用天气变量时间序列估计天气的可预报性

本文从非线性系统的吸引子概念出发,用单个气象时间序列重构维数较高的相空间并嵌入天气吸引子,根据相轨道上初始时刻紧邻的点随时间的演化来估计吸引子的维数和天气的可预报性。用500hPa亚洲环流指数和北京冬季气温的逐日资料计算表明,天气吸引子的维数分别为3.8和5.4;可预报时间尺度约6—14天,考虑相空间e指数膨胀因素后为4—9天。     

Climate predictability on interannual to decadal time scales: the initial value problem

Any initial value forecast of climate will be subject to errors originating from poorly known initial conditions, model imperfections, and by "chaos" in the sense that, even if the initial conditions were perfectly known, infinitesimal errors can amplify and spoil the forecast at some lead time. Here the latter source of error is examined using a "perfect model" approach whereby small perturbations are made to a coupled atmosphere-ocean general circulation model and the spread of nearby model trajectories, on time and space scales appropriate to seasonal-decadal climate variability, is measured to assess the lead time at which the error saturates. The study therefore represents an estimate of the upper limit of the predictability of climate (appropriate to the initial value problem) given a perfect model and near perfect knowledge of the initial conditions. It is found that, on average, surface air temperature anomalies are potentially predictable on seasonal to interannual time scales in the tropical regions and are potentially predictable on decadal time scales over the ocean in the North Atlantic. For mid-latitude surface air temperature anomalies over land, model trajectories rapidly diverge and there is little sign of any potential predictability on time scales greater than a season or so. For mean sea level pressure anomalies, there is potential predictability on seasonal time scales in the tropics, and for some global scale annual-decadal anomalies, although not those associated with the North Atlantic Oscillation. For precipitation, the only potential for predictability is for seasonal time anomalies associated with the El-Niño Southern Oscillation. For the majority of the highly populated regions of the world, climate predictability on interannual to decadal time scales based in the initial value approach is likely to be severely limited by chaotic error growth. It is found however that there can be cases in which the potential predictability can be higher than average indicating that there is perhaps some utility in making initial value forecasts of climate in those regions which show low predictability on average.     

LOCALIZED FEATURES OF CHAOTIC SYSTEM AND ATMOSPHERIC PREDICTABILITY

The localized features on chaotic attractor in phase space and predictability are investigated in thepresent study.It will be suggested that the localized features in phase space have to be considered indetermining the predictability.The notions of the local instability including the finite-time and local-time instabilities which determine the growth rate of error are introduced,and the calculation methodsare discussed in detail.The results from the calculation of the 3-component Lorenz model show thatsuch instability,correspondingly the growth rate of error,varies dramatically as the trajectoriesevolve on the chaotic attractor.The region in which the growth rate of error is small is localizedconsiderably,and is separable from the region in which the growth rate is large.The localpredictability is of important interest.It is also suggested that such localized features may be the maincause for a great deal of case-to-case variability of the predictive skill in the operational forecasts.     

误差非线性的增长理论及可预报性研究

对非线性系统的误差发展方程不作线性化近似,直接用原始的误差发展方程来研究初始误差的发展,提出了误差非线性的增长理论。首先,在相空间中定义一个非线性误差传播算子,初始误差在这个算子的作用下,可以非线性发展成任意时刻的误差;然后,在此基础上,引入了非线性局部Lyapunov指数的概念。由平均非线性局部Lyapunov指数可以得到误差平均相对增长随时间的演变情况;对于一个混沌系统,误差平均相对增长被证明将趋于一个饱和值,利用这个饱和值,混沌系统的可预报期限可以被定量地确定。误差非线性的增长理论可以应用于有限尺度大小初始扰动的可预报性研究,较误差的线性增长理论有明显的优越性。     

Application of Backward Nonlinear Local Lyapunov Exponent Method to Assessing the Relative Impacts of Initial Condition and Model Errors on Local Backward Predictability

