Application of the Conditional Nonlinear Local Lyapunov Exponent to Second-Kind Predictability

In order to quantify the influence of external forcings on the predictability limit using observational data,the author introduced an algorithm of the conditional nonlinear local Lyapunov exponent(CNLLE)method.The effectiveness of this algorithm is validated and compared with the nonlinear local Lyapunov exponent(NLLE)and signal-to-noise ratio methods using a coupled Lorenz model.The results show that the CNLLE method is able to capture the slow error growth constrained by external forcings,therefore,it can quantify the predictability limit induced by the external forcings.On this basis,a preliminary attempt was made to apply this method to measure the influence of ENSO on the predictability limit for both atmospheric and oceanic variable fields.The spatial distribution of the predictability limit induced by ENSO is similar to that arising from the initial conditions calculated by the NLLE method.This similarity supports ENSO as the major predictable signal for weather and climate prediction.In addition,a ratio of predictability limit(RPL)calculated by the CNLLE method to that calculated by the NLLE method was proposed.The RPL larger than 1 indicates that the external forcings can significantly benefit the long-term predictability limit.For instance,ENSO can effectively extend the predictability limit arising from the initial conditions of sea surface temperature over the tropical Indian Ocean by approximately four months,as well as the predictability limit of sea level pressure over the eastern and western Pacific Ocean.Moreover,the impact of ENSO on the geopotential height predictability limit is primarily confined to the troposphere.

Determining the Spectrum of the Nonlinear Local Lyapunov Exponents in a Multidimensional Chaotic System

For an n-dimensional chaotic system, we extend the definition of the nonlinear local Lyapunov exponent （NLLE） from one- to n-dimensional spectra, and present a method for computing the NLLE spectrum. The method is tested on three chaotic systems with different complexity. The results indicate that the NLLE spectrum realistically characterizes the growth rates of initial error vectors along different directions from the linear to nonlinear phases of error growth. This represents an improvement over the traditional Lyapunov exponent spectrum, which only characterizes the error growth rates during the linear phase of error growth. In addition, because the NLLE spectrum can effectively separate the slowly and rapidly growing perturbations, it is shown to be more suitable for estimating the predictability of chaotic systems, as compared to the traditional Lyapunov exponent spectrum.

Quantitative Comparison of Predictabilities of Warm and Cold Events Using the Backward Nonlinear Local Lyapunov Exponent Method

The backward nonlinear local Lyapunov exponent method(BNLLE)is applied to quantify the predictability of warm and cold events in the Lorenz model.Results show that the maximum prediction lead times of warm and cold events present obvious layered structures in phase space.The maximum prediction lead times of each warm(cold)event on individual circles concentric with the distribution of warm(cold)regime events are roughly the same,whereas the maximum prediction lead time of events on other circles are different.Statistical results show that warm events are more predictable than cold events.

Quantifying local predictability of the Lorenz system using the nonlinear local Lyapunov exponent

The nonlinear local Lyapunov exponent（NLLE） can be used as a quantification of the local predictability limit of chaotic systems. In this study, the phase-spatial structure of the local predictability limit over the Lorenz-63 system is investigated. It is found that the inner and outer rims of each regime of the attractor have a high probability of a longer than average local predictability limit, while the center part is the opposite. However, the distribution of the local predictability limit is nonuniformly organized, with adjacent points sometimes showing quite distinct error growth.The source of local predictability is linked to the local dynamics, which is related to the region in the phase space and the duration on the current regime.

非线性局部Lyapunov指数方法在目标观测中的应用初探

本文初步探讨了非线性局部Lyapunov指数方法(NLLE)在目标观测中的应用。首先,在NLLE理论基础上研究了非线性动力系统内局部平均误差相对增长(LAGRE)特征,证明了在误差发展进入随机状态前,LAGRE与初始误差大小无关而是与初始状态有关;在演化进入随机状态后,LAGRE的饱和值由初始误差大小决定这一特征。同时利用三个变量的常微分方程模型Lorenz63验证了这一结论。其次,从非线性局部误差增长理论出发,在局部动力演化相似方法(LDA)的基础上提出向前局部动力演化相似方法(FLDA)的概念,并通过两个混沌个例来说明LDA和FLDA方法能够有效的利用历史资料还原任意初始状态的LAGRE。这些方法的提出为NLLE理论应用于观测资料研究目标观测问题提供了依据。

