条件非线性最优扰动(CNOP):简介与数值求解

介绍了条件非线性最优扰动(Conditional Nonlinear Optimal Perturbation,CNOP)的定义及其在大气和海洋等可预报性研究中的应用。根据研究对象不同,CNOP分为与初始扰动有关的CNOP(CNOP-I)方法、与模式参数扰动有关的CNOP(CNOP-P)方法和同时考虑初始扰动和模式参数扰动的CNOP方法。目前,CNOP-I方法已经应用于ENSO、黑潮和阻塞可预报性以及热盐环流和草原生态系统稳定性的研究。此外,CNOP-I方法也被应用于探讨台风目标观测的研究,利用CNOP-I方法能够识别出台风预报的初值敏感区,通过观测系统模拟试验表明在初值敏感区增加观测能够有效改进台风的预报技巧。CNOP-P方法也在ENSO和黑潮可预报性以及热盐环流和草原生态系统稳定性研究中得到了应用。为了将CNOP方法应用于更多的领域,本文利用一个简单的Burgers方程,介绍了如何通过建立Burgers方程的切线性模式和伴随模式,从而利用非线性最优化算法计算获得CNOP。这一数值试验为将CNOP方法应用于更多的领域提供了借鉴。     

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。     

数值天气预报和气候预测的可预报性问题

考察由初始状态误差和模式中参数误差所引起的预报结果的不确定性。提出了数值天气预报与气候预测中三类可预报性问题，即，最大可预报时间，最大预报误差，初值与参数的最大允许误差。然后将这三类问题化成了对应的非线性优化问题，给出了处理此类非线性优化问题的思路，并且有数值方法对Lorenz模型研究了这三类问题。     

从Nio-3指数看MJO不确定性对ENSO可预报性的影响

通过在Zebiak Cane数值模式中引入参数化MJO随机外强迫,着重从Nio 3指数的演变发展探讨了MJO不确定性对ENSO可预报性的影响。结果表明,对Zebiak Cane模式而言,MJO不确定性对由条件非线性最优扰动(CNOP)导致的ENSO事件最大预报误差影响较小;与初始误差相比,由MJO不确定性产生的模式误差在ENSO预报不确定性的产生中具有较小作用,对ENSO可预报性的影响不显著。该结果强调了初始误差在ENSO预报不确定性中的主要作用,从而为ENSO预测的资料同化提供了理论基础。     

从Nio-3指数看MJO不确定性对ENSO可预报性的影响北大核心CSCD

通过在Zebiak Cane数值模式中引入参数化MJO随机外强迫,着重从Nio 3指数的演变发展探讨了MJO不确定性对ENSO可预报性的影响。结果表明,对Zebiak Cane模式而言,MJO不确定性对由条件非线性最优扰动(CNOP)导致的ENSO事件最大预报误差影响较小;与初始误差相比,由MJO不确定性产生的模式误差在ENSO预报不确定性的产生中具有较小作用,对ENSO可预报性的影响不显著。该结果强调了初始误差在ENSO预报不确定性中的主要作用,从而为ENSO预测的资料同化提供了理论基础。     

条件非线性最优扰动在可预报性问题研究中的应用

本文总结了近年来条件非线性最优扰动方法的应用研究的主要进展.主要包括四个方面:(1)将条件非线性最优扰动(CNOP)方法拓展到既考虑初始扰动又考虑模式参数扰动,形成了拓展的CNOP方法.拓展的CNOP方法不仅能够应用于研究分别由初始误差和模式参数误差导致的可预报性问题,而且能够用于探讨初始误差和模式参数误差同时存在的情形;(2)将拓展的CNOP方法分别应用于ENSO和黑潮可预报性研究,考察了初始误差和模式参数误差对其可预报性的影响,揭示了初始误差在导致ENSO和黑潮大弯曲路径预报不确定性中的重要作用;(3)考察了阻塞事件发生的最优前期征兆(OPR)以及导致其预报不确定性的最优增长初始误差(OGR),揭示了OPR和OGR空间模态及其演变机制的相似性及其局地性特征;(4)研究了台风预报的目标观测问题,用CNOP方法确定了台风预报的目标观测敏感区,通过观测系统模拟试验(OSSEs)和/或观测系统试验(OSEs),表明了CNOP敏感区在改进台风预报中的有效性.具体地,台风OGR的主要误差分布在某一特定区域,空间分布具有明显的局地性特征,在台风OGR的局地性区域增加观测,有效改进了台风的预报技巧,该区域代表了台风预报的初值敏感区.事实上,上述El Ni(n)o事件、黑潮路径变异以及阻塞事件的OGR的空间分布也具有明显的局地性特征,这些事件的OGR刻画的局地性区域可能也代表了初值敏感区.     

