一个求解泊桑方程差分法的比较试验

一、方法概述 在动力气象数值计算中,经常要解椭圆型方程,泊桑方程ΔZ=F很有代表性。用差分求解这种方程,以往大都采用迭代法或局地格林函数法。迭代法的一般公式为:     

基于缓变地形的远程海啸传播模拟的改进

在Sung(2002)所提出方案的基础上,考虑了非线性的作用,使得浅水差分方程组的数值解可以近似代替精度较高但却难以计算的Boussinesq方程的理论解,通过实验将新方案与Sung(2002)所提方案进行了比较,表明新方案适用于地形变化的情况。     

正压大气非线性波的二级近似(二)——位涡度方程的求解

本文是用摄动法求取正压原始方程一类非线性大尺度慢波解的二级近似的第二部分,提出了一种在电子计算机上进行某些初等函数非数值运算的方法,由此求得了二级问题中位涡度方程的解.数值试验的结果表明,所求得的二级近似是合理的.     

自适应网格在大气海洋问题中的初步应用

自适应网格法是80年代兴起的通过求解椭圆型方程的边值问题来数值生成网格的一种新方法。它是在任意形状的区域上求偏微分方程的数值解的一种非常有效的工具。该方法抛弃了等距均匀的差分网格，代之以能够自动地适应所研究问题中解的特征的疏密程度不均的曲线网格。如在边界上计算网格与实际边界相重合，在区域内部可任意调节网格点的疏密程度等。本文扼要地介绍了自适应网格的原理及其构造方法。并将其应用于生成南海区域的计算网格以及数值预报台风路径的自适应网格。     

泊松方程分解解法迭代参数的一种选择

在大气动力学的研究过程中,常用位涡度方程讨论大尺度天气过程的动力学特征,一般可把位涡度写成~2φ λ~(-2)φ=ω(其中φ表示流函数)。因此求位涡方程数值解的过程中,除了要解决所谓“非线性”问题之外,还要考虑椭圆型方程的数值方法。李荣凤等说     

Interval of effective time-step size for the numerical computation of nonlinear ordinary differential equations

由于满足计算的不确定性原理,需适当选取时间步长以保证非线性常微分方程组数值解的可靠性,目前尚未见关于有效步长区间的理论结果。本文对于给定的误差限,将方法截断误差与机器舍入误差的相关曲线分别进行平移,从而得到一种确定有效步长近似区间的方法,并推导出近似区间相比于原区间的相对误差公式。另外,研究了有效步长区间随积分时间的变化规律,并对已有的数值结果给出解释。本文所得结论可为数值求解常微分方程组选取有效步长并得到可靠的数值解提供理论支持。     

论数值预报发展的途径

数值预报作为预报天气的一种方法,发展至今已有二十多年的历史了。认真总结二十多年来正反两方面的经验,对于指导今后的工作是有益的。里查逊(1922)首先近似地求流体—热力学方程的数值解,进行了六小时的预报,结果失败了。后来罗斯贝(1939)总结了天气分析预报的实践经验,提     

第四讲 数值方法(一)——差分法及线性计算不稳定

数值天气预报就是对在给定的初始条件和边界条件下这组控制方程求解的问题。一般很难求得解析解,而只能采用数值积分方法近似求解。数值求解微分方程的方法有:有限差分法,有限元法,谱展开法等。本讲仅介绍有限差分法。     

砧状积雨云发展的一个动力学分析

本文在考虑了水汽凝结的反馈影响后,用近似方法求出了积云动力学方程组的解。对解的一个数值计算表明:积云强烈的发展最后必然会使云顶出现砧状结构,从而发展成砧状积雨云。     

Fjφrtoft图解方法的一些统计研究

在缺乏电子计算机的条件下,如果要把数值预报方法应用到日常天气预报工作中去的话,Fjφrtoft(1952)提出的正涡度方程的图解积分方法是有实际意义的.这方法的主要精神是:在计算涡度平流时,引进平均流场(?场)代替实际流场(H 场)作推移场而一次推12或24小时,因而大大减少了所需的运算,在由涡度变化计算变高时,应用所推得的 Poisson 方程的近似图解积分公式,这方法按其简便迅速来说,已可以供日常预报应用,并且还可以推广到许多别的情形,不过在这样做之前,我们有必要先弄清楚这个方法究竟近似到何种程度.     

