气溶胶动力学与热力学平衡预测的有效算法
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摘要
气溶胶是液态或固态微粒在空气中的悬浮体系。这些空气中悬浮颗粒的相关直径可以从几纳米到几十乃至几千微米。由于大气气溶胶在全球气候变化以及人类健康的问题中有着突出的作用,气溶胶模型也在大气环境预测中扮演了越来越重要的角色。气溶胶能够散射以及吸收外来的太阳日照和辐射,影响云层的热效应,进而影响到全球的气候。与此同时,由于气溶胶是非常微小的颗粒物质,那些直径在微米级别的细小颗粒容易被人体吸入,并进入人体内部,从而损害到人类的健康。除了这些不利的影响,另一方面,气溶胶微粒物质被广泛应用于工业生产中。这些微粒物质主要指的是细颗粒,它们在生产染料、黑炭、光纤、硅制品以及陶瓷粉等工业产品中都发挥了非常积极的作用。而这些消极和积极的作用都跟气溶胶颗粒的大小以及成分有着密不可分的关系。考虑气溶胶动力学以及热力学平衡问题时,模拟以及预测气溶胶的矢径分布和气溶胶的成分分布是研究气溶胶的两个关键性的工具(Moya et al.,2002).
首先,在第一个研究课题中,越来越多的目光集中到了研究在气溶胶化学反应和动力过程中如何预测其矢径分布。从气溶胶渐渐受到人们重视发展至今,学者们提出了非常多的数值方法以及化学方法来求解气溶胶动力学方程,其中比较经典的方法有区域方法(sectional method) (Gelbard et al.,1980)、动量法(moment method) (Brock et al.,1987;Seo et al.,1990)、模式法(modal method)(Ackermann et al.,1998; Whitby et al.,1997)、随机法(stochastic approach)(Debry et al.,2003)等等。在这些方法中,传统的区域方法对扩散项有着较大的限制并且数值精度比较低；模式方法虽然有着较高的数值精度但是缺乏物理意义并且不能有效解决多气溶胶成分时的矢径分布问题；动量方法其本质是一种化学分析方法,它是基于单种气溶胶成分的物理以及化学属性所得到的微粒大小分布,但是它也同样不适用于多成分的气溶胶矢径分布预测问题；而随机方法的最大限制就是最终得到数值结果的误差精度得不到一个满意的保证。在近期的研究成果中,Sandu和Borden(Sandu et al.,2003)构建出关于气溶胶动力学方程有限元方法的框架。有限元方法在解决此类偏微分方程无论在时间方向还是在颗粒大小方向上都有着较高的精度。进一步地,Sandu(Sandu et al.,2006)成功的将Runge-Kutta方法和分片线性多项式相结合提出时间方向更高阶的数值算法。Liang(Liang etal.,2008)在其发表的论文中提出了分片小波函数应用于气溶胶动力方程的时间以及颗粒大小这两个方向,得到了很好的精度和效果。但是上述方法都忽略了气溶胶动力方程的重要特点：第一,气溶胶的增长过程产生了方程中的对流项；第二,气溶胶的凝并过程在方程中是一个非线性项。这两项的结合构成了气溶胶动力学方程最大的特征。除此之外,根据气溶胶的观测结果,气溶胶颗粒大小的分布在欧拉坐标下变化非常剧烈而在对数坐标下呈正态分布。众所周知,对具有陡峭峰值的对流占优问题进行数值逼近是一件较为困难的事情。很多求解对流占优问题的数值算法虽然达到了相应的精度,但是是条件收敛的,不能计算所有的增长因子和凝并因子,或者对时间步长有所限制；而另一些算法则是精度得不到很好的保证。因此,高阶有效的数值方法去处理气溶胶动力学方程对流项以及非线性凝并项成为学者们热衷研究的问题。
由于气溶胶动力学方程是非线性的积分-微分方程,并且由于对流项和非线性凝并项的作用,在欧拉坐标下分布变化剧烈,呈对数正态分布,因此我们考虑特征线方法。由于偏微分方程中的对流项具有双曲特性,Douglas提出的修正特征线方法(MMOC)(Douglas et al.,1982)能够很好的求解对流扩散问题。特征线方法主要用于解决对流占优问题,它的主要思路是沿特征线方向由前一层网格的结果推出后一层网格的数值结果,这样就可以很好的解决让人难以接受的非物理震荡以及过多的数值弥散,从而磨平了尖锐的移动前沿。这种修正特征有限元方法被成功并且广泛的应用于多孔介质流问题的计算中如油藏数值模拟、海水侵入数值模拟等(Ewing et al.,1983; Russell et al.,1985).但是Douglas所提出的特征方法(Douglas et al.,1982)在时间方向只有一阶精度。为了使时间方向精度有所提高,Bermudez在他的论文中(Bermudez et al.,2006)关于线性的对流-扩散-反应问题提出了特征有限元方法,该方法在时间方向达到了二阶精度。因此提出新的改进的有效高阶特征算法,成为模拟求解气溶胶动力方程非常有意义的研究课题。
