Numerical integration of a linear barotropic model using three methods of treating meteorological and gravitational modes separately

Summary The analytical solution of a linear barotropic model is derived, including details of the quasi-geostrophic initialization procedure. The prognostic equations are integrated using three different methods of treating the meteorological and gravitational modes separately. These are a semi-Eulerian, semi-implicit (EI) technique, a semi-Lagrangian, semi-implicit (LI) procedure, and a split-explicit (SE) method. The stability criteria and phase speeds are derived for each of the three techniques.The following theoretical conclusions are derived. Of course, in actual numerical integrations particularly those using more complex models, the results are not so unequivocal.The stability of the EI procedure is governed by the CFL criterion for the meteorological mode. Gravity waves have no effect on the timestep but move more slowly than the analytical waves. The LI method is unconditionally stable with respect to both meteorological and gravitational modes. There is thus no timestep restriction. However, the gravity waves have the same reduced phase speed as in the EI technique. The SE procedure has CFL timestep criteria for both the meteorological and gravitational calculations. However, its gravity wave phase speeds are relatively accurate. Moreover, it is the only one of the three methods that handles the nearly-compensating pressure gradient and Coriolis forces together. From the point of computational efficiency, the LI technique is probably the best.     

非线性正规模初值化原则在有限区模式的实现

本文在普遍情况下导出了与非线性正规模初值化方法等价的、用模式变量表达的初值化准则与运算迭代公式.由于模式的正规模态在初值化计算中不明显地出现,故本文的结果对于正规模态难以求得的有限区模式的初值化有重要意义_试验实例表明,按本文的方案作初值处理,对于消除初值中的不平衡是十分有效的.     

The skill of atmospheric linear inverse models in hindcasting the Madden–Julian Oscillation

A suite of statistical atmosphere-only linear inverse models of varying complexity are used to hindcast recent MJO events from the Year of Tropical Convection and the Cooperative Indian Ocean Experiment on Intraseasonal Variability/Dynamics of the Madden–Julian Oscillation mission periods, as well as over the 2000–2009 time period. Skill exists for over two weeks, competitive with the skill of some numerical models in both bivariate correlation and root-mean-squared-error scores during both observational mission periods. Skill is higher during mature Madden–Julian Oscillation conditions, as opposed to during growth phases, suggesting that growth dynamics may be more complex or non-linear since they are not as well captured by a linear model. There is little prediction skill gained by including non-leading modes of variability.     

The application of non‐linear normal mode initialization to an operational forecast model

Abstract

The non‐linear normal mode initialization technique used in shallow water equation models by Baer (1977) and Machenhauer (1977) is now applied to a full baroclinic primitive equations forecast model. The initialization procedure is shown to be capable of completely removing high frequency oscillations from model integrations, even in the presence of topography. The procedure also produces a consistent and physically realistic initial vertical motion field.     

线性与非线性发展方程计算稳定性的比较分析

针对性与非线性发展方程的几种差分格式，以一维线性和非线性平流方程为例，对线性与非线笥发展方程差分格式的计算稳定性进行了比较分析，揭示了差分格式结构和初值形式与计算稳定性的关系。理论分析和数值试验证明，线性与非线性发展方程差分格式的计算稳定性在本质上是完全不同的。     

中尺度扰动不稳定的数值研究

利用一个二维Boussinesq流体的绝热无粘非静力数值模式,将中尺度不稳定问题作为一个初值问题进行数值研究.线性情况下数值试验的结果基本与采用特征值方法研究得到的结论一致.非线性情况的数值试验表明,其不稳定发生的范围可与线性情况不一致;非线性不稳定的增长率一般较线性不稳定的增长率要小;非线性作用会造成波型的陡凸,从而造成流函数正负环流的不对称和环流流线的密集;非线性情形下的流型有些与强对流系统的流型相像.     

