In conclusion, many of the presented results are in agreement with similar studies.
The most striking difference is the fact that negative local largest Lyapunov expo-nents were not previously observed for quasi-geostrophic experiments, even though
4.8 Conclusions 63
they are from a theoretical point of view simpler than the primitive equation model used here. It seems that the primitive equations have more potential to produce such behaviour.
65
Chapter 5
Summary and outlook
5.1 Summary
In this study, the dynamical systems theory is employed to assess predictability in the global atmospheric circulation Model PUMA. The general experiment setup fea-tures the model run in two parallel instances such that one of the instances serves as a reference run, while the other run is the main run that is used for the investigation.
The focus of the predictability analysis lies on the assessment of the growth of errors.
For this purpose the two model runs are initialised with slightly different initial con-ditions and the evolution of this difference is subsequently analysed. The mean rate with which this difference grows with time is the global Lyapunov exponent. Larger values for these exponents indicate lower predictability and fast error growth. In non-chaotic systems where errors are not amplified by the dynamics this exponent vanishes. A sensitivity study is conducted where the Lyapunov exponent is calcu-lated for different values of the pole-equator temperature gradient that drives the model. Very low gradients result in vanishing Lyapunov exponents and consequently non-chaotic dynamics, while larger gradients show increasing Lyapunov exponents indicating a dependence of predictability on the temperature gradient. Furthermore, a clear dependence on the model resolution is found as Lyapunov exponents in the T42 resolutions were approximately twice as large as the corresponding exponents in the T21 case. In addition to the largest Lyapunov exponent, the Lyapunov spec-trum provides further insight into the dynamics of the system. Since the specspec-trum is not available directly, an indirect method to approximate the number of positive Lyapunov exponents is applied. This method utilises the link between different at-tractor dimensions and the Lyapunov spectrum to give a first guess for the number of positive Lyapunov exponents.
This global assessment of predictability is supplemented by a local analysis. Here the growth rate of errors is analysed for shorter time intervals in the order of days.
The result is a local largest Lyapunov exponent that is dependent on the state of the system. The mean value of the local largest Lyapunov exponent is identical to the global largest Lyapunov exponent, however, individual values fluctuate and there are periods found where the exponent becomes negative. These periods are
especially common for low resolution experiments. A statistical evaluation of these fluctuations shows that they were nearly Gaussian and that the probability to ob-serve negative values decreases with the length of the time interval for which the growth is assessed. Negative values of error growth become increasingly unlikely with increasing horizontal resolution and therefore increasing degrees of freedom.
The fluctuations are checked whether they obey the relations of the fluctuation the-orem. Therefore a relation between error growth and the entropy production of the system is established and subsequently the growth rates are used as surrogates for the actual entropy production that is not available. Consequently the applicability of the fluctuation theorem can only be shown in an approximated way. The analysis confirms that the fluctuations are in accordance with the fluctuation theorem in this context. Additionally it is investigated whether larger initial errors show the same growth characteristics than the infinitesimal errors used before. According to the-ory the whole analysis is only valid for small errors, however, no different behaviour is found even for clearly non-infinitesimal errors. The deviations of the considered distributions from Gaussianity is assessed and skewness-kurtosis relations, common in many system, are investigated.
While these methods provide a unique insight into the model dynamics they provide very little information about the spatial distribution of error growth. For this reason the difference itself is analysed and its distribution evaluated. The mid-latitudes are identified as the regions with the largest mean error growth but also with the largest fluctuations. To put this result into perspective traditional methods to assess predictability are used for comparison. The Eady growth rate shows distinct differences as the region of largest growth rates is shifted equator-wards, while the analysis of the deviations of potential vorticity from the zonal mean show very similar distributions with regard to the error growth pattern. The reasons for the differences are discussed and most probably it is due to the different definitions of instability.
The study concludes with the attempt to find a link between the error growth rates and blocking. Blocking is of general importance due to its persistent nature.
During a blocking period the atmospheric circulation is very stable in the vicinity of the block and it is theorised that this would be visible in low or even negative values of error growth. However, the link can not be established as the global analysis of the error growth shows no correlation with large, but on a global scale still local phenomena like blocking.