其次,在第二个研究课题中,过去很多的文章研究并发展了气溶胶热力学平衡模型的建立,比如MARS (Saxena et al.,1986), SEQUILIB (Pilinis and Seinfeld,1987), SCAPE (Kim et al.,1993),以及ISORROPIA (Nenes et al.,1998,1999).以上所有的气溶胶热力学平衡模型都是基于热力学平衡方程,因此所有的计算这些平衡模型都是用了迭代算法。与其它模型所不同的,由于考虑了共同潮解度(mutual deliquescencehumidity) ISORROPIA方法(Nenes et al.,1998,1999;Makar et al.,2003;Metzger et al.,2002)被认为是使用最广泛并且预测气溶胶热力学平衡较为准确的一个模型方法。而该模型方法的不足之处与其它方法类似,在模型计算中使用了迭代技巧去解决非线性气／液气溶胶平衡方程。其中每一次迭代求解气溶胶热力学平衡方程都依赖于这次迭代时气溶胶的成分以及外部环境(如温度,相对湿度),而这些迭代计算将耗费预测气溶胶热力学平衡模型时的大量CPU时间。由于ISORROPIA模型的计算占用大量CPU时间,因此提出关于多态多相气溶胶的热平衡学输入输出算法在预测空气质量AQ (air quality)问题中显得尤为重要。
高维数值模拟方法(high dimensional model representation (HDMR))是处理、评估、分析大量数据模型的一个有效方法,它的主要思想是捕捉高维输入以及输出变量之间的关系,文献(Rabitz et al.,1999; Li et al.,2003a,b,2004; Rabitz and Alis, 1999; Alis and Rabitz,2001)都对该方法进行了研究和扩展。HDMR方法在最近的研究成果中,被广泛应用于输入输出模型(input-output (10) system)用以减轻CPU负担提高计算效率减少计算时间中去。事实上,高维数值模拟HDMR方法的技巧类似于黑盒效应,通过实验数据或者观测数据建立高维数学模型中的输入与输出量之间的关系。而该技巧所得到的优势是,能够很好的避免严重的计算迭代并且能够大大的降低计算时间。HDMR方法主要包括随机抽样HDMR方法(random sampling HDMR (RS-HDMR)) (Rabitz et al.,1999; Li et al.,2003a,b)以及切割点HDMR方法(cut point HDMR (cut-HDMR)) (Rabitz and Alis,1999;Li et al.,2004).根据近几年的文献,HDMR的计算技巧已经被广泛应用于各种化学以及物理模拟实验计算中,成为一种解决多维问题的行之有效的办法。因此,利用改进后的HDMR方法模拟气溶胶热力学平衡模型成为一种新的尝试。
在导师王文洽教授和梁栋教授的悉心指导下,本文作者关于气溶胶动力学和热力学平衡模型做了部分研究工作。在气溶胶动力学的方面,我们提出了有效的二阶特征有限元方法解决气溶胶动力方程。高阶的特征线方法被用于处理气溶胶的增长过程,同时二阶沿特征线方向的外推格式被用于解决气溶胶的非线性凝并过程。与标准的特征有限元方法比较而言,数值算例很好地证明了该方法在时间方向达到了二阶精度；在气溶胶体积方向(空间方向)达到了最优阶。在另一方面,我们提出了一种有效的高维方法预测气溶胶热力学平衡。结合移动切割点的技巧,我们利用HDMR方法来解决多态多相的高维气溶胶输入输出平衡模型。该方法能够在大范围的气溶胶浓度环境下(10-10(mol／m3)-10-6(mol／m3)),很好地模拟大气气溶胶平衡模型。数值算例显示这两种方法在不同的气溶胶预测领域中都有非常好的表现。