季节内印度洋-西太平洋对流涛动对次季节-季节尺度大气可预报性的影响

利用非线性局部Lyapunov指数和条件非线性局部Lyapunov指数定量估计了季节内印度洋-西太平洋对流涛动(IPCO)和实时多变量Madden-Julian指数(RMM指数)可预报期限,量化了季节内IPCO对S2S尺度大气可预报性的贡献,深入研究了季节内IPCO演变下S2S尺度可预报期限空间分布的变化规律。结果表明:(1)与RMM指数相比,季节内IPCO指数可预报性更强,可预报期限达到31天左右,比RMM指数高出2周以上;(2)印度洋-西太平洋区域S2S尺度大气可预报性最强,可预报期限达到30天以上,其中季节内IPCO是该地区的主要可预报性来源之一,其贡献达到6天,占总可预报期限的25%以上;(3)随着季节内IPCO的演变,印度洋-西太平洋地区S2S尺度大气可预报性有空间结构变化,表现为可预报期限异常的传播和振荡。S2S尺度大气可预报期限正负异常沿季节内IPCO传播路径,一支以赤道中西印度洋为起点北传至印度半岛,一支向东传播,经过海洋性大陆到赤道西太平洋后向北传播,到达日本南部。同时,可预报性异常的传播在在东印度洋和西太平洋表现出反向变化的特征,形成东西两极振荡,当季节内IPCO向正位相发展时,东印度洋具有更强的可预报性,西太平洋具有更弱的可预报性,反之亦然。季节内IPCO的发展(衰退)可使东印度洋(西太平洋)S2S尺度大气可预报性更强,表明模式预报技巧对此具有更大的提升空间。

The Predictability Limit of Oceanic Mesoscale Eddy Tracks in the South China Sea

Employing the nonlinear local Lyapunov exponent (NLLE) technique, this study assesses the quantitative predictability limit of oceanic mesoscale eddy (OME) tracks utilizing three eddy datasets for both annual and seasonal means. Our findings reveal a discernible predictability limit of approximately 39 days for cyclonic eddies (CEs) and 44 days for anticyclonic eddies (AEs) within the South China Sea (SCS). The predictability limit is related to the OME properties and seasons. The long-lived, large-amplitude, and large-radius OMEs tend to have a higher predictability limit. The predictability limit of AE (CE) tracks is highest in autumn (winter) with 52 (53) days and lowest in spring (summer) with 40 (30) days. The spatial distribution of the predictability limit of OME tracks also has seasonal variations, further finding that the area of higher predictability limits often overlaps with periodic OMEs. Additionally, the predictability limit of periodic OME tracks is about 49 days for both CEs and AEs, which is 5-10 days higher than the mean values. Usually, in the SCS, OMEs characterized by high predictability limit values exhibit more extended and smoother trajectories and often move along the northern slope of the SCS.