Initial condition and model errors both contribute to the loss of atmospheric predictability. However, it remains debatable which type of error has the larger impact on the prediction lead time of specific states. In this study, we perform a theoretical study to investigate the relative effects of initial condition and model errors on local prediction lead time of given states in the Lorenz model. Using the backward nonlinear local Lyapunov exponent method, the prediction lead time,also called local backward predictability limit(LBPL), of given states induced by the two types of errors can be quantitatively estimated. Results show that the structure of the Lorenz attractor leads to a layered distribution of LBPLs of states. On an individual circular orbit, the LBPLs are roughly the same, whereas they are different on different orbits. The spatial distributions of LBPLs show that the relative effects of initial condition and model errors on local backward predictability depend on the locations of given states on the dynamical trajectory and the error magnitudes. When the error magnitude is fixed, the differences between the LBPLs vary with the locations of given states. The larger differences are mainly located on the inner trajectories of regimes. When the error magnitudes are different, the dissimilarities in LBPLs are diverse for the same given state.     

混沌系统的局域特征与可预报性

讨论了混沌系统的时间和空间的局域特征。首先分析了研究时间和空间局域特征的必要性。接着引进了有限时间不稳定和局域时间不稳定的概念，并对有关的计算问题进行了研究。对Ｌｏｒｅｎｚ系统的具体计算表明，随着轨线在混沌吸引子上的演变，局域不稳定特征有很大的变化，相应误差增长也有很大的变化。相应于误差迅速增长的轨线部分局限于很有限的相空间范围内，而且同误差增长缓慢的轨线部分占据的相空间区域截然可分。每一个例的可预报性依赖于轨线在相空间中所处的区域。混沌系统的这种局域特征可以是导致个例业务预报技巧之间有很大差别的主要原因。     

一维气候时间序列的拓展及其相空间中浑沌吸引子维数的确定

本文利用上海和广州近百年(1873—1980)月平均气温时间序列资料,将一维气候时间序列拓展到多维相空间上去。计算结果表明,月平均气温所表示的我国季风区短期气候演化,在相空间中存在吸引子,具有分维结构,其维数分别是d=3.4和d=2.3,为奇怪吸引子。由此推论,就我国季风区气候短期变化而言,为了能在多维相空间支撑起上述奇怪吸引子,最好选取四个变量或者建立最低为四阶的动力学模式来进行描述。      

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.     

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.     

Characteristics of tropical Pacific SST predictability in coupled GCM forecasts using the NCEP CFS

The limits of predictability of El Niño and the Southern Oscillation (ENSO) in coupled models are investigated based on retrospective forecasts of sea surface temperature (SST) made with the National Centers for Environmental Prediction (NCEP) coupled forecast system (CFS). The influence of initial uncertainties and model errors associated with coupled ENSO dynamics on forecast error growth are discussed. The total forecast error has maximum values in the equatorial Pacific and its growth is a strong function of season irrespective of lead time. The largest growth of systematic error of SST occurs mainly over the equatorial central and eastern Pacific and near the southeastern coast of the Americas associated with ENSO events. After subtracting the systematic error, the root-mean-square error of the retrospective forecast SST anomaly also shows a clear seasonal dependency associated with what is called spring barrier. The predictability with respect to ENSO phase shows that the phase locking of ENSO to the mean annual cycle has an influence on the seasonal dependence of skill, since the growth phase of ENSO events is more predictable than the decay phase. The overall characteristics of predictability in the coupled system are assessed by comparing the forecast error growth and the error growth between two model forecasts whose initial conditions are 1 month apart. For the ensemble mean, there is fast growth of error associated with initial uncertainties, becoming saturated within 2 months. The subsequent error growth follows the slow coupled mode related the model’s incorrect ENSO dynamics. As a result, the Lorenz curve of the ensemble mean NINO3 index does not grow, because the systematic error is identical to the same target month. In contrast, the errors of individual members grow as fast as forecast error due to the large instability of the coupled system. Because the model errors are so systematic, their influence on the forecast skill is investigated by analyzing the erroneous features in a long simulation. For the ENSO forecasts in CFS, a constant phase shift with respect to lead month is clear, using monthly forecast composite data. This feature is related to the typical ENSO behavior produced by the model that, unlike the observations, has a long life cycle with a JJA peak. Therefore, the systematic errors in the long run are reflected in the forecast skill as a major factor limiting predictability after the impact of initial uncertainties fades out.     

