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     

非均匀网格下的高阶精度中央差分格式:理论推导和理想试验

传统的高阶精度有限差分格式通常是在均匀网格的基础上推导得到的,在非均匀网格的情况下它会出现精度退化的问题。基于泰勒展开方法构造了一种适用于非均匀网格的2阶、4阶和6阶精度中央有限差分方案,利用Burgers方程和一维平流方程对新方案的性能进行测试,着重分析新方案对其误差大小及分布形态的改进效果。数值模拟结果表明:在非均匀网格下,提高差分方案的精度可明显减小数值解误差(降低了70%~88%),特别是当差分精度从2阶提高到4阶的时候。同时,高阶精度方案在梯度变化较大或者网格距较粗区域的模拟结果更有优势,4阶和6阶精度方案在以上区域的误差远小于2阶精度方案。方案可用于提高数值天气预报模式中非均匀分层模式的垂直差分计算精度。      

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.     

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。     

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。     

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...     

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.     

一种提高数值模式时间差分计算精度的新格式———回溯时间积分格式

为了使用以制作数值预报的微分方程中包含更多的信息量、以期提高预报准确度, 该文提出了一种基于记忆动力学的时间积分格式。以天气预报为实例, 计算表明, 平流方程的回溯时间积分格式所得的预报准确度大大高于传统的蛙跃差分格式。此方法在海洋、水文、环境、航空等应用平流方程计算的多种科学中亦是有效的。     

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.     

Numerical simulation of plume-advection over a hill with coupled models. Part I: Numerical solution of the transport-equation

Summary Four numerical methods suitable for the calculation of transport are compared on a linear two-dimensional advection equation. It is found that a mass-compensation algorithm which removes negative values has also a beneficial effect on the accuracy of schemes which are of higher order but are not positive. It is shown by numerical experiments, that the upstream spline advection scheme combined with a mass compensation is prefereable for some applications rather than other high accurate positive schemes like the flux-corrected transport scheme.With 3 Figures     

A Two-Step Shape-Preserving Advection Scheme

This paper proposes a new two-step non-oscillatory shape-preserving positive definite finite difference advection transport scheme, which merges the advantages of small dispersion error in the simple first-order upstream scheme and small dissipation error in the simple second-order Lax-Wendroff scheme and is completely different from most of present positive definite advection schemes which are based on revising the upstream scheme results. The proposed scheme is much less time consuming than present shape-preserving or non-oscillatory advection transport schemes and produces results which are comparable to the results obtained from the present more complicated schemes. Elementary tests are also presented to examine the behavior of the scheme.     

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。     

Application of variational algorithms in semi- lagrangian framework

1.IntroductionWeatherforecastingmodelisasetofpartialdifferentialequations(PDE).Itshouldsatisfycertaininitial--boundaryconditions.Clearly,oneofthemostimportantimprovementsinweatherforecastingmodelistooptimizeitsinitialandboundaryconditions.Forthesimplicityofdiscussion,wemightregardthespatialboundaryconditionasakindofinitialconditioninspatialdimension.Aneffectivedataassimilationprocedureisrequiredinordertoutilizewidelycollectivedatafortheweatherforecastingmodelandfindanoptimalinitialcondition.…     

Inland and Offshore Propagation Speeds of a Sea Breeze from Simulations and Measurements

