高分辨率数值模式气压梯度力算法研究
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摘要
使用非静力完全弹性的动力框架进行数值模拟和数值天气预报正成为大气科学研究和业务数值天气预报模式发展的主流，陡峭地形是困扰高分辨模式发展的主要问题之一。通过对气压梯度力误差分析和数值模式动力框架的发展研究得到的主要结论是：
(1)现有模式中适合于天气尺度的气压梯度力算法在陡峭地形附近都存在较大的计算误差。在中尺度理想场典型参数设置下Corby方案计算的气压梯度力误差最小，精度为10~(-4)m／s~2，但即便如此，其气压梯度力算法也不收敛，即随着模式垂直分辨率的提高，计算精度不但没有提高，反而有恶化的趋势。
(2)提出了基于静力方程订正的气压回插法来改进气压梯度力的计算方法。研究表明：1)改进的气压梯度力算法比Corby方案的计算精度有较大提高，在典型中尺度模式参数设置下计算精度可达10~(-6)m／s~2，当垂直分层为60时，可比Corby方案减少95％的误差。2)改进的气压梯度力算法是收敛的，即随着模式分辨率的提高，计算精度不断提高。特别是当现有模式的垂直分辨率提高时，水平气压梯度力计算误差将快速减小。3)当水平网格距为5km、垂直分层为100时，来自差分近似截断误差和静力方程截断误差基本相当，数量级都为10~(-6)至10~(-5)m／s~2，即由于差分近似造成的计算误差和静力方程截断导致的误差在该模式参数下基本相当，再要提高计算精度应同时考虑提高模式的水平分辨率和垂直分辨率。
(3)发展了非静力完全弹性的数值模式动力框架，该动力框架采用两时间层的半隐式半拉格朗目的时间积分方案，使用的非跳点A网格比通常C跳点网格减少了拉格朗日插值的计算量。对气压梯度力采用水平显式的后向积分，对垂直气压梯度力采用半隐式的时间积分方案，水平显式、垂直隐式时间积分计算方案比全隐式简单。与一般非静力弹性模式动力框架不同的是，由于没有使用参考廓线，减少了使用不同廓线可能对模式性能的影响。理想场的试验表明发展的非静力全可压完全弹性数值模式动力框架具有正确模拟典型中小尺度天气系统特征的潜力，这些中小尺度天气过程和大尺度天气过程相比具有可压缩和非静力的特点。
(4)从中国气象局发展科学研究和业务实践数值天气模式出发，给出了GRAPES模式动力框架的出发方程组、计算方案，边界条件和差分方法，讨论了GRAPES模式设计中物理模式选择、垂直坐标、隐式和显式时间积分方案等几个动力框架发展过程中的重要问题以及采用的3维Helmholtz方程气压方程解法、拉格朗日插值和矢量离散化等关键技术。通过进行平衡流、密度流和钟型地形扰动和应用牛顿松弛到纬向对称的温度场和动量的表面拖曳这两个简单物理过程进行长时间积分试验等一系列典型的理想场试验，表明使用GRAPES模式动力框架作中小尺度预报和大气环流的大气动力框架是基本可行的，同时也将为进一步改进GRAPES模式动力框架提供了线索和依据。
(5)将改进的气压梯度力计算方案运用到非静力完全弹性模式动力框架中，通过陡峭地形试验分析其试验效果。采用新气压梯度力计算方法可望得到比较好的模拟精度，改进后的气压梯度力计算误差仅为改进前的3％左右。新气压梯度力计算方案计算稳定，计算结果比原方案有一定改进。在此基础上改进了GRAPES模式的气压梯度力算法；改进后的模式，其地形平衡流试验，运行稳定，运算结果基本合理。
The trends are witnessed to use the nonhydrostatic and fully elastic dynamicalcore for simulation and numerical weather forecast. The high resolution simulation ofatmospheric flows over complex orography is the main difficult numerical problem.Through the error analysis for pressure gradient force (PGF) and the development fornumerical model dynamical framework, the main conclusions are summarized asfollows:
(1) The errors in several kinds of PGF computing methods have been diagnosedunder very steep slope with high-resolution used in meso-scale model by ideal fieldtests. Tests show that PGF errors can reach 10~(-2) m/s~2 for classical method, 10~(-3) m/s~2for average classical method and the highest precision 10~(-4)m/s~2 for Corby's schemeunder ideal meso-scale parameters. It is also indicated that computing results of PGFin theσterrain-following vertical coordinates will be deteriorated slightly and theerrors will not be converged under the conditions of steep slopes if the modelresolutions increase in vertical direction.