The synoptic meaning of helicity

Summary Helicity, a scalar quantity resulting from the inner product of velocity and vorticity, has until now mostly been used in the field of mesoscale meteorology and boundary layer meteorology. Goals of this paper are the derivation of the flux form of the helicity equation in general form without neglection of friction and Coriolis force and the examination of helicity patterns of larger scales. The general helicity equation is approximated for the synoptic and frontal scale by use of the scale analysis. High helicity values are bounded to fronts, such that helicity determines their positions. Finally the helicity patterns at different heights and helicity sources and sinks are discussed for a case study of a cyclogenesis over the Atlantic.With 6 Figures     

弱频散条件下大气运动的非线性演变过程

研究非线性过程对大气运动影响的一个常用方法是在一定条件下,将大气运动力学方程化为一些熟知的典型非线性演变方程,再利用数学中已有的结果进行讨论。Long、Redekopp等从地转涡度方程出发,证明了迭加在带状切变气流中的长的Rossby波的演变可以用著名的Kdv方程来描述。此     

Iterative Methods for Solving the Nonlinear Balance Equation with Optimal Truncation

Two types of existing iterative methods for solving the nonlinear balance equation(NBE)are revisited.In the first type,the NBE is rearranged into a linearized equation for a presumably small correction to the initial guess or the subsequent updated solution.In the second type,the NBE is rearranged into a quadratic form of the absolute vorticity with the positive root of this quadratic form used in the form of a Poisson equation to solve NBE iteratively.The two methods are rederived by expanding the solution asymptotically upon a small Rossby number,and a criterion for optimally truncating the asymptotic expansion is proposed to obtain the super-asymptotic approximation of the solution.For each rederived method,two iterative procedures are designed using the integral-form Poisson solver versus the over-relaxation scheme to solve the boundary value problem in each iteration.Upon testing with analytically formulated wavering jet flows on the synoptic,sub-synoptic and meso-αscales,the iterative procedure designed for the first method with the Poisson solver,named M1a,is found to be the most accurate and efficient.For the synoptic wavering jet flow in which the NBE is entirely elliptic,M1a is extremely accurate.For the sub-synoptic wavering jet flow in which the NBE is mostly elliptic,M1a is sufficiently accurate.For the meso-αwavering jet flow in which the NBE is partially hyperbolic so its boundary value problem becomes seriously ill-posed,M1a can effectively reduce the solution error for the cyclonically curved part of the wavering jet flow,but not for the anti-cyclonically curved part.     

THE INVERSE PROBLEM OF KLEIN-GORDON EQUATION BOUNDARY VALUE PROBLEM AND ITS APPLICATION IN DATA ASSIMILATION

Since the solution of elliptic partial differential equations continuously depends on the boundary condition, the Euler equation derived from variational method cannot be solved without boundary condition. It is often difficult to provide the exact boundary condition in the practical use of variational method. However, in some application problems such as the remote sensing data assimilation, the values can be easily obtained in the inner region of the domain. In this paper, the boundary condition is tried to be retrieved by using part solutions in the inner area. Firstly, the variational problem of remote sensing data assimilation within a circular area is established. The Klein-Gordon elliptic equation is derived from the Euler method of variational problems with assumed boundary condition. Secondly, a computer-friendly Green function is constructed for the Dirichlet problem of two-dimensional Klein-Gordon equation, with the formal solution according to Green formula. Thirdly, boundary values are retrieved by solving the optimal problem which is constructed according to the best approximation between formal solutions and high-accuracy measurements in the interior of the domain. Finally, the assimilation problem is solved on substituting the retrieved boundary values into the Klein-Gordon equation. It is a type of inverse problem in mathematics. The advantage of this method lies in that it overcomes the inherent instability of the inverse problem of Fredholm integral equation and alleviates the error introduced by artificial boundary condition in data fusion using variational method in the past.     

