高阶正定守恒的中心约束多矩有限体积平流模式

采用新的均匀三点中心约束多矩有限体积方法（3-point Multi-moment Constrained finite-Volume scheme for Uniform Points with Center Constraints, MCV3_UPCC）,发展了一个三阶正定守恒的平流模式。三点多矩有限体积方法在单网格内定义等距的3个自由度,采用多矩约束条件并通过控制方程获得时间演变方程。新的三点中心约束多矩方法能在单网格内采用等距的3个点值及中心一阶、二阶导数作为约束条件进行空间4次多项式数值重构,获得3个自由度的时间演变方程;所构建的新数值方案具有三阶精度,边界通量连续性保证了其数值严格守恒。为了抑制该方法的非物理数值振荡,引入了边界保型限制器技术,它能够把数值解控制在既定物理场最小值（最小值为0时则保持数值正定）与最大值之间。数值试验表明新发展的三阶平流模式具有良好的计算精度,能够严格保持数值解的正定性和守恒性,同其他高精度平流模式相当,在实际大气模式水汽等平流输送应用中具备良好的发展潜力。      

A 3D Nonhydrostatic Compressible Atmospheric Dynamic Core by Multi-moment Constrained Finite Volume Method

A 3D compressible nonhydrostatic dynamic core based on a three-point multi-moment constrained finite-volume (MCV) method is developed by extending the previous 2D nonhydrostatic atmospheric dynamics to 3D on a terrain-following grid. The MCV algorithm defines two types of moments: the point-wise value (PV) and the volume-integrated average (VIA). The unknowns (PV values) are defined at the solution points within each cell and are updated through the time evolution formulations derived from the governing equations. Rigorous numerical conservation is ensured by a constraint on the VIA moment through the flux form formulation. The 3D atmospheric dynamic core reported in this paper is based on a three-point MCV method and has some advantages in comparison with other existing methods, such as uniform third-order accuracy, a compact stencil, and algorithmic simplicity. To check the performance of the 3D nonhydrostatic dynamic core, various benchmark test cases are performed. All the numerical results show that the present dynamic core is very competitive when compared to other existing advanced models, and thus lays the foundation for further developing global atmospheric models in the near future.     

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.     

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.     

守恒型有理函数插值半拉格朗日平流方案及性能分析

通过多种理想试验对正定、保形守恒型有理函数插值半拉格朗日平流方案分别在平面直角坐标以及阴阳网格球面坐标中进行了计算性能分析,并采用多种误差模对守恒型有理函数插值半拉格朗日平流方案的网格收敛性进行评估。结果表明,采用分段有理函数插值的守恒型半拉格朗日平流方案可以有效消除不连续分布处的数值振荡、保证正定性,物理场平滑分布时维持1-2阶收敛速度;而在不连续点或大梯度区域以及应用分维技术的多维算法都会通过有理函数的降阶特性,影响平流计算的收敛阶数,并且,在球面坐标中受球面曲率的影响,守恒型有理函数插值半拉格朗日平流算法的网格收敛速度有所降低。     

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.     

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。     

完全非内插半拉格朗日格式在一维 Burgers方程及二维浅水波方程上的应用

在作者过去提出的完全非内插半拉格朗日格式的基础上,针对半拉格朗日格式由于内插带来预报场人为的光滑性问题,进一步发展了这种计算格式,证明了此格式的计算稳定性。为检验这种新的计算格式的性能,在一维和二维问题上进行了应用。在一维问题中采用了一维无粘Burgers方程（方程中有突变点）;二维问题采用了浅水波方程,同时将这些计算结果与Ritchie方案及欧拉方案或一般半拉格朗日内插方案的计算结果进行了比较,发现新格式消除了内插和预报场的人为光滑,并且计算精度有一定程度的提高,这为以后将此格式推广到全球谱模式打下了基础。     

Modelling Boundary-Layer Clouds With a Statistical Cloud Scheme and a Second-Order Turbulence Closure

