二阶Adams-Bashforth格式的垂直敏感性分析

蛙跳(leapfrog)时间差分格式采用Asselin-Robert时间滤波方案去除计算解能够降低原始方程组的时间差分格式的计算精度,采用二阶Adams-Bashforth格式构造的欧拉前差方案可弥补蛙跳格式的不足,即:在不存在计算解的条件下去除滤波的影响,更大程度的保持方程组的计算准确性。本文基于NCAR CAM3.0(Community Atmosphere Model 3.0)完善的软件平台,将原模式的三时间层蛙跳时间差分方案修改为两时间层二阶Adams-Bashforth时间差分格式,对与重力波有关项使用中央差隐式处理,以此构建半隐式大气环流谱模式。利用动力检验的方法探讨模式对垂直分辨率的敏感性,从而寻找模式在较小计算代价下提高计算效果的可能。通过斜压波实验发现,提高垂直分辨率使模式具有更强的斜压波模拟能力,其模拟效果甚至已经与提高水平分辨率的效果相当,可以作为一种弥补模式运算效率不足的可行方案。     

蛙跳格式的替代方案及其在大气环流模式中的应用

蛙跳（Leapfrog）时间差分格式采用AsselinRobert时间滤波方案去除计算解能够降低原始方程组的时间差分格式的计算精度,采用二阶Runge-Kutta格式构造欧拉前差方案可弥补蛙跳格式的不足。即在不存在计算解的条件下去除滤波影响,更大程度保持方程组的计算准确性。作者基于NCAR CAM3.0(Community Atmosphere Model 3.0)完善的软件平台,将原模式的三时间层蛙跳时间差分方案修改为两时间层二阶Runge-Kutta时间差分格式,对与重力波有关项使用中央差隐式处理,以此构建半隐式大气环流谱模式。通过斜压波实验比较不同格式在保持初值稳定性上的表现,从而发现,二阶Runge-Kutta方案能够更好的保证方案的初值稳定性。同时在纬向对称平衡场中加入扰动的情况下,二阶Runge-Kutta方案模拟的斜压波动发展演变的特征具有良好的收敛性,对波动发展的描述能力更强。存在这种优势的可能原因可归结为格式自身的优势和摆脱了时间滤波的负面影响,通过加入不同滤波系数的比较实验可以看到,滤波的平滑作用对模式结果的影响显著,但格式自身的优势也是改进模拟结果的主要因素。通过非绝热条件下20 年（1980～1999年）气候态全模式模拟考察模式在气候模拟中的表现,结果表明,此方案在长期的气候模拟中同样可降低预报变量及诊断变量的模拟误差,具有更好的模拟能力。     

正压原始方程半隐式样条格式

本文把样条函数应用于半隐式正压原始方程模式,分析了这种格式的精度、线性稳定性条件,讨论了一种用三次样条函数求解Helmholtz方程的迭代法,并证明了迭代的收敛性。实例分析表明,半隐式样条格式计算稳定,预报场光滑完整,与一般差分格式相比,预报系统移速偏慢现象有较明显改进,用于台风路径预报,其结果有参考价值。     

半隐式半拉格朗日平方守恒计算格式的构造

在显式半拉格朗日完全平方守恒格式基础上, 构造出半隐式半拉格朗日完全平方守恒计算格式, 它继承了显式半拉格朗日完全平方守恒格式的优点, 并突破计算不稳定柯朗条件对时间步长的约束, 使时间步长大为增长.此外, 还给出这种新的计算格式在一维原始方程上的应用.     

正压原始方程模式分离半隐式积分方案的试验

本文设计了一个正压原始方程模式的分离半隐式积分方案,并考察了它的稳定性和精确性。试验结果表明,分离半隐式积分方案是非常稳定而有效的。用这种方案制作有限区域正压原始方程模式24小时预报所需要的计算时间是分离显式方案的二分之一,是显式方案的四分之一。     

拉格朗日法在数值天气预报中的应用(二)

