Advanced Numerical Methods for Complex Environmental Models: Needs and Availability

An overview is presented regarding the development, structure and problems of applications of complex numerical atmospheric environmental models. Different model types based on the mass balance equation for minor reactive atmospheric constituents are briefly described. General aspects of process treatment in aerosol chemistry transport models are addressed. They include chemical mechanisms, particle dynamics and chemistry, emission of anthropogenic and natural compounds and wet and dry deposition. The role of the atmospheric boundary layer is highlighted.

Anisotropic diffusion is a common physical phenomenon and describes processes where the diffusion of some scalar quantity is directionally dependent. Anisotropic diffusive processes are for instance Darcy’s flow for porous media, large scale turbulence where turbulence scales are anisotropic in size, and heat conduction and momentum dissipation in fusion plasmas. In fusion plasmas there is extreme anisotropy due to the high temperature and large magnetic field strength. This causes diffusive processes, heat diffusion and energy/momentum loss due to viscous friction, to effectively be aligned with the magnetic field lines. This alignment leads to different values for the respective diffusive coefficients in the magnetic field direction and in the perpendicular direction, to the extent that heat diffusion coefficients can be up to 10¹² times larger in the parallel direction than in the perpendicular direction. This anisotropy puts stringent requirements on the numerical methods used to approximate the MHDequations since any misalignment of the grid may cause the perpendicular diffusion to be polluted by the numerical error in approximating the parallel diffusion. Currently the common approach is to apply magnetic field-aligned coordinates, an approach that automatically takes care of the directionality of the diffusive coefficients. This approach runs into problems in the case of crossing field lines, e.g., x-points and points where there is magnetic reconnection. It is therefore useful to consider numerical schemes that are more tolerant to the misalignment of the grid with the magnetic field lines, both to improve existing methods and to help open the possibility of applying regular nonaligned grids. To investigate this, several discretization schemes are applied to the unsteady anisotropic heat diffusion equation on a cartesian grid. All methods presented are generic and carry over to any other anisotropic diffusion problem.

The mathematical terms describing the chemical reactions introduce nonlinearity in the systems of partial differential equations (PDEs), by which most of the large-scale air pollution models are described. The numerical treatment of these nonlinear terms causes normally great difficulties. Several algorithms for handling the chemical part of a particular air pollution model, the Unified Danish Eulerian Model (UNI-DEM), are described and discussed. The ideas used in the development of these methods are rather general and can also be applied in connection with other air pollution models as well as in the numerical treatment of some systems of PDEs that arise in other areas of science and engineering.

Environmental models are usually described with systems of linear or non-linear differential equations. Due to the complexity of these equations, one cannot usually find an off-the-shelf numerical method which could provide a sufficiently accurate numerical solution, while taking reasonable integration time. Moreover, for such complicated problems it is not easy to formulate the conditions which guarantee the preservation of the different qualitative properties of the true solution. Operator splitting is a powerful tool to decompose a complex time-dependent problem into a sequence of simpler subproblems. During this procedure, the spatial differential operator of an evolutionary equation is split (decomposed) into a sum of different sub-operators having simpler forms. Then, instead of the original problem, the simpler sub-problems obtained in this way are solved successively. The application of operator splitting raises some important questions to be addressed when we aim to develop a robust modeling algorithm. Several splitting methods exist, which have different properties, and it may be rather difficult to choose the best way to perform splitting. In this chapter first we introduce the principle of operator splitting and the basic definitions related to them through the example of the simplest splitting methods with two sub-operators. Then we give the mathematical background of operator splitting in a more general framework. As we will see, the application of a splitting procedure usually gives rise to the so-called splitting error. Keeping the size of this error within reasonable bounds is extremely important for the successful application of splitting. Therefore, we analyse the order of the splitting error for the classical splitting schemes. Then we introduce further splitting methods, namely, the weighted splittings and the iterated splittings. We also examine the convergence of the different splitting methods, first in the matrix case, and then in a more general setting. In the next part some applications of operator splitting in real-life environmental models are presented.

