Applications of conditional nonlinear optimal perturbation to the study of the stability and sensitivity of the Jovian atmosphere

A two-layer quasi-geostrophic model is used to study the stability and sensitivity of motions on smallscale vortices in Jupiter’s atmosphere. Conditional nonlinear optimal perturbations (CNOPs) and linear singular vectors (LSVs) are both obtained numerically and compared in this paper. The results show that CNOPs can capture the nonlinear characteristics of motions in small-scale vortices in Jupiter’s atmosphere and show great difference from LSVs under the condition that the initial constraint condition is large or the optimization time is not very short or both. Besides, in some basic states, local CNOPs are found. The pattern of LSV is more similar to local CNOP than global CNOP in some cases. The elementary application of the method of CNOP to the Jovian atmosphere helps us to explore the stability of variousscale motions of Jupiter’s atmosphere and to compare the stability of motions in Jupiter’s atmosphere and Earth’s atmosphere further.     

Nonlinearity and Finite-Time Instability in a T21L3 Quasigeostrophic Model

In this paper,a nonlinear optimization method is used to explore the finite-time instability of the atmospheric circulation with a three-level quasigeostrophic model under the framework of the conditional nonlinear optimal perturbation (CNOP).As a natural generalization of linear singular vector (SV),CNOP is defined as an initial perturbation that makes the cost function the maximum at a prescribed forecast time under certain physical constraint conditions.Special attentions are paid to the different structures and energy evolutions of the optimal perturbations.The results show that the most instable region of the global atmospheric circulation lies in the midlatitude Eurasian continent.More specially,SV and CNOP in the total energy norm with an optimization time of 2 days both present localness:they are mainly located in the midlatitude Asian continent and its east coast.With extension of the optimization time,SVs are more upstream and less localized in the zonal direction,and CNOPs differ essentially from SVs with broader zonal and meridional coverages; as a result,CNOPs acquire larger kinetic and available potential energy amplifications than SVs in the nonlinear model at the corresponding optimization time.For the climatological wintertime flow,it is seen that the baroclinic terms remain small over the entire time evolution,and the energy production comes essentially from the eddy kinetic energy,which is induced by the horizontal shear of the basic flow.In addition,the effects of SVs and CNOPs on the Eurasian atmospheric circulation are explored.The results show that the weather systems over the Eurasian continent in the perturbed fields by CNOPs are stronger than those by SVs at the optimization time.This reveals that the CNOP method is better in evaluating the instability of the atmospheric circulation while the SV method underestimates the possibility of extreme weather events.     

Algorithm Studies on How to Obtain a Conditional Nonlinear Optimal Perturbation (CNOP)

The conditional nonlinear optimal perturbation (CNOP), which is a nonlinear generalization of the linear singular vector (LSV), is applied in important problems of atmospheric and oceanic sciences, including ENSO predictability, targeted observations, and ensemble forecast. In this study, we investigate the computational cost of obtaining the CNOP by several methods. Differences and similarities, in terms of the computational error and cost in obtaining the CNOP, are compared among the sequential quadratic programming (SQP) algorithm, the limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm, and the spectral projected gradients (SPG2) algorithm. A theoretical grassland ecosystem model and the classical Lorenz model are used as examples. Numerical results demonstrate that the computational error is acceptable with all three algorithms. The computational cost to obtain the CNOP is reduced by using the SQP algorithm. The experimental results also reveal that the L-BFGS algorithm is the most effective algorithm among the three optimization algorithms for obtaining the CNOP. The numerical results suggest a new approach and algorithm for obtaining the CNOP for a large-scale optimization problem.     