广州“5.7”暴雨预报的模式不确定性研究

2017年5月7日,在弱天气尺度强迫下,广州发生了暖区特大暴雨,局地发展迅速,降水强度极端,多家业务模式出现了漏报情况。为了探究此次降水过程模式预报的不确定性,采用条件非线性最优参数扰动（Conditional Nonlinear Optimal Perturbation related to Parameters,CNOP-P）方法筛选出最能体现中小尺度系统非线性误差增长特征的关键物理参数,以此构造CNOP-P-RP模式扰动方案,并基于CMA-Meso模式进行对流尺度集合预报试验,最后探究了CNOP-P关键参数影响局地对流发生、发展不同阶段的物理机理。结果显示,不同降水阶段的CNOP-P敏感参数主要与垂直扩散、云雨自动转换或其他水成物向雨滴的转换有关。与业务上常用的随机物理倾向扰动（Stochastically Perturbed Parameterization Tendencies,SPPT）方案相比,在本次降水过程中,基于CNOP-P-RP方案构造的集合预报试验具有更高的降水和地面要素的概率预报技巧,集合预报系统可靠性也占优。进一步分析发现,垂直扩散不确定性导致的山前温度梯度和...     

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。     

用非线性最优化方法研究El Nio可预报性的进展与前瞻

综述用非线性优化方法研究厄尔尼诺(El Ni~no)南方涛动(ENSO)事件可预报性的进展。针对ENSO可预报性研究中的热点问题———“前期征兆”、“春季可预报性障碍”,以及如何量化研究ENSO可预报性和ENSO的不对称性问题,作者在近年来的工作中先后用理论模式和中等复杂程度ENSO模式研究了ENSO可预报性的动力学,揭示了ENSO的若干重要非线性特征。主要结果如下:(1)条件非线性最优扰动(CNOP)(局部CNOP)比线性奇异向量更易发展成ENSO事件,扮演了ENSO的最优前期征兆。这些ENSO事件关于气候平均态是不对称的。理论分析表明,非线性温度平流过程是造成这种不对称性的重要原因。1980~2002年的海洋再分析资料验证了上述理论结果。(2)ENSO事件CNOP型初始误差的发展有明显的季节依赖性,该误差导致了ENSO事件最显著的春季可预报性障碍(SPB)现象。ENSO事件SPB的发生不仅依赖于气候平均态,而且依赖于ENSO事件本身及其初始误差模态,是三者综合作用的结果。(3)建立了关于ENSO可预报性的最大可预报时间下界、最大预报误差上界和最大允许初始误差下界的三类可预报性问题,分别从三个方面揭示了ENSO事件的春季可预报性障碍现象,比较有效地量化了其可预报性。(4)通过CNOP方法,揭示了非线性温度平流在年代际尺度ENSO不对称性研究中的重要作用,解释了ENSO不对称性的年代际变化,基于所用ENSO模式给出了ENSO不对称性年代际变化的机制。最后,展望了非线性优化方法在ENSO可预报性中应用的前景,并期望该方法能拓展到ENSO第二类可预报性问题的研究中。     

混沌系统单变量可预报性研究

对于n维的混沌系统, 不同变量的可预报性是不同的。为了研究混沌系统中单个变量的可预报性, 本文在以前提出的混沌系统整体的非线性局部Lyapunov指数基础上(李建平等, 2006), 引入了单变量的非线性局部Lyapunov指数及其相关统计量, 进一步完善了非线性误差增长理论。通过应用到几个混沌个例, 结果表明单变量的非线性局部Lyapunov指数及其相关统计量可以用来定量地研究多维混沌系统中不同变量的可预报性, 系统不同变量的可预报性之间不是相互独立的, 而是单个变量的可预报期限与系统整体的可预报期限之比都近似保持一个常数, 但各个变量的常数值有所不同。     

Application of the Conditional Nonlinear Optimal Perturbation Method to the Predictability Study of the Kuroshio Large Meander