UNI-DEM (the Unified Danish Eulerian Model) is a large-scale environmental model designed to study the transport of pollutants in Europe and the levels of pollution in different European areas. It is described mathematically by a system of partial differential equations. The discretization of the derivatives leads to huge computational tasks (involving the solution of systems containing many millions of equations), which have to be handled during many thousands of time-steps. Furthermore, long sequences of runs with different scenarios have often to be carried out. The application of different splitting procedures is very useful in the efforts to resolve successfully the extremely difficult computational tasks, because it allows us to;

(a) decompose the original huge problem into a series of small problems,

(b) select the most suitable numerical methods to every class of the involved small problems and

(c) facilitate the utilization of various techniques for parallel computations.

In many cases, the splitting procedures provide the only means for overcoming the computational difficulties related to the particular study that is to be performed by the model.

Large-scale air pollution models, which are normally described mathematically as systems of partial differential equations, must very often be run efficiently on high-speed computer architectures. The requirement for efficiency is especially important when some fine discretization of the spatial domain is to be applied. In practice, this means that an efficient implementation of such a model on fast modern computers must nearly always be achieved, because as a rule fine grids are needed in the efforts to avoid the appearance of numerical errors that are comparable with or even larger than the errors which are caused by other reasons (uncertainties of the meteorological data, of the emission data, of the rates of the involved chemical reactions, etc.). The organization of the parallel computations will be discussed in this chapter of the eBook. The major principles, on which the parallelization is based, are rather general and, therefore, some of the discussed techniques can also be applied in connection with some large-scale models arising in other areas of science and engineering.

The subjects of this chapter are the adjoint methods, which are widely used in environmental sciences. The adjoint methods are based on contemporary results of mathematical analysis, like variational calculus and functional analysis. The first nonmathematician users of the technique were the great generation of nuclear physicists in the 20th century. Actually the method was first transferred to Earth sciences by them, and has been used in this field successfully since the 1970's. The earliest Earth science applications appeared in meteorology, but for today it is a widespread technique in all branches of Earth science like oceanography or geophysics. In the 21st century its use widened to the field of almost all natural sciences, like chemistry, biology, etc. It is basically an inverse method, which utilizes the notion of adjoint operator of the considered model operator. The adjoint operator provides a duality between a model inputs and outputs, this way it is an efficient tool for sensitivity studies or for optimization problems. Probably the greatest success of the method in Earth sciences, at least in meteorology, is the basis of the so called ensemble forecasting, which is considered as the numerical forecasting method of the future. The adjoint functions act like backward signal transmitters, they can reveal the sensitive or unstable parts of a considered dynamical system. Following from this feature they have definite physical meaning, and give an insight how the given dynamical system is functioning. In this paper the most important mathematical formulations of the method are described and also the most important applications are introduced like sensitivity analysis, variational data assimilation, and finally the use of singular vectors in ensemble forecasting and in the method of targeted observations.

This chapter presents the mathematical framework to evaluate the sensitivity of a model forecast aspect to the input parameters of a nonlinear four-dimensional variational data assimilation system (4D-Var DAS): observations, prior state (background) estimate, and the error covariance specification. A fundamental relationship is established between the forecast sensitivity with respect to the information vector and the sensitivity with respect to the DAS representation of the information error covariance. Adjoint modeling is used to obtain first- and second-order derivative information and a reduced-order approach is formulated to alleviate the computational cost associated with the sensitivity estimation. Numerical results from idealized 4D-Var experiments performed with a global shallow water model are used to illustrate the theoretical concepts.

Systematic changes of the human-made emissions in Europe were simulated by applying a carefully chosen series of appropriate scenarios and the impact of these changes on the pollution levels in different parts of the model domain was studied. It was established that, while the changes of the sulphur pollutants correspond in a nearly perfect way to the changes of the emissions, for the most of the other pollutants this was not true. Furthermore, the experiments also indicate that the changes in the different part of Europe can be rather different although the emissions were reduced with the same factor. The conclusions are illustrated by many results presented in tables and plots. Several ideas for future research in this direction are briefly discussed in the end of this chapter.