Intrinsic modulation of ENSO predictability viewed through a local Lyapunov lens

The presence of rich ENSO variability in the long unforced simulation of GFDL’s CM2.1 motivates the use of tools from dynamical systems theory to study variability in ENSO predictability, and its connections to ENSO magnitude, frequency, and physical evolution. Local Lyapunov exponents (LLEs) estimated from the monthly NINO3 SSTa model output are used to characterize periods of increased or decreased predictability. The LLEs describe the growth of infinitesimal perturbations due to internal variability, and are a measure of the immediate predictive uncertainty at any given point in the system phase-space. The LLE-derived predictability estimates are compared with those obtained from the error growth in a set of re-forecast experiments with CM2.1. It is shown that the LLEs underestimate the error growth for short forecast lead times (less than 8 months), while they overestimate it for longer lead times. The departure of LLE-derived error growth rates from the re-forecast rates is a linear function of forecast lead time, and is also sensitive to the length of the time series used for the LLE calculation. The LLE-derived error growth rate is closer to that estimated from the re-forecasts for a lead time of 4 months. In the 2,000-year long simulation, the LLE-derived predictability at the 4-month lead time varies (multi)decadally only by 9–18 %. Active ENSO periods are more predictable than inactive ones, while epochs with regular periodicity and moderate magnitude are classified as the most predictable by the LLEs. Events with a deeper thermocline in the west Pacific up to five years prior to their peak, along with an earlier deepening of the thermocline in the east Pacific in the months preceding the peak, are classified as more predictable. Also, the GCM is found to be less predictable than nature under this measure of predictability.     

Extended Range(10–30 Days) Heavy Rain Forecasting Study Based on a Nonlinear Cross-Prediction Error Model

Extended range(10–30 d) heavy rain forecasting is difficult but performs an important function in disaster prevention and mitigation. In this paper,a nonlinear cross prediction error(NCPE) algorithm that combines nonlinear dynamics and statistical methods is proposed. The method is based on phase space reconstruction of chaotic single-variable time series of precipitable water and is tested in 100 global cases of heavy rain. First,nonlinear relative dynamic error for local attractor pairs is calculated at different stages of the heavy rain process,after which the local change characteristics of the attractors are analyzed. Second,the eigen-peak is defined as a prediction indicator based on an error threshold of about 1.5,and is then used to analyze the forecasting validity period. The results reveal that the prediction indicator features regarded as eigenpeaks for heavy rain extreme weather are all reflected consistently,without failure,based on the NCPE model; the prediction validity periods for 1–2 d,3–9 d and 10–30 d are 4,22 and 74 cases,respectively,without false alarm or omission. The NCPE model developed allows accurate forecasting of heavy rain over an extended range of 10–30 d and has the potential to be used to explore the mechanisms involved in the development of heavy rain according to a segmentation scale. This novel method provides new insights into extended range forecasting and atmospheric predictability,and also allows the creation of multi-variable chaotic extreme weather prediction models based on high spatiotemporal resolution data.     