全文共分三章。
在第一章中,我们主要介绍了气溶胶相关背景知识,包括气溶胶动力学以及气溶胶热力学平衡模型。首先给出气溶胶的简单定义以及在大气中发挥的主要作用。在1.2节中,我们给出了气溶胶动力方程的具体表达形式,其中每一项的气溶胶过程我们都作出了详细解释。在1.3节中,我们给出了气溶胶热力学平衡方程。同时,我们列出了气溶胶热力学平衡模型所涉及到的气溶胶成分输入量以及输出量,其中考虑了内陆地区和海洋地区两种情况。
在第二章中,根据在模拟气溶胶矢径分布时所遇到的实际问题和需要,我们关于非线性积分-微分气溶胶动力学方程提出了有效的二阶特征有限元算法。我们所提出的二阶特征有限元方法分为两步,首先,将时间方向导数与对流项相结合,把它们转换为沿特征线方向的方向导数,然后沿特征线方向做差分,这样完成气溶胶动力方程半离散有限元格式。最后,在处理方程右边的非线性凝并项上,我们提出了沿特征线方向的二阶外推格式,其中当前层的数值解由沿特征线方向前两层的数值解所得到。结合上述的两种数值技巧以及有限元方法,我们提出了关于时间方向高阶的特征有限元方法。这种方法能够很好的适应气溶胶动力方程的特点,在数值算例中也显示出其特定的优势。由于这种方法的最大特点是将传统的特征有限元格式在时间方向的精度进行了提高,因此在模拟气溶胶矢径分布时可以在时间上采用大步长并且同时保证数值逼近的精度。同时该方法继承了特征线法的优点,最大程度的减少了过多的数值弥散以及很好的克服了在尖锐解处的非物理震荡。第二章给出的数值算例包括存在精确解的数值问题以及实际气溶胶矢径分布问题,与此同时我们也在对数坐标下给出了相应的数值算例。所有的数值算例都表明,我们所提出的二阶特征有限元方法能够很好的求解对数正态分布的气溶胶动力学方程。第二章中的内容已发表在高水平SCI杂志International Journal for Numerical Methods in Engineering中(2009年影响因子2.229)。
在第三章中,对于多态多相高维的气溶胶热力学平衡模型预测,我们提出了结合移动切割点技术的有效的高维数值模拟HDMR方法。在经典的cut-HDMR方法中,有且仅有一个切割点被应用。但是,当输入值的数据偏离这个切割点较远的时候,那么预测结果可能会出现比较大的误差。为了解决计算误差的问题,相应地,多切割点高维数值模拟方法(multicut-HDMR method (Li et al.,2004))在最近的研究成果中被提出来。但是,在一个非常庞大的高维输入定义系统中,如何去选取多切割点(multi-cut points)使得误差精度得到较大的提高成为一项困难的研究课题。在这一章中,我们结合了移动切割点,根据气溶胶热力学平衡物理以及化学特性,提出了移动切割点HDMR方法去模拟和预测气溶胶热力学平衡结果。这种方法的主要思路是结合多移动切割点解决高维的输入输出系统。与经典的cut-HDMR方法比较而言,我们所提出的方法能够很大程度上的提高数值模拟的误差精度。数值实验表明我们的方法不仅在计算误差上能够达到较为满意的效果,并且在CPU计算时间上也能得到非常大的改善。在与经典的气溶胶热力学平衡模型算法ISORROPIA的比较中,我们所用的计算时间要远远低于ISORROPIA所用的时间。对于实际的气溶胶热力学平衡预测,我们的方法也取得了比较好的效果.首先,我们在大气溶胶摩尔浓度范围内10-10到10-6 mol／m3,计算结果能够得到较好的保证；而经典的HDMR方法一般输入量最多涉及到两个量级的变化。进一步地,我们利用移动切割点HDMR方法去做了关于PM (particulate matter)浓度的预测,在与其它气溶胶热力学平衡模型的比较中比如MARS、SEQUILIB、EQUISOLV以及ISORROPIA,也取得了很好的效果。北京地区一天之中的气溶胶热力学平衡模型成分浓度预测也将在出现在我们的模拟实验中。第三章中的内容已发表在国际顶尖环境SCI期刊Atmospheric Environment中(2009年影响因子2.89)。
本文研究课题为国家基础研究973项目的子课题,课题编号2006CB403703。文中所涉及到部分气溶胶实际预测的数据由973项目子课题组提供。
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.
