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.     

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。     

物理守恒律保真格式构造与数值预报斜压原始方程传统谱模式改进研究

文中构造并证明了一般二次和三次物理守恒律时间差分保真格式两个构造定理，以往一些主要时间离散守恒格式构造方案可作为两个定理特例给出。它们不仅可为解决更加广泛类别的时间离散保真格式构造基本问题提供适用数学基础，而且也为结合已有瞬时空间离散守恒格式，解决更加广泛类别的时－空离散意义下保真格式构造基本问题提供适用的数学基础。此外，文中两个定理还可解决两大类问题的线性和非线性计算不稳定性问题。斜压原始方程传统半隐式全球谱－垂直有限差分模式目前是世界上许多国家的业务预报和大气环流模式。本工作利用文中新构定理，构造并且实现了斜压原始方程全球谱－垂直有限差分模式半隐式高阶全能量守恒方案。以往该项基本问题无论在理论还是实践上长期以来一直都未能得到解决。该项全能量守恒半隐式全球谱模式方案适用于实测资料的长时间数值预报积分。使用ＦＧＧＥ夏季资料进行的１３个个例３０ｄ数值积分实验表明：新型全能量半隐式保真方案可以有效地改进传统预报方案中关于能量质量守恒性质的系统性偏差。值得注意的是，实验统计分析还显示：在本文实验条件下，传统方案中由于时间离散过程中原物理守恒律性质破坏导致的系统误差（简称Ｚ类误差），对于实验总体均方根系统误差的贡献     

LONG VALID TIME ENERGY PERFECT CONSERVATIVE FIDELITY SPECTRAL SCHEMES OF BAROTROPIC PRIMITIVE EQUATIONS*

In accordance with a new compensation principle of discrete computations,the traditional meteorological global (pseudo-) spectral schemes of barotropic primitive equation (s) are transformed into perfect energy conservative fidelity schemes,thus resolving the problems of both nonlinear computational instability and incomplete energy conservation,and raising the computational efficiency of the traditional schemes.As the numerical tests of the new schemes demonstrate,in solving the problem of energy conservation in operational computations,the new schemes can eliminate the (nonlinear) computational instability and,to some extent even the (nonlinear) computational diverging as found in the traditional schemes,Further contrasts between new and traditional schemes also indicate that,in discrete operational computations,the new scheme in the case of nondivergence is capable of prolonging the valid in-tegral time of the corresponding traditional scheme,and eliminating certain kind of systematical computational "climate drift",meanwhile increasing its computational accuracy and reducing its amount of computation.The working principle of this paper is also applicable to the problem concerning baroclinic primitive equations.     

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。     

长时效的正压原始方程能量完全守恒(拟)谱模式

遵循误差反演补偿新计算原理,对正压原始方程传统气象全球拟谱模式方案进行了改造,构造了正压原始方程能量完全守恒全球拟增模式新计算方案,解决了正压原始方程的（非线性）计算稳定性问题和能量守恒整体性质保持问题,改进了相应正压原始方程传统气象全球拟谱模式方案的计算效能。新方案的数值试验表明：在计算实践上,新方案在解决能量守恒问题的同时,可解决（非线性）计算稳定性问题,并在一定条件下可解决非线性计算收敛性问题。进一步的比较数值试验还表明：在计算实践上,新方案具有在提高相应传统气象方案的计算精度,减少其计算量的同时,延长其计算时效,解决其中一类特定“气候漂移”问题方面的效用。本工作原理也适用于斜压原始方程情形。     

地转风适应过程的数值试验研究

本文用C—网格模拟了地转风适应过程,给出了能量及位涡守恒关系和微分方程相一致的一种差分格式,并讨论了时间差分对重力惯性波频散特性的影响。为了保持计算稳定性,提出了一种改进的向前向后时间积分方法。通过实例计算,说明了有限区域各种边界条件对重力惯性波传播的影响。     

A NUMERICAL SIMULATION OF THE GEOSTROPHIC ADJUSTMENT PROCESS

In this paper,a numerical simulation of the geostrophic adjustment process with C-grid network is illustrated.A difference scheme which has the energy and potential vorticity conserving relation consistent with the differential equations is given,and the effect of some time difference schemes on dispersion of the gravity-inertia wave is discussed.An improved forward-backward time integration scheme is proposed for keeping the computational stability.The effect of various boundary conditions for a finite region model On the gravity-inertia wave is shown by some calculated results.     