We present a second-order turbulence model for the cloudy planetary boundary layer (PBL), which includes a statistical scheme of the sub-grid scale condensation. The model contains prognostic equations for the turbulent kinetic energy, total water, and liquid water temperature, the latter two being assumed to be conservative variables. Using these conservative thermodynamic variables the condensation process is formulated as a function of the departure of the total water from saturation and its variance. The computation of the variance requires second moment correlations which are modelled through the parameterization of the third-order moments using a convective mass-flux formulation. The inclusion of these third moments and new assumptions on heat flux transport lead to a nonlocal turbulence scheme with counter-gradient effects. The final form for the heat flux turns out to be a linearized version of a previously established result. For the statistical cloud formulation, a linear combination of a Gaussian and a positively skewed distribution function is used with a modified liquid water flux expression to account fornon-Gaussian behaviour.The effect of the turbulence scheme on the boundary-layer cloud structure is discussed and the performance of the model is tested by comparing it against the large eddy simulation (LES) of the undisturbed period of the Atlantic Stratocumulus Transition Experiment (ASTEX). The model is able to produce both mean and turbulent quantities that are in reasonable agreement with the LES output of ASTEX.     

ATTILA: atmospheric tracer transport in a Lagrangian model

The model ATTILA has been developed to treat the global-scale transport of passive trace species in the atmosphere within the framework of a general circulation model (GCM). ATTILA runs online within the GCM ECHAM4 and advects the centroids of 80.000 to 190.000 constant mass air parcels. Each trace constituent is thereby represented by a mass mixing ratio in each parcel. ATTILA contains state-of-the-art parameterizations of convection, turbulent boundary layer mixing and inter-parcel transport, and provides an algorithm to map the tracer concentrations from the trajectories to the ECHAM model grid. The transport characteristics of ATTILA are evaluated against observations and the standard semi-Lagrangian transport scheme of ECHAM by two experiments. (1) We simulate the distribution of the short-lived tracer radon (²²²Rn) in order to examine fast vertical transport over continents, and long-range transport from the continents to remote areas. (2) We simulate the distribution of radiocarbon (¹⁴C) from nuclear weapon tests in order to examine upper tropospheric and stratospheric transport characteristics. Contrary to the semi-Lagrangian scheme, ATTILA shows a greatly reduced meridional transport in the upper troposphere and lower stratosphere, and a reduced downward flux from the stratosphere to the troposphere, especially in mid-latitudes. Since ATTILA is a numerically non-diffusive scheme, it is able to maintain steep gradients, which compare better to the observations than the rather smooth gradients produced by the semi-Lagrangian scheme.     

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...     

Multispectrum Method and the Computation of Vapor Equation

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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.     

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.     

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.     

基于极大似然算法的风廓线雷达谱矩估计

在风廓线雷达谱数据处理中,雷达低层取样体积探测的谱数据通常出现地物杂波、间歇性杂波、地磁杂波与大气回波谱混杂交叠的情况。为了有效抑制和去除杂波干扰,并且提高风廓线雷达探测范围和测量精度,必须对大气回波谱的谱矩进行有效的估计。通过对风廓线雷达多普勒回波功率谱分析,提出基于极大似然算法估计雷达回波谱谱数据的方法,并且进行了实际数据仿真分析,通过与常规方法进行对比分析,验证了算法的可行性和有效性,在低层取样体积谱矩估计得到了显著的改善。     

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.     

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。     

Simulation of orographically generated waves in a nonhydrostatic model of an adiabatic atmosphere

Results of numerical experiments on the simulation of a flow moving around an isolated mountain are presented. The influence of the sizes of a barrier and of the flow velocity on characteristics of wave oscillations is discussed. All calculations are carried out with the authors’ two-dimensional (in the vertical plane) version of a nonhydrostatic dynamic scheme, in which equations of the dry quasi-incompressible atmosphere are solved with a semi-implicit semi-Lagrangian method. This method uses large time steps as compared to explicit-implicit Eurlerian methods. The results of calculations agree with results obtained by other authors, which gives hope for finding physically correct solutions in the simulation of nonhydrostatic processes in the atmosphere.     

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