Nonlinear local Lyapunov exponent and atmospheric predictability research

Because atmosphere itself is a nonlinear system and there exist some problems using the linearized equations to study the initial error growth, in this paper we try to use the error nonlinear growth theory to discuss its evolution, based on which we first put forward a new concept: nonlinear local Lyapunov exponent. It is quite different from the classic Lyapunov exponent because it may characterize the finite time error local average growth and its value depends on the initial condition, initial error, variables, evolution time, temporal and spatial scales. Based on its definition and the at-mospheric features, we provide a reasonable algorithm to the exponent for the experimental data, obtain the atmospheric initial error growth in finite time and gain the maximal prediction time. Lastly, taking 500 hPa height field as example, we discuss the application of the nonlinear local Lyapunov exponent in the study of atmospheric predictability and get some reliable results: atmospheric predictability has a distinct spatial structure. Overall, predictability shows a zonal distribution. Prediction time achieves the maximum over tropics, the second near the regions of Antarctic, it is also longer next to the Arctic and in subtropics and the mid-latitude the predictability is lowest. Particularly speaking, the average prediction time near the equation is 12 days and the maximum is located in the tropical Indian, Indonesia and the neighborhood, tropical eastern Pacific Ocean, on these regions the prediction time is about two weeks. Antarctic has a higher predictability than the neighboring latitudes and the prediction time is about 9 days. This feature is more obvious on Southern Hemispheric summer. In Arctic, the predictability is also higher than the one over mid-high latitudes but it is not pronounced as in Antarctic. Mid-high latitude of both Hemispheres (30°S―60°S, 30°―60°N) have the lowest predictability and the mean prediction time is just 3―4 d. In addition, predictability varies with the seasons. Most regions in the Northern Hemisphere, the predictability in winter is higher than that in summer, especially in the mid-high latitude: North Atlantic, North Pacific and Greenland Island. However in the Southern Hemisphere, near the Antarctic regions (60°S―90°S), the corresponding summer has higher predictability than its winter, while in other areas especially in the latitudes of 30°S―60°S, the prediction does not change obviously with the seasons and the average time is 3―5 d. Both the theoretical and data computation results show that nonlinear local Lyapunov exponent and the nonlinear local error growth really may measure the predictability of the atmospheric variables in different temporal and spatial scales.

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 approx- imately 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 calcu- lated 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.

耦合Lorenz模型的吸引子特性及其可预报性分析

通过改变耦合Lorenz模型中控制快、慢子系统之间耦合强度的参数,本文探究了耦合强度对该系统的混沌吸引子特性及可预报性的影响。结果表明:随着耦合增强,快系统中显示出与慢系统类似的低频变化特征,其吸引子也随之变大;而慢系统高频分量变大,导致其变率增强,吸引子轨道变得更加密集。在此基础上,利用非线性局部Lyapunov指数方法定量分析了耦合强度对耦合Lorenz系统可预报性的影响。具体来说,在耦合之后,耦合系统的对数误差增长曲线包含前后两段不同的误差增长率,分别代表快速和慢速误差增长过程。此外,各子系统的可预报性对耦合强度变化响应并不一致,随着对快系统的耦合强度增加,快/慢两个不同尺度系统的可预报上限均减少。然而,增加对慢系统的耦合强度却只能提高快系统的可预报上限,对慢系统的可预报性改变不大。

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。

数值天气预报和气候预测可预报性研究的若干动力学方法

简要回顾了数值天气预报和气候预测可预报性研究的若干动力学方法,包括用于研究第一类可预报性问题的线性奇异向量(LSV)和条件非线性最优初始扰动(CNOP-I)方法,以及Lyapunov指数和非线性局部Lyapunov指数方法。前两种方法用于研究预报或预测的预报误差问题,可以用于估计天气预报和气候预测的最大预报误差,而且根据导致最大预报误差的初始误差结构的信息,这两种方法可以用于确定预报或预测的初值敏感区。应该指出的是,LSV是基于线性化模式,对于描述非线性大气和海洋的运动具有局限性。因而,对于非线性模式,应该选择使用CNOP-I估计最大预报误差。Lyapunov指数和非线性局部Lyapunov指数可以用于研究第一类可预报性问题中的预报时限问题,前者是基于线性模式,不能解释非线性对预报时限的影响,而非线性局部Lyapunov指数方法则考虑了非线性的影响,能够较好地估计实际天气和气候的预报时限。第二类可预报性问题的研究方法相对较少,本文仅介绍了由我国科学家提出的关于模式参数扰动的条件非线性最优参数扰动(CNOP-P)方法,该方法可以用于寻找到对预报有最大影响的参数扰动,并可以进一步确定哪些参数最应该利用观测资料进行校准。另一方面,通过对比CNOP-I和CNOP-P对预报误差的影响,可以判断导致预报不确定性的主要误差因子,进而指导人们着力改进模式或者初始场。