(2) A revised scheme has been put forward in the conditions of high-resolutionmeso-scale model. It is based on the techniques of interpolating to isobaric level tocalculate PGF after finding precise geopotentials through integrating hydrostaticequation calculations with high vertical resolution. The ideal field tests show that thePGF errors decreased greatly under the steep topography slopes in the meso-scalemodel and the highest precision can reach 10~(-6) m/s~2 under the typical meso-scaleparameters. The calculation errors come mainly from integrals of truncation errors ofhydrostatic equation for geopotentials in vertical direction when number of verticallayers is less than 100. When the number of vertical layers is more than 100,truncation errors of differencing in horizontal discretizations will also play animportant role. The most significant feature is that the errors of PGF will reducedramatically with vertical resolution increase and the revised scheme on PGF will beconverged with the increasing vertical resolutions under the conditions of the idealatmospheric field.
(3) A dynamical core of numerical model of nonstagger (A-grid) and no referenceprofile is introduced with the nonhydrostatic and fully elastic atmosphere equations. Asemi lagrangian time discretization is used for time integration of the advection termswhile the horizontal explicit and vertical implicit are adopted for time integration ofthe pressure gradient force. The advantages of the scheme are: 1) the reference profileimpact on dynamical core is taken away otherwise than most of the nonhydrostaticdynamical cores using different reference profile. 2) the horizontal explicit andvertical implicit time stepping makes the scheme more simple than 3D implicit timediscretization because there are no need to solve 3D Helmholtz equations. 3)nonstagger A-grid mesh is much more simple than stagger mesh (such as C-grid)when the equations are carded out for spatial discretization. At the same time, thecalculation is also reduced for semi lagrangian interpolations under A-grid mesh. Theprimary tests with idealized field show that the dynamical core has the ability to simulate nonlinear flow motions, such as density flow and rising thermal, as well aspropagation of gravity waves in horizontal and topographic mountain waves.
(4) GRAPES (Global/Regional Assimilation and PrEdiction System) modelis a new generation numerical weather model with non-hydrostatic multi-scalecharacterization. The GRAPES dynamical core is fully compressible withnon-hydrostatic/hydrostatic switch and large time step by semi-implicit andsemi-lagrangian technique. This paper reviews briefly the basic prognostic equations,the scheme for time step integration and differencing discretizations used byGRAPES. It also describes the key technique methods in GRAPES framework andproblems during the model development, including GCR solvers for pressure equation,lagrangian interpolation and vector discretizations, as well as designing philosophy ofthe dynamical core. In addition, the important issues during developing dynamicalcore, such as physical model used in framework, vertical coordinate, implicit orexplicit time stepping, are discussed. The three idealized tests, which are calledbalanced flow, density flow and the bell-pattern mountain separately, have beencarded out in order to understand and verify the GRAPES model's ability to simulateand forecast weather systems, epically for middle and small systems. The balancedflow test shows that the GRAPES dynamical core preserve pattern and stable duringtime integration under the condition of geostrophic balance. The density currentsuggests that the model framework has the capability to simulate nonlinear stream andits continual evolution in case of fine scale. And the last case, bell-pattern mountaintrial, verifies that the model has the ability to simulate propogation in horizontal andvertical direction when the flow runs across an isolate mountain. The idealized testsabove imply primarily that the GRAPES multi-scale dynamical core has the potentialto simulate synoptic-scale and meso-scale systems. To verify whether the GRAPESdynamical core can take as a framework of AGCM, the long time integration has beencarried out with GRAPES dynamical core following benchmark test similar to thetechnique introduced by Held and Suarez. And the results indicate that it is feasibleusing the GRAPES frame as a dynamical core for AGCM and climate investigation.
(5) The revised PGF scheme has been applied to the nonhydrostatic and fullyelastic dynamical core to test the impact under the steep slope. The results show thenew PGF scheme is better than before with 3% errors of the old scheme and theperformances are stable. The idea applying the revised PGF scheme to GRAPESmodel has been brought forward and a balanced flow test has been carded out withreasonable results.