The enhanced understanding of the symbiosis between urban planning, traffic dynamics and local air quality is a needed goal towards the ultimate objective of promoting healthier and sustainable cities. This chapter is focused on the most powerful tool currently available for the assessment of air quality at the street-scale (i.e., at the pedestrian level): the Computational Fluid Dynamics (CFD) models. A brief overview of the numerical concept applied in the simulation of turbulent flows and dispersion processes is given. Major questions in the CFD modeling of urban environments are highlighted: the role of vehicle emissions estimation, the contribution of background concentrations to local air quality levels, and the capability of these tools to assess the impacts of urban planning strategies and scenarios. Examples of the application of a CFD model to real urban cases are given for the cities of Lisbon (Portugal) and Helsinki (Finland).

A progressive interest has been found in the recent years on real-time simulations of air quality forecasting systems for environmental policy and management. The advances on air quality modeling have been substantial in the last decade. Nowadays, air quality modeling systems are capable to provide accurate information on the impact of different sources in relation to the total air pollution concentrations in real-time and forecasting mode. Large industrial emissions located in the surrounding areas of cities are a substantial and important part of air concentrations in the surrounding areas of the city and industrial plant. In this chapter we describe an operational forecast system used for impact of industrial sources on the air quality. The system has been implemented by using an emission model which includes anthropogenic and biogenic emissions. The system is tested in real-time and forecasting mode by implementing the “base case” on routine operational mode which includes emissions of the large industrial plants operating in an area. The system runs for specific domain architecture in ON/OFF mode (two simultaneous simulations) to extract the impact in time and space of the pollution emissions of a large industrial plant. The system is based on the TEAP (EUREKA) project, a tool to evaluate the air quality impact of industrial plants funded by EUREKA. The ON/OFF methodology helps to run the system over the different domain architecture with the anthropogenic and biogenic emissions from traffic, industrial, tertiary and domestic sectors including the targeted large industrial source (combined cycle power station, oil refinery, incinerator, etc.). This will be the ON mode. The OFF mode is exactly the same than the ON mode but switching off the emissions from the targeted large industrial source. The sensitive cases include the impact of disconnecting the emissions of each of the industrial plants completely and also other possible scenarios, as for example of disconnecting 50 % of the total emissions for each industrial plant. The results show that the modeling system is capable to determine the impact of the different emission scenarios in real-time and forecasting mode. The system can be used as an excellent tool for a possible future inmission trading EU Directive since the system identifies in time and space the percentage of inmission concentrations due to the industrial plant (or any other emission source). The system has been used in Spain as an advanced air pollution system for determination of industrial emission impacts (inmissions) of future or planed industrial or electric power plants.

In this chapter, we study the effect of initial and lateral boundary conditions (IC&LBC) on regional climate change simulations. To achieve this, different sets of IC&LBC were used: data from the ERA40 reanalysis, and global climate model HadCM3 (versions HadAM3P and HadCM3Q0) for the reference period 1961–1990, and for the future. Emission scenarios A2 and B2 were considered for 2071–2100, and A1B for 1961–2100. In case of temperature a clear connection can be found between regional climate change and the applied emission scenarios: the higher the estimated CO₂ concentration level in the scenario, the larger the projected seasonal mean warming rate. Significant warming is projected at 0.05 level for any of the A2, A1B, and B2 scenarios; the largest warming is estimated in summer. Not only the mean is likely to change, but also the distribution of daily mean temperature. Projections for precipitation involve much more uncertainty than for temperature and do not depend linearly on the estimated CO₂ change. By the end of the century the annual precipitation in the Carpathian Basin is likely to decrease, and the annual distribution of monthly mean precipitation is expected to change. Significant drying is projected in the region in summer, while in winter the precipitation is estimated to increase.