The term ‘numerical model’ means different thing to different people. To a mathematician it provides a numerical solution to a problem which cannot be solved analytically; the numerical solution can be obtained to any required degree of accuracy-given sufficient computing power and assuming a finite number of solutions. The classic example is the problem of the motion of three point masses in space, form specified initial conditions. (The motion of two masses has an analytic solution).
To a scientist, a model is a mathematical approximation of part of the real world which attempts to evaluate a few parameters describing that world, wither deterministically or in statistical terms. Thus, the scientist's model is a simplification of reality, so while we might solve the model with increasing accuracy, we cannot expect it to give increasingly accurate answers to the real problem. Validations is necessary. The scientist's model has two further problems: he initial conditions and other input values are not known, and they can only estimated from measurements, and as with the mathematical model, computing power may be inadequate.
Environmentalists have especial difficulty with validation and input values for their models. Controlled, repeatable experiments are not possible (except with an analogue model such as a wave tank), and measurements are often difficult and expensive to make-particularly oceanic measurements, including meteorological measurements over the sea.
With all these difficulties and problems, why do we use models? Essentially they have two roles:
To further scientific investigation and understanding.
To provide answers for immediate application
An example of (a) is the use of a climate model to investigate the significance of atmospheric carbon dioxide. Suppose carbon dioxide continues its present rate of increase. What would be the effect upon our climate by the middle of the next century? What change of air temperature or sea level does the model predict for the increase in CO2 since the Industrial Revolution, and are these changes large enough to measure? If the model prediction and observations do not agree, how can the model- and our understanding of the processes involved-be improved? A model might suggest where measurements could usefully be made. For example, models of the Equatorial Pacific have shown how ocean waves trapped at the Equator might be involved in the El Nino which affects the climate and fisheries of Peru; measurements have confirmed the existence and importance of these waves (Philander, 1986).
This use of models, in furtherance of scientific investigation, is perfectly sound. The model is the hypothesis which can be disproved by measurements.
Design engineer and offshore operators generally use a model in role (b), and this is more questionable. They require estimates of environmental parameters-such as tomorrow's wave height or the 50-year surge level-which can often only be obtained from models which simplify the problem and which have to be chosen to utilize the available data. These estimates cannot be correct and it is important to appreciate the order of the possible error.