Predictions of Temperature Changes by IPCC Specialists and the Importance of Climatic Scenarios

1. Predictionsabout the changes of the temperature made by the IPCC specialists

2. Why is it useful to use climatic scenarios?

3. IPCC – Intergovernmental Panel on Climate Change 1. Duringthe last years everybody or at least nearly everybody is discussing climatic changes, but it is remarkable that IPCC wascreated by the UNmany years ago. 2. The global warming is the major and perhaps the most important consequence of the climatic changes. 3. Predictions of possible changes of the temperature were prepared and discussed in many reports of IPCC. 4. It is worthwhile to answer the question: will the changes of the temperature lead to changes of some pollution levels? The answer is “Yes” because the pollutants are not only transported by the wind. The pollutants also participate in many chemical reactions during the transport and some of the chemical reactions depend on the temperature. This is why it is important to study the influence of the climatic changes on pollution levels which might be increased and, therefore, might becomepotentiallydangerous for human health.

4. Contents of the talk 1. Mathematical models and difficulties related to their treatment: (A) Non-linear systems of PDEs (B) Several other tasks have to be treated after the discretization 2. Additional difficulties when climatic scenarios are used: (A) Need to run the model during many consecutive years (B) Need to use many scenarios and to compare the results (C) Need to handle huge input and output data files 3. Using classical tools during the treatment of the models (A) Fast and efficient numerical methods (B) Splitting procedures 4. It is necessary to run the models on high performance computers (A) Parallel tasks (B) Exploiting efficiently the cache memories 5. Numerical results (A) At a given point (points) of the space domain of the model (B) In the whole space domain or in selected parts of it

5. Non-linear systems of partial differential equations (PDEs) + or 168

6. Splitting procedure or 168 Major advantages: the large problem is replaced by small problems no need for artificial boundary conditions different methods for the different sub-models many parallel tasks after the space discretization Major disadvantage: low order of accuracy “The choice of a numerical method is nearly always a question of achieving an acceptable compromise when large problems are solved”

7. Space discretization or 168 The first system is obtained by applying central differences. It is not stiff and can be handled by using explicit methods for solving ODEs. The second system can be obtained directly (there are no spatial derivatives in the second sub-model), but it is stiff, badly scaled and ill-conditioned, which causes the major problems during the numerical treatment of the model. The third system is obtained by using backward differences. It is linear and mildly stiff. The Trapezoidal Rule works sufficiently well.

8. Number of equations The number of equations depends on the four parameters: . The numbers of equations for the different used by us values of these parameters are given in Table 1. Table 1 Numbers of equations for different choices horizontal cells Fine spatial resolution is necessary if we wish to study the situation in a small part of the space region (say, Denmark) or even to get some more reliable information about the pollution in a given city (say, Paris). However, this implies the use of a small time-step: seconds was consistently used, which results in time-steps in a one-year run.

9. NO2 pollution in Denmark (coarse grid) Red colours: indicate high pollution levels Blue colours: low polluted regions In this case, the coarse grid does not reproduce in a very realistic way the actual situation

10. NO2 pollution in Denmark (fine grid) Red colours: indicate high pollution levels The fine grid produces qualitatively correct results Why? Because the high pollution levels are located where the emissions are also high

11. Parallel tasks (A) The computations for each of the horizontal planes and for each of the chemical species can be carried out in parallel at every time-step when (1) is handled. This means that there are parallel tasks and each of these tasks contains equations. (B) The computations at each of the spatial grid-points can be carried in parallel at every time-step when (2) is solved. This means that there are parallel tasks and each of these tasks contains equations. (C) The computations at each of the vertical grid-lines and for each of the chemical species can be carried in parallel at every time-step when (3) is solved. This means that there are parallel tasks and each of these tasks contains equations.

12. Numbers and sizes of the parallel tasks

13. Efficient exploitation of the cache memory The parallel tasks are different in size. In order to exploit better the cache memory, it is necessary either (1) to split the large tasks in several smaller tasks or (2) to combine several small tasks in a larger task. The second problem is relatively easy, while the first one can cause difficulties. Main purpose: Organize the computations so that a “small” amount of data is used as long as possible in the computations. The actual size of“small” amount of data depends on the cache memories of the available computer architecture. The idea is quite simple, but it may be difficult to implement it.

14. Optimal size of the chunks

15. Speed for the optimal chunks

16. Numerical algorithms for the first system of ODEs The parallel tasks in (1) are non-stiff and can be handled by explicit methods for solving ODEs. It is necessary to be careful and to attempt to preserve the stability during the computations. The eigenvalues of the shifted Jacobian matrix of are close to the imaginary axis and can cause difficulties when the wind velocity coefficients are large. Some predictor-corrector schemes with several different correctors work sufficiently well. Other explicit methods for solving ODEs can also be used. Basic linear algebra operations (mainly matrix-vector computations) are the most time consuming part of the computational work during the treatment of the first sub-model when explicit methods for solving ODEs are used. Efficient kernels for performance of such operations are normally available and can be used (also to divide the large tasks into smaller tasks).