首先,在第一个研究课题中,越来越多的目光集中到了研究在气溶胶化学反应和动力过程中如何预测其矢径分布。从气溶胶渐渐受到人们重视发展至今,学者们提出了非常多的数值方法以及化学方法来求解气溶胶动力学方程,其中比较经典的方法有区域方法(sectional method) (Gelbard et al.,1980)、动量法(moment method) (Brock et al.,1987;Seo et al.,1990)、模式法(modal method)(Ackermann et al.,1998; Whitby et al.,1997)、随机法(stochastic approach)(Debry et al.,2003)等等。在这些方法中,传统的区域方法对扩散项有着较大的限制并且数值精度比较低；模式方法虽然有着较高的数值精度但是缺乏物理意义并且不能有效解决多气溶胶成分时的矢径分布问题；动量方法其本质是一种化学分析方法,它是基于单种气溶胶成分的物理以及化学属性所得到的微粒大小分布,但是它也同样不适用于多成分的气溶胶矢径分布预测问题；而随机方法的最大限制就是最终得到数值结果的误差精度得不到一个满意的保证。在近期的研究成果中,Sandu和Borden(Sandu et al.,2003)构建出关于气溶胶动力学方程有限元方法的框架。有限元方法在解决此类偏微分方程无论在时间方向还是在颗粒大小方向上都有着较高的精度。进一步地,Sandu(Sandu et al.,2006)成功的将Runge-Kutta方法和分片线性多项式相结合提出时间方向更高阶的数值算法。Liang(Liang etal.,2008)在其发表的论文中提出了分片小波函数应用于气溶胶动力方程的时间以及颗粒大小这两个方向,得到了很好的精度和效果。但是上述方法都忽略了气溶胶动力方程的重要特点：第一,气溶胶的增长过程产生了方程中的对流项；第二,气溶胶的凝并过程在方程中是一个非线性项。这两项的结合构成了气溶胶动力学方程最大的特征。除此之外,根据气溶胶的观测结果,气溶胶颗粒大小的分布在欧拉坐标下变化非常剧烈而在对数坐标下呈正态分布。众所周知,对具有陡峭峰值的对流占优问题进行数值逼近是一件较为困难的事情。很多求解对流占优问题的数值算法虽然达到了相应的精度,但是是条件收敛的,不能计算所有的增长因子和凝并因子,或者对时间步长有所限制；而另一些算法则是精度得不到很好的保证。因此,高阶有效的数值方法去处理气溶胶动力学方程对流项以及非线性凝并项成为学者们热衷研究的问题。
由于气溶胶动力学方程是非线性的积分-微分方程,并且由于对流项和非线性凝并项的作用,在欧拉坐标下分布变化剧烈,呈对数正态分布,因此我们考虑特征线方法。由于偏微分方程中的对流项具有双曲特性,Douglas提出的修正特征线方法(MMOC)(Douglas et al.,1982)能够很好的求解对流扩散问题。特征线方法主要用于解决对流占优问题,它的主要思路是沿特征线方向由前一层网格的结果推出后一层网格的数值结果,这样就可以很好的解决让人难以接受的非物理震荡以及过多的数值弥散,从而磨平了尖锐的移动前沿。这种修正特征有限元方法被成功并且广泛的应用于多孔介质流问题的计算中如油藏数值模拟、海水侵入数值模拟等(Ewing et al.,1983; Russell et al.,1985).但是Douglas所提出的特征方法(Douglas et al.,1982)在时间方向只有一阶精度。为了使时间方向精度有所提高,Bermudez在他的论文中(Bermudez et al.,2006)关于线性的对流-扩散-反应问题提出了特征有限元方法,该方法在时间方向达到了二阶精度。因此提出新的改进的有效高阶特征算法,成为模拟求解气溶胶动力方程非常有意义的研究课题。
其次,在第二个研究课题中,过去很多的文章研究并发展了气溶胶热力学平衡模型的建立,比如MARS (Saxena et al.,1986), SEQUILIB (Pilinis and Seinfeld,1987), SCAPE (Kim et al.,1993),以及ISORROPIA (Nenes et al.,1998,1999).以上所有的气溶胶热力学平衡模型都是基于热力学平衡方程,因此所有的计算这些平衡模型都是用了迭代算法。与其它模型所不同的,由于考虑了共同潮解度(mutual deliquescencehumidity) ISORROPIA方法(Nenes et al.,1998,1999;Makar et al.,2003;Metzger et al.,2002)被认为是使用最广泛并且预测气溶胶热力学平衡较为准确的一个模型方法。而该模型方法的不足之处与其它方法类似,在模型计算中使用了迭代技巧去解决非线性气／液气溶胶平衡方程。其中每一次迭代求解气溶胶热力学平衡方程都依赖于这次迭代时气溶胶的成分以及外部环境(如温度,相对湿度),而这些迭代计算将耗费预测气溶胶热力学平衡模型时的大量CPU时间。由于ISORROPIA模型的计算占用大量CPU时间,因此提出关于多态多相气溶胶的热平衡学输入输出算法在预测空气质量AQ (air quality)问题中显得尤为重要。
