IMPLICIT NONLINEAR NORMAL MODE INITIALIZATION: A MULTIGRID APPROACH

The implicit nonlinear normal mode initialization (INMI) is applied to a tropical limited area shallow water model in spherical coordinates.The boundary condition for the INMI scheme is based on the boundary formulation of the model.The INMI scheme is found to be very efficient in suppressing spurious gravity wave oscillation and providing a well balanced initial data set for the model.The INMI scheme involves solving a number of elliptic type equations with varying complexity.and hence an efficient numerical technique is required for solving such equations.In order to make INMI computationally more attractive,we are employing the multigrid method for solving all the elliptic type equations in the INMI scheme.The numerical procedures for the development of such multigrid solvers are briefly described.     

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。     

Solitary waves of two-layer quasi-geostrophic flow and analytical solutions with scalar nonlinearity

In this paper we investigate the instability of two-layer quasi-geostrophic model with scalar nonlinearity, which is a geophysical fluid dynamics model, and give a nonlinear stability criterion. Furthermore, the analytical solutions of the two-layer quasi-geostrophic model are obtained via the Jacobi elliptic periodic function transform method and symbolic computation. It is efficient in solving a range classes of nonlinear partial differential equations for this method. The result shows that the structure of Rossby waves is influenced by many factors such as base flow and scalar nonlinearity.     

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.     

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.     

天气雷达测定区域降水量方法的改进与比较

讨论变分校准法用于雷达－雨量计系统联合探测降水。由雷达反射率因子Ｚ和地面降水强度Ｉ实时地获得最优Ｚ－Ｉ关系，在求解欧拉方程时采用多重网格法，不仅可提高计算结果的精度，还可大大提高计算速度。     

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.     

气象问题中椭圆型狄氏边值问题数值解的迭代法

本文从天气学的事实出发,应用自共轭椭圆型边值问题解的简单格林函数表达式,建立适合解动力气象学中椭圆型方程狄氏边界值问题的数值迭代解法,这个方法有较普遍的意义。目前,气象中常用的近似方法——方法、方法、Fjortoft方法以及外推Liebmann方法都是本文所提方法的特殊情形。在本文所提方法的一般形式基础上,还可以对上述各种近似方法的准确度、收敛情况以及改进途径得到明确的了解。作者将公式Ⅱ_1用在数值解平衡方程的计算中,作为本文所提方法的数值计算的检验,试用结果表明本文所提方法有理论概括意义和实用前途。     

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.     

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 unique,well posed reduced system for atmospheric flows: Robustness in the presence of small scale surface irregularities

It is well known that the primitive equations (the atmospheric equations of motion under the additional assumption of hydrostatic equilibrium for large scale motions) are ill posed when used in a limited area on the globe. Yet the atmospheric equations of motion for large scale motions are essentially a hyperbolic system that with appropriate boundary conditions should lead to a well posed system in a limited area. This apparent paradox was resolved by Kreiss through the introduction of the mathematical Bounded Derivative Theory (BDT) for any symmetric hyperbolic system with multiple time scales (as is the case for the atmospheric equations of motion). The BDT uses norm estimation techniques from the mathematical theory of symmetric hyperbolic systems to prove that if the norms of the spatial and temporal derivatives of the ensuing solution are independent of the fast time scales (thus the concept of bounded derivatives), then the subsequent solution will only evolve on the advective space and time scales (slowly evolving in time in BDT parlance) for a period of time. The requirement that the norm of the time derivatives of the ensuing solution be independent of the fast time scales leads to a number of elliptic equations that must be satisfied by the initial conditions and ensuing solution. In the atmospheric case this results in a 2D elliptic equation for the pressure and a 3D equation for the vertical component of the velocity.Utilizing those constraints with an equation for the slowly evolving in time vertical component of vorticity leads to a single time scale (reduced) system that accurately describes the slowly evolving in time solution of the atmospheric equations and is automatically well posed for a limited area domain. The 3D elliptic equation for the vertical component of velocity is not sensitive to small scale perturbations at the lower boundary so the equation can be used all of the way to the surface in the reduced system, eliminating the discontinuity between the equations for the boundary layer and troposphere and the problem of unrealistic growth in the horizontal velocity near the surface in the hydrostatic system.     

An application of an efficient non-hydrostatic mesoscale model

This paper deals with a non-hydrostatic mesoscale model that achieves full vectorization on computers like the CYBER 205. The model formulation ensures the conservation of all fluxes and takes into account the terrain inhomogeneities by the aid of suitable transformations. The diagnostic equation for the pressure change is solved using a very efficient vectorized elliptic solver. By imposing appropriate boundary conditions no additional precautions at the boundaries are necessary to achieve meaningful results. As an application, the steady-state inviscid flow over a single mountain is simulated.     

A radiative-conductive model for the prediction of radiation fog

A conductive-radiative model is used to predict the formation and growth of radiation fog. This is accomplished by solving numerically the heat and mass transport equations in conjunction with an approximate form of the radiative transfer equation. The equations of motion in simplified form are included in the physical system to make use of Blackadar's (1962) formulation of the exchange coefficient of the boundary layer.It is found for a number of hypothetical test cases that the model gives results which appear to represent real physical conditions. One actual situation is tested. Results show that the model reproduces better than qualitatively those parameters which are obtained from routine observations.     

