GRAPES模式中三维科氏力计算及其效果评估

作为一种连续可压缩流体,大气具有分层流体特性,其状态变化可由牛顿第二定律、热力学第一定律、连续方程和理想大气状态方程组成的偏微分方程组描述。为了更加精细地描述GRAPES全球模式的动力过程,使模式大气更接近真实大气,在全球非静力模式GRAPES中考虑三维科氏力作用,通过重新构建半隐式半拉格朗日求解大气动力方程组的亥姆霍兹方程系数,在不改变求解方案的前提下实现对GRAPES模式动力过程的更新。然后采用在静力平衡基础上建立的三维大气理想试验对新的动力过程进行数值试验,检验其计算效果和数值稳定性。结果显示,考虑三维科氏力的模式动力框架计算稳定,提高了三维标量和矢量场的计算精度,在水平1°×1°分辨率模式中,平衡流试验第15天计算结果标量场Π'的l₁和l₂误差分别为0.00023和0.0004,而三维矢量场 V 的l₁和l₂误差分别为0.002和0.003,均较原模式误差小一个数量级。在罗斯贝-豪威兹波、地形罗斯贝波和斜压波试验中,新框架均表现出很高的计算稳定性和良好的计算效果。      

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。     

Properties of High-Order Finite Difference Schemes and Idealized Numerical Testing

Construction of high-order difference schemes based on Taylor series expansion has long been a hot topic in computational mathematics, while its application in comprehensive weather models is still very rare. Here, the properties of high-order finite difference schemes are studied based on idealized numerical testing, for the purpose of their application in the Global/Regional Assimilation and Prediction System(GRAPES) model. It is found that the pros and cons due to grid staggering choices diminish with higher-order schemes based on linearized analysis of the one-dimensional gravity wave equation. The improvement of higher-order difference schemes is still obvious for the mesh with smooth varied grid distance. The results of discontinuous square wave testing also exhibits the superiority of high-order schemes. For a model grid with severe non-uniformity and non-orthogonality, the advantage of high-order difference schemes is inapparent, as shown by the results of two-dimensional idealized advection tests under a terrain-following coordinate. In addition, the increase in computational expense caused by high-order schemes can be avoided by the precondition technique used in the GRAPES model. In general, a high-order finite difference scheme is a preferable choice for the tropical regional GRAPES model with a quasi-uniform and quasi-orthogonal grid mesh.     

基于大涡模拟评估GRAPES模式对对流边界层的模拟性能

为检验GRAPES半拉格朗日动力框架在大涡尺度上的模拟性能,为未来发展千米及其以下高分辨尺度的数值模式奠定基础,并构造GRAPES大涡模式以检验和发展边界层湍流参数化提供科学工具。通过在GRAPES模式中加入Smagorinsky-Lilly小尺度湍涡参数化,并将模式分辨率提高至50 m,构建GRAPES大涡模式(GRAPES_LES),以便分析GRAPES模式在大涡尺度上的适用性。同时利用广泛应用的已有大涡模式UCLA_LES作为参考,通过对干对流边界层湍流的模拟分析及与UCLA_LES模拟结果的对比,得出如下主要结论:GRAPES半拉格朗日动力框架能够模拟出与已有的大涡模式相似的边界层湍流特征;同时,通过分析也证明GRAPES存在由于采用半拉格朗日平流计算而带来过度耗散的问题:当使用相同的滤波尺度(Smagorinsky 常数)时,GRAPES_LES模拟出的速度场更为平滑,小尺度湍流结构过于光滑,通过对湍流能量的能谱分析更清楚地表明了这一点。进一步,对不同的Smagorinsky常数(对应不同的滤波尺度)进行了敏感性试验,表明可以通过改变滤波尺度,有效地缓解半拉格朗日框架隐含的耗散问题,得到更接近UCLA_LES所模拟的湍流特征。     

阴阳网格守恒算法的设计和改进

阴阳网格上的质量守恒算法对于阴阳网格在全球模式构建和应用具有重要意义,是模式长期稳定积分和保证计算效果的重要性能指标。本研究在已有的质量均匀分布假定下阴阳网格守恒强迫算法的基础上,构建网格内质量的双线性分布和边界通量线性分布的质量守恒强迫算法,以提高阴阳网格平流计算的精度和模式积分的稳定性。运用CIP-CSLR平流方案对通量形式平流方程数值求解,分别通过"余弦钟"平流试验、正弦波试验和变形流试验对质量双线性分布、边界通量线性分布的新方案与质量和通量均匀分布的原方案进行了对比,标准化误差和标量场分布均表明新方案可有效提高阴阳网格守恒算法的计算效果,且计算负担没有明显增加,具有较好的实用价值。     

