Relationships between the Limit of Predictability and Initial Error in the Uncoupled and Coupled Lorenz Models

In this study, the relationship between the limit of predictability and initial error was investigated using two simple chaotic systems:the Lorenz model, which possesses a single characteristic time scale, and the coupled Lorenz model, which possesses two different characteristic time scales. The limit of predictability is defined here as the time at which the error reaches 95% of its saturation level; nonlinear behaviors of the error growth are therefore involved in the definition of the limit of predictability. Our results show that the logarithmic function performs well in describing the relationship between the limit of predictability and initial error in both models, although the coefficients in the logarithmic function were not constant across the examined range of initial errors. Compared with the Lorenz model, in the coupled Lorenz model-in which the slow dynamics and the fast dynamics interact with each other-there is a more complex relationship between the limit of predictability and initial error. The limit of predictability of the Lorenz model is unbounded as the initial error becomes infinitesimally small; therefore, the limit of predictability of the Lorenz model may be extended by reducing the amplitude of the initial error. In contrast, if there exists a fixed initial error in the fast dynamics of the coupled Lorenz model, the slow dynamics has an intrinsic finite limit of predictability that cannot be extended by reducing the amplitude of the initial error in the slow dynamics, and vice versa. The findings reported here reveal the possible existence of an intrinsic finite limit of predictability in a coupled system that possesses many scales of time or motion.     

Lorenz混沌的控制

在混炖的Ｌｏｒｅｎｚ系统中,相点坐标ｘ与ｙ之间的符号关系决定相点能否通过ｙ－ｚ面,从而决定了相点在吸引子的哪一“翼”中运动。通过在一定位置对ｙ施加微小的影响,使系统未来发展方向只剩下一种可能性,系统便成为周期性的,并且通过选择对ｙ实施影响的位置,可以选择系统的周期性运动的方式,从而达到对Ｌｏｒｅｎｚ混沌的控制,这就说明内在随机性这一对预测不利的因素,却为对Ｌｏｒｅｎｚ混饨的人为控制提供了有利的时机。     

Chaos of a simple coupled system generated by interaction and external forcing

Summary ?This paper is concerned with the chaotic behavior of a coupled system consisting of two components, one representing the atmosphere and the other representing the ocean. The system is expressed as a highly truncated spectral model and for each component, the spectral model is similar to that of Lorenz (1963). Interactions between the two components are permitted, which lead to the temporal variation of surface temperature and hence that of a critical model parameter (the Rayleigh number). The emphasis of the paper is placed upon the chaotic behavior arising from the interactions between the two components and from periodic external forcing. Numerical tests are carried out to show that through interactions, the chaotic behavior of one component may result in chaos of the other even if the latter is otherwise stationary or periodic. It is shown that chaos may also occur if the system is forced periodically at certain frequencies. This study indicates that a new mechanism for chaos exists for coupled systems which are subject not only to internal fluid dynamic nonlinear interactions, but also to interactions between different components and external forcing. Received July 24, 2001; revised March 25, 2002     

正压过滤模式大气中的对称运动和非对称运动

把流函数分成对赤道对称和对赤道反对称的两部分,可以把正压涡度方程分离为描写大气对称运动和反对称运动的两个方程。根据它们,讨论了在全球范围内几种基本物理量,如涡度和绝对角动量等在对称和反对称情况的守恒性。还给出了这两类运动的能量收支方程和能量转换表达式。可以看出,在大气中不仅有纬圈平均动能和扰动动能之间的转换,也有对称运动产生的和反对称运动产生的动能之间的转换。 还在涡度方程中考虑地形和水平扩散的情形下,提出了大气非对称性质产生的机制。结果表明:非对称的地形分布和水平扩散系数的分布可能是导致非对称运动的原因。 最后,还对全球预报和半球预报,从物理观点作了理论比较。     

SOME BASIC FORMS OF LARGE-SCALE ATMOSPHERIC MOTION AND DISSIPATIVE STRUCTURES

Recently, Chou (1983) puts forward a principle about the simplification of the full equationsgoverning the motion of the atmosphere. This principle is used to discuss the properties of Lorenzlow-order system. As a way of simplification, the low-order spectral method satisfies the principle.The quasi-geostrophic baroclinic waves and the orographic standing waves are two basic kinds ofthe forms for the large-scale atmospheric motions, they all have the characters of dissipative structures.     

