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

Advances in atmospheric sciences 2024年第9期41卷 1769-1786页

作者： Ming ZHANG ruiqiang ding Quanjia ZHONG {2. li Deyu LU State Key Laboratory of Earth Surface Processes and Resource Ecology(ESPRE)/Key Laboratory of Environmental Change and Natural Disasters of Chinese Ministry of EducationBeijing Normal UniversityBeijing 100875China Department of Ocean Science Hong Kong University of Science and TechnologyHong KongChina college of Oceanic and atmospheric {2./Frontiers Science Center for Deep Ocean Multispheres and Earth System(DOMES)/Key Laboratory of Physical Oceanography Ministry of Education/Academy of the Future Ocean/Center for Ocean Carbon NeutralityOcean University of ChinaQingdao 266100China Laoshan Laboratory Qingdao 266237China State Key Laboratory of Numerical Modeling for atmospheric {2. and Geophysical Fluid Dynamics(LASG) Institute of Atmospheric PhysicsChinese Academy of SciencesBeijing 100029China college of Earth Science University of Chinese Academy of SciencesBeijing 100049China

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)*** 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 *** 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 *** 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 *** spatial distribution of the predictability limit induced by ENSO is similar to that arising from the initial conditions calculated by the NLLE *** similarity supports ENSO as the major predictable signal for weather and climate *** addition,a ratio of predictability limit(RPL)calculated by the CNLLE method to that calculated by the NLLE method was *** RPL larger than 1 indicates that the external forcings can significantly benefit the long-term predictability *** 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 ***,the impact of ENSO on the geopotential height predictability limit is primarily confined to the troposphere.

关键词： conditional nonlinear local Lyapunov exponent second-kind predictability coupled Lorenz model ENSO

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Nonlinear local Lyapunov exponent and atmospheric predictability research

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Science china Earth sciences 2006年第10期49卷 1111-1120页

作者： chen baohua1, li jianping1,{2., lanzhou University, lanzhou 73.000, china 2. State Key Laboratory of Numerical Modeling for atmospheric Science and Geophysical Fluid Dynamics (LASG), Institute of atmospheric Physics, Chinese Academy of sciences, Beijing 10002., china 1. college of atmospheric {2. Lanzhou University Lanzhou 730000 China 2. State Key Laboratory of Numerical Modeling for atmospheric Science and Geophysical Fluid Dynamics (LASG) Institute of Atmospheric Physics Chinese Academy of Sciences Beijing 100029 China

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, predictab

关键词： nonlinear, local, Lyapunov exponent, atmospheric predictability, maximal prediction time.

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

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Advances in atmospheric sciences 2019年第6期36卷 669-677页

作者： Xuan li ruiqiang {2. {2. li State Key Laboratory of Numerical Modeling for atmospheric {2. and Geophysical Fluid Dynamics (LASG) Institute of Atmospheric Physics Chinese Academy of Sciences college of Earth Science University of Chinese Academy of Sciences Laboratory for Regional Oceanography and Numerical Modeling Qingdao National Laboratory for Marine Science and Technology college of Global Change and Earth System sciences (GCESS) Beijing Normal University

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 ***, 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.

关键词： nonlinear local Lyapunov exponent forward and backward predictability limit negative correlation average value

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

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Advances in atmospheric sciences 2017年第9期34卷 1027-1034页

作者： ruiqiang ding {2. li Baosheng li State Key Laboratory of Numerical Modeling for atmospheric {2. and Geophysical Fluid Dynamics Institute of Atmospheric PhysicsChinese Academy of Sciences Plateau Atmosphere and Environment Key Laboratory of Sichuan Province Chengdu University of Information Technology college of Global Change and Earth System sciences Beijing Normal University University of Chinese Academy of sciences

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.

关键词： Lyapunov exponent nonlinear local Lyapunov exponent predictability

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Influence of the Preceding Austral Summer Southern Hemisphere Annular Mode on the Amplitude of ENSO Decay

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Advances in atmospheric sciences 2017年第11期34卷 1358-1379页

作者： Fei ZHENG jianping li {2. {2. State Key Laboratory of Numerical Modeling for atmospheric {2. and Geophysical Fluid Dynamics Institute of Atmospheric PhysicsChinese Academy of SciencesBeijing 100029China college of Global Change and Earth System Science Beijing Normal UniversityBeijing 100875China Laboratory for Regional Oceanography and Numerical Modeling Qingdao National Laboratory for Marine Science and Technologyqingdao 266237China Plateau Atmosphere and Environment Key Laboratory of Sichuan Province Chengdu University of Information Technologychengdu 610103China

There is increasing evidence of the possible role of extratropical forcing in the evolution of ENSO. The Southern Hemi- sphere Annular Mode （SAM） is the dominant mode of atmospheric circulation in the Southern Hemisphere extratropics. This study shows that the austral summer （December-January-February; DJF） SAM may also influence the amplitude of ENSO decay during austral autumn （March-April-May; MAM）. The mechanisms associated with this SAM-ENSO relationship can be briefly summarized as follows： The SAM is positively （negatively） correlated with SST in the Southern Hemisphere middle （high） latitudes. This dipole-like SST anomaly pattern is referred to as the Southern Ocean Dipole （SOD）. The DJF SOD, caused by the DJF SAM, could persist until MAM and then influence atmospheric circulation, inclu{2. trade winds, over the Nifio3.4 area. Anomalous trade winds and SST anomalies over the Nifio3.4 area related to the DJF SAM are further developed through the Bjerkness feedback, which eventually results in a cooling （warming） over the Nifio3.4 area followed by the positive （negative） DJF SAM.

