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In statistical mechanics physical systems have probability distributions of characteristics. These distibutions have averages and variances. These are called ensemble averages and variances. Any one system with specified characteristics will evolve over time according to the dynamics of the system. Suppose the averages of the characteristics of that system are tabulated for its evolution. There are some systems such that the asymptotic time averages for any one member of an ensemble are equal to the corresponding ensemble averages. Such systems are called ergodic.
In predictive climatology as it is usually practiced a meteorological model which cannot accurately predict the weather beyond about ten days is run to give predictions a century into the future. The averages of these predicted future weathers are taken to be valid predictions of the future climate. In effect, the socalled climate modelers are assuming without a shred of theoretical or empirical justification that the Earth's weather system is ergodic. They are also assuming that a necessarily imperfect meteorological model is a valid predictor of the actual weather system will bring in the future.
Tens of billions of dollars have been poured into socalled research which may be no more than irrelevant arithmetic. Edward N. Lorenz discovered the notion of chaos for systems of nonlinear differential equations. What chaos amounts to is the infinite sensitivity of such dynamic systems to initial conditions. The notion of chaos is sometimes labeled the Butterfly Effect; i.e., even the flap of a butterfly's wings will change the weather drastically at some distant future time.
Before Lorenz's discovery meteorologist believed accurate long term weather forecasts were possible; after Lorenz they realized that such the results of running the models for distant times in the future would be completely meaningless as forecasts.
Some climatologist did examine the theoretical basis for climate predictions. Some of the results of their investigations are given in Predictability of weather and climate edited by Tim Palmer and Renate Hagedorn. These are not the climatologists whose projection get media attentions. People like James Hansen at the Goddard Space Science Institute, without much training in meteorology, unhesitatingly present century ahead prediction from nonvalidated meteorological models as though they were hard science.
Usually the weaknesses of the model predictions is supposedly handled by presenting the predictions as a range. A range sounds more scientific. But often the basis for the range is imbecilic. For example, when climate forecasting first started two forecasts of the effect of the doubling of the carbon dioxide level in the atmosphere were presented to a committee of the American Meteorological Association. The forecasts were in terms of the effect on average global temperature in the year 2100. One forecast was by Syukuro Manabe, a highly respected meteorologist at Princeton University. He said that his modeling efforts were resulting in a great variety of figures as he tried to refine his model. At the time of the presentation to the committee his forecast was 2°C. The other forecast was by James Hansen who formulated a model and got figure of 4°C. Manabe said with a bit of contempt that the AMA committee took his figure of 2°C and Hansen's figure of 4°C and tacked on a ½°C at either end of the range and announced to the world that the prediction for the effect of a doubling of the level of CO_{2} would be from 1½°C to 4½°C. The media picked that up and took the upper limit as being the prediction of the effect of a doubling of the level of CO_{2}. That socalled predicted range of from 1½°C to 4½°C seemed have a life of its own, appearing again and again over the years as somehow being hard science.
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