A Variant Constrained Genetic Algorithm for Solving Conditional Nonlinear Optimal Perturbations简

A variant constrained genetic algorithm （VCGA） for effective tracking of conditional nonlinear optimal perturbations （CNOPs） is presented. Compared with traditional constraint handling methods, the treatment of the constraint condition in VCGA is relatively easy to implement. Moreover, it does not require adjustments to indefinite pararneters. Using a hybrid crossover operator and the newly developed multi-ply mutation operator, VCGA improves the performance of GAs. To demonstrate the capability of VCGA to catch CNOPS in non-smooth cases, a partial differential equation, which has ＂on off＂ switches in its forcing term, is employed as the nonlinear model. To search global CNOPs of the nonlinear model, numerical experiments using VCGA, the traditional gradient descent algorithm based on the adjoint method （ADJ）, and a GA using tournament selection operation and the niching technique （GA-DEB） were performed. The results with various initial reference states showed that, in smooth cases, all three optimization methods are able to catch global CNOPs. Nevertheless, in non-smooth situations, a large proportion of CNOPs captured by the ADJ are local. Compared with ADJ, the performance of GA-DEB shows considerable improvement, but it is far below VCGA. Further, the impacts of population sizes on both VCGA and GA-DEB were investigated. The results were used to estimate the computation time of ~CGA and GA-DEB in obtaining CNOPs. The computational costs for VCGA, GA-DEB and ADJ to catch CNOPs of the nonlinear model are also compared.     

条件非线性最优扰动方法在适应性观测研究中的初步应用

针对适应性观测中敏感性区域的确定问题，考虑初始误差对预报结果的影响, 比较了条件非线性最优扰动(CNOP)与第一线性奇异向量(FSV)在两个降水个例中的空间结构的差异，考察了它们总能量范数随时间发展演变的异同。结合敏感性试验的分析，揭示了预报结果对CNOP类型的初始误差的敏感性要大于对FSV类型的初始误差的敏感性，因而减少初值中CNOP类型误差的振幅比减少FSV类型的收益要大。这一结果表明可以把CNOP方法应用于适应性观测来识别大气的敏感区。关于将CNOP方法有效地应用于适应性观测所面临的挑战及需要采取的对策等也进行了讨论。     

Nonlinearity and finite-time instability in a T21L3 quasigeostrophic model

In this paper, a nonlinear optimization method is used to explore the finite-time instability of the atmospheric circulation with a three-level quasigeostrophic model under the framework of the conditional nonlinear optimal perturbation (CNOP). As a natural generalization of linear singular vector (SV), CNOP is defined as an initial perturbation that makes the cost function the maximum at a prescribed forecast time under certain physical constraint conditions. Special attentions are paid to the different structures and energy evolutions of the optimal perturbations.     

A comparison study of the methods of conditional nonlinear optimal perturbations and singular vectors in ensemble prediction

The authors apply the technique of conditional nonlinear optimal perturbations (CNOPs) as a means of providing initial perturbations for ensemble forecasting by using a barotropic quasi-geostrophic (QG) model in a perfect-model scenario. Ensemble forecasts for the medium range (14 days) are made from the initial states perturbed by CNOPs and singular vectors (SVs). 13 different cases have been chosen when analysis error is a kind of fast growing error. Our experiments show that the introduction of CNOP provides better forecast skill than the SV method. Moreover, the spread-skill relationship reveals that the ensemble samples in which the first SV is replaced by CNOP appear superior to those obtained by SVs from day 6 to day 14. Rank diagrams are adopted to compare the new method with the SV approach. The results illustrate that the introduction of CNOP has higher reliability for medium-range ensemble forecasts.     

INTELLIGENT ALGORITHMS FOR SOLVING CNOP AND THEIR APPLICATIONS IN ENSO PREDICTABILITY AND TROPICAL CYCLONE ADAPTIVE OBSERVATIONS