SYMMETRIC AND ASYMMETRIC MOTIONS IN THE BAROTROPIC FILTERED MODEL ATMOSPHERE

By expressing the stream function in terms of the symmetric and the antisymmetric components with respect to the equator,the barotropic vorticity equation can be separated into two equations,one describing the behaviours of the atmospheric symmetric motion,and the other those of the atmospheric antisymmetric motion.From them,for the entire global surface at the equivalent barotropic level,the conservations of several basic quantities,such as vorticity,angular momentum etc.in the symmetric and antisymmetric cases,have been discussed.In addition,the energy budget equations and the energy conservation expressions for the two kinds of motion are given.It can be seen from them that there are not only the conversion between the zonal mean kinetic energy and the disturbed kinetic energy in the atmosphere,but also the conversion between the kinetic energy generated by the symmetric and antisymmetric motions.In the case of including orography and horizontal diffusion into the vorticity equation,a mechanism of the generation of asymmetric behaviours of the atmosphere is proposed.The results show that the asymmetric distribution of orography and that of horizontal diffusion coefficients are likely the causes leading to the asymmetric motion.Finally,a theoretical comparison between the global model and the hemispheric one from the physical point of view is made.     

平均流变化对波反馈的A-B混合方程理论

从两层模式的基本方程出发，推导了描述平均流对波振幅反馈的A－B混合方程，并通过适当变换把混合方程转化为洛仑兹系统，用于研究平均流对边缘波扰动振幅增长的反馈机制，以及在中性廓线附近边缘波的稳定性问题。这是对研究波流相互作用的E-P通量理论的一个重要扩充。同时，作者还指出了A－B混合方程理论在研究大气中的波流相互作用及对流方面有相当应用前景。     

平均流变化对波反馈的A-B混合方程理论及应用

从两层模式的基本方程出发,推导了描述平均流对波振幅反馈的A-B混合方程,并通过适当变换把混合方程转化为洛仑兹系统,用于研究平均流对边缘波扰动振幅增长的反馈机制,以及在中性廓线附近边缘波的稳定性问题。这是对研究波流相互作用的E-P通量理论的一个重要扩充。同时,作者还指出了A-B混合方程理论在研究大气中的波流相互作用及对流方面有相当应用前景。     

A spectral fully compressible nonhydrostatic mesoscale model in hydrostatic sigma coordinates: Formulation and preliminary results

Summary The set of fully compressible nonhydrostatic equations governing a broad spectrum of atmospheric motion was transformed fromz coordinates to sigma coordinates under a hydrostatic base state. The hydrostatic base state may be either time-independent, such as a hydrostatic balance with-out motion or with motion such as a thermal wind balance, or time-dependent such as might be obtained from the result of integrating a hydrostatic model. The transformed set of equations can be used to predict and study all scales of at mospheric phenomena.The set of perturbation equations was also derived under the same condition. The computational sensitivity in computation of pressure gradient force in sigma coordinate can be improved by computing the pressure gradient interms of perturbations under a certain hydrostatic state.The hydrostatic regional spectral model developed by Juang and Kanamitsu (1991) was modified to be a nonhydrostatic spectral model based on the nonhydrostatic equations in sigma coordinates with time-dependent hydrostatic base states. A semi-implicit time integration scheme was used. Two experiments were performed to test this nonhydrostatic spectral model with acceptable results.With 7 Figures     

The variation of the GPS dropwindsonde drag coefficient and its impact on the wind retrieval from dropwindsonde measurements

