Implementation of CMA-GFS 3D-Var System on the Yin-Yang Grid

The global three-dimensional variational (3D-Var) data assimilation is implemented on a new quasi-uniform overset (Yin-Yang) grid on sphere. As a quasi-uniform spherical grid, it covers the sphere by overlapping two perpendicularly oriented grid components which is nothing but low latitude region of the usual latitude-longitude grid. Based on this characteristic of the Yin-Yang grid, it enables us to implement the regional 3D-Var system efficiently and accurately on the Yin or Yang component grid, respectively. The global analysis could update directly from the regional analysis since they have the same configurations like the precondition of eigenvalue decomposition for vertical direction, recursive filtering for horizontal direction, minimization method and observation operator and so on. However, the balance equation and vector wind are needed to be paid more attention on the Yin grid analysis due to its coordinate transformation. How to spread the observation information near the boundary of Yin and Yang grid is a key to the 3D- Var analysis. Extending double the horizontal correlation length distance in the overset boundary of Yin and Yang grid has successfully solved the problem. The results show that the analysis on the Yin-Yang grid is reasonable and similar to the result on the latitude-longitude (LAT-LON) grid. This paper provides a promising strategy for the development of a 3D-Var global system for overset grids.     

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.     

下一代大气模式中的数值方法综述

数值预报是逐日天气预报、气候预测和气象防灾减灾的核心科技支撑。为进一步提高预报预测的准确度和延长预见期,甚高分辨率、多圈层耦合、多尺度嵌套、多尺度集合、数值地球系统模拟技术等是下一代数值预报的重要发展方向。异构众核高性能计算机和E级计算的高速发展为这一发展提供了契机,但也对现有业务数值预报中采用的数值方法提出了挑战。此文仅对国内外下一代大气模式涉及到的数值方法进行综述,着重于数值算法、准均匀球面网格和时间积分方案等3个方面,期望为相关研究者提供参考。     

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。     

AN INTEGRATION METHOD WITH FITTING CUBIC SPLINE FUNCTIONS TO A NUMERICAL MODEL OF 2ND-ORDER SPACE-TIME DIFFERENTIAL REMAINDER——FOR AN IDEAL GLOBAL SIMULATION CASE WITH PRIMITIVE ATMOSPHERIC EQUATIONS

In this paper,the forecasting equations of a 2nd-order space-time differential remainder are deduced from the Navier-Stokes primitive equations and Eulerian operator by Taylor-series expansion.Here we introduce a cubic spline numerical model(Spline Model for short),which is with a quasi-Lagrangian time-split integration scheme of fitting cubic spline/bicubic surface to all physical variable fields in the atmospheric equations on spherical discrete latitude-longitude mesh.A new algorithm of"fitting cubic spline—time step integration—fitting cubic spline—……"is developed to determine their first-and2nd-order derivatives and their upstream points for time discrete integral to the governing equations in Spline Model.And the cubic spline function and its mathematical polarities are also discussed to understand the Spline Model’s mathematical foundation of numerical analysis.It is pointed out that the Spline Model has mathematical laws of"convergence"of the cubic spline functions contracting to the original functions as well as its 1st-order and 2nd-order derivatives.The"optimality"of the 2nd-order derivative of the cubic spline functions is optimal approximation to that of the original functions.In addition,a Hermite bicubic patch is equivalent to operate on a grid for a 2nd-order derivative variable field.Besides,the slopes and curvatures of a central difference are identified respectively,with a smoothing coefficient of 1/3,three-point smoothing of that of a cubic spline.Then the slopes and curvatures of a central difference are calculated from the smoothing coefficient 1/3 and three-point smoothing of that of a cubic spline,respectively.Furthermore,a global simulation case of adiabatic,non-frictional and"incompressible"model atmosphere is shown with the quasi-Lagrangian time integration by using a global Spline Model,whose initial condition comes from the NCEP reanalysis data,along with quasi-uniform latitude-longitude grids and the so-called"shallow atmosphere"Navier-Stokes primitive equations in the spherical coordinates.The Spline Model,which adopted the Navier-Stokes primitive equations and quasi-Lagrangian time-split integration scheme,provides an initial ideal case of global atmospheric circulation.In addition,considering the essentially non-linear atmospheric motions,the Spline Model could judge reasonably well simple points of any smoothed variable field according to its fitting spline curvatures that must conform to its physical interpretation.     

