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。     

THE SENSITIVITY INFLUENCE OF NUMERICAL MODEL INITIAL VALUES ON FOUR-DIMENSIONAL ASSIMILATION—STUDY BASED ON LORENZ SYSTEM

Numerical experiments of adjoint variational assimilation have been performed using the knownLorenz system.With the increase of sensitivity of model's initial values,it is more and more difficultto use the adjoint method to get the initial values which are consistent with the dynamics of the fore-cast model.Under some circumstances the algorithm completely fails.This shows that four-dimen-sional assimilation is related to the limit of predictability.On the other hand.with the increase ofmodel equation's error,the result of variational assimilation may become worse and worse so that theprediction has no meaning.But if the model parameters are corrected when variational assimilation ismade,the forecast results can be greatly improved based on Lorenz model.     

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.     

THE SENSITIVITY INFLUENCE OF NUMERICAL MODEL INITIAL VALUES ON FOUR-DIMENSIONAL ASSIMILATION-STUDY BASED ON LORENZ SYSTEM*

Numerical experiments of adjoint variational assimilation have been performed using the known Lorenz system.With the increase of sensitivity of model's initial values,it is more and more difficult to use the adjoint method to get the initial values which are consistent with the dynamics of the forecast model.Under some circumstances the algorithm completely fails.This shows that four-dimensional assimilation is related to the limit of predictability.On the other hand.with the increase of model equation's error,the result of variational assimilation may become worse and worse so that the prediction has no meaning.But if the model parameters are corrected when variational assimilation is made,the forecast results can be greatly improved based on Lorenz model.     

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。     

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.     

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.     

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.     

Impact of Perturbation Schemes on the Ensemble Prediction in a Coupled Lorenz Model

Based on a simple coupled Lorenz model, we investigate how to assess a suitable initial perturbation scheme for ensemble forecasting in a multiscale system involving slow dynamics and fast dynamics. Four initial perturbation approaches are used in the ensemble forecasting experiments: the random perturbation(RP), the bred vector(BV), the ensemble transform Kalman filter(ETKF), and the nonlinear local Lyapunov vector(NLLV) methods. Results show that,regardless of the method used, the ensemble ave...     

Theoretical Basis and Application of an Analogue- Dynamical Model in the Lorenz System

The theoretical basis and application of an analogue-dynamical model (ADM) in the Lorenz system is studied. The ADM can effectively combine statistical and dynamical methods in which the small disturbance of the current initial value superimposed on the historical analogue reference state can be regarded as a prediction objective. Primary analyses show that under the condition of appending disturbances in model parameters, the model errors of ADM are much smaller than those of the pure dynamical model (PDM). The characteristics of predictability on the ADM in the Lorenz system are analyzed in phase space by conducting case studies and global experiments. The results show that the ADM can quite effectively reduce prediction errors and prolong the valid time of the prediction in most situations in contrast to the PDM, but when model errors are considerably small, the latter will be superior to the former. To overcome such a problem, the multi-reference-state updating can be applied to introduce the information of multi-analogue and update analogue and can exhibit exciting performance in the ADM.     

Overturning and wind-driven circulation in a low-order ocean–atmosphere model

A low-order ocean–atmosphere model is presented which combines coupling through heat exchange at the interface and wind stress forcing. The coupling terms are derived from the boundary conditions and the forcing terms of the constituents. Both the ocean and the atmosphere model are based on Galerkin truncations of the basic fluid dynamical equations. Hence, the coupled model can readily be extended to include more physics and more detail. The model presented here is the simplest of a hierarchy of low-order ocean–atmosphere models. The behaviour of the coupled model is investigated by means of geometric singular perturbation theory and bifurcation analysis. Two ways are found in which the slow time scales can play a role in the coupled dynamics. In the first scenario, a limit cycle on the overturning time scale is created. The associated oscillatory behaviour is governed by internal ocean dynamics. In the second scenario, intermittent behaviour occurs between periodic and chaotic regimes in parameter space.     

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.     

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。     

A Case Study of the Error Growth and Predictability of a Meiyu Frontal Heavy Precipitation Event

The Advanced Regional Eta-coordinate Model (AREM) is used to explore the predictability of a heavy rainfall event along the Meiyu front in China during 3-4 July 2003.Based on the sensitivity of precipitation prediction to initial data sources and initial uncertainties in different variables,the evolution of error growth and the associated mechanism are described and discussed in detail in this paper.The results indicate that the smaller-amplitude initial error presents a faster growth rate and its growth is characterized by a transition from localized growth to widespread expansion error.Such modality of the error growth is closely related to the evolvement of the precipitation episode,and consequcntly remarkable forecast divergence is found near the rainband,indicating that the rainfall area is a sensitive region for error growth.The initial error in the rainband contributes significantly to the forecast divergence,and its amplification and propagation are largely determined by the initial moisture distribution.The moisture condition also affects the error growth on smaller scales and the subsequent upscale error cascade.In addition,the error growth defined by an energy norm reveals that large error energy collocates well with the strong latent heating,implying that the occurrence of precipitation and error growth share the same energy source-the latent heat.This may impose an intrinsic predictability limit on the prediction of heavy precipitation.     