A reduced-gravity barotropic shallow-water model was used to simulate the Kuroshio path variations.The results show that the model was able to capture the essential features of these path variations.We used one simulation of the model as the reference state and investigated the effects of errors in model parameters on the prediction of the transition to the Kuroshio large meander (KLM) state using the conditional nonlinear optimal parameter perturbation (CNOP-P) method.Because of their relatively large uncertainties,three model parameters were considered:the interfacial friction coefficient,the wind-stress amplitude,and the lateral friction coefficient.We determined the CNOP-Ps optimized for each of these three parameters independently,and we optimized all three parameters simultaneously using the Spectral Projected Gradient 2 (SPG2) algorithm.Similarly,the impacts caused by errors in initial conditions were examined using the conditional nonlinear optimal initial perturbation (CNOP-I) method.Both the CNOP-I and CNOP-Ps can result in significant prediction errors of the KLM over a lead time of 240 days.But the prediction error caused by CNOP-I is greater than that caused by CNOP-P.The results of this study indicate not only that initial condition errors have greater effects on the prediction of the KLM than errors in model parameters but also that the latter cannot be ignored.Hence,to enhance the forecast skill of the KLM in this model,the initial conditions should first be improved,the model parameters should use the best possible estimates.     

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.     

Progress in Predictability Studies in China （2003-2006）

Since the last International Union of Geodesy and Geophysics General Assembly(2003),predictability studies in China have made significant progress.For dynamic forecasts,two novel approaches of conditional nonlinear optimal perturbation and nonlinear local Lyapunov exponents were proposed to cope with the predictability problems of weather and climate,which are superior to the corresponding linear theory.A possible mechanism for the"spring predictability barrier"phenomenon for the El Ni(?)o-Southern Oscillation (ENSO)was provided based on a theoretical model.To improve the forecast skill of an intermediate coupled ENSO model,a new initialization scheme was developed,and its applicability was illustrated by hindcast experiments.Using the reconstruction phase space theory and the spatio-temporal series predictive method, Chinese scientists also proposed a new approach to improve dynamical extended range(monthly)prediction and successfully applied it to the monthly-scale predictability of short-term climate variations.In statistical forecasts,it was found that the effects of sea surface temperature on precipitation in China have obvious spatial and temporal distribution features,and that summer precipitation patterns over east China are closely related to the northern atmospheric circulation.For ensemble forecasts,a new initial perturbation method was used to forecast heavy rain in Guangdong and Fujian Provinces on 8 June 1998.Additionally, the ensemble forecast approach was also used for the prediction of a tropical typhoons.A new downscaling model consisting of dynamical and statistical methods was provided to improve the prediction of the monthly mean precipitation.This new downsealing model showed a relatively higher score than the issued operational forecast.     

Progress in the Study of Nonlinear Atmospheric Dynamics and Predictability of Weather and Climate in China (2007--2011)

Recent progress in the study of nonlinear atmospheric dynamics and related predictability of weather and climate in China (2007-2011) are briefly introduced in this article. Major achievements in the study of nonlinear atmospheric dynamics have been classified into two types:(1) progress based on the analysis of solutions of simplified control equations, such as the dynamics of NAO, the optimal precursors for blocking onset, and the behavior of nonlinear waves, and (2) progress based on data analyses, such as the nonlinear analyses of fluctuations and recording-breaking temperature events, the long-range correlation of extreme events, and new methods of detecting abrupt dynamical change. Major achievements in the study of predictability include the following:(1) the application of nonlinear local Lyapunov exponents (NLLE) to weather and climate predictability; (2) the application of condition nonlinear optimal perturbation (CNOP) to the studies of El Nin o-Southern Oscillation (ENSO) predictions, ensemble forecasting, targeted observation, and sensitivity analysis of the ecosystem; and (3) new strategies proposed for predictability studies. The results of these studies have provided greater understanding of the dynamics and nonlinear mechanisms of atmospheric motion, and they represent new ideas for developing numerical models and improving the forecast skill of weather and climate events.     

Decreasing predictability as a precursor indicator for abrupt climate change

Abrupt climate change has an important impact on sustainable economic and social development, as well as ecosystem. However, it is very difficult to predict abrupt climate changes because the climate system is a complex and nonlinear system. In the present paper, the nonlinear local Lyapunov exponent (NLLE) is proposed as a new early warning signal for an abrupt climate change. The performance of NLLE as an early warning signal is first verified by those simulated abrupt changes based on four folding models. That is, NLLE in all experiments showed an almost monotonous increasing trend as a dynamic system approached its tipping point. For a well-studied abrupt climate change in North Pacific in 1976/1977, it is also found that NLLE shows an almost monotonous increasing trend since 1970 which give up to 6 years warning before the abrupt climate change. The limit of the predictability for a nonlinear dynamic system can be quantitatively estimated by NLLE, and lager NLLE of the system means less predictability. Therefore, the decreasing predictability may be an effective precursor indicator for abrupt climate change.