高维数值模拟方法(high dimensional model representation (HDMR))是处理、评估、分析大量数据模型的一个有效方法,它的主要思想是捕捉高维输入以及输出变量之间的关系,文献(Rabitz et al.,1999; Li et al.,2003a,b,2004; Rabitz and Alis, 1999; Alis and Rabitz,2001)都对该方法进行了研究和扩展。HDMR方法在最近的研究成果中,被广泛应用于输入输出模型(input-output (10) system)用以减轻CPU负担提高计算效率减少计算时间中去。事实上,高维数值模拟HDMR方法的技巧类似于黑盒效应,通过实验数据或者观测数据建立高维数学模型中的输入与输出量之间的关系。而该技巧所得到的优势是,能够很好的避免严重的计算迭代并且能够大大的降低计算时间。HDMR方法主要包括随机抽样HDMR方法(random sampling HDMR (RS-HDMR)) (Rabitz et al.,1999; Li et al.,2003a,b)以及切割点HDMR方法(cut point HDMR (cut-HDMR)) (Rabitz and Alis,1999;Li et al.,2004).根据近几年的文献,HDMR的计算技巧已经被广泛应用于各种化学以及物理模拟实验计算中,成为一种解决多维问题的行之有效的办法。因此,利用改进后的HDMR方法模拟气溶胶热力学平衡模型成为一种新的尝试。
在导师王文洽教授和梁栋教授的悉心指导下,本文作者关于气溶胶动力学和热力学平衡模型做了部分研究工作。在气溶胶动力学的方面,我们提出了有效的二阶特征有限元方法解决气溶胶动力方程。高阶的特征线方法被用于处理气溶胶的增长过程,同时二阶沿特征线方向的外推格式被用于解决气溶胶的非线性凝并过程。与标准的特征有限元方法比较而言,数值算例很好地证明了该方法在时间方向达到了二阶精度；在气溶胶体积方向(空间方向)达到了最优阶。在另一方面,我们提出了一种有效的高维方法预测气溶胶热力学平衡。结合移动切割点的技巧,我们利用HDMR方法来解决多态多相的高维气溶胶输入输出平衡模型。该方法能够在大范围的气溶胶浓度环境下(10-10(mol／m3)-10-6(mol／m3)),很好地模拟大气气溶胶平衡模型。数值算例显示这两种方法在不同的气溶胶预测领域中都有非常好的表现。
全文共分三章。
在第一章中,我们主要介绍了气溶胶相关背景知识,包括气溶胶动力学以及气溶胶热力学平衡模型。首先给出气溶胶的简单定义以及在大气中发挥的主要作用。在1.2节中,我们给出了气溶胶动力方程的具体表达形式,其中每一项的气溶胶过程我们都作出了详细解释。在1.3节中,我们给出了气溶胶热力学平衡方程。同时,我们列出了气溶胶热力学平衡模型所涉及到的气溶胶成分输入量以及输出量,其中考虑了内陆地区和海洋地区两种情况。
在第二章中,根据在模拟气溶胶矢径分布时所遇到的实际问题和需要,我们关于非线性积分-微分气溶胶动力学方程提出了有效的二阶特征有限元算法。我们所提出的二阶特征有限元方法分为两步,首先,将时间方向导数与对流项相结合,把它们转换为沿特征线方向的方向导数,然后沿特征线方向做差分,这样完成气溶胶动力方程半离散有限元格式。最后,在处理方程右边的非线性凝并项上,我们提出了沿特征线方向的二阶外推格式,其中当前层的数值解由沿特征线方向前两层的数值解所得到。结合上述的两种数值技巧以及有限元方法,我们提出了关于时间方向高阶的特征有限元方法。这种方法能够很好的适应气溶胶动力方程的特点,在数值算例中也显示出其特定的优势。由于这种方法的最大特点是将传统的特征有限元格式在时间方向的精度进行了提高,因此在模拟气溶胶矢径分布时可以在时间上采用大步长并且同时保证数值逼近的精度。同时该方法继承了特征线法的优点,最大程度的减少了过多的数值弥散以及很好的克服了在尖锐解处的非物理震荡。第二章给出的数值算例包括存在精确解的数值问题以及实际气溶胶矢径分布问题,与此同时我们也在对数坐标下给出了相应的数值算例。所有的数值算例都表明,我们所提出的二阶特征有限元方法能够很好的求解对数正态分布的气溶胶动力学方程。第二章中的内容已发表在高水平SCI杂志International Journal for Numerical Methods in Engineering中(2009年影响因子2.229)。
