Despite the substantial benefits o f numerical models it is also o f interest here to describe some of the difficulties which a modeller typically encounters. The value o f any numerical model is determined by its abihty to reproduce as accurately as possible measured parameters under as many different conditions as possible. During the development stage o f a model, comparisons with existing measurements are carried out. Once a model has been tested and verified it can be used to predict conditions which have not yet been measured. As long as the model successfully reproduces measurements within tolerable error boundaries it is assumed that the physical processes have been correctly described mathematically. Strictly, however, a deep understanding of the physics is not always provided by models. In reality, model output can only be compared to a limited number o f observations and will only match a selection o f these, while other measurements can differ substantially fi*om the simulated predictions. The modeller does not expect perfect agreement since he is often aware o f some o f the limitations o f his model. The most common sources o f error include ambiguous and over-simplified parameterizations o f certain variables, incomplete descriptions o f processes such as chemistry, controversial parameters such as chemical reaction rates and problematic numerical schemes. If a model delivers inaccurate results one key problem is to identify which o f the above listed categories is most likely to be responsible. This can often take long to identify, and essentially the question is whether the
Background______________________________________________________________Chapter I
problem is o f “scientific” or “technical” nature. The first three o f the above listed possible error sources would fall into the category o f scientific errors, while the last would be technical.
Technical difficulties in models often cause results which contradict expected behaviour. One particular category are problems with numerical stability. Finite difference techniques which are used when differentiating and integrating numerically (see also III.3.9) may become unreliable under various circumstances. When parameters change very little compared with their magnitude, gradients become very small and the digit-precision insufficient. Small gradients can easily fluctuate by factors o f 10 or more which in some calculations may produce similar or larger fluctuations o f other important parameters. These random noise errors either become evident immediately in the data output or they build up slowly and suddenly appear in the form of often localized “bursts” o f a parameter value. Occasionally, these cause a simulation to stop running. Numerical smoothing is necessary to remove these errors. When discovered, the modeller will have to trace back the source o f an error since the most likely case is that the variable which became unstable in the output was influenced by another variable which might have influenced many other model parameters even though they may not appear unrealistic at the time. Tracing back the sources o f numerical errors and correcting them is often a very time consuming process.
Identifying and reducing scientific errors can similarly be difficult if the exact source o f the faults is not known. Comprehensive atmosphere models, such as that used in this thesis, contain a large number o f coupled equations which ünphes that the results of one calculation depend on the output of another. If the variable passed between two sets of calculations is a measurable physical quantity its values can be verified and the error source more easily identified. This, however, is not always the case and the modeller may in some cases prefer to parameterize certain quantities in order to decouple as far as possible the calculations, thus making the code more easily traceable. Still, a certain amount o f ambiguity is often inevitable and models sometimes contain empirical factors which are introduced to match the observations better. The scientific reason behind such a factor is not always known.
From these general remarks it is evident that there is not only ambiguity in the sources of errors produced by numerical models but that a correct model output is no guarantee for the accuracy of calculations. If a systematic error is present it will usually be identified when compared to several measurements, preferably from different geographic locations, but it cannot fully be ruled out that
Background______________________________________________________________Chapter I
several errors compensate each other under many simulation conditions and might not be noticed by the modeller for a long time. It is therefore important to thoroughly test new models or improvements to existing code under various conditions. This has become easier nowadays with faster and cheaper computers, while previously modellers often had to work with a minimum num ber o f runs not only because o f long run times but also because high costs for computers constrained them to systems shared with other users. Chapter V is dedicated to validation o f the changes made to the CTIM code, as described in Chapter III, section III. 5
Classical Tidal Theory___________________________________________________Chapter II