八、显式和隐式格式所谓显式和隐式格式是指对时间积分所采取的两种不同的计算格式.显式格式,简言之就是都可以用前一个时间层的函数值,求出后一个时间层的函数值.这种格式,常用它在实际计算中比较方便而被普遍选用.但是,从时间积分稳定性角度论,这种格式只有在满足一定条件下计算才是稳定的,属条件性稳定,对网格模式讲,空间格距与时间步长之间相互制约.若取空间格距一定,在波速C较大时,Δt需取得较小,因而对一定的积分时间T,运算     

变网格一体化模式的研究进展

作者介绍了近年来在数值天气预报领域的一个新动向－变网格一体化模式的进展。这种模式可以替代目前业务数值天气预报的有限区模式和全球谱模式，降低业务数值天气预报的计算成本。网格距离的可变性，以及采用半隐式－半拉格朗日式时间差分方案是该模式的主要特点。法国的变网格一体化模式自１９９２年已投入业务使用，其可行性已得到了预报检验的初步证实。     

JFNK方法概述及其在大气全隐式非静力模式中的应用方案

首先介绍了近年来新发展的非线性方程全隐式数值求解的JFNK方法,及其在地球流体力学方面应用计算实例.可看到,无论在计算精度还是计算效率方面,全隐式数值求解远远超过常规的半隐式计算格式.其次,还讨论了JFNK方法在气象非静力模式中应用方案,并提出了用静力假定和半隐式差分格式来构造预条件处理器,变三维求解为二维求解,简化了方程组求解难度.该方案不仅可用于差分模式,也为用譜方法求解非静力模式提供可能.     

一类计算性系统误差消除与斜压原始方程天气气候模式改进

斜压原始方程半隐式全能量守恒格式的构造问题长期没有解决。本研究在成功地构造实现其全能量完全守恒的半隐式方案基础上，进行了此守恒方案与欧洲中期天气预报中心（ECMWF）的σ－坐标原始方程全球谱模式半隐式方案间的实际资料对比实验。实验表明，850hPa平均预报高度场RMS误差在积分一周以后得到明显改进，到第30天其预报误差降低达到了50％，进一步的对比实验表明，对流层中部和下部的月预报平均高度场RMS误差也显降低，而且一些明显的系统性误差也得到大幅度改进。更加详细的分析显示，这些收益的很大一部分是从超长波成分的改进中得到的。这说明，通过构造守恒性时间差分方案消除了响应的计算性系统误差源汇，进而能够使模式气候漂移得到显改进，而这种误差源汇存在于传统的，现仍被普遍采用的斜压原始方程天气气候模式中。     

GRAPES区域模式水汽平流方案的比较与改进

与欧拉显式时间差分方法相比,GRAPES区域模式采用半隐半拉格朗日时间差分方案可增加时间步长且不影响稳定性,而且模式积分可有较高的计算效率和准确性。半拉格朗日法需要用到内插算法来预测下一时刻的值,对于水汽场的内插值来说,常常会造成预报值的过饱和或者是负值,需要进行特殊处理。比较GRAPES模式的准单调半拉格朗日方案(QMSL)和高精度正定保形方案(PRM),分析模式的降水预报、形势预报,同时初步总结了两方案的优缺点。在参考LCSL(Linear Constraint Semi-Lagarangain)方案的基础上,改进QMSL方案,通过连续试验运行,表明新方案基本稳定可靠,对于降水预报、形势预报有一定的改进,在台风预报试验中也有良好的表现。     

AN IMPROVED SEMI-IMPLICIT TIME DIFFERENCE SCHEME OF SPECTRAL MODEL AND NUMERICAL APPLICATIONS

In fact,the popular semi-implicit time difference scheme of spectral model still includes someimportant linear terms using time explicit difference scheme,and the major terms are directlyrelated to fast internal-and external-gravity waves in the atmospheric forecasting equation.Additionally,due to using time difference on two terms at different time.the popular schemeartificially introduces unbalance between pressure gradient force and Coriolis force terms whilenumerically computing their small difference between large quantities.According to thecomputational stability analysis conducted to the linear term time difference scheme in simpleharmonic motion equation,one improved semi-implicit time difference scheme is also designed inour study.By adopting a kind of revised time-explicit-difference scheme to these linear terms thatstill included in spectral model governing equations,the defect of spectral model which only partlyusing semi-implicit integrating scheme can be overcome effectively.Moreover,besides all spectralcoefficients of prognostic equations,especially of Helmholtz divergence equation,can be workedout without any numerical iteration,the time-step (computation stability) can also be enlarged(enhanced) by properly introducing an adjustable coefficient.     