The inland and offshore propagation speeds of a sea breeze circulation cell are simulated using a three-dimensional hydrostatic model within a terrain-following coordinate system. The model includes a third-order semi-Lagrangian advection scheme, which compares well in a one-dimensional stand-alone test with the more complex Bott and Smolarkiewicz advection schemes. Two turbulence schemes are available: a local scheme by Louis (1979) and a modified non-local scheme based on Zhang and Anthes (1982). Both compare well with higher-order closure schemes using the Wangara data set for Day 33–34 (Clark et al., 1971).Two-dimensional cross-sections derived from airborne sea breeze measurements (Finkele et al. 1995) constitute the basis for comparison with two-dimensional numerical model results. The offshore sea breeze propagation speed is defined as the speed at which the seaward extent of the sea breeze grows offshore. On a study day, the offshore sea breeze propagation speed, from both measurements and model, is -3.4 m s^-1. The measured inland propagation speed of the sea breeze decreased somewhat during the day. The model results show a fairly uniform inland propagation speed of 1.6 m s^-1 which corresponds to the average measured value. The offshore sea breeze propagation speed is about twice the inland propagation speed for this particular case study, from both the model and measurements.The influence of the offshore geostrophic wind on the sea breeze evolution, offshore extent and inland penetration are investigated. For moderate offshore geostrophic winds (-5.0 m s^-1), the offshore and inland propagation speeds are non-uniform. The offshore extent in moderate geostrophic wind conditions is similar to the offshore extent in light wind conditions (-2.5 m s^-1). The inland extent is greater in light offshore geostrophic winds than in moderate ones. This suggests that the offshore extent of the sea breeze is less sensitive to the offshore geostrophic wind than its inland extent. However, these results hold only if it is possible to define an inland propagation speed. For stronger offshore geostrophic winds (-7.5 m s^-1), the sea breeze is completely offshore and the inland propagation speed is ill-defined.     

The Influence of Advection on the Short Term CO 2 -Budget in and Above a Forest Canopy

An experimental micrometeorological set-up was established at the CARBOEURO-FLUX site in Tharandt, Germany, to measure all relevant variables for the calculation of the vertical and horizontal advective fluxes of carbon dioxide. The set-up includes two auxiliary towers to measure horizontal and vertical CO₂ and H₂O gradients through the canopy, and to make ultrasonic wind measurements in the trunk space. In combination with the long-term flux tower an approximately even-sided prism with a typical side-length of 50 m was established. It is shown that under stable (nighttime) conditions the mean advective fluxes have magnitudes on the same order as the daily eddy covariance (EC) flux, which implies that they play a significant, but not yet fully understood, role in the carbon budget equation. The two advective fluxes are opposite and seem to cancel each other at night (at least for these measurements). During the day, vertical advection tends to zero, while horizontal advection is still present implying a flow of CO₂ out of the control volume. From our measurements, a mean daily gain of 2.2 gC m^–2 d^–1 for the horizontal advection and a mean daily loss of 2.5 gC m^–2d^–1 for the vertical advection is calculated for a period of 20 days. However the large scatter of the advective fluxes has to be further investigated. It is not clear yet whether the large variability is natural or due to measurement errors and conceptual deficiencies of the experiment. Similar results are found in the few comparable studies.     

A High-Order Spatiotemporal Precision-Matching Taylor–Li Scheme for Time-Dependent Problems

Based on the Taylor series method and Li's spatial differential method, a high-order hybrid Taylor–Li scheme is proposed.The results of a linear advection equation indicate that, using the initial values of the square-wave type, a result with thirdorder accuracy occurs. However, using initial values associated with the Gaussian function type, a result with very high precision appears. The study demonstrates that, when the order of the time integral is more than three, the corresponding optimal spatial difference order could be higher than six. The results indicate that the reason for why there is no improvement related to an order of spatial difference above six is the use of a time integral scheme that is not high enough. The author also proposes a recursive differential method to improve the Taylor–Li scheme's computation speed. A more rapid and highprecision program than direct computation of the high-order space differential item is employed, and the computation speed is dramatically boosted. Based on a multiple-precision library, the ultrahigh-order Taylor–Li scheme can be used to solve the advection equation and Burgers' equation.     

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-LAGRANGIAN ADVECTIVE SCHEMES AND THEIR NONLINEAR COMPUTATIONAL CHAOS*

In this paper,we have suggested several interpolating formulae which are convinient for vector computers to perform vector calculation.Analysis shows that for a simple advective problem,the semi-Lagrangian schemes constructed with these formulae are linearly unconditionally stable.Moreover,we have discussed the nonlinear computational chaos in the semi-Lagrangian schemes.It is found that for a simple nonlinear advective problem,we can introduce physically unreal and direct interaction of three or more waves.Numerical tests indicate that this phenomenon has certain influence on the simulated results of two-dimensional Benard convection.     

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.