阴阳网格上质量守恒计算性能分析

质量守恒数值计算是球面准均匀阴阳网格构造全球大气环流模式的重要条件,也是提高阴阳网格应用质量的重要技术手段。本文针对通量形式平流方程,在球面坐标上采用多种理想数值试验对阴阳网格上的三种守恒计算方案和边界插值非守恒计算方案进行了比较检验。发现,质量守恒方案不仅对全球数值积分重要,还影响数值计算精度,满足局地守恒条件的全球强迫守恒方法可以获得较高的精度;网格内质量均匀分布的阴阳网格边界通量一致性守恒强迫计算方案,实现了在不增加计算误差条件下保证局地和全球守恒的目的,且具有很小的计算负担,可以作为阴阳网格上全球质量强迫守恒的有效计算方案;而网格质量的线性分布可以有效提高阴阳网格的数值积分计算精度,但在一定程度上会增加计算负担。     

GRAPES模式动力框架的长期积分特征

通过考虑动量表面拖曳并利用牛顿松弛方法将温度松弛到纬向对称的温度场, 对GRAPES模式的动力框架进行了长期时间积分试验。通过统计分析其积分结果表明:GRAPES模式的动力框架可以模拟出大气环流的基本特征; 随着分辨率的提高, GRAPES模式动力框架的模拟结果显示出收敛的特性; 虽然GRAPES模式动力框架采用的是能量不守衡的半隐式半拉格朗日时间积分方案, 但长期时间积分试验表明其能量基本保持稳定。这些结果显示使用GRAPES模式动力框架作大气环流和气候研究的大气动力框架是可行的, 同时也为进一步改进GRAPES模式动力框架提供了线索和依据。     

阴阳网格上通量型非静力模式动力框架的理想试验

为改善球面经纬度网格在高分辨率应用的苛刻限制,提高全球大气动力模式的时间积分效率,选取以阴阳网格为基础构建通量型非静力大气模式动力框架,采用有限体积法和通量型平流显式算法积分方案,保证模式的守恒计算性能。该动力框架在标准三维大气理想试验中进行了中期积分试验,对动力框架的计算效果、性能进行检验。在三维平衡流试验,罗斯贝-豪威茨波试验和山脉罗斯贝波试验中均表现出很好的稳定性和三维计算效果,其中水平2.5°分辨率模式的平衡流垂直速度误差为10-5量级,而经向速度误差在10-2量级,罗斯贝-豪威茨波保持基本波形稳定传播,而地形罗斯贝波试验则给出背风坡激发低槽在发展过程中不断向下游和南半球传播。      

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.     

一个基于Eta垂直坐标的新WRF动力框架及其数值试验

针对我国陡峭地形数值预报难题,本文在国际先进WRF(Weather Research and Forecasting)模式动力框架中引入阶梯地形垂直坐标,以期为改进复杂地形区域数值天气预报提供模式发展可选方案.设计气柱质量变换方法,实现阶梯地形和追随地形两种垂直坐标下动力方程组的形式一致,从而简化方程组离散及程序实现的...     

二阶精度半隐式半拉格朗日轨迹计算和时间积分方案在GRAPES区域模式中的应用

将两时间层稳定外插方案(Stable Extrapolation Two-Time-Level Scheme,SETTLS)引入GRAPES区域模式,并将其用于上游点和非线性项的时间外插计算。对线性项采用二阶精度的非中央权重时间平均,并取等温参考大气的温度大于实际大气平均温度,以保证半隐式积分方案的稳定性。原GRAPES时间积分方案对线性项做一阶非中央权重时间平均,对参考大气的选择并无限制,而为保证稳定性,须取较大的非中央权重系数,但非中央权重系数会对低波数波动产生较强的衰减作用。理想试验结果表明,相比原GRAPES半隐式半拉格朗日(SISL)时间积分方案,新SISL时间积分方案计算稳定且对波动的衰减作用较弱。     

Implementation of the semi-Lagrangian advection scheme on a quasi-uniform overset grid on a sphere

