humanity)? Finally, what should we do if the results of the phenomenon that we need to know are not permissible in principle (for example, in the case where there is a problem of clarifying the consequences of a nuclear conflict)? In such situations there is only one answer - to use mathematical models of the phenomena to be investigated.

Mathematical models of the climatic system represent the complex of hydrothermodynamic equations describing the state of separate subsystems complemented by proper initial and boundary conditions and parametriza-tions for those physical processes which, whether owing to imperfections in our knowledge, or owing to the limitations of available technical and economical means, or both of these, cannot be solved explicitly. All the current models of the climatic system are divided into analytical and numerical, into deterministic and stochastic, into hydrothermodynamic and thermodynamic (energy balance), into zero-dimensional, 0.5-dimensional (box), one-dimensional, two-dimensional (including zonal) and three-dimensional. This, as any other, classification is conventional to a certain extent. Its purpose is only to classify models by method of solution (analytical and numerical), by whether an ensemble of climatic system states are taken into account or not (stochastic and deterministic), by the way of reproducing the ordered large-scale circulation (immediately, in hydrothermodynamical models, and in parametrized form, in thermodynamical ones), and, finally, by the number of independent variables - spatial coordinates (zero-dimensional, 0.5-dimensional, etc., models).

We explain in more detail the principle of model division into deterministic and stochastic. As has already been mentioned (see Section 1.2), the climatic system has an extremely wide spectrum of time scales from a fraction of a second, for turbulence, to hundreds of millions of years for the cyclic reorganization of convective motions in the Earth's mantle and for continental drift. Now there is no one acceptable model that can simulate this diversity of temporal variability of the climatic system and take into account the real geography of continents and oceans. Apparently, there will be no such model in the near future. We should take this into consideration and compromise by reducing the range of time scales studied or replacing a direct description of processes which cannot be resolved by a parametric one. This is precisely what happens in short-term processes. As for long-term processes, their determining parameters are usually fixed. This last circumstance can lead to distortion in the response of slow inertial links of the climatic system to external forcing.

But when rejecting the fixing of parameters of inertial links of the climatic system it not only creates the problem of closure, similarly to that in turbulence theory, but also leads to the appearance of additional terms in model equations that describe random disturbances. Let x = (x,) be a vector of a fast component state, and y = (y,) be a vector of a slow component state of the climatic system (say, the atmosphere and the ocean), and the first one of them is specified by the typical time scale t0x, the second by t0y connected with t0x by the inequality t0x « t0y. The latter means that in the spectrum of climatic variability the fast and slow components are separated relative to each other. In this case the evolution of the climatic set can be described by the following set of equations:

where f¡ and g¡ are known non-linear functions of x and y.

We consider the evolution of the climatic system on time scales t»t0x. It is natural in this case to attempt to reduce the set (5.1.1) and (5.1.2) to one equation for the slow variable y, averaging Equation (5.1.2) in time

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