The Dynamical and Climate Tests of an Atmospheric General Circulation Model Using the Second-Order Adams-Bashforth Method

The Asselin-Robert time filter used in the leapfrog scheme can degrade the accuracy of calculations. The second-order Adams-Bashforth method with the same accuracy as the leapfrog scheme is not subject to time splitting instability. A new semi-implicit atmospheric general circulation spectral model is developed on the basis of NCAR (National Center for Atmospheric Research) CAM3.0 (Community Atmosphere Model3.0). In this new model, the second-order Adams-Bashforth method is used as an alternative to the leapfrog scheme, and a Crank-Nicholson scheme is incorporated for the treatment of fast gravity modes. In this paper, the new model is tested by the Held-Suarez test and an idealized baroclinic wave test. Results of the Held-Suarez test show that the second-order Adams-Bashforth model has similar climate states to those of many other global models and it converges with resolutions. Based on the idealized baroclinic wave test, the capability of di?erent time di?erencing methods for keeping the initial steady-state are compared.This convinces a better ability of the second-order Adams-Bashforth method in maintaining the stability of the initial state. Furthermore, after the baroclinic wave is triggered through overlaying the steady-state initial conditions with the zonal perturbation, the second-order Adams-Bashforth method has an excellent property of convergence, and can represent the process of the baroclinic wave development much better than the original scheme in CAM3.0. A long-term integration of the new model during the period of 1980-1999 is also carried out and compared with that of CAM3.0. It is found that due to the reduction of simulation errors of prognostic variables, the second-order Adams-Bashforth method also has a better simulation ability for the diagnostic variables, such as precipitation.     

On the use of the Boussinesq equations,the reduced system,and the primitive equations for the computation of geophysical flows

Motivated in part by the mathematical problems associated with the application of open boundary conditions to the hydrostatic primitive equations (PE), Browning et al. (1990, Dyn. Atmos. Oceans, 14: 303–332) proposed the use of the reduced system (RS) of equations to replace PE for oceanographic problems. The RS are essentially the Boussinesq equations (BO) with the non-hydrostatic terms in the vertical momentum equation multiplied by a constant δ² ? 1. This artificially alters the physics (e.g. changing the intemal-inertial wave properties) to facilitate numerical integration, but the changes are assumed to have negligible effects on the dynamics of interest. We assess the accuracy and utility of the RS (following the guidelines for the choice of δ) by comparing numerical finite difference solutions of RS, PE and BO for initial-value problems involving three-dimensional instability of an ocean front and atmospheric frontal development in a two-dimensional Eady wave. Both explicit (BO, PE) and semi-implicit (BOSI, PESI) time-difference schemes are used for the Boussinesq and primitive equations. For RS, the same explicit scheme as for BO is used where δ ? 1 allows larger time steps than with the other explicit models. It is found that relative to BO solutions, the errors for RS are small but increase rapidly and monotonically with increasing δ (over a range consistent with the guidelines) and are greater than the errors for the other models. The use of BOSI allows time steps at least as large as those for RS and results in smaller errors than RS. For these problems, BOSI is the preferable model to replace PE.     

显式自由表面模型时间分裂格式的一个积分方案

讨论了有关显式自由表面模型的时间分裂格式中的分裂误差。在传统的积分方案中，由于采用蛙跳格式，斜压动量方程具有三层的时间离散结构，但是，正压动量方程在相应的时间层上只有二层的离散结构。因此，正压方程与斜压方程的时间离散结构是不相容的。分析结果表明，这是导致分裂误差的原因。本提出一种能保持时间离散结构相容的正压和斜压方程的时间积分方案。该积分方案的分裂误差很小因此特别适合于需要长期积分的气候研究。该方案在北太平洋的一个自由表面模型中得到了应用。     

Computed rotational energy spectra of two energy and enstrophy conserving schemes on semi-staggered grids

Summary Rotational energy spectra are examined in experiments with two energy and enstrophy conserving schemes defined on the semi-staggered grid. One of the schemes conserved enstrophy as defined on an equivalent staggered grid, and the other conserved enstrophy as defined on the semi-staggered grid.As predicted by the theory (Janji, 1984), the rotational energy corresponding to the scheme conserving enstrophy as defined on an equivalent staggered grid decreases faster with increasing intensity of the wave number vector than that of the scheme conserving enstrophy as defined on the semistaggered grid. Moreover, the spectrum obtained with the scheme conserving the equivalent staggered grid enstrophy resembles the observed atmospheric spectra in a large part of the admissible wave number range.With 2 Figures     

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.     