The semi-Lagrangian advection scheme is implemented on a new quasi-uniform overset （Yin-Yang） grid on the sphere. The Yin-Yang grid is a newly developed grid system in spherical geometry with two perpendicularly-oriented latitude-longitude grid components （called Yin and Yang respectively） that overlapp each other, and this effectively avoids the coordinate singularity and the grid convergence near the poles. In this overset grid, the way of transferring data between the Yin and Yang components is the key to maintaining the accuracy and robustness in numerical solutions. A numerical interpolation for boundary data exchange, which maintains the accuracy of the original advection scheme and is computationally efficient, is given in this paper. A standard test of the solid-body advection proposed by Williamson is carried out on the Yin-Yang grid. Numerical results show that the quasi-uniform Yin-Yang grid can get around the problems near the poles, and the numerical accuracy in the original semi-Lagrangian scheme is effectively maintained in the Yin-Yang grid.     

Semi-Implicit Scheme for the DWD Lokal-Modell

Summary  The fourth generation of numerical weather prediction (NWP) models is currently under development at the Deutscher Wetterdienst (DWD) consisting of a global grid point model (GME) and limited-area Lokal-Modell (LM). The nonhydrostatic fully compressible LM has been designed to meet high-resol ution regional fore-cast requirements at meso-β and meso-γ scales. The initial LM implementation is based on the NCAR/Penn State MM5 with the addition of a novel generalized terrain-following coordinate and rotated lat-lon grid. A fully 3D semi-implicit time-stepping scheme has been implemented by retaining the full buoyancy term instead of the approximate form found in MM5. In contrast with earlier schemes, mass-lumping is not applied to simplify the elliptic operator on an Arakawa-C/Lorenz grid. The resulting variable-coefficient elliptic problem is solved using a minimal residual Krylov iterative method with line relaxation preconditione rs. The new semi-implicit scheme is compared with a variant of the Klemp–Wilhelmson split-explicit scheme (horizontal explicit, vertical implicit) on the basis of computational efficiency and accuracy at resolutions ranging from 7 km to 400 m. Both idealized 3D mountain wave flows and naturally occuring flows are analyzed. Below the tropopause, the 3D semi-implicit scheme can be more efficient for low Mach number M ≪ 1 flows when the number of small time steps Δt _s of the split-explicit approach increases with the sound-speed Courant number. Revised October 7, 1999     

GRAPES全球切线性和伴随模式的调优

伴随技术是四维变分同化（4DVar）系统中计算代价函数梯度的最佳办法,切线性和伴随模式的效果和效率直接影响着4DVar系统的发展。基于GRAPES（Global and Regional Assimilation PrEdiction System）全球切线性和伴随模式1.0版本,利用GRAPES全球模式2.0版本在并行框架和性能等方面的改善,重新优化和设计了GRAPES全球切线性伴随模式2.0版本,提高了GRAPES全球切线性和伴随模式的效果和效率,优化了切线性模式程序结构,使其计算时间最优可控制在非线性模式的1.2倍以内;采用在切线性模式中保存基态的方法,重构了伴随模式的程序结构,使其计算时间最优控制在非线性模式的1.5倍以内;在GRAPES全球切线性物理过程的设计中,将线性物理过程的轨迹基态计算和切线性扰动计算解耦,提高了GRAPES全球切线性和伴随模式的计算效果和效率。     

GRAPES中地形重力波拖曳物理过程的引进和应用试验

在中国新一代全球中期/区域中尺度同化与预报系统(GRAPES)模式中引进了ECMWF地形重力波拖曳物理过程,填补了GRAPES全球中期数值预报系统中物理过程的空白。重新计算了地形重力波过程需要的地形静态资料数据,并与原ECMWF模式的地形静态参数进行了对比分析,验证了模式地形参数的正确性。利用GRAPES模式,进行了地形重力波拖曳物理过程影响的敏感性数值试验;结果表明:引进地形重力波拖曳过程以后,在存在大地形的区域,风场会发生变化,当纬向风遇到青藏高原时,一部分气流会产生爬坡效应而越过高原,使高原上空的西风气流减弱;另一部分气流会绕过高原,在高原的南侧产生绕流;随着模式积分时间的延长,风场变化会越来越明显,地形越复杂,风场的变化也越复杂;连续的模式积分试验结果显示,引进地形重力波过程,可以延长GRAPES模式的可用预报时效,提高了全球形势预报的准确率。通过对一次降水过程的模拟,对地形重力波过程影响降水预报的原因进行了简单分析。结果显示:引进地形重力波拖曳过程后,改变了大气流场的分布,使预报的流场更接近于大气真实状态,从而提高了降水预报的准确率。     