The wind finding equations currently used to retrieve horizontal winds from Global Positioning System dropwindsonde wind measurements are derived based on a point object model in which the drag coefficient of the dropwindsonde is assumed to be a constant. The wind tunnel tests performed as part of this study showed, however, that the dropwindsonde aerodynamic coefficients vary appreciably with angles of attack. To investigate the impact of this finding, the dropwindsonde motion in a pseudo-stochastic wind field has been simulated using a motion model more sophisticated than the point object model. The results showed that, although the constant drag coefficient assumption is not supported by the wind tunnel test results, the wind finding equations still correctly calculate both the mean and the turbulence intensity profiles. In addition, a revised method to calculate the vertical wind was proposed based on the derivation of the improved motion model, which enhanced the accuracy of vertical wind estimates by including the real-time dropwindsonde drag coefficient and the dropwindsonde vertical acceleration into calculation.     

A THREE-DIMENSIONAL CLOUD-SCALE MODEL SUITABLE FOR COMPRESSIBLE ATMOSPHERE

A three-dimensional cloud-scale model has been designed.The governing equations of the model arecomposed of two groups of equations:one group includes compressible motion equations,continuity equation,pressure equation and thermodynamic equation,which are of Eulerian type,and the other consists of cloud-precipitation microphysics equations which are of Lagrangian type.Since the degree of influence of sound waveon the air motion is quite different from that on the temperature or hydrometeors,the time splitting procedureis used in solving governing equations.Both unstaggered and staggered meshes have been utilized.Integra-tion schemes adopted are the Eulerian backward difference method for the unstaggered mesh and semi-implicitmethod for staggered mesh.Several experiments of modelling have been conducted and a reasonable three-dimensional image of deep convection is obtained.With this model the horizontal and vertical vortex circula-tions are simulated.Furthermore,the effects of horizontal vortex on the formation and development ofdowndraft within cloud have also been studied.     

Intelligent optimization algorithms to VDA of models with on/off parameterizations

Some variational data assimilation (VDA) problems of time- and space-discrete models with on/off parameterizations can be regarded as non-smooth optimization problems. Same as the sub-gradient type method, intelligent optimization algorithms, which are widely used in engineering optimization, can also be adopted in VDA in virtue of their no requirement of cost functions gradient (or sub-gradient) and their capability of global convergence. Two typical intelligent optimization algorithms, genetic algorithm (GA) and particle swarm optimization (PSO), are introduced to VDA of modified Lorenz equations with on-off parameterizations, then two VDA schemes are proposed, that is, GA based VDA (GA-VDA) and PSO based VDA (PSO-VDA). After revealing the advantage of GA and PSO over conventional adjoint methods in the ability of global searching at the existence of cost functions discontinuity induced by on-off switches, sensitivities of GA-VDA and PSO-VDA to population size, observational noise, model error and observational density are detailedly analyzed. Its shown that, in the context of modified Lorenz equations, with proper population size, GA-VDA and PSO-VDA can effectively estimate the global optimal solution, while PSO-VDA consumes much less computational time than GA-VDA with the same population size, and requires a much lower population size with nearly the same results, both methods are not very sensitive to observation noise and model error, while PSO-VDA shows a better performance with observational noise than GA-VDA. It is encouraging that both methods are not sensitive to observational density, especially PSO-VDA, using which almost the same perfect assimilation results can be obtained with comparatively sparse observations.     

COMPARATIVE EXPERIMENTS OF DYNAMICAL FORECAST AND STATISTICAL FORECAST

After analyzing the advantages and disadvantages of dynamical system and statistical system.some simple models with the equations of Lorenz system and auto-regresslon model have beenrespectively set up,and comparative experiments conducted.The result of the research demonstratesthat in chaotic parametric domain,the accuracy of statistical forecast is higher than that of dynamicalforecast,while in non-chaotic parametric domain,dynamical forecast is more accurate than statisticalforecast.     