AN IMPROVED SEMI-IMPLICIT TIME DIFFERENCE SCHEME OF SPECTRAL MODEL AND NUMERICAL APPLICATIONS*

In fact,the popular semi-implicit time difference scheme of spectral model still includes some important linear terms using time explicit difference scheme,and the major terms are directly related to fast internal-and external-gravity waves in the atmospheric forecasting equation.Additionally,due to using time difference on two terms at different time.the popular schemear tificially introduces unbalance between pressure gradient force and Coriolis force terms while numerically computing their small difference between large quantities.According to the computational stability analysis conducted to the linear term time difference scheme in simple harmonic motion equation,one improved semi-implicit time difference scheme is also designed in our study.By adopting a kind of revised time-explicit-difference scheme to these linear terms that still included in spectral model governing equations,the defect of spectral model which only partlyusing semi-implicit integrating scheme can be overcome effectively.Moreover,besides all spectral coefficients of prognostic equations,especially of Helmholtz divergence equation,can be worked out without any numerical iteration,the time-step (computation stability) can also be enlarged (enhanced) by properly introducing an adjustable coefficient.     

THE FORMULATION OF FIDELITY SCHEMES OF PHYSICAL CONSERVATION LAWS AND IMPROVEMENTS ON A TRADITIONAL SPECTRAL MODEL OF BAROCLINIC PRIMITIVE EQUATIONS FOR NUMERICAL PREDICTION

In this paper,two formulation theorems of time-difference fidelity schemes for generalquadratic and cubic physical conservation laws are respectively constructed and proved,with earliermajor conserving time-discretized schemes given as special cases.These two theorems can providenew mathematical basis for solving basic formulation problems of more types of conservative time-discrete fidelity schemes,and even for formulating conservative temporal-spatial discrete fidelityschemes by combining existing instantly conserving space-discretized schemes.Besides.the twotheorems can also solve two large categories of problems about linear and nonlinear computationalinstability.The traditional global spectral-vertical finite-difference semi-implicit model for baroclinicprimitive equations is currently used in many countries in the world for operational weatherforecast and numerical simulations of general circulation.The present work,however,based onTheorem 2 formulated in this paper,develops and realizes a high-order total energy conservingsemi-implicit time-difference fidelity scheme for global spectral-vertical finite-difference model ofbaroclinic primitive equations.Prior to this,such a basic formulation problem remains unsolved forlong,whether in terms of theory or practice.The total energy conserving semi-implicit schemeformulated here is applicable to real data long-term numerical integration.The experiment of thirteen FGGE data 30-day numerical integration indicates that the newtype of total energy conserving semi-implicit fidelity scheme can surely modify the systematicdeviation of energy and mass conserving of the traditional scheme.It should be particularly notedthat,under the experiment conditions of the present work,the systematic errors induced by theviolation of physical laws of conservation in the time-discretized process regarding the traditionalscheme designs(called type Z errors for short)can contribute up to one-third of the totalsystematic root-mean-square(RMS)error at the end of second week of the integration and exceedone half of the total amount four weeks afterwards.In contrast,by realizing a total energyconserving semi-implicit fidelity scheme and thereby eliminating corresponding type Z errors,roughly an average of one-fourth of the RMS errors in the traditional forecast cases can be reducedat the end of second week of the integration,and averagely more than one-third reduced at integraltime of four weeks afterwards.In addition,experiment results also reveal that,in a sense,theeffects of type Z errors are no less great than that of the real topographic forcing of the model.The prospects of the new type of total energy conserving fidelity schemes are very encouraging.     