关键词： Southern Hemisphere Annular Mode ENSO Southern Ocean Dipole

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Comparison of Nonlinear Local Lyapunov Vectors with Bred Vectors, Random Perturbations and Ensemble Transform Kalman Filter Strategies in a Barotropic Model

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Advances in atmospheric sciences 2016年第9期33卷 1036-1046页

作者： Jie FENG ruiqiang ding {2. li Deqiang liU State Key Laboratory of Numerical Modeling for atmospheric {2. and Geophysical Fluid Dynamics Institute of Atmospheric PhysicsChinese Academy of SciencesBeijing 100029 college of Earth Science University of Chinese Academy of SciencesBeijing 100049 Global Systems Division Earth System Research Laboratory/Oceanic and Atmospheric Research/National Oceanic and Atmospheric AdministrationBoulderCO80305USA Plateau Atmosphere and Environment Key Laboratory of Sichuan Province Chengdu University of Information TechnologyChengdu 610225 college of Global Change and Earth System sciences Beijing Normal UniversityBeijing 100875 Joint Center for Global Change Studies Beijing 100875 Fujian Meteorological Observatory Fuzhou 350001

The breeding method has been widely used to generate ensemble perturbations in ensemble forecasting due to its simple concept and low computational cost. This method produces the fastest growing perturbation modes to catch the growing components in analysis errors. However, the bred vectors （BVs） are evolved on the same dynamical flow, which may increase the dependence of perturbations. In contrast, the nonlinear local Lyapunov vector （NLLV） scheme generates flow-dependent perturbations as in the breeding method, but regularly conducts the Gram-Schmidt reorthonormalization processes on the perturbations. The resulting NLLVs span the fast-growing perturbation subspace efficiently, and thus may grasp more com- ponents in analysis errors than the BVs. In this paper, the NLLVs are employed to generate initial ensemble perturbations in a barotropic quasi-geostrophic model. The performances of the ensemble forecasts of the NLLV method are systematically compared to those of the random pertur- bation （RP） technique, and the BV method, as well as its improved version--the ensemble transform Kalman filter （ETKF） method. The results demonstrate that the RP technique has the worst performance in ensemble forecasts, which indicates the importance of a flow-dependent initialization scheme. The ensemble perturbation subspaces of the NLLV and ETKF methods are preliminarily shown to catch similar components of analysis errors, which exceed that of the BVs. However, the NLLV scheme demonstrates slightly higher ensemble forecast skill than the ETKF scheme. In addition, the NLLV scheme involves a significantly simpler algorithm and less computation time than the ETKF method, and both demonstrate better ensemble forecast skill than the BV scheme.

关键词： ensemble forecasting bred vector nonlinear local Lyapunov vector ensemble transform Kalman filter

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Predictability of Ensemble Forecasting Estimated Using the Kullback–Leibler Divergence in the Lorenz Model

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Advances in atmospheric sciences 2019年第8期36卷 837-846页

作者： ruiqiang ding Baojia {2.U Bin GU {2. {2. Xuan {2. State Key Laboratory of Numerical Modeling for atmospheric {2. and Geophysical Fluid Dynamics Institute of Atmospheric PhysicsChinese Academy of SciencesBeijing 100029China college of Earth sciences University of Chinese Academy of SciencesBeijing 100049China Institute of Space Weather School of Math and StatisticsNanjing University of Information Science and TechnologyNanjing 210044China college of Physics and Optoelectronic Engineering Nanjing University of Information Science and TechnologyNanjing 210044China college of Global Change and Earth System sciences Beijing Normal UniversityBeijing 100875China

A new method to quantify the predictability limit of ensemble forecasting is presented using the Kullback–Leibler(KL)divergence(also called the relative entropy), which provides a measure of the difference between the probability distributions of ensemble forecasts and local reference(true) states. The KL divergence is applicable to a non-normal distribution of ensemble forecasts, which is a substantial improvement over the previous method using the ensemble spread. An example from the three-variable Lorenz model illustrates the effectiveness of the KL divergence, which can effectively quantify the predictability limit of ensemble forecasting. On this basis, the KL divergence is used to investigate the dependence of the predictability limit of ensemble forecasting on the initial states and the magnitude of initial errors. The local predictability limit of ensemble forecasting varies considerably with the initial states, as well as with the magnitude of initial errors. Further research is needed to examine the real-world applications of the KL divergence in measuring the predictability of ensemble weather forecasts.