月尺度气温可预报性对资料长度的依赖及可信度

利用全国518个站1960—2011年逐日气温观测资料和160个站1983—2012年月尺度气温客观预测数据,基于非线性局部Lyapunov指数和非线性误差增长理论,研究中国区域月尺度气温可预报性期限对资料序列长度的依赖性。结果表明:气温可预报性期限对资料序列的长度有一定程度的依赖性,在西北、东北及华中地区尤为明显。平均而言,45年的资料序列长度才能够得到稳定合理的可预报性期限。为了验证气温可预报期限计算结果的可信度,将月尺度气温的可预报性期限与客观气候预测方法的预报评分技巧进行对比,发现两者结果非常一致。其中,由观测资料得到的1月气温的可预报性期限明显低于7月,1月客观气候预测方法的预报评分技巧也明显低于7月,且1月(7月)预报评分的空间分布型与1月(7月)气温可预报性期限的空间分布型较为一致。因此,利用非线性局部Lyapunov指数和台站逐日观测资料分析气温的可预报性期限结果是可信的。

采样间隔和插值对混沌系统可预报期限估计的影响

利用非线性局部Lyapunov指数的方法研究了采样间隔和插值对混沌系统可预报性的影响,结果表明:在一定范围内,采样间隔对系统可预报期限估计的影响基本是随采样间隔的增大而逐渐减小。但当采样间隔超过一定大小时,所得序列已不能找到真实的局地动力相似,无法得到其真实的误差增长情况,也无法得到系统准确的可预报期限的估计。本文通过插值的方法试图还原同样长度的时间序列,结果表明不论是在采样间隔较大还是较小的情况下,插值都不能有效地改善对系统可预报期限的估计。此外,在采样间隔固定的情况下,随着插值个数的增多,系统的可预报期限的估计反而更低。以上结果提示我们在利用实际海洋观测资料估计其可预报期限时,选用较高分辨率较长时间序列的资料可以得到更接近真实的可预报期限。

强迫Lorenz系统的可预报性研究

根据非线性局部Lyapunov指数方法,分别以常数强迫Lorenz系统和准周期强迫Lorenz系统为例,研究了在外强迫存在的条件下混沌系统可预报性的改变.结果表明:外强迫会影响混沌系统的可预报性,两种不同类型的强迫Lorenz系统的可预报期限都随着外强迫的增强而增加,但是大小相等方向相反的外强迫对系统可预报性的影响不同,其中正值强迫比负值强迫作用下的可预报期限更长,并且这种差异随着强度的增加而增大;不同形式的外强迫对可预报性的影响也不同,常数强迫的影响主要体现在误差增长的线性阶段,准周期强迫的影响除了线性阶段还必须考虑到非线性阶段;当强度相等的常数强迫和准周期强迫驱动Lorenz系统时,常数强迫作用下的系统可预报性更高.本文基于混沌理论模型的研究,对于实际大气的可预报性研究具有一定的启示意义.

Estimating the Predictability Limit of Tropical Cyclone Tracks over the Western North Pacific Using Observational Data

In this study,the nonlinear local Lyapunov exponent(NLLE)approach was used to quantitatively determine the predictability limit of tropical cyclone(TC)tracks based on observed TC track data obtained from the Joint Typhoon Warning Center.The results show that the predictability limit of all TC tracks over the whole western North Pacific(WNP)basin is about 102 h,and the average lifetime of all TC tracks is about 174 h.The predictability limits of the TC tracks for short-,medium-,and long-lived TCs are approximately 72 h,120 h,and 132 h,respectively.The predictability limit of the TC tracks depends on the TC genesis location,lifetime,and intensity,and further analysis indicated that these three metrics are closely related.The more intense and longer-lived TCs tend to be generated on the eastern side of the WNP(EWNP),whereas the weaker and shorter-lived TCs tend to form in the west of the WNP(WWNP)and the South China Sea(SCS).The relatively stronger and longer-lived TCs,which are generated mainly in the EWNP,have a longer travel time before they curve northeastwards and hence tend to be more predictable than the relatively weaker and shorter-lived TCs that form in the WWNP region and SCS.Furthermore,the results show that the predictability limit of the TC tracks obtained from the best-track data may be underestimated due to the relatively short observational records currently available.Further work is needed,employing a numerical model to assess the predictability of TC tracks.