A case study of the error growth and predictability of a Meiyu frontal heavy precipitation event

The Advanced Regional Eta-coordinate Model (AREM) is used to explore the predictability of a heavy rainfall event along the Meiyu front in China during 3-4 July 2003. Based on the sensitivity of precipitation prediction to initial data sources and initial uncertainties in different variables, the evolution of error growth and the associated mechanism are described and discussed in detail in this paper. The results indicate that the smaller-amplitude initial error presents a faster growth rate and its growth...     

数值模式预报时效对计算精度和时间步长的依赖关系

通过数值计算研究Lorenz非线性动力系统,探讨了非线性动力系统中初值问题的解对时间步长和计算精度的依赖关系,从新的角度研究动力系统的预报时效问题,讨论了评价舍入误差对预报时效影响程度的方法。实验结果表明:动力系统的预报时效不仅与初值误差有关,而且在一定条件下敏感地依赖于计算采用的时间步长和计算精度。     

基于Lorenz系统提取数值模式可预报分量的初步试验

针对数值预报模式中存在的非线性混沌特性, 从提取可预报分量的思路出发, 阐述了在数值模式中提取可预报分量的方法, 并利用Lorenz系统进行了相关数值试验。研究发现, Lorenz系统初始误差在相空间中的增长速度是不同的, 某些方向的误差增长速度较慢, 即存在对初值扰动不敏感、相对稳定的可预报分量。根据数值模式切线性误差算子的特征值演化规律, 提取出数值模式的可预报分量, 并将模式变量在其基底上进行投影变换, 建立了可预报分量数值模式。在此基础上, 研究了Lorenz系统的混沌状态、模式参数误差及外部随机噪声对提取可预报分量的影响, 发现基于可预报分量的数值模式, 具有更好的预报技巧。     

变分同化方法在Lorenz系统中的简单应用研究

利用Lorenz模式作变分同化数值试验，通过对一个简单系统的讨论，介绍四维变分同化方法。对初值敏感性和观测点的个数及观测值作了对比试验，发现随着模式对初值敏感性的增加，同化效果会越来越差；观测点越少，观测值误差越大，这些都会影响同化效果，甚至导致同化失败。     

Initial-value predictability of prominent modes of North Pacific subsurface temperature in a CGCM

Three 40-member ensemble experiments and a 700?year control run are used to study initial value predictability in the North Pacific in Community Climate System Model version 3 (CCSM3). Our focus is on the leading two empirical orthogonal functions (EOFs) of subsurface temperature variability, which together produce an eastward propagating mode. Predictability is measured by relative entropy, which compares both the mean and spread of predictions of ensembles to the model??s climatological distribution of states. Despite the fact that EOF1, which is structurally similar to the observational Pacific Decadal Oscillation (PDO), has pronounced spectral peaks on decadal time scales, its predictability is less than 6?years. Additional predictability resides in the tendency of EOF1 to evolve to EOF2, primarily through simple advective processes. The propagating mode represented by the combination of EOF1 and EOF2 is predictable for about a decade. Information in both the mean and spread of predicted ensembles contribute to this predictability. Among the leading 15 EOFs, EOF1 is the least predictable mode in terms of the rate at which the corresponding principal component disperses in the ensemble experiments. However, it can produce enhanced predictability of the whole system by inducing EOF2, which is one of the two EOFs with the slowest dispersion rate. The first two EOFs can also enhance the ensemble mean (or ??signal??) component of predictability of the entire system. For typical amplitude initial states, this component contributes to predictability for about 6?years. For initial states with unusually high amplitude projections onto these two EOFs, this contribution can last much longer. The major findings from the three ensemble experiments are replicated and generalized when the initial condition predictability for each of many hundreds of different initial states is estimated. These estimates are derived from the behavior of a linear inverse model (LIM) that is based on the intrinsic variability present in the control run.     

Comparisons of Two Ensemble Mean Methods in Measuring the Average Error Growth and the Predictability

In this paper,taking the Lorenz system as an example,we compare the influences of the arithmetic mean and the geometric mean on measuring the global and local average error growth.The results show that the geometric mean error (GME) has a smoother growth than the arithmetic mean error (AME) for the global average error growth,and the GME is directly related to the maximal Lyapunov exponent,but the AME is not,as already noted by Krishnamurthy in 1993.Besides these,the GME is shown to be more appropriate than the AME in measuring the mean error growth in terms of the probability distribution of errors.The physical meanings of the saturation levels of the AME and the GME are also shown to be different.However,there is no obvious difference between the local average error growth with the arithmetic mean and the geometric mean,indicating that the choices of the AME or the GME have no influence on the measure of local average predictability.