THE FORMULATION OF FIDELITY SCHEMES OF PHYSICAL CONSERVATION LAWS AND IMPROVEMENTS ON A TRADITIONAL SPECTRAL MODEL OF BAROCLINIC PRIMITIVE EQUATIONS FOR NUMERICAL PREDICTION*

In this paper,two formulation theorems of time-difference fidelity schemes for general quadratic and cubic physical conservation laws are respectively constructed and proved,with earlier major conserving time-discretized schemes given as special cases.These two theorems can provide new mathematical basis for solving basic formulation problems of more types of conservative time-discrete fidelity schemes,and even for formulating conservative temporal-spatial discrete fidelity schemes by combining existing instantly conserving space-discretized schemes.Besides.the two theorems can also solve two large categories of problems about linear and nonlinear computational instability.The traditional global spectral-vertical finite-difference semi-implicit model for baroclinic primitive equations is currently used in many countries in the world for operational weather forecast and numerical simulations of general circulation.The present work,however,based on Theorem 2 formulated in this paper,develops and realizes a high-order total energy conserving semi-implicit time-difference fidelity scheme for global spectral-vertical finite-difference model of baroclinic primitive equations.Prior to this,such a basic formulation problem remains unsolved for long,whether in terms of theory or practice.The total energy conserving semi-implicit scheme formulated here is applicable to real data long-term numerical integration.The experiment of thirteen FGGE data 30-day numerical integration indicates that the new type of total energy conserving semi-implicit fidelity scheme can surely modify the systematic deviation of energy and mass conserving of the traditional scheme.It should be particularly noted that,under the experiment conditions of the present work,the systematic errors induced by the violation of physical laws of conservation in the time-discretized process regarding the traditional scheme designs(called type Z errors for short) can contribute up to one-third of the total systematic root-mean-square(RMS) error at the end of second week of the integration and exceed one half of the total amount four weeks afterwards.In contrast,by realizing a total energy conserving semi-implicit fidelity scheme and thereby eliminating corresponding type Z errors,roughly an average of one-fourth of the RMS errors in the traditional forecast cases can be reduced at the end of second week of the integration,and averagely more than one-third reduced at integral time of four weeks afterwards.In addition,experiment results also reveal that,in a sense,the effects of type Z errors are no less great than that of the real topographic forcing of the model.The prospects of the new type of total energy conserving fidelity schemes are very encouraging.     

SPLIT SEMI-IMPLICIT INTEGRATION OF TROPICAL LIMITED AREA MULTI-LEVEL PRIMITIVE EQUATION MODEL

A tropical limited area multi-level primitive equation model,in which the time integral is performed with a split semi-implicit scheme,is presented.The operating time is shortened to one-third of the time the explicit scheme needs.The results of operational experiments show that the model is stable in computation and capable of producing satisfactory forecast for tropical systems.Besides synoptic reports,data derived from other sources are used so as to improve the forecast,especially over tropical oceans where conventional data are scarce.     

热带有限区分裂半隐式斜压初始方程预报模式

本文提出一个采用分裂半隐式时间积分方案的热带有限区多层初始方程数值预报模式。业务预报试验表明模式的计算稳定性较好,预报作业时间较显式方案缩短约三分之二,对热带天气系统有一定预报能力。为了弥补热带大洋上探空资料不足,还试验了插补其他来源的资料,效果也较好。     

A FINITE VOLUME SEMI-IMPLICIT SEMI-LAGRANGIAN SCHEME ON THE YIN-YANG MESH

Designed for grid point systems, the traditional semi-Lagrangian semi-implicit scheme is not mass-conserving and can lead to significant solution errors. In the present study, a finite-volume semi-Lagrangian semi-implicit scheme (hereafter “FVSLSI”) is designed for the Yin-Yang mesh and tested in a barotropic shallow water model in the spherical coordinate system. Three test cases, i.e. the advection of a solid body, a steady state nonlinear zonal geostrophic flow and the deformation flow, are simulated to compare the performance of the FVSLSI with that of the traditional semi-Lagrangian scheme (hereafter “SL”) from perspectives of shape preservation, mass conservation, normalized bias, and convergence rate. Results indicate that the FVSLSI performs better than the SL in mass conservation and shape preservation. The bias by the FVSLSI is smaller than that by the SL, while the rate of convergence by the FVSLSI is larger than that by the SL. The FVSLSI also allows large time step. Therefore, the FVSLSI is suggested to be distributed to communities that are developing atmospheric/oceanic models.