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.     

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。     

基于新型LE水平跳点网格的阴阳网格平流方案设计

为了模拟球面平流传输过程,本文基于球面阴阳重叠网格设计了一种两时间层半拉格朗日平流方案.该方案在球面坐标下采用新型的LE水平跳点网格,同时针对阴阳网格重叠区,采用了不同插值方法进行比较分析,且进行了相关的理想数值试验对方案设计效果进行评估.数值试验表明方案设计是成功的,阴阳网格重叠区平流对插值方案比较敏感;半拉格朗日方案能较好地模拟球面刚体平流和变形涡旋的结构、位置及演变过程,并具有较好的数值稳定性和较高的数值精度.     

An improved dynamic core for a non-hydrostatic model system on the Yin-Yang grid

A 3D dynamic core of the non-hydrostatic model GRAPES(Global/Regional Assimilation and Prediction System) is developed on the Yin-Yang grid to address the polar problem and to enhance the computational efficiency. Three-dimensional Coriolis forcing is introduced to the new core, and full representation of the Coriolis forcing makes it straightforward to share code between the Yin and Yang subdomains. Similar to that in the original GRAPES model, a semi-implicit semi-Lagrangian scheme is adopted for temporal integration and advection with additional arrangement for cross-boundary transport. Under a non-centered second-order temporal and spatial discretization, the dry nonhydrostatic frame is summarized as the solution of an elliptical problem. The resulting Helmholtz equation is solved with the Generalized Conjugate Residual solver in cooperation with the classic Schwarz method. Even though the coefficients of the equation are quite different from those in the original model, the computational procedure of the new core is just the same. The bi-cubic Lagrangian interpolation serves to provide Dirichlet-type boundary conditions with data transfer between the subdomains. The dry core is evaluated with several benchmark test cases, and all the tests display reasonable numerical stability and computing performance. Persistency of the balanced flow and development of both the mountain-induced Rossby wave and Rossby–Haurwitz wave confirms the appropriate installation of the 3D Coriolis terms in the semi-implicit semi-Lagrangian dynamic core on the Yin-Yang grid.     

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。     

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.     

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。     

Improvement of the semi-Lagrangian advection scheme in the GRAPES model: Theoretical analysis and idealized tests

ABSTRACT The Global/Regional Assimilation and PrEdiction System （GRAPES） is the newgeneration numerical weather predic- tion （NWP） system developed by the China Meteorological Administration. It is a fully compressible non-hydrostatical global/regional unified model that uses a traditional semi-Lagrangian advection scheme with cubic Lagrangian interpola tion （referred to as the SL_CL scheme）. The SL_CL scheme has been used in many operational NWP models, but there are still some deficiencies, such as the damping effects due to the interpolation and the relatively low accuracy. Based on Reich＇s semi-Lagrangian advection scheme （referred to as the R2007 scheme）, the Re_R2007 scheme that uses the low- and high-order B-spline function for interpolation at the departure point, is developed in this paper. One- and two-dimensional idealized tests in the rectangular coordinate system with uniform grid cells were conducted to compare the Re..R2007 scheme and the SL_CL scheme. The numerical results showed that： （1） the damping effects were remarkably reduced with the Re_R2007 scheme; and （2） the normalized errors of the Re_R2007 scheme were about 7.5 and 3 times smaller than those of the SL_CL scheme in one- and two-dimensional tests, respectively, indicating the higher accuracy of the Re..R2007 scheme. Furthermore, two solid-body rotation tests were conducted in the latitude-longitude spherical coordinate system with non uniform grid cells, which also verified the Re_R2007 scheme＇s advantages. Finally, in comparison with other global advection schemes, the Re_R2007 scheme was competitive in terms of accuracy and flow independence. An encouraging possibility for the application of the Re_R2007 scheme to the GRAPES model is provided.     

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.     

GRAPES模式中高精度正定保形物质平流方案的研究Ⅰ:理论方案设计与理想试验

平流计算的精度对数值模式的结果有着重要影响.如何在半拉格朗日模式中发展高阶精度的标量平流计算方案是提高半拉格朗日数值模式精度的重要问题.文中采用计算流体力学中一个新的高精度正定保形的物质平流方案,通过映射单元格方法将其与半拉格朗日模式结合起来,既保留了半拉格朗日时间积分方案中积分时间步长大、计算效率高的特点,又发挥新方...     