Modeling heavy precipitation in complex terrain

Summary A numerical prediction model is described which uses the full set of prognostic equations on a domain roughly the size of the United States with a 96 km horizontal grid resolution and six sigma-coordinate levels. Within this grid resides a nested domain of approximately 1000×1000 km with 24 km horizontal resolution. In this nested grid only modifications to the wind field by the better resolved terrain are considered on the lowest two sigma levels. The terrain effects necessitate adjustments in the location of these two sigma levels. Adjusted wind fields cause modifications in the mass and moisture divergence fields, hence in precipitation. These modifications are averaged into the appropriate meteorological fields on the larger grid.The algorithms used by our model allow continuous interaction between both grids with high computational efficiency.The relative importance of synoptic forcing and terrain is demonstrated for the cases of the Big Thompson, Colorado, flood of 1976 and the Cheyenne, Wyoming, flood of 1985.With 15 Figures     

垂直变量配置对静力适应过程的影响

为了研究垂直变量配置对静力适应过程的影响,本文从描写静力适应过程的方程组出发,分别在将所有变量置于整层上的非跳点N网格;将垂直速度和温度放置在整层,水平速度、气压和密度等变量放置在半层的Charney Phillips跳点网格(CP网格);将水平速度、气压和温度放置在整层,将垂直速度和密度放置在半层的Lorenz跳点网格(L网格);将密度变量放置在整层的Charney Phillips跳点网格(CP_N网格);将密度放置在整层的Lorenz跳点网格(L_N网格)上进行离散,垂直格距分1 km、0.5 km、0.2 km和0.01 km,研究了在这5种网格上产生的频率和垂直群速的相对误差。结果表明:(1)L_N网格和CP网格是完全等效的两种网格。(2)不论垂直格距为多少,CP网格和L网格的误差都是最小,N网格次之,CP_N网格的误差最大。(3)随着垂直格距的减少,在这几种网格上产生的误差都在减小。对于CP网格、L网格和N网格,在水平长波和垂直短波处产生的误差较大。而CP_N网格对水平波长变化不敏感,垂直波长越短,误差越大。(4)当垂直格距为0.01 km时,这几种网格都对水平波长的变化不敏感了,仅对垂直波长敏感。(5)CP网格、L_N网格和L网格在描写静力适应过程和斜压地转适应过程都是误差最小的垂直变量配置方案,因此在非静力完全可压缩深层大气数值预报模式中应优先选择这3种方案。     

多尺度非静力通用模式框架的设计策略

针对目前国际上数值预报模式的最新发展趋势,综述了新形势下多尺度非静力通用数值预报模式框架的设计策略。目前世界上开发通用模式主要有两个途经,一是建立一个离散化方案和源程序代码共享的、全球和有限区通用的模式,二是建立一个单一的全球可变分辨率的模式。还从通用模式方程组的选取策略、模式网格属性的构造,时间积分方案、空间离散方案,垂直坐标的选取等方面进行了分析。     

Experiments on numerical modeling of intense convection

Results of numerical simulations using the WRF-ARW nonhydrostatic model are presented for eight episodes of intense convection over European Russia in the summer of 2007. The calculations were performed on four nested grids with horizontal grid meshes of 27, 9, 3, and 1 km. Convection was parametrized on the first two grids and explicitly resolved on the other two. It has been found that simulations on finer grids with explicit calculation of convective flows make it possible to reproduce heavy rainfalls and strong-wind zones in the areas of intense convection. A preliminary verification of the short-range predictions of convective systems shows that the maximum 12-h precipitation totals and the maximum winds at 10 m are close, in the order of magnitude, to the observed values. Prediction of convection centers is the weakest point. Difficulties in the model verification associated with the absence of data with high space-time resolution are discussed.     

Solving the map overlay problem with a fuzzy approach

The map overlay problem occurs when mismatched gridded data need to be combined, the problem consists of determining which portion of grid cells in one grid relates to the partly overlapping cells of the target grid. This problem contains inherent uncertainty, but it is an important and necessary first step in analysing and combining data; any improvement in achieving a more accurate relation between the grids will positively impact the subsequent analysis and conclusions. Here, a novel approach using techniques from fuzzy control and artificial intelligence is presented to provide a new methodology. The method uses a fuzzy inference system to decide how data represented in one grid can be distributed over another grid using any additionally available knowledge, thus mimicking the higher reasoning that we as humans would use to consider the problem.     