在第三章中,对于多态多相高维的气溶胶热力学平衡模型预测,我们提出了结合移动切割点技术的有效的高维数值模拟HDMR方法。在经典的cut-HDMR方法中,有且仅有一个切割点被应用。但是,当输入值的数据偏离这个切割点较远的时候,那么预测结果可能会出现比较大的误差。为了解决计算误差的问题,相应地,多切割点高维数值模拟方法(multicut-HDMR method (Li et al.,2004))在最近的研究成果中被提出来。但是,在一个非常庞大的高维输入定义系统中,如何去选取多切割点(multi-cut points)使得误差精度得到较大的提高成为一项困难的研究课题。在这一章中,我们结合了移动切割点,根据气溶胶热力学平衡物理以及化学特性,提出了移动切割点HDMR方法去模拟和预测气溶胶热力学平衡结果。这种方法的主要思路是结合多移动切割点解决高维的输入输出系统。与经典的cut-HDMR方法比较而言,我们所提出的方法能够很大程度上的提高数值模拟的误差精度。数值实验表明我们的方法不仅在计算误差上能够达到较为满意的效果,并且在CPU计算时间上也能得到非常大的改善。在与经典的气溶胶热力学平衡模型算法ISORROPIA的比较中,我们所用的计算时间要远远低于ISORROPIA所用的时间。对于实际的气溶胶热力学平衡预测,我们的方法也取得了比较好的效果.首先,我们在大气溶胶摩尔浓度范围内10-10到10-6 mol／m3,计算结果能够得到较好的保证；而经典的HDMR方法一般输入量最多涉及到两个量级的变化。进一步地,我们利用移动切割点HDMR方法去做了关于PM (particulate matter)浓度的预测,在与其它气溶胶热力学平衡模型的比较中比如MARS、SEQUILIB、EQUISOLV以及ISORROPIA,也取得了很好的效果。北京地区一天之中的气溶胶热力学平衡模型成分浓度预测也将在出现在我们的模拟实验中。第三章中的内容已发表在国际顶尖环境SCI期刊Atmospheric Environment中(2009年影响因子2.89)。
本文研究课题为国家基础研究973项目的子课题,课题编号2006CB403703。文中所涉及到部分气溶胶实际预测的数据由973项目子课题组提供。
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.
引文
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[3]Arias-Zugasti M, Application of orthogonal collocation in aerosol science:Fast calculation of the coagulation tensor. Journal of Aerosol Science,37 (2006),1336-1369.
[4]Ansari, S., Pandis, S.N., An analysis of four models predicting the partitioning of semevolatile inorganic aerosol components, Aerosol Science and Technology,31 (1999), 129-153.
[5]Ansari, A.G., Pandis, S.N., Prediction of multicomponent inorganic atmospheric aerosol behavior, Atmospheric Environment,33 (1999),745-757.
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[9]Brock JR, Oates J, Moment simulation of aerosol evaporation, Journal of Aerosol Science,18 (1987),59-64.
[10]Capaldo, K., Pilinis, C., Pandis, S.N., A computationally efficient hybrid approach for dynamic gas/aerosoltransfer in air quality models, Atmospheric Environment,34 (2000),3617-3627.