关键词： predictability ensemble forecasting Kullback–Leibler divergence

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The Relationship between Deterministic and Ensemble Mean Forecast Errors Revealed by Global and Local Attractor Radii

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Advances in atmospheric sciences 2019年第3期36卷 271-278,339页

作者： Jie FENG jianping li Jing ZHANG Deqiang liU ruiqiang {2. School of Meteorology University of Oklahoma college of Global Change and Earth System Science (GCESS) Beijing Normal University Laboratory for Regional Oceanography and Numerical Modeling Qingdao National Laboratory for Marine Science and Technology Cooperative Institute for Research in the Atmosphere GSD/ESRL/OAR/NOAA Fujian Meteorological Observatory Wuyishan National Park Meteorological Observatory State Key Laboratory of Numerical Modeling for atmospheric {2. and Geophysical Fluid Dynamics Institute of Atmospheric Physics Chinese Academy of Sciences Plateau Atmosphere and Environment Key Laboratory of Sichuan Province Chengdu University of Information Technology

It has been demonstrated that ensemble mean forecasts, in the context of the sample mean, have higher forecasting skill than deterministic(or single) forecasts. However, few studies have focused on quantifying the relationship between their forecast errors, especially in individual prediction cases. Clarification of the characteristics of deterministic and ensemble mean forecasts from the perspective of attractors of dynamical systems has also rarely been involved. In this paper, two attractor statistics—namely, the global and local attractor radii(GAR and LAR, respectively)—are applied to reveal the relationship between deterministic and ensemble mean forecast errors. The practical forecast experiments are implemented in a perfect model scenario with the Lorenz96 model as the numerical results for verification. The sample mean errors of deterministic and ensemble mean forecasts can be expressed by GAR and LAR, respectively, and their ratio is found to approach2.(1/2. with lead time. Meanwhile, the LAR can provide the expected ratio of the ensemble mean and deterministic forecast errors in individual cases.

关键词： attractor radius ensemble forecasting ensemble mean forecast error saturation

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Relationships between the Extratropical ENSO Precursor and Leading Modes of atmospheric Variabi{2.ty in the Southern Hemisphere

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Advances in atmospheric sciences 2017年第3期34卷 360-370页

作者： Jianhuang QIN ruiqiang ding Zhiwei WU {2. li Sen ZHAO Key Laboratory of Meteorological Disaster of Ministry of Education and Earth System Modeling Center Nanjing University of Information Science and Technology State Key Laboratory of Numerical Modeling for atmospheric {2. and Geophysical Fluid Dynamics Institute of Atmospheric PhysicsChinese Academy of Sciences Plateau Atmosphere and Environment Key Laboratory of Sichuan Province Chengdu University of Information Technology Institute of atmospheric sciences Fudan University college of Global Change and Earth System sciences Beijing Normal University

Previous studies suggest that the atmospheric precursor of E1 Nifio-Southern Oscillation （ENSO） in the extratropical Southern Hemisphere （SH） might trigger a quadrapole sea surface temperature anomaly （SSTA） in the South Pacific and subsequently influence the following ENSO. Such a quadrapole SSTA is referred to as the South Pacific quadrapole （SPQ）. The present study investigated the relationships between the atmospheric precursor signal of ENSO and lea{2. modes of atmospheric variability in the extratropical SH [inclu{2. the SH annular mode （SAM）, the first Pacific-South America （PSA1） mode, and the second Pacific-South America （PSA2. mode]. The results showed that the atmospheric precursor signal in the extratropical SH basically exhibits a barotropic wavenumber-3 structure over the South Pacific and is significantly correlated with the SAM and the PSA2.mode during austral summer. Nevertheless, only the PSA2.mode was found to be a precursor for the following ENSO. It leads the SPQ-like SSTA by around one month, while the SAM and the PSA1 mode do not show any obvious linkage with either ENSO or the SPQ. This suggests that the PSA2.mode may provide a bridge between the prece{2. circulation anomalies over the extratropical SH and the following ENSO through the SPQ-like SSTA.

关键词： Pacific-South America atmospheric variability ENSO South Pacific quadrapole

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

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Advances in atmospheric sciences 2018年第12期35卷 1491-1504页

作者： Quanjia ZHONG lifeng ZHANG jianping li ruiqiang {2. Jie FENG State Key Laboratory of Numerical Modeling for atmospheric {2. and Geophysical Fluid Dynamics Institute of Atmospheric Physics Chinese Academy of Sciences college of Meteorology and Oceanography National University of Defense Technology college of Earth Science University of Chinese Academy of Sciences Laboratory for Regional Oceanography and Numerical Modeling Qingdao National Laboratory for Marine Science and Technology college of Global Change and Earth System sciences Beijing Normal University School of Meteorology University of Oklahoma

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 *** 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 *** predictability limits of the TC tracks for short-,medium-,and long-lived TCs are approximately 72.h,12. h,and 132.h,*** 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 *** 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 ***,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 *** work is needed,employing a numerical model to assess the predictability of TC tracks.

关键词： predictability tropical cyclone tracks nonlinear local Lyapunov exponent

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