非线性大气化学动力学方程组数值解法的比较

大气化学模式中描述化学反应动力学的是一组高度耦合、刚性、非线性的常微分方程组,高效精确的计算方法在空气质量模拟预测、气候化学相互作用研究中尤为重要。本文介绍了求解此类非线性常微分方程组(ODEs)的七种常见的数值计算方法,包括QSSA、MQSSA、HYBRID、MHYBRID、LSODE、GongCho以及TWOSTEP方法,并基于同一反应机理进行24 h的数值模拟,通过对模拟结果的讨论分析,比较了各种方法的优劣。研究结果表明,就计算效率而言,QSSA方法最为省时。就计算精度而言,LSODE方法最优。若综合考虑计算精度和效率,HYBRID、MHY-BRID是比较适中的数值解法。     

On stochastic stability of regional ocean models to finite-amplitude perturbations of initial conditions

We consider error propagation near an unstable equilibrium state (classified as an unstable focus) for spatially uncorrelated and correlated finite-amplitude initial perturbations using short- (up to several weeks) and intermediate (up to 2 months) range forecast ensembles produced by a barotropic regional ocean model. An ensemble of initial perturbations is generated by the Latin Hypercube design strategy, and its optimal size is estimated through the Kullback–Liebler distance (the relative entropy). Although the ocean model is simple, the prediction error (PE) demonstrates non-trivial behavior similar to that existing in 3D ocean circulation models. In particular, in the limit of zero horizontal viscosity, the PE at first decays with time for all scales due to dissipation caused by non-linear bottom friction, and then grows faster than (quasi)-exponentially. Statistics of a prediction time scale (the irreversible predictability time (IPT)) quickly depart from Gaussian (the linear predictability regime) and becomes Weibullian (the non-linear predictability regime) as amplitude of initial perturbations grows. A transition from linear to non-linear predictability is clearly detected by the specific behavior of IPT variance. A new analytical formula for the model predictability horizon is introduced and applied to estimate the limit of predictability for the ocean model.     

The transition of a multi-dimensional lorenz system

A multi-dimensional Lorenz system, which includes thirty-three amplitude equations describing time evolu-tion of convection, is derived from two-dimensional Boussinesq equations by using the Galerkin method. Its transition is studied by numerical solution. It is found that, with Rayleigh number increasing from zero to one hundred, five different types of motion appear one after another as follows: stationary motion, periodic motion, quasiperiodic motion with two-fundamental frequencies, quasi-periodic motion with three fundamental frequencies, and chaotic motion. By comparing with the Lorenz model and Curry's fourteen-dimensional model, it is shown that as retained modes increase, the critical values of transition become larger and the types of bi-furcation change. The results of dynamic behavior happen to be in agreement with the IIIa route of the Gollub and Benson experiments.     

中高纬度地区500 hPa高度场动力预测统计订正

利用DEMETER多模式集合研究计划中Météo France模式的预报资料集,在分析其对冬季北半球中高纬度地区 (20°~90°N)500 hPa高度场预报效果的基础上,针对模式预测较差的模态分别运用最优子集回归修正方案和回归-相似相结合的修正方案对其进行订正。结果表明:数值模式对观测模态的预测能力并非随模态数的增加而递减,方差贡献较小的模态的预报效果可能好于方差贡献较大的模态;基于最优子集的回归订正方法未能改进原模式预报技巧;在最优子集回归基础上再经相似订正的方法 (DAP-OSR) 能够改进预测效果,独立试报的距平相关系数平均每年提高0.1。     

The study of in-orbit calibration accuracy of NOAA satellite infrared sounder and its effect on temperature profile retrievals

The calibration accuracy of High Resolution Infrared Radiation Sounder Mod. 2 (H1RS / 2) on NOAA-10 sat-ellite is analyzed in this paper. The non-linear effect in the linear calibration curve induces a deviation of 1.5 degrees (k) of brightness temperature in the tenth channel (8.3 urn, water vapor absorption) of the HIRS / 2 and the non-linear effect affects the other channels to a different extent. Based on analyzing non-linearity in two-point calibration curve, a tri-point calibration equation is given. A numerical test of effects of the linear and non-linear calibration models on the accuracy of atmospheric temperature retrievals is carried out.     

