Our climate prediction schemes, therefore, contain well-tried equations combined with simpler descriptions of poorly-understood phenomena. They are solved on massive digital computers. The outputs are probabilities of various statistics, like average summer and winter temperatures, rainfall, windspeeds and so on. Obviously, there are similarities with the simple-plume theory, but also strong differences. In fact the difference is emphasized by calling our climate prediction scheme a `model', not a `theory'. Originally synonymous, these terms have now acquired distinct meanings. This is illustrated best in the way we test our predictions. For the smoke-plume, it is a relatively simple matter to measure the precise quantities predicted by the theory, the statistics of plume concentration and spread. For the `Global Climate Model' it is not so straightforward.
To confirm the accuracy of the climate model we compare its output with the observed climate. Of course we have no control over our `experimental' data; we cannot rerun the last hundred years of climate over and over again, to get accurate statistics. We are reduced to comparing the statistics of our model output with the one set of climate measurements that we do have. Personal judgement plays a much greater rôle than scientists would like, in deciding whether the model has been `validated' in these circumstances. Indeed, there is an implicit reliance on Dirac's second principle: we assume that if the central equations upon which the model is based are close to the exact equations, then their solution--approximated on the digital computer--will behave like the real climate. In short, a reliance on the `` unreasonable effectiveness of mathematics in describing the physical world'' is bound up in our acceptance of models that cannot be completely tested, but only partially validated before being used for crucial predictions.
Lest anyone think I am being too hard on Climate models, let me say that they do contain some of the best environmental science we have assembled. Our confidence in their basic structure is bolstered by the fact that they perform extremely well when used to predict weather a few days in advance, rather than climate a hundred years hence. To put it in perspective, their predictive power is at least an order of magnitude better than the economic models that our Governments use daily, to set the country's financial course.
What lessons can we learn from this comparison? Most models used for environmental prediction fall between the extremes of the smoke-plume and the Climate model. They include descriptions of crop growth or vegetation change, water yield from catchments, salinisation and urban air pollution. Often, just as in the climate models, we cannot rigorously test but only validate them. If this is the case, then the model must be based on differential equations that have been proved to encapture the key processes at work. If, on the other hand, the model is based only on empirical correlations and rigorous testing has been replaced by validation, then alarm bells should be ringing for practitioners and the users of the predictions alike.
Models nowadays often come as glossy software packages and their provenence may not be obvious.
The canny practitioner must put himself in a position to critically assess their validation before using them for prediction. Ultimately there is no substitute for a sound knowledge of the field.