Some intelligent algorithms (IAs) proposed by us, including swarm IAs and single individual IAs, have been applied to the Zebiak-Cane (ZC) model to solve conditional nonlinear optimal perturbation (CNOP) for studying El Ni?o – Southern Oscillation (ENSO) predictability. Compared to the adjoint-based method (the ADJ-method), which is referred to as a benchmark, these IAs can achieve approximate CNOP results in terms of magnitudes and patterns. Using IAs to solve CNOP can avoid the use of an adjoint model and widen the application of CNOP in numerical climate and weather modeling. Of the proposed swarm IAs, PCA-based particle swarm optimization (PPSO) obtains CNOPs with the best patterns and the best stability. Of the proposed single individual IAs, continuous tabu search algorithm with sine maps and staged strategy (CTS-SS) has the highest efficiency. In this paper, we compare the validity, stability and efficiency of parallel PPSO and CTS-SS using these two IAs to solve CNOP in the ZC model for studying ENSO predictability. The experimental results show that CTS-SS outperforms parallel PPSO except with respect to stability. At the same time, we are also concerned with whether these two IAs can effectively solve CNOP when applied to more complicated models. Taking the sensitive areas identification of tropical cyclone adaptive observations as an example and using the fifth-generation mesoscale model (MM5), we design some experiments. The experimental results demonstrate that each of these two IAs can effectively solve CNOP and that parallel PPSO has a higher efficiency than CTS-SS. We also provide some suggestions on how to choose a suitable IA to solve CNOP for different models.     

遗传算法求解最大预报误差的可行性研究

利用条件非线性最优扰动（conditional nonlinear optimal perturbation,CNOP）可以实现最大预报误差的上界估计。CNOP通常由基于梯度信息的约束优化算法进行求解,且其中的梯度信息由伴随模式提供。然而当非线性模式中含不连续＂开关＂时,传统伴随方法不能为优化过程提供正确的梯度方向,从而导致优化失败。为此,采用自适应变异和混合交叉的遗传算法,联赛选择机制和小生境技术的约束处理方法来求解最大预报误差上界。为检验新方法的有效性,以修改的Lorenz模型作为预报模式,对3个初始态分别用新方法和传统伴随方法进行比较,数值试验结果显示新方法求解出的最大预报误差的上界更加精确。     

THE EFFECTIVENESS OF GENETIC ALGORITHM IN CAPTURING CONDITIONAL NONLINEAR OPTIMAL PERTURBATION WITH PARAMETERIZATION "ON-OFF" SWITCHES INCLUDED BY A MODEL

In the typhoon adaptive observation based on conditional nonlinear optimal perturbation (CNOP), the ‘on-off’ switch caused by moist physical parameterization in prediction models prevents the conventional adjoint method from providing correct gradient during the optimization process. To address this problem, the capture of CNOP, when the on-off switches are included in models, is treated as non-smooth optimization in this study, and the genetic algorithm (GA) is introduced. After detailed algorithm procedur...     

条件非线性最优扰动方法在黑潮目标观测研究中的应用

对近年来利用条件非线性最优扰动（Conditional Nonlinear Optimal Perturbation,CNOP）方法开展的黑潮目标观测研究进行了总结,主要包括日本南部黑潮路径变异的目标观测研究、黑潮延伸体模态转变的目标观测研究和源区黑潮流量变化的目标观测研究。通过计算这些事件的CNOP型扰动,发现这些事件的CNOP型扰动具有局地特征,可以作为实施目标观测的敏感区。理想回报试验结果表明,如果在由CNOP方法识别的敏感区内实施目标观测,则会大幅度提高上述事件的预报技巧。     

On the Application of a Genetic Algorithm to the Predictability Problems Involving "On-Off" Switches

The lower bound of maximum predictable time can be formulated into a constrained nonlinear opti- mization problem, and the traditional solutions to this problem are the filtering method and the conditional nonlinear optimal perturbation (CNOP) method. Usually, the CNOP method is implemented with the help of a gradient descent algorithm based on the adjoint method, which is named the ADJ-CNOP. However, with the increasing improvement of actual prediction models, more and more physical processes are taken into consideration in models in the form of parameterization, thus giving rise to the on-off switch problem, which tremendously affects the effectiveness of the conventional gradient descent algorithm based on the ad- joint method. In this study, we attempted to apply a genetic algorithm (GA) to the CNOP method, named GA-CNOP, to solve the predictability problems involving on-off switches. As the precision of the filtering method depends uniquely on the division of the constraint region, its results were taken as benchmarks, and a series of comparisons between the ADJ-CNOP and the GA-CNOP were performed for the modified Lorenz equation. Results show that the GA-CNOP can always determine the accurate lower bound of maximum predictable time, even in non-smooth cases, while the ADJ-CNOP, owing to the effect of on-off switches, often yields the incorrect lower bound of maximum predictable time. Therefore, in non-smooth cases, using GAs to solve predictability problems is more effective than using the conventional optimization algorithm based on gradients, as long as genetic operators in GAs are properly configured.     