Characteristics Of Chaotic Attractors In Atmospheric Boundary-Layer Turbulence

In this paper, the attractors of turbulent flows in phase space are reconstructed by the time delay technique using observed data of atmospheric boundary-layer turbulence, which include high resolution temperature, humidity andthree-dimensional wind speed measurements in Gansu province and Beijing, China. The correlation dimensions and largest Lyapunov exponents have been computed. The results indicate that all the largest Lyapunov exponents in different conditions of time, site and atmospheric stability are greater than zero. This means that the atmospheric boundary-layer turbulence system is really chaotic and has appropriate low-dimensional strange attractors whose dimension numbers range from 3 to 7 and vary with different variables (dynamical variables or non-dynamical variables) and atmospheric stability. Turbulent kinetic energy is first applied to reconstruct the attractor of turbulence, and is found to be feasible.     

Estimation of the monthly precipitation predictability limit in China using the nonlinear local Lyapunov exponent

By using the nonlinear local Lyapunov exponent and nonlinear error growth dynamics, the predictability limit of monthly precipitation is quantitatively estimated based on daily observations collected from approximately 500 stations in China for the period 1960–2012. As daily precipitation data are not continuous in space and time, a transformation is first applied and a monthly standardized precipitation index (SPI) with Gaussian distribution is constructed. The monthly SPI predictability limit (MSPL) is quantitatively calculated for SPI dry, wet, and neutral phases. The results show that the annual mean MSPL varies regionally for both wet and dry phases: the MSPL in the wet (dry) phase is relatively higher (lower) in southern China than in other regions. Further, the pattern of the MSPL for the wet phase is almost opposite to that for the dry phase in both autumn and winter. The MSPL in the dry phase is higher in winter and lower in spring and autumn in southern China, while the MSPL values in the wet phase are higher in summer and winter than those in spring and autumn in southern China. The spatial distribution of the MSPL resembles that of the prediction skill of monthly precipitation from a dynamic extended-range forecast system.     

随机误差对混沌系统可预报性的影响

根据非线性局部Lyapunov指数的方法, 以Logistic映射和Lorenz系统的试验数据序列为例, 研究了在初始误差存在的情况下, 随机误差对混沌系统可预报性的影响。结果表明: 初始误差和随机误差对可预报期限影响所起的作用大小主要取决于两者的相对大小。当初始误差远大于随机误差时, 系统的可预报期限主要由初始误差决定, 可以不考虑随机误差对预报模式可预报性的影响; 反之, 当随机误差远大于初始误差时, 系统的可预报期限主要由随机误差决定; 当初始误差和随机误差量级相当时, 两者都对系统的可预报期限起重要作用。在后两种情况下, 在考虑初始误差对可预报性影响的同时还必须考虑随机误差的作用。此外, 我们在已知系统精确的控制方程和误差演化方程的条件下, 研究了随机误差对可预报性的影响, 理论所得结果与试验数据所得结果相似。这表明在随机误差较小的情况下, 对系统可预报期限的估计相对准确, 但在随机误差较大的情况下, 可预报期限的估计误差也较大。本文利用三种不同的滤波方法对序列进行了试验, 结果表明, Lanczos高通滤波得到的高频序列与原始加入的噪声序列无论是在强度上还是在演变趋势上都表现得相当一致, 其能有效地去除高频噪音继而提高对系统的可预报期限的估计, 这对实际气象观测资料如何有效地去除噪音具有一定的启发意义。     

气候噪声和气候系统的分维

根据相空间嵌入定理，按照Grassberger和Procaccia提出的计算分数维的方法，利用近百年来南、北半球地面气温资料，估算了气候吸引子的分数维，计算结果表明:气候吸引子的分数维南半球为3.3～3.7，北半球为3.2～3.7。它提供了气候吸引子的自相似结构的基本信息，表明模似气候系统最少需要4个独立变量。另外，还讨论了气候噪声对估算维数的影响。     

