Efficient Methods For Aerosol Dynamics And Thermodynamic Equilibrium Predictions

Atmospheric aerosols are suspended particles. The diameters of these particles range from a few nanometers to tens of micrometers. Aerosol modeling has recently become a significantly important application in atmospheric environmental prediction due to the major environmental impacts of aerosols on climate change and human health. Aerosols scatter and absorb the incoming solar radiation, and thus decrease the precipitation efficiency of warm clouds, thereby cause an indirect radiative forcing associated with changes in cloud properties. Meanwhile, it has also been recognized that the particles of aerosols in the sub-micrometer size range can be inhaled and thus pose certain health hazards. On the other aspects, aerosols are widely used in industry for the production of fine particles including pigments, carbon black, optical fibers, sil-icon and ceramic powders. Simulating the aerosol size and composition distribution is an invaluable tool in increasing our understanding of aerosol behavior and in determin-ing its role in atmospheric processes(Moya et al.,2002), considering aerosol dynamics and thermodynamic equilibrium models respectively.Firstly as mentioned above, more and more interest is recently being focused on the study of prediction of aerosol distributions of different chemical and dynamic processes. Many numerical methods have been studied to solve the aerosol dynamic equations such as sectional method(see Gelbard et al.,1980), moment method(see Brock et al.,1987; Seo et al.,1990), modal method(see Ackermann et al.,1998; Whitby et al.,1997), sto-chastic approach(see Debry et al.,2003), etc. The conventional sectional approach has some limitations such as numerical diffusion and lower accuracy, while the modal ap-proach has the high numerical efficiency but less physical representation of real aerosol distribution and overlap of various models. The moment method based on the aerosol physical properties tends to the chemical or physical method with single modal distrib-ution but is not suitable to practical multi-modal distribution processes. The limitation of the stochastic method is that it cannot get a satisfied error accuracy. More recently, Sandu and Borden(see Sandu et al.,2003) developed a framework of finite element methods for numerical solutions of the aerosol dynamics equations and proposed some high-order methods in time and particle size, and further, Sandu(see Sandu et al., 2006) successfully studied the piecewise polynomial approximations by combining the Runge-Kutta technique. Liang et al. The paper of Liang(see Liang et al.,2008) devel-oped a splitting wavelet method for solving the general aerosol dynamic equations on time, particle size and vertical spatial coordinate. However, one important feature of the non-linear aerosol dynamic equations is the joint effects of the advection process caused by condensation growth and the non-linear coagulation process. Meanwhile, the aerosol distribution varies highly and normally obey the very sharp log-normal distrib-utions. As we know, numerical approximations to advection-dominated problems with sharp fronts present serious difficulties. Many standard numerical methods for solving advection-dominated problems exhibit some combination of difficulties ranging from non-physical oscillations to excessive numerical diffusions at sharp fronts of solutions. Therefore, efficient high-order methods of treating the condensation advection process and the non-linear coagulation process are required for solving the aerosol dynamic equations.Since the aerosol dynamic equations are non-linear integral-differential equations, which normally have very sharp log-normal distribution solutions and are dominated by both the condensation advection and the non-linear coagulation, we consider the characteristic method. Because of the hyperbolic nature of the advection process, the modified method of characteristics was developed in the paper by Douglas(see Douglas et al.,1982) to solve convection-diffusion equations, which follow the flow by track-ing the characteristics backward from the current level grid. The method avoids the grid distortion, greatly reduces temporal errors and eliminates the excessive numerical dispersion. The method has been successfully applied in many applications such as in computation of fluid flows in porous media (see, for example, Ewing et al.,1983; Russell et al.,1985). However, the characteristic method in the paper by Douglas(see Douglas et al.,1982) is only of first-order accuracy in time. In order to improve the accuracy, recently, second-order characteristics/finite element methods were studied for linear convection-diffusion-reaction problems in the paper by Bermudez(see Bermudez et al.,2006). Thus, it is very important to develop high-order characteristic methods for efficiently and accurately simulating the aerosol dynamic equations.Secondly, to predict the concentrations of aerosol components, several aerosol thermodynamic equilibrium modules have been built such as MARS (see Saxena et al.,1986), SEQUILIB (see Pilinis and Seinfeld,1987), SCAPE (see Kim et al.,1993), and ISORROPIA (see Nenes et al.,1998,1999) based on solving the thermodynamic equilibrium equations by iterative methods. Considering the mutual deliquescence hu-midity, ISORROPIA (Nenes et al.,1998,1999;Makar et al.,2003;Metzger et al.,2002) is regarded as the most widely used module for prediction of the aerosol thermodynamic equilibrium model, in which the numerical thermodynamic model schemes generally solve the system of nonlinear gas/aerosol equilibrium equations using iterative tech-niques. The number of iterations needed for solving the equilibrium equations strongly depend on the aerosol compositions and the meteorological conditions, which actually involve huge computations in the solution process of thermodynamic equilibrium sys-tems. Because of the large cost of computation of the ISORROPIA, it is important to develop efficient methods to predict the multi-phases and multi-components aerosol thermodynamic input-output systems in order to be used in AQ forecasting models.The high dimensional model representation (HDMR) method is a new technique of quantitative model assessment and analysis tools for capturing the high-dimensional relationships between sets of input and output variables (Rabitz et al.,1999; Li et al., 2003a,b,2004; Rabitz and Alis,1999; Alis and Rabitz,2001). The method has re-cently been developed for improving the efficiency of