(1)现有模式中适合于天气尺度的气压梯度力算法在陡峭地形附近都存在较大的计算误差。在中尺度理想场典型参数设置下Corby方案计算的气压梯度力误差最小，精度为10~(-4)m／s~2，但即便如此，其气压梯度力算法也不收敛，即随着模式垂直分辨率的提高，计算精度不但没有提高，反而有恶化的趋势。
(2)提出了基于静力方程订正的气压回插法来改进气压梯度力的计算方法。研究表明：1)改进的气压梯度力算法比Corby方案的计算精度有较大提高，在典型中尺度模式参数设置下计算精度可达10~(-6)m／s~2，当垂直分层为60时，可比Corby方案减少95％的误差。2)改进的气压梯度力算法是收敛的，即随着模式分辨率的提高，计算精度不断提高。特别是当现有模式的垂直分辨率提高时，水平气压梯度力计算误差将快速减小。3)当水平网格距为5km、垂直分层为100时，来自差分近似截断误差和静力方程截断误差基本相当，数量级都为10~(-6)至10~(-5)m／s~2，即由于差分近似造成的计算误差和静力方程截断导致的误差在该模式参数下基本相当，再要提高计算精度应同时考虑提高模式的水平分辨率和垂直分辨率。
(3)发展了非静力完全弹性的数值模式动力框架，该动力框架采用两时间层的半隐式半拉格朗目的时间积分方案，使用的非跳点A网格比通常C跳点网格减少了拉格朗日插值的计算量。对气压梯度力采用水平显式的后向积分，对垂直气压梯度力采用半隐式的时间积分方案，水平显式、垂直隐式时间积分计算方案比全隐式简单。与一般非静力弹性模式动力框架不同的是，由于没有使用参考廓线，减少了使用不同廓线可能对模式性能的影响。理想场的试验表明发展的非静力全可压完全弹性数值模式动力框架具有正确模拟典型中小尺度天气系统特征的潜力，这些中小尺度天气过程和大尺度天气过程相比具有可压缩和非静力的特点。
(4)从中国气象局发展科学研究和业务实践数值天气模式出发，给出了GRAPES模式动力框架的出发方程组、计算方案，边界条件和差分方法，讨论了GRAPES模式设计中物理模式选择、垂直坐标、隐式和显式时间积分方案等几个动力框架发展过程中的重要问题以及采用的3维Helmholtz方程气压方程解法、拉格朗日插值和矢量离散化等关键技术。通过进行平衡流、密度流和钟型地形扰动和应用牛顿松弛到纬向对称的温度场和动量的表面拖曳这两个简单物理过程进行长时间积分试验等一系列典型的理想场试验，表明使用GRAPES模式动力框架作中小尺度预报和大气环流的大气动力框架是基本可行的，同时也将为进一步改进GRAPES模式动力框架提供了线索和依据。
(5)将改进的气压梯度力计算方案运用到非静力完全弹性模式动力框架中，通过陡峭地形试验分析其试验效果。采用新气压梯度力计算方法可望得到比较好的模拟精度，改进后的气压梯度力计算误差仅为改进前的3％左右。新气压梯度力计算方案计算稳定，计算结果比原方案有一定改进。在此基础上改进了GRAPES模式的气压梯度力算法；改进后的模式，其地形平衡流试验，运行稳定，运算结果基本合理。
The trends are witnessed to use the nonhydrostatic and fully elastic dynamicalcore for simulation and numerical weather forecast. The high resolution simulation ofatmospheric flows over complex orography is the main difficult numerical problem.Through the error analysis for pressure gradient force (PGF) and the development fornumerical model dynamical framework, the main conclusions are summarized asfollows:
(1) The errors in several kinds of PGF computing methods have been diagnosedunder very steep slope with high-resolution used in meso-scale model by ideal fieldtests. Tests show that PGF errors can reach 10~(-2) m/s~2 for classical method, 10~(-3) m/s~2for average classical method and the highest precision 10~(-4)m/s~2 for Corby's schemeunder ideal meso-scale parameters. It is also indicated that computing results of PGFin theσterrain-following vertical coordinates will be deteriorated slightly and theerrors will not be converged under the conditions of steep slopes if the modelresolutions increase in vertical direction.
(2) A revised scheme has been put forward in the conditions of high-resolutionmeso-scale model. It is based on the techniques of interpolating to isobaric level tocalculate PGF after finding precise geopotentials through integrating hydrostaticequation calculations with high vertical resolution. The ideal field tests show that thePGF errors decreased greatly under the steep topography slopes in the meso-scalemodel and the highest precision can reach 10~(-6) m/s~2 under the typical meso-scaleparameters. The calculation errors come mainly from integrals of truncation errors ofhydrostatic equation for geopotentials in vertical direction when number of verticallayers is less than 100. When the number of vertical layers is more than 100,truncation errors of differencing in horizontal discretizations will also play animportant role. The most significant feature is that the errors of PGF will reducedramatically with vertical resolution increase and the revised scheme on PGF will beconverged with the increasing vertical resolutions under the conditions of the idealatmospheric field.