Uncertainty of the numerical solution of a nonlinear system’s long-term behavior and global convergence of the numerical pattern

The computational uncertainty principle in nonlinear ordinary differential equations makes the numerical solution of the long-term behavior of nonlinear atmospheric equations have no meaning. The main reason is that, in the error analysis theory of present-day computational mathematics, the non-linear process between truncation error and rounding erroris treated as a linear operation. In this paper, based on the operator equations of large-scale atmospheric movement, the above limitation is overcome by using the notion of cell mapping. Through studying the global asymptotic characteristics of the numerical pattern of the large-scale atmospheric equations, the definitions of the global convergence and an appropriate discrete algorithm of the numerical pattern are put forward. Three determinant theorems about the global convergence of the numerical pattern are presented, which provide the theoretical basis for constructing the globally convergent numerical pattern. Further, it is pointed out that only a globally convergent numerical pattern can improve the veracity of climatic prediction.     

Uncertainty of the Numerical Solution of a Nonlinear System‘s Long-term Behavior and Global Convergence of the Numerical Pattern

1. Introduction The partial di?erential equations for atmosphericmovement are often nonlinear and often very complex;we usually cannot obtain an analytic solution but must?nd a numerical solution. The general method is ?rstto discretize the space varia…     

非线性平衡方程初值化方法及其在中期数值天气预报试验中的应用

本文给出了一种求解非线性平衡方程的新的有效的方法及有关的数值试验结果。和以往的求解方法相比，本方法的优点是:收敛速度快，不需要冗长的迭代计算，也不需要对初始高度场的某些记录作修改，并能节省大量的计算时间。文中利用北半球七层原始方程谱模式，使用了1982年的客观分析资料，进行中期数值天气预报试验。试验结果表明，用非线性平衡方程初值化方法制作中期数值预报比其他的如线性平衡方程初值化方程的更佳。后者因去掉了非线性项的作用，天气系统的强度预报结果偏弱且偏平滑。     

非线性正规模初值化及其对资料同化和预报的影响

文章介绍了国家气象中心中期数值预报谱模式（T42L9）的非线性正规模初值化方案（NNMI）。讨论了该方案对资料同化和预报的影响。试验表明，应用前5个垂直模，经2次迭代能够得到一个对分析资料修正较小的平衡初始场。它成功地消除了预报中虚假的高频重力波振荡，对改进资料同化和模式预报起了重要作用。     

Simulation and Regionalization of Daily Global Solar Radiation: A Case Study in Quebec,Canada

Global solar radiation (GSR) is essential for agricultural and plant growth modelling, air and water heating analyses, and solar electric power systems. However, GSR gauging stations are scarce compared with stations for monitoring common meteorological variables such as air temperature and relative humidity. In this study, one power function, three linear regression, and three non-linear models based on an artificial neural network (ANN) are developed to extend short records of daily GSR for meteorological stations where predictors (i.e., temperature and/or relative humidity) are available. The seven models are then applied to 19 meteorological stations located across the province of Quebec (Canada). On average, the root-mean-square errors (RMSEs) for ANN-based models are 0.33–0.54?MJ?m^?2?d^?1 smaller than those for the power function and linear regression models for the same input variables, indicating that the non-linear ANN-based models are more efficient in simulating daily GSR. Regionalization potential of the seven models is also evaluated for ungauged stations where predictors are available. The power function and the three linear regression models are tested by interpolating spatially correlated at-site coefficients using universal kriging or by applying a leave-one-out calibration procedure for spatially uncorrelated at-site coefficients. Regional ANN-based models are also developed by training the model based on the leave-one-out procedure. The RMSEs for regional ANN models are 0.08–0.46?MJ?m^?2?d^?1 smaller than for other models using the same input conditions. However, the regional ANN-based models are more sensitive to new station input values compared with the other models. Maps of interpolated coefficients and regional equations of the power function and the linear regression models are provided for direct application to the study area.     