带TVB限制器的RKDG方法与浸入边界方法在Euler方程中的应用

在求解Euler方程时,带TVB限制器的Runge-Kutta间断有限元(RKDG)方法是一种高精度、高并行效率的方法,而浸入边界方法是一种较新颖且对网格要求较低的方法,适用于处理复杂几何外形的边界.尝试了将上述2种方法结合起来求解Euler方程在笛卡尔网格上具有复杂几何外形的物体绕流问题,数个经典算例的数值结果验证了该方法的有效性.     

Numerical simulation of the boundary layer above waves

A new approach to investigations of the structure of the boundary layer above waves is discussed. The approach is based on direct numerical simulation of wave motions in the boundary layer produced by a moving curved surface. Model equations are derived, which are the Reynolds equations in a curvilinear nonstationary system of co-ordinates, evolution equations for turbulent kinetic energy, and Kolmogorov's approximate similarity formulae relating the coefficient of turbulent viscosity to the dissipation of turbulent energy; the length scale is assumed to grow linearly with increasing distance from the surface. Principles of constructing the model numerical scheme are described. Results are given of modelling the structure of the boundary layer above a nonsteady surface, which, in a general case, is a superposition of progressive waves with assigned dispersion relations and amplitudes. Mechanisms of energy and momentum transfer to the surface, effects of density stratification and energy structure in the boundary layer are studied. Merits and demerits of the approach are discussed.     

基于球面阴阳网格数值计算研究进展

对球面阴阳网格的转换关系、优缺点及边界数据插值交换方法的相关知识进行了较为详细介绍。同时对应用球面阴阳网格的3种数值计算方法进行了回顾总结,包括优化的Schwarz方法、CIP-CSLR平流数值计算方法、多离散矩有限体积法。针对优化的Schwarz方法,从浅水方程组的离散入手,讨论了其在求解球面椭圆型问题的优势;而对CIP-CSLR平流数值计算方法和多离散矩有限体积法,主要从如何在网格单元内构造插值函数的角度对其进行分析。最后对开发全球非静力阴阳网格模式进行展望。     

On no-slip boundary conditions for the incompressible Navier-Stokes equations

Two local implementations of no-slip boundary conditions are investigated for both the vorticity—streamfunction and momentum—pressure formulations of the time-dependent planar incompressible Navier-Stokes equations, as applied to barotropic ocean circulation modelling. The objective is to determine the extent to which the local accuracy and numerical consistency of these conditions affects the global solution. The effects of a non-local implementation of no-slip conditions for the vorticity—streamfunction equations are also studied. In all cases, boundary condition effects are measured by comparing time-averaged dynamics of turbulent solutions of numerical models based on the two formulations.In the model interior, the energy and enstrophy conserving Arakawa Jacobian is used for the vorticity—streamfunction equations while an extension of the energy and potential enstrophy conserving Arakawa and Lamb finite difference scheme is used for the momentum-pressure equations.Numerical experiments performed with a non-linear model similar to Bryan's barotropic ocean reveal no significant differences between the time-averaged solutions obtained with either of the two formulations, with each using either of the two local boundary conditions. A simple one-dimensional analogue of the vorticity—streamfunction equations is solved algebraically to explain the experimental results. A similar analogue suggests that an apparent inconsistency in the no-slip boundary conditions within the Cox stratified, primitive equation, ocean circulation model should not affect the accuracy or convergence of the global solution.     

在简化的简单生物圈模式中结合新的雪盖模型

本文提出了将一个新的雪盖模型（ＳＡＳＴ）与简化的简单生物圈模型（ＳＳＩ）相结合后的ＳＳＩＢ模型的改进形式，并利用俄罗斯及法国的观测数据对改进后的模型进行了检验。文中描述了改进后的ＳＳＩＢ模型中空气－植被－雪盖－土壤复杂系统之间有关能量和质量交换的过程，并给出为求解这一组非线性方程组所发展的有效计算方法。利用俄罗斯及法国的观测结果对这一新的改进模型功能进行了验证。结果表明改进后ＳＳＩＢ的数值模拟结果能较好地重现现场观测数据，说明这个新的改进形式可以推广用于气候研究。     

有限区域风速场求解流函数和速度势场的有效方案

流函数和速度势是表示风场的一种变量, 在数值天气预报模式和分析、同化方案中经常使用, 通常可以用风速分量场求解Poisson方程得到。对于有限区域系统, 往往采用差分方法, 但由于存在边界问题, 用计算所得到的流函数和速度势场重建风速场, 在边界附近经常出现明显的偏差。基于差分方法、利用有限区域风速场求解流函数和速度势场的基本方法和特点的分析, 在Arakawa A网格分布的有限区域, 设计了一种用差分方法求解流函数和速度势场的有效方案。在该有效方案中, 通过将有限区域向外扩展二圈, 风速场线性外推, 改进计算边界风速值和边界定解条件的效果; 尽可能使用协调、一致的差分格式, 提高求解精度; 最后利用一种增量订正迭代方法, 迭代2~3次就可以获得令人满意的结果。实例试验的对比、检验显示, 用该方案计算求得的流函数和速度势场重建风速场, 具有非常高的精度。     

正压原始方程的一个数值试验

本文为应用正压原始方程作数值预报的初步试验。采用时间空间中央差分进行数值求解。通过试验确定了一种简单可用的边界条件及初始条件。 应用此方案对四个天气实例进行了24小时预报。计算结果尚好。对这四个实例中所包括的正压天气过程一般能较好的反映出来。