(3) A dynamical core of numerical model of nonstagger (A-grid) and no referenceprofile is introduced with the nonhydrostatic and fully elastic atmosphere equations. Asemi lagrangian time discretization is used for time integration of the advection termswhile the horizontal explicit and vertical implicit are adopted for time integration ofthe pressure gradient force. The advantages of the scheme are: 1) the reference profileimpact on dynamical core is taken away otherwise than most of the nonhydrostaticdynamical cores using different reference profile. 2) the horizontal explicit andvertical implicit time stepping makes the scheme more simple than 3D implicit timediscretization because there are no need to solve 3D Helmholtz equations. 3)nonstagger A-grid mesh is much more simple than stagger mesh (such as C-grid)when the equations are carded out for spatial discretization. At the same time, thecalculation is also reduced for semi lagrangian interpolations under A-grid mesh. Theprimary tests with idealized field show that the dynamical core has the ability to simulate nonlinear flow motions, such as density flow and rising thermal, as well aspropagation of gravity waves in horizontal and topographic mountain waves.
(4) GRAPES (Global/Regional Assimilation and PrEdiction System) modelis a new generation numerical weather model with non-hydrostatic multi-scalecharacterization. The GRAPES dynamical core is fully compressible withnon-hydrostatic/hydrostatic switch and large time step by semi-implicit andsemi-lagrangian technique. This paper reviews briefly the basic prognostic equations,the scheme for time step integration and differencing discretizations used byGRAPES. It also describes the key technique methods in GRAPES framework andproblems during the model development, including GCR solvers for pressure equation,lagrangian interpolation and vector discretizations, as well as designing philosophy ofthe dynamical core. In addition, the important issues during developing dynamicalcore, such as physical model used in framework, vertical coordinate, implicit orexplicit time stepping, are discussed. The three idealized tests, which are calledbalanced flow, density flow and the bell-pattern mountain separately, have beencarded out in order to understand and verify the GRAPES model's ability to simulateand forecast weather systems, epically for middle and small systems. The balancedflow test shows that the GRAPES dynamical core preserve pattern and stable duringtime integration under the condition of geostrophic balance. The density currentsuggests that the model framework has the capability to simulate nonlinear stream andits continual evolution in case of fine scale. And the last case, bell-pattern mountaintrial, verifies that the model has the ability to simulate propogation in horizontal andvertical direction when the flow runs across an isolate mountain. The idealized testsabove imply primarily that the GRAPES multi-scale dynamical core has the potentialto simulate synoptic-scale and meso-scale systems. To verify whether the GRAPESdynamical core can take as a framework of AGCM, the long time integration has beencarried out with GRAPES dynamical core following benchmark test similar to thetechnique introduced by Held and Suarez. And the results indicate that it is feasibleusing the GRAPES frame as a dynamical core for AGCM and climate investigation.
(5) The revised PGF scheme has been applied to the nonhydrostatic and fullyelastic dynamical core to test the impact under the steep slope. The results show thenew PGF scheme is better than before with 3% errors of the old scheme and theperformances are stable. The idea applying the revised PGF scheme to GRAPESmodel has been brought forward and a balanced flow test has been carded out withreasonable results.
引文
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钱永甫，周天军，1994，陡峭地形区气压梯度力的误差扣除法，热带气象学报，4，358-368
胡江林，王盘兴，2007，地形跟随坐标下的中尺度模式气压梯度力计算误差分析及其改进方案，大气科学，31，109-118
杨晓娟，钱永甫，2003，P-σ坐标系数值模式中气压梯度力的递推算法，大气科学，27，171-181
曾西平，任阵海，1995，水平气压梯度力差分格式误差的定量分析，大气科学，19，722-732
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Bauer, M. B., G. J. Mayr, I. Vergeiner and H. Pichler, 2000: Strongly nonlinear flow over and around a three-dimension mountain as a function of the horizontal aspect ratio. J. Atmos. Sci.,2000,3971-3991
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Zhang Xuehong, Liang Xinzhong. Comparison and examination of dynamic frameworks of IAP and OSU AGCM. Advances in Atmospheric Sciences, 1989, 6: 265-274.