轨线不稳定与误差增长

从一般的谱展开方程出发，详细推导了误差增长方程。结果表明误差增长率主要由准确解的切线性方程所决定，扰动非线性平流作用不产生方差意义下的误差增长，而只起分配误差的作用。轨线不稳定是产生误差增长的根本原因。文中提出了计算轨线不稳定增长率的方法。这一方法也适合于时间演变状态不稳定问题的讨论，对Ｌｏｒｅｎｚ系统的轨线不稳定计算表明了理论分析的正确及其意义。     

COMPARATIVE EXPERIMENTS OF DYNAMICAL FORECAST AND STATISTICAL FORECAST*

After analyzing the advantages and disadvantages of dynamical system and statistical system.some simple models with the equations of Lorenz system and auto-regresslon model have been respectively set up,and comparative experiments conducted.The result of the research demonstrates that in chaotic parametric domain,the accuracy of statistical forecast is higher than that of dynamical forecast,while in non-chaotic parametric domain,dynamical forecast is more accurate than statistical forecast.     

集合平均所构成的动力系统及其特性研究

以Lorenz系统为例,推导出集合平均所定义的完整动力方程(均值方程),将初值的集合平均问题作为一个广义的动力系统问题来进行研究;对于双初值和多初值的均值方程,利用定性理论分析了其吸引中心的位置和个数,并使用数值试验进行了验证,结果表明平均值的吸引子的结构与原解的吸引子位置、数量和结构均有不同。对均值方程的特征矩阵分析表明,定点附近的稳定性与原方程相同,而且特征方程所对应的特征值也与原方程相同。均值方程对应的相流散度为负值且数值上与原系统相同,因此其在相空间中的体积收缩速度和原系统相同,最终趋向一个低纬曲面,均值方程的这个性质使得Lorenz系统的集合平均解趋于一个吸引子。均值方程可以保持原方程的耗散特性、吸引子特性,但稳定点位置和个数发生了变化,非定点处的Jacobian矩阵特征值与原系统也有不同。简而言之,一旦使用了集合平均方法,那么集合数值解并不是原系统的解,仅保持了原系统的部分特征,因而集合平均是否有效需要根据具体问题和其他外部限定条件才能确定。     

The atmospheric boundary layer over a wavy surface

The wavy area of north-west Bohemia (Czechoslovakia) is simulated by a cylindrical surface model. A curvilinear orthogonal system of coordinates along the model surface is introduced. The hydrodynamical equations of motion are transformed into this system of coordinates. By applying boundary-layer assumptions, the equations of motion for the atmospheric boundary layer (ABL) above the model are derived. The equations and boundary conditions show an equivalence of the ABL above the model with that above a flat surface with external pressure gradient.     

Bred vectors of the Lorenz63 system

The breeding method has been widely used in studies of data assimilation, predictability and instabilities. The bred vectors(BVs), which are the nonlinear difference between the control and perturbed runs, represent the time-evolving rapidly growing errors in dynamic systems. The Lorenz(1963) model(hereafter Lorenz63 model) has chaotic dynamics similar to weather and climate. This study investigates the features of BVs of the Lorenz63 model and its impact on regime prediction of the Lorenz63 model. The results show that the Lorenz63 model has two different BVs for each breeding cycle, and the two BVs approach being identical when growth rate is high. The duration of the current and next regime is associated with the relative directions between the BV with high growth rate and the model trajectory.     

数值模式初值的敏感性程度对四维同化的影响

用著名的Ｌｏｒｅｎｚ系统作了共轭变分同化的数值试验。发现随着模式对初值敏感性程度的增加，用这种方法得到和模式相协调的初始场愈来愈困难，直到某些情况下的完全失败。这表明四维同化和可预报期限是联系在一起的。另一方面，随着方程不精确程度的增加，变分同化的效果愈来愈差，直到所做的预报无任何意义可言。如果在做变分同化的同时对模式参数也进行反演，就可使得基于Ｌｏｒｅｎｚ系统所做的预报效果大大提高。     

从气旋和反气旋谈正、负阻尼

在定常条件下,利用简化的大气运动控制方程,分析了气旋与反气旋的基本特征。在此基础上,利用常微分方程的定性分析与求解,获得了气旋和反气旋轨道的解析解。接合气旋与反气旋的基本特征,论证了正负阻尼在气旋和反气旋运动中具体体现,并从物理机理上说明产生气旋和反气旋运动中正负阻尼的差异在于水平辐合在低层的不同。