Determination of the Backward Predictability Limit and Its Relationship with the Forward Predictability Limit

In this work, two types of predictability are proposed—forward and backward predictability—and then applied in the nonlinear local Lyapunov exponent approach to the Lorenz63 and Lorenz96 models to quantitatively estimate the local forward and backward predictability limits of states in phase space. The forward predictability mainly focuses on the forward evolution of initial errors superposed on the initial state over time, while the backward predictability is mainly concerned with when the given state can be predicted before this state happens. From the results, there is a negative correlation between the local forward and backward predictability limits. That is, the forward predictability limits are higher when the backward predictability limits are lower, and vice versa. We also find that the sum of forward and backward predictability limits of each state tends to fluctuate around the average value of sums of the forward and backward predictability limits of sufficient states.Furthermore, the average value is constant when the states are sufficient. For different chaotic systems, the average value is dependent on the chaotic systems and more complex chaotic systems get a lower average value. For a single chaotic system,the average value depends on the magnitude of initial perturbations. The average values decrease as the magnitudes of initial perturbations increase.

Influence of Sea Surface Temperature on the Predictability of Idealized Tropical Cyclone Intensity

The role of sea surface temperature(SST)forcing in the development and predictability of tropical cyclone(TC)intensity is examined using a large set of idealized numerical experiments in the Weather Research and Forecasting(WRF)model.The results indicate that the onset time of rapid intensification of TC gradually decreases,and the peak intensity of TC gradually increases,with the increased magnitude of SST.The predictability limits of the maximum 10 m wind speed(MWS)and minimum sea level pressure(MSLP)are~72 and~84 hours,respectively.Comparisons of the analyses of variance for different simulation time confirm that the MWS and MSLP have strong signal-to-noise ratios(SNR)from 0-72 hours and a marked decrease beyond 72 hours.For the horizontal and vertical structures of wind speed,noticeable decreases in the magnitude of SNR can be seen as the simulation time increases,similar to that of the SLP or perturbation pressure.These results indicate that the SST as an external forcing signal plays an important role in TC intensity for up to 72 hours,and it is significantly weakened if the simulation time exceeds the predictability limits of TC intensity.

混沌系统可预报期限随初始误差变化规律研究

利用非线性误差增长理论计算了Logistic映射和Lorenz系统可预报期限随初始误差的变化,发现Logistic映射等简单混沌系统的可预报期限与初始误差的对数存在线性关系.在非线性误差增长理论的框架下,理论分析表明,平均误差增长达到一定值时,误差增长进入明显的非线性增长阶段,最终达到饱和;对于一个确定的混沌系统,在控制参数固定的情况下误差增长的饱和值也是固定的,因此可预报期限只依赖于初始误差.在可预报期限与初始误差对数存在的线性函数关系式中,线性系数与最大Lyapunov指数有关,在已知混沌系统的最大Lyapunov指数和某个固定初始误差的可预报期限的条件下,利用可预报期限与初始误差对数的线性函数关系可以外推得到任意固定初始误差的可预报期限.

Spatiotemporal distribution and decadal change of the monthly temperature predictability limit in China

Based on the nonlinear Lyapunov exponent and nonlinear error growth dynamics, the spatiotemporal distribution and decadal change of the monthly temperature predictability limit(MTPL) in China is quantitatively analyzed. Data used are daily temperature of 518 stations from 1960 to 2011 in China. The results are summarized as follows:(1) The spatial distribution of MTPL varies regionally. MTPL is higher in most areas of Northeast China, southwest Yunnan Province, and the eastern part of Northwest China. MTPL is lower in the middle and lower reaches of the Yangtze River and Huang-huai Basin.(2)The spatial distribution of MTPL varies distinctly with seasons. MTPL is higher in boreal summer than in boreal winter.(3) MTPL has had distinct decadal changes in China, with increase since the 1970 s and decrease since2000. Especially in the northeast part of the country, MTPL has significantly increased since 1986. Decadal change of MTPL in Northwest China, Northeast China and the Huang-huai Basin may have a close relationship with the persistence of temperature anomaly. Since the beginning of the 21 st century, MTPL has decreased slowly in most of the country, except for the south. The research provides a scientific foundation to understand the mechanism of monthly temperature anomalies and an important reference for improvement of monthly temperature prediction.