左瑞亭，张铭，张东凌等．21层大气环流模式IAP AGCM-Ⅲ的设计及气候数值模拟Ⅰ动力框架，大气科学，2004，28(5)：659-674．
梁丹青，张铭，曾庆存．分块地形坐标大气环流模式框架的计算稳定性及数值试验，大气科学，2005，29(3)：354-362．
王斌，季仲贞．大气科学中的数值新方法及其应用，北京：科学出版社，2006：200-205．
Gates, W. L. The atmospheric model intercomparison project. Bull. Amer. Meteor. Soc., 1992, 73:1962-1970.
Galewsky, J., Richard, K. S., Polvani, L. M., 2004:An initial-value problem for testing numerical models of the global shallow-water equations. Tellus 56A, 429-440
Held, I. H., M. J. Suarez. A proposal for the intercomparison of the dynamical cores of atmospheric general circulation models. Bull. Amer. Meteror. Soc.,1994, 75:1825-1830
Polvani L. M., R. K. Scott, S. J. Thomas. Numerical converged solutions of the global primitive equations for testing the dynamical core of atmospheric GCMs. Mon. Wea. Rev.,2004,132: 2539~2552.
Trenberth, K. E., and J. G. Olson, ECMWF Global Analyses 1979-1986: Circulation statistics and data evalution, NCAR/TN -300+STR, NCAR Technical Note, 1988.
Smith, R. B.,1980: Linear theory of stratified hydrostatic flow past an isolated mountain. Tellus, 32,348-364
Wang Bin, Wan Hui, Ji Zhongzhen, et al. Design of a new dynamical core for global atmospheric models based on some efficient numerical methods. Science in China Ser. A Mathematics, 2004, 47:4-21.
Williamson, D. L., J. B. Drake, J. J. Hack, et al. A standard test for numerical approximations to the shallow water equations in spherical geometry. J. Comput. P HYS.,1992,102:211-224.
Gal-Chen T And. Sommerville R. C. J. On the use of a coordinate transformation for the solution of the Navier-Stokes equations. J Comput Phys, 1975,17: 209～228.
Tanguay M., Robert A., Laprise R. A Semi-implicit Semi-Lagrangian fully compressible regional forecast model. Mon. Wea Rev,1990, 118: 1970～1980.
Davies, H. C. A Lateral Boundary Formulation for Multi-Level Prediction Models. Quart. J. Roy. Meteor. Soc., 1976, 102, 405～418.
Eisenstat S. C, Elman H. C, Schultz M. H. Variational iterative methods for nonsymmetric systems of linear equations. SIAM J Numer Anal, 1983,20,346～357
Staniforth A. and Cote J:Semi-Lagrangian integration scheme for atmospheric models-A review. Mon Wea Rev, 1991,119, 2206～2223.
Bates J R, Moorthi S, and Higgins R W. A global multilevel atmosphereic model using a vector semi-lagrangian finite-difference scheme. Part Ⅰ: Adiabatic Formulation. Mon Wea Rev, 1993, 121:244～263
Clark T J. A small scale numerical model using a terrain following coordinate system. J Comput Phys, 1977,24, 186～215
Janjic Z I. A nonhydrostatic model based on a new approach. Meteor Atmos Phy, 2003,82:271～285
Golding B W. An efficient non-hydrostatic forecast model. Meteorol Atmos Phys, 1992,50:89～103
Laprise R. The Euler equation of motion with hydrostatic pressure as an independent variable. Mon Wea Rev, 1992, 120:197～207
Thompsin J F, Warsi U A and Mastin C W. Numerical grid generation:foundation and applications. North-Holland, 1985, 483 pp
胡江林，王盘兴．地形跟随坐标下的中尺度模式气压梯度力计算误差分析及其改进方案，大气科学，2007，31，109～118
胡江林，沈学顺，张红亮等，GRAPES模式动力框架的长期积分特征，应用气象学报，2007(3)
Klemp J B, Wilhelmson R B and Fuhrer O. Numerical consistency of Metric terms in terrain-following coordinates. Mon Wea Rea, 2003, 131:1229～1239
纪立人，陈嘉滨，张道民等．数值预报模式动力框架发展的若干问题综述，大气科学，2005，29：120～130
Gallus W A and Rancic M. A non-hydrostatic version of NMC' s regional Eta model. Q J R Meteorol Sci, 1996, 122: 495～513
Gallus W A, Jr and Klemp J B. Behavior of flow over step orography. Mon Wea Rev, 2000, 128: 1153～1164
Klemp J B and Willhemson R B. The simulation of three-dimensional convective storm dynamics. J Atmos Sci, 1978, 35: 1070～1096
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