Application of the Characteristic CIP Method to a Shallow Water Model on the Sphere

Semi-implicit algorithms are popularly used to deal with the gravitational term in numerical models. In this paper, we adopt the method of characteristics to compute the solutions for gravity waves on a sphere directly using a semi-Lagrangian advection scheme instead of the semi-implicit method in a shallow water model, to avoid expensive matrix inversions. Adoption of the semi-Lagrangian scheme renders the numerical model always stable for any Courant number, and which saves CPU time. To illustrate the efficiency of the characteristic constrained interpolation profile (CIP) method, some numerical results are shown for idealized test cases on a sphere in the Yin-Yang grid system.     

Long-Term Integration of a Global Non-Hydrostatic Atmospheric Model on an Aqua Planet

A global non-hydrostatic atmospheric model, i.e., GRAPES_YY (Global/Regional Assimilation and Prediction System on the Yin–Yang grid), with a semi-implicit semi-Lagrangian (SISL) dynamical core developed on the Yin–Yang grid was coupled with the physical parameterization package of the operational version of GRAPES. A 3.5-yr integration was carried out on an aqua planet to assess the numerical performance of this non-hydrostatic model relative to other models. Specific aspects of precipitation and general circulation under two different sea surface temperature (SST) conditions (CONTROL and FLAT) were analyzed. The CONTROL SST peaked at the equator. The FLAT SST had its maximum gradient at about 20° latitude, giving a broad equatorial SST maximum in the tropics and flat profile approaching the equator. The tropical precipitation showed different propagation features in the CONTROL and FLAT simulations. The CONTROL showed tropical precipitation bands moving eastward with some envelopes of westward convective-scale disturbance. Less organized westward-propagating rainfall cells and bands were seen in the FLAT and the propagation of the tropical wave varied with the SST gradient. The Inter Tropical Convergence Zone (ITCZ), Hadley cell, and westerly jet core were weaker and more poleward as the SST profile flattened from the CONTROL to FLAT. The climatological structures simulated by GRAPES_YY, such as the distribution of precipitation and the large-scale circulation, fell within the bounds from other models. The stronger ITCZ precipitation, accompanied with stronger Hadley cells and convective heating in the CONTROL simulation, may be summed up as a result of stronger parameterized convection and the non-hydrostatic effects in GRAPES_YY. In addition, mechanism of the zonal mean circulation maintaining is analyzed for the different SST patterns referring the transient eddy flux.     

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区域扰动预报模式动力框架设计及检验

设计了适用于四维变分同化系统的扰动预报模式GRAPES_PF。根据GRAPES的地形追随坐标非静力原始方程组,采用小扰动分离方法推导微分形式的线性扰动预报方程组,并利用与GRAPES非线性模式相似的数值求解方案求解线性扰动微分方程组。在设计扰动预报模式时采用了两个时间层半隐式半拉格朗日方案对动量方程、热力学方程、水汽方程和连续方程进行时间差分,空间差分方案的变量分布水平方向采用Arakawa C跳点网格,垂直方向采用Charney/Phillips跳层。利用代数消元法进一步推导得到只包含未来时刻扰动Exner气压的亥姆霍兹方程,进而通过广义共轭余差法（GCR）求解,在此基础上得到未来时刻扰动量的预报值。基于所开发的扰动模式开展了数值试验。首先在非线性模式中施加一个中尺度初始扰动高压,得到初始扰动在非线性模式中的后续演变,然后将相同的初始扰动作为扰动模式的初值进行时间积分,将扰动模式预报的结果与非线性模式的结果做了对比。结果表明,所开发的扰动模式GRAPES_PF较好地模拟了惯性重力内波的传播过程:初始高压扰动激发了一个迅速向外传播的惯性重力内波,在气压场向风场适应的过程中,水平风场、垂直运动、位温和湿度等变量均出现了扰动增量,与非线性模式得到的结果相当接近。GRAPES_PF作为GRAPES非线性模式的合理线性模式为建立基于线性扰动预报的区域四维变分同化系统奠定了科学基础。