An accurate multimoment constrained finite volume transport model on Yin-Yang grids

A global transport model is proposed in which a multimoment constrained finite volume (MCV) scheme is applied to a Yin-Yang overset grid. The MCV scheme defines 16 degrees of freedom (DOFs) within each element to build a 2D cubic reconstruction polynomial. The time evolution equations for DOFs are derived from constraint conditions on moments of line-integrated averages (LIA), point values (PV), and values of first-order derivatives (DV). The Yin-Yang grid eliminates polar singularities and results in a quasi-uniform mesh. A limiting projection is designed to remove nonphysical oscillations around discontinuities. Our model was tested against widely used benchmarks; the competitive results reveal that the model is accurate and promising for developing general circulation models.     

A Positivity-preserving Conservative Semi-Lagrangian Multi-moment Global Transport Model on the Cubed Sphere

A positivity-preserving conservative semi-Lagrangian transport model by multi-moment finite volume method has been developed on the cubed-sphere grid. Two kinds of moments(i.e., point values(PV moment) at cell interfaces and volume integrated average(VIA moment) value) are defined within a single cell. The PV moment is updated by a conventional semi-Lagrangian method, while the VIA moment is cast by the flux form formulation to assure the exact numerical conservation. Different from the spatial approximation used in the CSL2(conservative semi-Lagrangian scheme with second order polynomial function) scheme, a monotonic rational function which can effectively remove non-physical oscillations is reconstructed within a single cell by the PV moments and VIA moment. To achieve exactly positive-definite preserving, two kinds of corrections are made on the original conservative semi-Lagrangian with rational function(CSLR)scheme. The resulting scheme is inherently conservative, non-negative, and allows a Courant number larger than one.Moreover, the spatial reconstruction can be performed within a single cell, which is very efficient and economical for practical implementation. In addition, a dimension-splitting approach coupled with multi-moment finite volume scheme is adopted on cubed-sphere geometry, which benefitsthe implementation of the 1 D CSLR solver with large Courant number.The proposed model is evaluated by several widely used benchmark tests on cubed-sphere geometry. Numerical results show that the proposed transport model can effectively remove nonphysical oscillations and preserve the numerical nonnegativity, and it has the potential to transport the tracers accurately in a real atmospheric model.     

Variable resolution on quasi-uniform grids: linear advection experiments

Summary A couple of different techniques for implementing the variable resolution within numerical models of the atmosphere that use quasi-uniform grids are estimated and compared based on their merit in the simple linear advection experiments. These techniques are: grid stretching, Schmidt transforming and grid overlapping. The quasi-uniform grids are: a conformal cubic and an octagonal grid. The techniques for the variable resolution are evaluated from the perspective of their effects on numerical solution and the practicality of their application. Our analysis suggests that the grid stretching is for now the most optimal choice. However, a special form of grid overlapping is presented that has potential to become a viable alternative in the future.     

PRM标量平流方案在GRAPES全球预报系统中的应用

如何更好地模拟水物质的分布,对于数值天气预报效果的改进,特别是对于更好地模拟降水过程,具有重要的意义。半拉格朗日模式中的标量平流计算要求做到高精度、守恒、正定和保形,但GRAPES_GFS (Global-Regional Assimilation and PrEdiction System, Global Forecast System) 中采用的QMSL(Quasi-Monotone Semi-Lagrangian)平流方案在水汽的强梯度、不连续区域计算精度较低,且不能做到严格守恒。本研究借鉴计算流体力学领域的研究进展,将一个基于分段有理函数的物质平流方案PRM(Piecewise Rational Method)引入GRAPES_GFS中,按照通量形式求解水汽方程,并对极区进行了混合等技术处理。通过一系列理想试验对两种平流方案进行了对比,证明了PRM方案精度较高,特别是在水汽梯度大的区域优势明显,频散、耗散误差较小,守恒、保形性也要好于QMSL方案。通过对GRAPES_GFS中批量预报试验效果的检验,验证了PRM方案可以有效地改进模式对水物质分布的模拟,提高了降水的预报效果,对模式综合预报性能的提升也有明显作用。     