The Sensitive Regions Identified by CNOPs of Three Typhoon Events

In this paper, several sets of observing system simulation experiments (OSSEs) were designed for three typhoon cases to determine whether or not the additional observation data in the sensitive regions identified by conditional nonlinear optimal perturbations (CNOPs) could improve the short-range forecast of typhoons. The results show that the CNOPs capture the sensitive regions for typhoon forecasts, which implies that conducting additional observation in these specific regions and eliminating initial errors could reduce forecast errors. It is inferred from the results that dropping sondes in the CNOP sensitive regions could lead to improvements in typhoon forecasts.     

Initial Error-induced Optimal Perturbations in ENSO Predictions,as Derived from an Intermediate Coupled Model

The initial errors constitute one of the main limiting factors in the ability to predict the El Nio–Southern Oscillation(ENSO) in ocean–atmosphere coupled models. The conditional nonlinear optimal perturbation(CNOP) approach was employed to study the largest initial error growth in the El Nio predictions of an intermediate coupled model(ICM). The optimal initial errors(as represented by CNOPs) in sea surface temperature anomalies(SSTAs) and sea level anomalies(SLAs) were obtained with seasonal variation. The CNOP-induced perturbations, which tend to evolve into the La Nia mode, were found to have the same dynamics as ENSO itself. This indicates that, if CNOP-type errors are present in the initial conditions used to make a prediction of El Nio, the El Nio event tends to be under-predicted. In particular, compared with other seasonal CNOPs, the CNOPs in winter can induce the largest error growth, which gives rise to an ENSO amplitude that is hardly ever predicted accurately. Additionally, it was found that the CNOP-induced perturbations exhibit a strong spring predictability barrier(SPB) phenomenon for ENSO prediction. These results offer a way to enhance ICM prediction skill and, particularly,weaken the SPB phenomenon by filtering the CNOP-type errors in the initial state. The characteristic distributions of the CNOPs derived from the ICM also provide useful information for targeted observations through data assimilation. Given the fact that the derived CNOPs are season-dependent, it is suggested that seasonally varying targeted observations should be implemented to accurately predict ENSO events.     

关于线性奇异向量和条件非线性最优扰动差别的一个注记

奇异向量(singular vectors,SVs)和条件非线性最优扰动(conditional nonlinear optimal perturbation,CNOP)已广泛应用于研究大气—海洋系统的不稳定性以及与其相关的可预报性、集合预报和目标观测问题研究。本文首先回顾了SVs和CNOP的发展历史,并简单描述了它们的基本原理;然后针对二维正压准地转模式,使用不同的范数组合,分析了第一线性奇异向量(first singular vector,FSV)和CNOP之间的异同。结果表明,当优化时间较短时,度量SVs和CNOP大小的范数不同也将导致FSV和CNOP相差很大,而当度量SVs和CNOP大小的范数相同时,FSV和CNOP之间的差别则主要是由非线性物理过程作用的结果。因此,针对不同的物理问题,应该选取合适的度量范数研究FSV和CNOP以及其所引起的大气或海洋动力学的异同,从而揭示非线性物理过程的影响机理。     

Behaviors of nonlinearities modulating the El Niño events induced by optimal precursory disturbances