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[15]Domilovskii ER, Lushnikov AA, Piskunov VN, Monte Carlo simulation of coagulation processes, Dokl. Akad. Nauk SSSR Ser. Phys. Chem.,240 (1) (1978).
[16]Debry E, Sportisse B, Numerical simulation of the general dynamic equation(GDE) for aerosols with two collocation methods, Applied Numerical Mathematics,57 (2007), 885-898.
[17]Debry E. Jourdain SB, A stochastic approach for the numerical simulation of the general dynamics equation for aerosols, Journal of Computational Physics,184 (2003), 649-669.
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[10]Capaldo, K., Pilinis, C., Pandis, S.N., A computationally efficient hybrid approach for dynamic gas/aerosoltransfer in air quality models, Atmospheric Environment,34 (2000),3617-3627.
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[14]Dhaniyala S, Wexler A, Numerical schemes to model condensation and evaporation of aerosols, Atmos.Environ.30 (1996),919-928.
[15]Domilovskii ER, Lushnikov AA, Piskunov VN, Monte Carlo simulation of coagulation processes, Dokl. Akad. Nauk SSSR Ser. Phys. Chem.,240 (1) (1978).
[16]Debry E, Sportisse B, Numerical simulation of the general dynamic equation(GDE) for aerosols with two collocation methods, Applied Numerical Mathematics,57 (2007), 885-898.
[17]Debry E. Jourdain SB, A stochastic approach for the numerical simulation of the general dynamics equation for aerosols, Journal of Computational Physics,184 (2003), 649-669.
[18]Douglas J, Russell TF, Numerical methods for convection dominated diffusion prob-lems based on combinning the method of characteristics with finite element or finite difference procedures, SIAM J.Numer.Anal., Vol 19 (1982),871-885.
[19]Ewing RE, Russell TF, Wheeler MF, Simulation of miscible displacement using mixed methods and a modified method of characteristics, SPE 12241, Reservoir Simulation Symposium, San Francisco, CA, (1983),71-151.
[20]Fernndez-Daz JM, Gonzlez-Pola Muiz C, M.A. Rodrguez Braa, B. Arganza Garca, P.J. Garca Nieto, A modified semi-implicit method to obtain the evolution of an aerosol by coagulation, Atmos. Environ.,34 (2000),4301-4314.
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[22]Fitzgerald, J.W., Marine aerosols:a review. Atmospheric Environment,25A (1991), 533-545.
[23]Gelbard FM, Seinfeld JH, Numerical solution of the dynamic equation for particulate systems, Journal of Computational Physics,28 (1978),357-375.
[24]Gelbard F, Tambour Y, Seinfeld JH, Sectional representations for simulating aerosol dynamics, Journal of Colloid and Interface Science,76 (1980),541-556.
[25]Gray, H.A., Cass, G.R., Huntzicker, J.J., Heyerdahl, E.K., Rau, J.A., Characteristics of atmospheric organic and elemental carbon particle concentrations in Los Angeles, EnvironmentalScience and Technology,20 (1986),580-589.
[26]Kalani A, Christofides P, Nonlinear control of spatially inhomogenous aerosol process, Chemical Engineering Science,54 (1999),2669-2678.
[27]Kim YP, Seinfeld JH, Saxena P, Atmospheric gas aerosol equilibrium. Ⅰ:thermody-namic model, Aerosol Science and Tecknology,19 (1993),157-181.
[28]Kim YP, Seinfeld JH, Saxena P, Atmospheric gas aerosol equilibrium, Ⅱ:analysis of common approximations and activity coefficient calculation methods, Aerosol Science and Tecknology,19 (1993),182-198.
[29]Kim, Y.P., Seinfeld, J.H., Atmospheric gas-aerosol equilibrium. Ⅲ. Thermodynamics of crustal elements Ca2+, K+ and Mg2+, AerosolScience and Technology,22 (1995), 93-110.
[30]Kulmala, M. Condensational growth and evaporation in the transition regime:An analytic expression, Aerosol Science and Technology,19 (1993),381-388.
[31]Jacobson, M.Z., Studying the effects of calcium and magnesium on size-distributed nitrate and ammonium with EQUISOLV II, Atmospheric Environment.33 (1999), 3635-3649.