     

Applications of Conditional Nonlinear Optimal Perturbation in Predictability Study and Sensitivity Analysis of Weather and Climate

Considering the limitation of the linear theory of singular vector (SV), the authors and their collaborators proposed conditional nonlinear optimal perturbation (CNOP) and then applied it in the predictability study and the sensitivity analysis of weather and climate system. To celebrate the 20th anniversary of Chinese National Committee for World Climate Research Programme (WCRP), this paper is devoted to reviewing the main results of these studies. First, CNOP represents the initial perturbation that has largest nonlinear evolution at prediction time, which is different from linear singular vector (LSV) for the large magnitude of initial perturbation or/and the long optimization time interval. Second, CNOP, rather than linear singular vector (LSV), represents the initial anomaly that evolves into ENSO events most probably. It is also the CNOP that induces the most prominent seasonal variation of error growth for ENSO predictability; furthermore, CNOP was applied to investigate the decadal variability of ENSO asymmetry. It is demonstrated that the changing nonlinearity causes the change of ENSO asymmetry. Third, in the studies of the sensitivity and stability of ocean’s thermohaline circulation (THC), the nonlinear asymmetric response of THC to finite amplitude of initial perturbations was revealed by CNOP. Through this approach the passive mechanism of decadal variation of THC was demonstrated; Also the authors studies the instability and sensitivity analysis of grassland ecosystem by using CNOP and show the mechanism of the transitions between the grassland and desert states. Finally, a detailed discussion on the results obtained by CNOP suggests the applicability of CNOP in predictability studies and sensitivity analysis.     

Baseline predictability of daily east Asian summer monsoon circulation indices

The nonlinear local Lyapunov exponent (NLLE) method is adopted to quantitatively determine the predictability limit of East Asian summer monsoon (EASM) intensity indices on a synoptic timescale. The predictability limit of EASM indices varies widely according to the definitions of indices. EASM indices defined by zonal shear have a limit of around 7 days, which is higher than the predictability limit of EASM indices defined by sea level pressure (SLP) difference and meridional wind shear (about 5 days). The initial error of EASM indices defined by SLP difference and meridional wind shear shows a faster growth than indices defined by zonal wind shear. Furthermore, the indices defined by zonal wind shear appear to fluctuate at lower frequencies, whereas the indices defined by SLP difference and meridional wind shear generally fluctuate at higher frequencies. This result may explain why the daily variability of the EASM indices defined by zonal wind shear tends be more predictable than those defined by SLP difference and meridional wind shear. Analysis of the temporal correlation coefficient (TCC) skill for EASM indices obtained from observations and from NCEP’s Global Ensemble Forecasting System (GEFS) historical weather forecast dataset shows that GEFS has a higher forecast skill for the EASM indices defined by zonal wind shear than for indices defined by SLP difference and meridional wind shear. The predictability limit estimated by the NLLE method is shorter than that in GEFS. In addition, the June-September average TCC skill for different daily EASM indices shows significant interannual variations from 1985 to 2015 in GEFS. However, the TCC for different types of EASM indices does not show coherent interannual fluctuations.     

Role of Parameter Errors in the Spring Predictability Barrier for ENSO Events in the Zebiak–Cane Model

ABSTRACT The impact of both initial and parameter errors on the spring predictability barrier （SPB） is investigated using the Zebiak Cane model （ZC model）. Previous studies have shown that initial errors contribute more to the SPB than parameter errors in the ZC model. Although parameter errors themselves are less important, there is a possibility that nonlinear interactions can occur between the two types of errors, leading to larger prediction errors compared with those induced by initial errors alone. In this case, the impact of parameter errors cannot be overlooked. In the present paper, the optimal combination of these two types of errors [i.e., conditional nonlinear optimal perturbation （CNOP） errors] is calculated to investigate whether this optimal error combination may cause a more notable SPB phenomenon than that caused by initial errors alone. Using the CNOP approach, the CNOP errors and CNOP-I errors （optimal errors when only initial errors are considered） are calculated and then three aspects of error growth are compared： （1） the tendency of the seasonal error growth; （2） the prediction error of the sea surface temperature anomaly; and （3） the pattern of error growth. All three aspects show that the CNOP errors do not cause a more significant SPB than the CNOP-I errors. Therefore, this result suggests that we could improve the prediction of the E1 Nifio during spring by simply focusing on reducing the initial errors in this model.