近年来中国统计气象学的新进展

近年来，统计气象学在中国取得了长足的进展。其中主要有：将熵原理用于气象学，从而建立了熵气象学；引进了忆及过去时次资料的记忆函数，导出了大气运动的自忆性方程；将模糊数学引入气象学；将非线性动力学用于气候学研究，提出了一系列相空间预报模式；将车贝雪夫多项式推广到不规则格点，提出了一种新的时间序列预报的迭代算法；应用子波分析方法进行气候学研究；将Ｌｏｇｉｓｔｉｃ判别分析用于气象预报，研究了二次判别及逐步判别等问题；将中国科学家提出的灰色系统理论和多层递阶方法引入气象预报。此外，还引进了复经验正交分解、奇异值分解、投影追踪、主振荡模态分析等较新的统计学方法。这些方法都已在气象业务预报中发挥了作用     

中期延伸天气预报方法研究

针对未来8—17 d中期预报方法短缺的现状,将数值预报产品与历史资料有效结合和应用,以天气学、统计学为基础,通过关键区相似预报等方法,研制出了一种通过计算环流相似和气候概率实现中期延伸预报的方法。结果表明:中期延伸天气预报方法能够反映未来天气演变和降水、气温的总体变化趋势。经过检验,认为可以应用于中期天气预报业务。     

ADVANCES IN STATISTICAL METEOROLOGY IN CHINA IN RECENT YEARS

Great advances in statistical meteorology have been made in recent years in China.The mainpoints are as follows:Introducing entropy principle into meteorology,entropy meteorology isfounded;Introducing memory function,self-memorization equation of atmospheric motion isderived;The fuzzy reasoning is introduced into meteorology;Using the method of nonlineardynamics in researches of climatology,some forecasting schemes of phase space are proposed;Chebyshev polynomial is generalized at irregular grids and an iterative scheme for forecast of timeseries is proposed:The wavelet transform is used in researches in climatology;Logisticdiscrimination is used in meteorology and quadratic discrimination and stepwise discrimination areinvestigated;The theory on grey system and the multilevel recursion proposed by Chinesescientists are introduced into meteorological forecast.In addition,complex empirical orthogonalfunction,singular value decomposition,projection pursuit,principal oscillation patterns and so onare also introduced.All the above methods have played great roles in operational weatherforecasts.     

自忆性建模及其气候应用

对制约系统运动的含时间微商项的方程,引进记忆函数后可导得一个差分—积分方程,它能包容含初值的多个往值。构建和求解系统的差分—积分方程的原理,称为自忆性原理,相应导得的数学模型称为自忆性模型。自忆性模型现已应用到气象、海洋、水文、市场、农业、交通和能源等多个领域的建模、计算和预报,尤其在预报方面取得了显著成果,提高了预报准确率。文中还论述了自忆性—界门模型,讨论了自忆性模型应用效果好的原因,给出了气候预测的实例。     

Group Velocity of Anisotropic Waves-Part Ⅰ: Mathematical Expression

The group velocity used in meteorology in the last 30 years was derived in terms of conservation of wave energy or crests in wave propagation. The conservation principle is a necessary but not a sufficient condition for deriving the mathematical form of group velocity, because it cannot specify a unique direction in which wave energy or crests propagate. The derived mathematical expression is available only for isotropic waves. But for anisotropic waves, the traditional group velocity may have no a definite direction, because it varies with rotation of coordinates. For these reasons, it cannot be considered as a general expression of group velocity. A ray defined by using this group velocity may not be the trajectory of a reference point in an anisotropic wave train. The more general and precise expression of group velocity which is applicable for both isotropic and anisotropic waves and is independent of coordinates will be derived following the displacement of not only a wave envelope phase but also a     

Group velocity of anisotropic waves—Part I: Mathematical expression

The group velocity used in meteorology in the last 30 years was derived in terms of conservation of wave energy or crests in wave propagation. The conservation principle is a necessary but not a sufficient condition for deriving the mathematical form of group velocity, because it cannot specify a unique direction in which wave energy or crests propagate. The derived mathematical expression is available only for isotropic waves. But for anisotropic waves, the traditional group velocity may have no a definite direction, because it varies with rotation of coordinates. For these reasons, it cannot be considered as a general expression of group velocity. A ray defined by using this group velocity may not be the trajectory of a reference point in an anisotropic wave train. The more general and precise expression of group velocity which is applicable for both isotropic and anisotropic waves and is independent of coordinates will be derived following the displacement of not only a wave envelope phase but also a wave reference point on the phase.