相变修正方案在GRAPES模式标量平流中的应用

如何更好地模拟水物质的空间分布和小尺度变化,对于数值天气预报效果的改进,特别是对于更好地模拟降水过程,具有重要的意义.计算机的飞速发展使数值模式的分辨率不断提高,云的显式计算成为可能,这样就要求水物质在平流的过程中必须要做到高精度、守恒、保形.水物质场是正定标量的场,具有空间和时间变化幅度大、存在强梯度甚至不连续的特点,水物质场的合理模拟一直是数值预报中的一个难题.GRAPES模式中的标量平流方案采用PRM分段有理函数方法,比较好地解决了该半拉格朗日模式中水物质平流的高精度、守恒、保形问题,但是当有凝结潜热发生时,由于半拉格朗日平流方案求解上游点时的插值,在云边缘区域会造成虚假的云水,进而导致不合理的相变过程.为了解决以上问题,本研究在GRAPES模式中PRM平流方案的案础上,加入了非线性半拉格朗日相变潜热的修正方案,旨在改进GRAPES模式对水物质平流问题的模拟,提高降水的预报效果.该研究通过理想试验,验证了非线性半拉格朗口相变修正方案可以有效地限制云边缘由于半拉格朗日平流方案插值产生的虚假柑变;然后将该方案加入GRAPES模式的PRM水物质平流方案中,通过实际个例模拟验证了加入非线性半拉格朗日方案以后,模式可以更好地模拟水物质的平流过程,且对云中热力场及水物质分布地模拟更加合理,同时预报出的雨带中心区与实况更加符合.     

On the forward-in-time upstream advection scheme for non-uniform and time-dependent flow

Summary The classical forward-in-time upstream advection scheme for uniform flow field has been extended to include non-uniform and time-dependent advective flow. This generalised scheme is described in one dimension for an advective flow which varies both in time and in space. The classical upstream advection scheme is only first-order accurate both in time and in space if the advective flow is not uniform. Higherorder accuracy in both time and space, however, can be easily obtained in the generalised scheme.This generalised scheme with third-order accuracy is applied to the one-dimensional inviscid Burgers equation (socalled self-advection problem), two-dimensional steady flow, and to a time-split shallow water equation model. The results are compared with those obtained from the Takacs' (1985) scheme and from a standard third-order semi-Lagrangian scheme, and also with those obtained from the fourth-order Lax-Wendroff scheme of Crowley (1968) in the time-split shallow water equation model. It is shown that the generalised scheme performs as well as, but is more efficient than, the standard semi-Lagrangian scheme with same order. It is much more accurate than the Takacs' scheme which has large dissipation errors, especially for the flow with strong deformation. In contrast, the generalised scheme has very weak dissipation and has much better dispersion and shapeconserving properties. Although the fourth-order Lax-Wendroff scheme has higher accuracy and can give more accurate numerical solutions for uniform advective flow or solid rotational flow (Crowley, 1968), it is inferior to the generalised third-order scheme for non-uniform flow with strong deformation or large spatial gradients. This generalised scheme, therefore, has considerable application potential in different numerical models, especially for the models using time-split algorithms.With 8 Figures     

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.     

An efficient non-hydrostatic forecast model

Summary A semi-implicit non-hydrostatic mesoscale forescasting model, which is used operationally in the United Kingdom Meteorological Office, has been reformulated using two time levels and semi-Lagrangian advection so as to permit longer timesteps. The boundaries are found to be an important influence on the model's stability and two methods of suppressing wave reflection are described with the aid of a one-dimensional experiment. The model's responses to two dimensional flow over a ridge and to three-dimensional flow over an isolated mountain are then demonstrated. The reformulated model is more efficient than the original provided a timestep at least twice as long can be used. It is also superior both in having minimal damping (except at the boundaries), and in the accuracy of the advection scheme.     

GRAPES_YY模式发展及其对梅雨锋降水模拟的性能检验

GRAPES_YY是在球面准均匀阴阳网格动力框架基础上发展起来的全球非静力模式，目前已耦合全套物理过程参数化方案，具备了中期天气预报能力。为了解模式对环流和降水的实际预报效果，检验模式对多尺度天气系统的预报能力和误差分布特征，以ERA-interim资料和逐时地面融合降水产品为参考，对1个月（2018年6月中旬—7月中旬）批量试验结果和一次长江流域梅雨个例（7月4—7日）模拟结果进行分析。结果表明:在改进上边界条件基础上，模式有较好的稳定性，短期预报时效内对全球环流和降水刻画准确，模式24 h累计降水纬向平均成功再现了低纬度和中纬度地区的两个降水峰值，但对低纬度对流性降水和中纬度格点尺度降水预报偏强。其次，模式成功模拟出7月4—6日中国东部梅雨雨带位置、走向及南北摆动，整体上能正确反映主要天气系统的移动演变，但受分辨率限制以及湿物理过程影响，暴雨以上量级降水还存在强度预报偏弱、位置偏北的问题。GRAPES_YY模式模拟结果基本合理可信，对中低纬度梅雨锋降水的预报能力较GRAPES_GFS略有改善。      

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