17. Sparse matrices appearing in the second system of ODEs The parallel tasks in (2) are many, but each of these tasks is small. Moreover, these tasks are very stiff, badly-scaled and ill-conditioned. Therefore, implicit methods for solving ODEs must be used and systems of non-linear algebraic equations must be handled, normally by applying some version of the well-known Newton Iterative Method at every time-step. Systems of linear algebraic equations are to be solved in an inner loop during the Newton Iterative Procedure. The coefficient matrices of these systems are sparse, but the application of general methods for solving linear algebraic systems with sparse matrices is not efficient, because the involved matrices are too small. Therefore, it was necessary to develop a special algorithm for the sparse matrices arising in the treatment of the chemical scheme.

18. Special algorithm for the small matrices related to the chemical scheme Main problems in the treatment of general sparse matrices: 1. Indirect addressing 2. Many integer arrays 3. Pivotal search 4. Pivotal interchanges If the matrix is large, then the cost of the performance of these four actions is small in relation to the total computational work, but if the matrix is small, then this is not the case. This is why it was necessary to develop a special algorithm, applicable only for the sparse matrices related to the chemical scheme. In this algorithm there is no indirect addressing, no use of integer arrays, no pivotal search and no pivotal interchanges. The obtained by applying this algorithm LU-factorization is used as a preconditioner in the solution of the system of linear algebraic equations.

19. Numerical algorithms for the second sub-model Several numerical have been tried in the treatment of the second sub-model. These methods are listed in the table given below. The Backward Euler Formula is used in the runs results of which will be presented in this talk. This method is both very simple and has excellent stability properties.

20. Numerical algorithms for the third system of ODEs The parallel tasks in (3) are linear but mildly-stiff and, therefore must be handled by implicit methods for solving ODEs. This causes in general problems because systems of linear algebraic equations are to be handled at each time-step. However, in our case the coefficient matrices of the systems of linear algebraic equations are tri-diagonal and efficient kernels for handling such problems are easily available. These kernels can also be used when several small tasks are to be combined in a larger task. The well-known Trapezoidal Rule is suitable for the treatment of the parallel tasks related to the third sub-model

21. Calculating different model results 1. Comparing measurements with model results: (A) by using scatter plots (B) by presenting results at some given sites 2. Sensitivity scenarios: (A) systematic changes of human-made emissions (B) systematic variations of chemical rates 3. Climatic scenarios (A) predicted by the IPCC specialists temperature changes (B) development of three climatic scenarios (C) results obtained at different sites in Europe (D) results obtained in the whole European area (E) results obtained in several European countries 4. Concluding remarks

22. Calculated versus measured concentrations: scatter plots

23. Calculated versus measured sulphate concentrations at eight stations

24. Sensitivity scenarios 1. Sensitivity of the concentrations of some chemical species to changes of the human-made emissions. 2. Sensitivity of the concentrations of some chemical species to changes of reaction rates. Two approaches: (A) Running scenarios with different values of the related parameters (B) Using more advanced approaches based on application of some stochastic algorithms

25. Sensitivity of concentrations to changes of human-made emissions

26. Sensitivity of concentrations to changes of rates of somechemicalreactions

27. Three climatic scenarios 1. Climatic Scenario 1 - it is assumed that the temperatures are changed as predicted in the IPCC reports. 2. Climatic Scenario 2 - (a) the temperatures are varied as in Climatic Scenario 1, (b) more hot days will appear in land areas, (c) the numbers of cold and frost days will be decreased, (d) the diurnal ranges of the temperatures will be reduced. 3. Climatic Scenario 3 – (a) the same assumptions as in Climatic Scenario 2 are used, (b) more intense precipitation events, (c) increased summer drying and associated risks of drought. Most of the recommendations in the IPCC reports were taken into account in Climatic Scenario 3. IPCC – Intergovernmental Panel on Climate Change

28. Predicted climatic changes

29. AOT40 values for crops and forests

30. Bad days in eight major European cities: Climatic Scenario 3 and Basic Scenario

31. Bad days in eight major European cities: (Climatic Scenario 3) – (Basic scenario)

32. Bad days in the United Kingdom and Ireland

33. Bad days in Germany

34. Bad days in Italy

35. Bad days in Estonia

36. Bad days in Portugal

37. Bad days in 2004 and increases when the third climatic scenario is used

38. Bad days in 1989 and increases when the third climatic scenario is used

39. Bad days in Hungary (1989)

40. Bad days in Hungary (2000)

41. Bad days in Hungary (2004)

42. Bad days in Hungary (2003)

43. Major conclusions I have shown that it is possible to perform systematic investigations with a large-scale air pollution model on a fine grid over a time-period of sixteen consecutive years also when the model is used together with climatic scenarios. Many different numerical methods and procedures were tested and/or used during the treatment of the methods. It was necessary to ensure good coordination and well-balanced performance when all these methods were implemented. It is desirable to run the model over longer time-periods in order to investigate some trends in a better way.