[32]Jacobson, M.Z., Tabazadeh, A., Turco, R.P., Simulating equilibrium within aerosols and nonequilibrium between gases and aerosols, Journal of Geophysical Research,101 (1996),9079-9091.
[33]Jacobson, M.Z., Development and application of a new air pollution modeling sys-temFⅡ:aerosolmodul e structure and design. Atmospheric Environment,31A (1997), 131-144.
[34]Jacobson, M.Z., Numericaltechniques to solve condensationaland dissolutional growth equations when growth is coupled to reversible reactions, Aerosol Science and Technol-ogy,27 (1997),491-498.
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[36]Jacobson M.Z., Turco, Jensen, Toon, Modeling coagulation among particles of different composition and size, Atmos. Environ.,28 (7) (1994),1327-1338.
[37]Jung CH, Park S.H, Kim P.Y, Size distribution of polydispersed aerosols during condensation in the continuum regime:Analytic approach using the lognormal moment method, Journal of Aerosol Science,37 (2006),1400-1406.
[38]Lee, K.W. Change of particle size distribution during Brownian coagulation, Journal of Colloid Interface Science,92 (1983),315-325.
[39]Liang D, Guo Q, Gong S, A new splitting wavelet method for solving the general aerosol dynamics equations, Journal of Aerosol Science,39 (2008),467-487.
[40]Li G, Rosenthal C, Rabitz H, High dimensional model representations, Journal of Physical Chemistry A,105 (2001),7765-77.
[41]Li G, Artamonov M, Rabitz H, Wang SW, Georgopoulos PG, and Demiralp M, High-dimensional model representation generated, from low order terms-lp-RS-HDMR, J Comput Chem,24 (2003),647-656.
[42]Li G, Rabitz H, Wang SW, and Georgopoulos PG, Correlation method for reduction of Monte Carlo integration in RS-HDMR, J. Comput. Chem.,24 (2003),277-283.
[43]Li G, Schoendorf, Ho TS, and Rabitz H, multicut-HDMR with application to an Ionospheric model, J. Comput. Chem.,25 (2004),1149-1156.
[44]Li G, Wang SW, Rosenthal C, and Rabitz H, High dimensional model representions generated from low dimensional data samples. I.mp-Cut-HDMR, Journal of Math-emetical Chemistry,30 (2001),1-30.
[45]Lorentz G, Golitschek M, and Makovoz Y, Constructive approximation, Springer. New York, (1996).
[46]Makar P, Bouchet V, Nenes A, Inorganic chemistry calculations using HETV-a vectorized solver for the sulfate-nitrate-ammonium system based on the ISORROPIA algorithms, Atmospheric Environment,37 (2003),2279-2294.
[47]Meng, Z.Y., Seinfeld, J.H., Time scales to achieve atmospheric gas-aerosol equilibrium for volatile species, Atmospheric Environment,30 (1996),2889-2900.
[48]Meng, Z.Y., Dabdub, D., Seinfeld, J.H., Size-resolved and chemically resolved model of atmospheric aerosoldynamics, Journalof Geophysical Research,103 (1998),3419-3435.
[49]Meng, Z.Y., Seinfeld, J.H., Saxena, P., Kim, Y.P., Atmospheric gas-aerosolequil ibrium. Ⅲ. Thermodynamics of carbonates, AerosolScience and Technology,23 (1995),131-154.
[50]Metzger S, Dentener F, Pandis S, and Lelieveld J, Gas/aerosol pratitioning:a computatinally efficient model, Journal of Geophysical research,107 D16(2002),1-24.
[51]Moya M, Pandis SN, Jacobson, Is the size distribution of urban aerosols determined by thermodynamic equilibrium? An application to Southern California, Atmospheric Environment,36 (2002),2349-2365.
[52]Moya, M., Ansari, A., Pandis, S.N., Partitioning of nitrate and ammonium between the gas and particulate phases during the 1997 IMADA-AVER study in Mexico City, Atmospheric Environment,35 (2002),1791-1804.
[53]Nenes A, Pandis S, and Pilinis C, ISORROPIA:A new thermodynamic equilibrium model for multiphase multicomponent inorganic aerosols, Aquatic Geochemistry,4 (1998),123-152.
[54]Nenes A, Pandis S, and Pilinis C, Continued development and testing of a new thermodynamic aerosol module for urban and regional air quality models, Atmospheric Environment,33 (1999),1553-1560.
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