伴随系统及非线性优化方法在REM模式可预报性研究中的实际个例应用

利用REM模式的伴随系统和非线性优化方法，通过三个实际天气个例，对REM模式的可预报性问题进行了研究。结果表明，REM模式在给定的实际应用中可接受的预报误差范围内，对三个天气个例都具有预报能力。对于个例一，利用现有的常规报文初始观测场，进行简单的插值处理（最优插值等），REM数值模式就可以得到比较满意的预报结果； 对于个例二和个例三，对现有的报文初始观测场进行处理（如四维变分资料同化）后，REM模式在给定的误差允许范围内，对这两个天气个例仍得到满意的预报。研究结果不仅对改进数值模式具有一定的指导意义，而且对如何改进数值模式的初值问题，特别是在中尺度天气预报中如何改进具有一定的参考价值。     

Precipitable water over Canada

Numerical experiments have been performed to determine the way in which initial errors of relatively small horizontal extent propagate in a barotropic primitive equations model. Six‐day forecasts are made with the model, starting from initial conditions which are assumed to be free from errors. The forecasts are then repeated using the same initial data, except for a small area near the Gulf of Alaska where an error in the form of a low pressure system is added. The difference between the two forecasts, or error, is then examined as a function of time. The results obtained from sixteen cases run with winter data indicate that on the average the largest value in the error pattern travels, in six days, from the Gulf of Alaska to the western tip of the Great Lakes and decreases in amplitude by a factor slightly greater than 2 for an initial amplitude of 8.4 dam at 500 mb. The root mean square error computed over the entire forecast area, on the other hand, is found to remain nearly constant for the first 24 hours and to increase systematically thereafter, with a doubling time of 5 days.     

Lyapunov exponents of a simple stochastic model of the thermally and wind-driven ocean circulation

A reformulation of the simple model of the thermally and wind-driven ocean circulation introduced by Maas [Tellus 46A (1994) 671] is considered. Under a realistic range of forcing parameters, this model displays multiple attractors, corresponding to thermally direct and indirect circulations. The fixed point associated with the thermally direct circulation is unstable for a broad range of parameters, leading to limit cycles and chaotic behaviour. It is demonstrated that if weather variability is parameterised as stochastic perturbations to the mechanical and buoyancy fluxes, then the leading Lyapunov exponent of the circulation can become positive for sufficiently strong fluctuations in parameter ranges where it is deterministically zero. If the fluctuations are sufficiently small that the stochastic trajectories are not too far from the deterministic attractor, it is demonstrated that the sign of the leading Lyapunov exponent can have a substantial effect on the predictability of the system.     

Optimally growing initial errors of El Niño events in the CESM

The optimally growing initial errors (OGEs) of El Niño events are found in the Community Earth System Model (CESM) by the conditional nonlinear optimal perturbation (CNOP) method. Based on the characteristics of low-dimensional attractors for ENSO (El Niño Southern Oscillation) systems, we apply singular vector decomposition (SVD) to reduce the dimensions of optimization problems and calculate the CNOP in a truncated phase space by the differential evolution (DE) algorithm. In the CESM, we obtain three types of OGEs of El Niño events with different intensities and diversities and call them type-1, type-2 and type-3 initial errors. Among them, the type-1 initial error is characterized by negative SSTA errors in the equatorial Pacific accompanied by a negative west–east slope of subsurface temperature from the subsurface to the surface in the equatorial central-eastern Pacific. The type-2 initial error is similar to the type-1 initial error but with the opposite sign. The type-3 initial error behaves as a basin-wide dipolar pattern of tropical sea temperature errors from the sea surface to the subsurface, with positive errors in the upper layers of the equatorial eastern Pacific and negative errors in the lower layers of the equatorial western Pacific. For the type-1 (type-2) initial error, the negative (positive) temperature errors in the eastern equatorial Pacific develop locally into a mature La Niña (El Niño)-like mode. For the type-3 initial error, the negative errors in the lower layers of the western equatorial Pacific propagate eastward with Kelvin waves and are intensified in the eastern equatorial Pacific. Although the type-1 and type-3 initial errors have different spatial patterns and dynamic growing mechanisms, both cause El Niño events to be underpredicted as neutral states or La Niña events. However, the type-2 initial error makes a moderate El Niño event to be predicted as an extremely strong event.