An Adaptive Nonhydrostatic Atmospheric Dynamical Core Using a Multi-Moment Constrained Finite Volume Method

An adaptive 2 D nonhydrostatic dynamical core is proposed by using the multi-moment constrained finite-volume(MCV) scheme and the Berger-Oliger adaptive mesh refinement(AMR) algorithm. The MCV scheme takes several pointwise values within each computational cell as the predicted variables to build high-order schemes based on single-cell reconstruction. Two types of moments, such as the volume-integrated average(VIA) and point value(PV), are defined as constraint conditions to derive the updating formulations of the unknowns, and the constraint condition on VIA guarantees the rigorous conservation of the proposed model. In this study, the MCV scheme is implemented on a height-based, terrainfollowing grid with variable resolution to solve the nonhydrostatic governing equations of atmospheric dynamics. The AMR grid of Berger-Oliger consists of several groups of blocks with different resolutions, where the MCV model developed on a fixed structured mesh can be used directly. Numerical formulations are designed to implement the coarsefine interpolation and the flux correction for properly exchanging the solution information among different blocks. Widely used benchmark tests are carried out to evaluate the proposed model. The numerical experiments on uniform and AMR grids indicate that the adaptive model has promising potential for improving computational efficiency without losing accuracy.     

New precipitation and temperature grids for northern Patagonia: Advances in relation to global climate grids

Climate data of mean monthly temperature and total monthly precipitation compiled from different sources in northern Patagonia were interpolated to 20-km resolution grids over the period 1997–2010. This northern Patagonian climate grid (NPCG) improves upon previous gridded products in terms of its spatial resolution and number of contributing stations, since it incorporates 218 and 114 precipitation and temperature records, respectively. A geostatistical method using surface elevation from a Digital Elevation Model (DEM) as the ancillary variable was used to interpolate station data into even spaced points. The maps provided by NPCG are consistent with the broad spatial and temporal patterns of the northern Patagonian climate, showing a comprehensive representation of the latitudinal and altitudinal gradients in temperature and precipitation, as well as their related patterns of seasonality and continentality. We compared the performance of NPCG and various other datasets available to the climate community for northern Patagonia. The grids used for the comparison included those of the Global Precipitation Climatology Project, ERAInterim, Climate Research Unit (University of East Anglia), and University of Delaware. Based on three statistics that quantitatively assess the spatial coherence of gridded data against available observations (bias, MAE, and RMSE), NPCG outperforms other global grids. NPCG represents a useful tool for understanding climate variability in northern Patagonia and a valuable input for regional models of hydrological and ecological processes. Its resolution is optimal for validating data from the general circulation models and working with raster data derived from remote sensing, such as vegetation indices.     

Variable resolution techniques for weather prediction

Summary This paper presents a brief overview of a few variable resolution techniques in the context of the horizontal discretization of the meteorological equations. These are the grid refinement method, the static and dynamic coordinate transformation methods and the variable resolution in physical space method. The latter is illustrated by a variable resolution reformulation of the popular C-grid discretization suitable for use in a limited area model.With 5 Figures     

不同降水方案对"03.7"一次暴雨过程模拟的影响

观测和数值模拟研究已经表明,潜热释放对中国东部夏季梅雨锋系统及其锋面降水的维持和发展发挥着非常重要的作用.然而,目前对于梅雨锋降水模拟中各种降水方案的相互协调和系统评估方面的工作仍不多见,为了增进对梅雨锋暴雨模拟中降水过程的认识,作者针对2003年7月4～5日一次梅雨锋暴雨过程,构造了四组试验,利用MM5模式考察了两种分辨率(36 km、12 km),各种隐/显式方案搭配下,对所生成的雨带、雨量和降水类型的配置进行了仔细的研究,得到了一些有意义的结论,为今后更好地使用模式、利用数值模式来认识中尺度降水过程中的气象问题打下基础.主要结论包括:模拟总降水的水平分布和强度,以及显式降水和隐式降水的划分对积云参数化方案的选择非常敏感.但对特定积云参数化方案而言,降水的模拟对36 km、12 km水平分辨率不敏感(除Betts-Miller方案外);在中尺度网格分辨率10～50 km范围内,不同积云参数化方案对梅雨锋降水分布和降水量模拟的影响比不同显式方案带来的变化大得多.     