Rayleigh-Bénard convection as a tool for studying dust devils

Intense columnar vortices in a convecting layer are explored with direct numerical simulations that are otherwise similar to the large-eddy simulations of6, Q. J. R. Meteorol. Soc. 126, 2789–2810). With free-slip boundaries and a Rayleigh number of 10⁶(4096 times critical), vortices similar to large dust devils are readily produced. The genesis, intensity and life cycle of these intense vortices (dust devils) are studied. The simulated dust devils last for the order of the over-turning time of the largest eddies. The intensity is limited by the hydrostatic pressure drop supported by the buoyancy confined in the core. The genesis of a simulated dust devil requires not only tilting of the baroclinically generated vorticity, but also a symmetry-breaking event that allows one sign of vorticity to become concentrated in an updraft. Such symmetry breaking is the rule with random initialization in the simulations. However, when initialization is restricted to certain Fourier modes, exceptions are found that produce only symmetric vortex couplets that are relatively weak.     

Nesting Turbulence in an Offshore Convective Boundary Layer Using Large-Eddy Simulations

The applicability of the one-way nesting technique for numerical simulations of the heterogeneous atmospheric boundary layer using the large-eddy simulation (LES) framework of the Weather Research and Forecasting model is investigated. The focus of this study is on LES of offshore convective boundary layers. Simulations were carried out using two subgrid-scale models (linear and non-linear) with two different closures [diagnostic and prognostic subgrid-scale turbulent kinetic energy (TKE) equations]. We found that the non-linear backscatter and anisotropy model with a prognostic subgrid-scale TKE equation is capable of providing similar results when performing one-way nested LES to a stand-alone domain having the same grid resolution but using periodic lateral boundary conditions. A good agreement is obtained in terms of velocity shear and turbulent fluxes, while velocity variances are overestimated. A streamwise fetch of 14 km is needed following each domain transition in order for the solution to reach quasi-stationary results and for the velocity spectra to generate proper energy content at high wavelengths, however, a pile-up of energy is observed at the low-wavelength portion of the spectrum on the first nested domain. The inclusion of a second nest with higher resolution allows the solution to reach effective grid spacing well within the Kolmogorov inertial subrange of turbulence and develop an appropriate energy cascade that eliminates most of the pile-up of energy at low wavelengths. Consequently, the overestimation of velocity variances is substantially reduced and a considerably better agreement with respect to the stand-alone domain results is achieved.     

Consistent balanced models in bounded and periodic domains

An analysis is made of a family of balanced models which are intermediate in physical content and complexity between the primitive equations and quasigeostrophy. The family is based on the linear balance equations and balance equations, and is characterized by truncations of the full vorticity and divergence equations that retain a global energy invariant. Consistent initial boundary value problems for this family of models are derived, and various aspects of numerical balanced model design are discussed.     

GRAPES全球切线性和伴随模式的调优

伴随技术是四维变分同化（4DVar）系统中计算代价函数梯度的最佳办法,切线性和伴随模式的效果和效率直接影响着4DVar系统的发展。基于GRAPES（Global and Regional Assimilation PrEdiction System）全球切线性和伴随模式1.0版本,利用GRAPES全球模式2.0版本在并行框架和性能等方面的改善,重新优化和设计了GRAPES全球切线性伴随模式2.0版本,提高了GRAPES全球切线性和伴随模式的效果和效率,优化了切线性模式程序结构,使其计算时间最优可控制在非线性模式的1.2倍以内;采用在切线性模式中保存基态的方法,重构了伴随模式的程序结构,使其计算时间最优控制在非线性模式的1.5倍以内;在GRAPES全球切线性物理过程的设计中,将线性物理过程的轨迹基态计算和切线性扰动计算解耦,提高了GRAPES全球切线性和伴随模式的计算效果和效率。