中纬度夏季一次深厚对流过程的数值模拟研究:高空热力层结和风切变影响

针对一次发生在中纬度夏季的深厚对流过程进行的数值模拟研究的结果,引入一个较详细描述深厚对流状态下云微物理过程的参数化方案,对一个有限差分三维非静力弹性大气数值模式进行了改进.数值模拟得到了与天气雷达观测一致的结果.结果还显示,对于这次穿透高度超过了对流层顶的深厚对流过程,与高空斜压扰动有关的上对流层弱稳定热力层结对系统的维持起到了重要的作用.另外,还发现强烈的对流发展到对流层顶的高度时,受上曳体的驱动,在对流层顶附近的稳定层结中产生浮力性的扰动;它们与风切变相结合,进而对系统的水平移动产生控制作用.     

THREE-DIMENSIONAL VARIABLES ALLOCATION IN MESOSCALE MODELS

Forecasts and simulations are varied owing to different allocation of 3-dimensional variables in mesoscale models. No attempts have been made to address the issue of optimizing the simulation with a 3-dimensional variables distribution that should come …     

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.     

数字滤波初值化方案的数值实验

本文以一个有限区域16层斜压原始方程模式为基础,对数字滤波初值化方案进行了数值试验,并与平衡初值及实测风初值等方案的积分结果进行了比较、结果表明：数字滤波初值化能有效地滤掉初始场中的高频重力波和小尺度扰动,使客观分析质量得到明显改进;该方法易于引入非绝热过程,能有效抑制积分初期的Spinup现象;形势预报和降水预报均较其它方案有所改进。     

Integration of the shallow water equations in a plane geometry using semi-Lagrangian and Eulerian schemes

Summary Extended integrations of semi-Lagrangian and Eulerian shallow water primitive equation models are performed. The semi-Lagrangian model used the semi-implicit two-time-level scheme. The Eulerian model used a conserving nonlinear advection scheme.For low resolution and longer integrations, difficulties were encountered with the semi-Lagrangian model which were absent in the Eulerian model. These difficulties are discussed.With 14 Figures     

北半球中高纬月平均环流正压斜压动能的年变化特征

本文利用多年月平均资料计算了北半球中高纬月平均环流正压、斜压动能的年变化特征。结果表明无论正压、斜压动能都具有明显的年变化,而且两者的变化趋势也是一致的,但是正压动能要比斜压动能对总动能的贡献大得多。环流异常具有相当正压的垂直结构。有关结论可为长期数值预报模式的简化提供依据。     

A new general circulation model: formulation and preliminary results in a single- and multi-processor environment

This article describes a new general circulation model (GCM) developed jointly by The University of New South Wales (UNSW) and the University of Hamburg. The model is versatile in that it can be run as a medium-range (1 to 15 days) global numerical weather prediction (NWP) model; as an extended range (15 to 30 days) NWP model; and as a GCM for periods extending from seasons, through annual and decadal periods, and beyond. The model can be coupled with ocean models that vary in complexity from simple "swamp" oceans to complex ocean GCMs. The atmospheric GCM also has a number of novel features, particularly in the numerical integration scheme which is a high-order, mass-conserving, semi-implicit semi-Lagrangian scheme, thereby removing the stability restriction on the time-step and allowing efficient long-term integrations. The emphasis here will be on demonstrating that the new model performs effectively on the usual measures of skill (statistics such as mean errors, root-mean-square errors and anomaly correlations) in several standard applications upon which new models usually are assessed. These applications include medium range weather forecasts out to 10 days on a daily basis over a one year period; a limited 10-year simulation climatology, prediction of atmospheric anomalies using SST anomalies in an El Nino year; and an alternative two-way approach to regional modelling (the "down-scaling problem") made possible because the unconditional stability of the semi-implicit, semi-Lagrangian formulation permits large variations in grid spacing without changing the time step size. Finally, the model is run on a variety of parallel computing platforms and it is shown that near-linear speed-up can be attained. This is significant for both medium range NWP and very long-term GCM integrations. Received: 28 February 1996 / Accepted: 30 July 1996