deducing high dimensional input-output (IO) system behaviors and for relieving the computational burdens. The HDMR method is similar to a black box technique, which can be constructed from lab/field data and can efficiently predict high dimensional relationships between input variables and output variables. As a result, it avoids heavy iterations and greatly reduces the CPU-time of computation. The HDMR methods include the random sampling HDMR (RS-HDMR) (Rabitz et al.,1999; Li et al.,2003a,b) and the cut point HDMR (cut-HDMR) ((Rabitz and Alis,1999; Li et al.,2004). The technique of HDMR has been recently used in different kinds of chemical and physical simulations as an efficient mod-eling technique. Therefore, simulating the aerosol thermodynamic equilibrium model by developed HDMR is a totally new attempt.Under the aborative guidance of Professor Wenqia Wang and Dong Liang, the author has finished this dissertation consisting of some work on efficient methods for aerosol dynamics and thermodynamic equilibrium models. On the aspect of aerosol dynamics, we develop an efficient second-order characteristic finite element method for solving the problem. A high accurate characteristic method is proposed to treat the condensation advection while a second-order extrapolation along the characteristics is proposed to approximate the non-linear coagulation. The method has second order accuracy in time and the optimal-order accuracy of finite element spaces in particle size, which improves the first-order accuracy in time of the classical characteristic method. On the other aspect of aerosol thermodynamic equilibrium, we develop a new and efficient approach for high dimensional atmospheric aerosol thermodynamic equilibrium predictions. The multi-phase and multi-component aerosol thermodynamic input-output systems are solved by the high dimensional model representation (HDMR) method combining with the moving multiple cut points. It can simulate efficiently the atmospheric aerosol thermodynamic equilibrium problems in a large range of aerosol concentrations (10-10(mol/m3) - 10-6(mol/m3)). Numerical experiments show the efficient performance of our method for these two problems.The dissertation is divided into three chapters.In Chapter 1, we introduce the background of aerosol including aerosol dynamics and thermodynamic equilibrium models. The definition and effects of aerosol are in-terpreted firstly. We give the expression of aerosol dynamic equation in Section 1.2. Each process of aerosol dynamic are discussed in detail. In Section 1.3, we represent the equilibrium equation of aerosol. Meanwhile, all components involved in aerosol thermodynamic equilibrium models are listed. Two systems consisting of inland case and sea case are taken into consideration.In Chapter 2, we consider the non-linear aerosol dynamic equation on time and particle size, which contains the advection process of condensation growth and the process of non-linear coagulation. We develop an efficient second-order characteristic finite element method for solving the problem. In this chapter, first, the time derivative and the condensation advection are transferred to the directional derivative along the characteristics and then discrete it by the difference along the characteristics which are accurate characteristic solutions of the characteristic equations. Second, for treating the non-linear coagulation on the right side of the equation, we propose second-order extrapolation along the characteristics where two previous level values are used along the characteristics. Combining these two efficiently treating techniques and the finite element method, we develop a high-order characteristic time-stepping procedure for the aerosol dynamic equations. The developed method has second-order accuracy in time and allows for large time steps in a simulation of high accuracy. It eliminates the exces- sive numerical dispersion and overcomes the oscillation at the sharp fronts of solutions. Numerical experiments show the efficient performance of our method for problems of log-normal distribution aerosols in both the Euler coordinates and the logarithmic co-ordinates. The results in this chapter have been published on the high level SCI journal "International Journal for Numerical Methods in Engineering"(Impact Factor:2.229).In Chapter 3, we develop an efficient HDMR approach combining with moving cut points for high dimensional atmospheric aerosol thermodynamic equilibrium pre-dictions on multi-phases and multi-components. In a standard cut-HDMR method, one cut point is used. But, if the inputs are far away from the cut point, the prediction results are not accurate. Consequently, the multicut-HDMR method (Li et al.,2004) was introduced by using multi-cut points and numerical errors depend on the multi-cut points. However, it is a difficult task of determining the multi-cut points for obtaining highly accurate approximations for large high dimensional domains. In this chapter, we propose the HDMR approach by combining the moving cut points for modeling and predicting the aerosol thermodynamic relationships based on the chemical and physical features of aerosols. This approach is a HDMR method combining with moving mul-tiple cut points for high dimensional input-output systems. The proposed approach improves the accuracy of numerical simulations for general high dimensional systems comparing with the standard cut-HDMR method. The numerical experiments show that the approach has great computational efficiency and the CPU-time of the ap-proach is much less than that of the ISORROPIA aerosol thermodynamic equilibrium module. For the actual example, the method obtains very accurate results in a high dimensional domain with a large range of aerosol concentrations from 10-10 to 10-6 mol/m3 in the area of Beijing, China, which are compared with those computed by ISORROPIA. Moreover, the approach also produces accurate particulate matter (PM) concentrations compared with those predicted by MARS, SEQUILIB, EQUISOLV and ISORROPIA aerosol thermodynamic modules. One whole day numerical prediction of aerosol thermodynamic equilibrium system in the Beijing area will also be simulated by the approach. The results in this chapter have been published on the top level SCI journal "Atmospheric Environment"(Impact Factor:2.89).The published chapters in this dissertation were supported by the National Basic Research Program (973) of China under the grant 2006CB403703. The data of the aerosol prediction in the numerical tests were offered by the 973 sub-project group.