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。     

自适应网格技术在数值模式中的应用研究 II.二维问题

用一个具有解析解的二维动力锋生成过程算例,对比自适应网格方案和固定网格方案的优劣,结果表明:同等误差要求下,固定网格方案的网格点数为自适应网格方案的3倍。从时间演化上看,自适应网格对温带气旋的预报,在同等误差要求下,可比固定网格的预报延长10小时以上。文章对加权系数选取及对解的影响进行了分析,从几个切面的计算结果展示了自适应网格对网格的优良安排并能抓住锋面特征结构。文章分析了光滑性、正交性对结果的影响。结果表明:网格的光滑性影响有一个优化值;在网格适当安排情况下,要适度考虑正交性。     

A grid nesting method for large-eddy simulation of planetary boundary-layer flows

A method for performing nested grid calculations with a large-eddy simulation code is described. A common numerical method is used for all meshes, and the grid architecture consists of a single outer or coarse grid, and nested or fine grids, which overlap in some common region. Inter-grid communication matches the velocity, pressure and potential temperature fields in the overlap region. Resolved and sub-grid scale (SGS) turbulent fluxes and kinetic energy on the fine grid are averaged to the coarse grid using a conservation rule equivalent to Germano's identity used to develop dynamic SGS models.Simulations of a slightly convective, strong shear planetary boundary layer were carried out with varying surface-layer resolutions. Grid refinements in the (x, y, z) directions of up to (5, 5, 2) times were employed. Two-way interaction solutions on the coarse and fine meshes are successfully matched in the overlap region on an instantaneous basis, and the turbulent motions on the fine grid blend smoothly into the coarse grid across the grid interface. With surface-layer grid nesting, significant increases in resolved eddy fluxes and variances are found. The energy-scale content of the vertical velocity, and hence vertical turbulent fluxes, appear to be most influenced by increased grid resolution. Vertical velocity spectra show that the dominant scale shifts towards higher wavenumbers (smaller scales) and the magnitude of the peak energy is increased by more than a factor of 3 with finer resolution. Outside of the nested region the average heat and momentum fluxes and spectra are slightly influenced by the fine resolution in the surface layer. From these results we conclude that fine resolution is required to resolve the details of the turbulent motions in the surface layer. At the same time, however, increased resolution in the surface layer does not appreciably alter the ensemble statistics of the resolved and SGS motions outside of the nested region.     

Towards Adaptive Grids for Atmospheric Boundary-Layer Simulations

We present a proof-of-concept for the adaptive mesh refinement method applied to atmospheric boundary-layer simulations. Such a method may form an attractive alternative to static grids for studies on atmospheric flows that have a high degree of scale separation in space and/or time. Examples include the diurnal cycle and a convective boundary layer capped by a strong inversion. For such cases, large-eddy simulations using regular grids often have to rely on a subgrid-scale closure for the most challenging regions in the spatial and/or temporal domain. Here we analyze a flow configuration that describes the growth and subsequent decay of a convective boundary layer using direct numerical simulation (DNS). We validate the obtained results and benchmark the performance of the adaptive solver against two runs using fixed regular grids. It appears that the adaptive-mesh algorithm is able to coarsen and refine the grid dynamically whilst maintaining an accurate solution. In particular, during the initial growth of the convective boundary layer a high resolution is required compared to the subsequent stage of decaying turbulence. More specifically, the number of grid cells varies by two orders of magnitude over the course of the simulation. For this specific DNS case, the adaptive solver was not yet more efficient than the more traditional solver that is dedicated to these types of flows. However, the overall analysis shows that the method has a clear potential for numerical investigations of the most challenging atmospheric cases.     

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.     

Conformal octagon: An attractive framework for global models offering quasi-uniform regional enhancement of resolution

Summary With the increasingly widespread adoption of massively parallel processing (MPP) computers for applications in computational fluid dynamics it becomes appropriate to reconsider the geometrical configuration of the computational grid that best suits the problem. In the case of global numerical weather prediction we have recently advocated a conformal spherical-cubic geometry. Among its merits, this grid lends itself naturally to simple domain-decomposition and obviates the need for polar filtering.Here we extend the same principles, but with an emphasis on the problem of regional forecasting. In this case we observe that it is possible to cover the global domain with a conformal grid geometry based on the mapping to the sphere of a back-to-back pair of octagonal regions. In the most symmetrical case, each octagon maps to a hemisphere. By compounding this mapping with a nonhomogeneous conformal mapping of the sphere to itself, one can also arrange to have quasi-uniform enhanced resolution of the resulting grid inside any chosen circle on the sphere, at the expense of relatively coarse resolution degrading gradually with distance outside the circle of interest.With appropriate grid dimensions, the new conformal octagon decomposes naturally into several identical square subdomains for efficient distribution over the nodes of an MPP computer.With 11 Figures