In the present paper, we explore the manner in which nonlinearities modulate El Niño events by investigating the optimal precursory disturbance for El Niño events in the Zebiak-Cane model. The initial anomalies of conditional nonlinear optimal perturbations (CNOPs) and linear singular vectors (LSVs) are investigated. The CNOPs evolve into stronger El Niño events than the LSVs and act as the optimal precursor for El Niño events. By examining the role of nonlinearities in El Niño events induced by CNOPs and LSVs, we determined that, when the initial anomalies of the CNOP and LSV structures are large, the nonlinearities enhance CNOP-El Niño events but suppress LSV-El Niño events. Nonlinearities in the Zebiak-Cane model arise from nonlinear temperature advection (NTA), sub-surface temperature parameterization (STP), and wind stress anomalies (WSA). Using these types of nonlinearities, we trace the approach of the nonlinearities modulating the CNOP- and LSV-El Niño events. The results demonstrate that nonlinearities that originate from NTA enhance both CNOP-El Niño events and LSV-El Niño events, while nonlinearities originating from STP and WSA suppress these events. For the CNOP-El Niño events, the enhancement effect of NTA is larger than the suppression effect of STP and WSA, resulting in the combined effect of the nonlinearities in the Zebiak-Cane model being an enhancement of the CNOP-El Niño events. However, for the LSV-El Niño events, the enhancement effect of NTA is smaller than the suppression effect of WSA and STP. Consequently, the combined effect of the nonlinearities in the Zebiak-Cane model suppresses the LSV-El Niño events.     

CONDITIONAL NONLINEAR OPTIMAL PERTURBATIONS: APPLICATION IN TROPICAL CYCLONE FORECASTS

Considering the feature of tropical cyclones (TCs) that strong positive vorticity exists in the lower layers of troposphere, this study proposed to use vorticity at 850 hPa as cost function to find the conditional nonlinear optimal perturbation (CNOP), which was largely different from those previous studies using total energy of perturbed forecast variables. The CNOP was obtained by an ensemble-based approach. All of the sensitive areas determined by CNOP with vorticity at 850 hPa as cost function for the three cases were located over the TC core region and its vicinity. The impact of the CNOP-based adaptive observations on TC forecasts was evaluated with three cases via observational system simulation experiments (OSSEs). Results showed obvious improvements in TC intensity or track forecasts due to the CNOP-based adaptive observations, which were related to the main error source of the verification area, i.e., intensity error or location error.     

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.     

Exploring the phase-strength asymmetry of the North Atlantic Oscillation using conditional nonlinear optimal perturbation

Negative-phase North Atlantic Oscillation(NAO) events are generally stronger than positive-phase ones, i.e., there is a phase-strength asymmetry of the NAO. In this work, we explore this asymmetry of the NAO using the conditional nonlinear optimal perturbation(CNOP) method with a three-level global quasi-geostrophic spectral model. It is shown that, with winter climatological flow forcing, the CNOP method identifies the perturbations triggering the strongest NAO event under a given initial constraint. Meanwhile, the phase-strength asymmetry characteristics of the NAO can be revealed. By comparing with linear results, we find that the process of perturbation self-interaction promotes the onset of negative NAO events, which is much stronger than during positive NAO onset. Results are obtained separately using the climatological and zonal-mean flows in boreal winter(December–February) 1979–2006 as the initial basic state. We conclude, based on the fact that NAO onset is a nonlinear initial-value problem, that phase-strength asymmetry is an intrinsic characteristic of the NAO.     

基于条件非线性最优扰动的目标观测中瞄准区 不同引导性变量的影响试验研究

基于GRAPES区域业务预报模式,采用一种快速算法计算出来的条件非线性最优扰动对实际台风个例麦莎(No.0509)开展了目标观测研究,应用数值模式,进行一系列的敏感性试验,讨论了与目标观测设计相关的一些问题,包括确定瞄准区时使用不同的引导性变量对目标观测效果的影响、及瞄准区范围变化对预报效果的影响。文中分别以提高麦莎在检验区(20.125°—35.3125°N,116.8125°—129.75°E)内的24 h海平面气压预报和24 h累积降水量预报为目的,基于条件非线性最优扰动使用了3种不同的引导性变量寻找敏感区(又称瞄准区),对这些敏感区的分布特点和有效性进行了比较和讨论。试验结果表明,在使用的3种引导性变量中,用不同的引导性变量识别的敏感区是有差别的,总体上说,文中使用的3种引导性变量识别的瞄准区对提高预报都是有效的,特别是第2和第3种的效果更好些,且两者识别的瞄准区常显示出类似的特点。文中进一步针对检验区内24 h累积降水量预报误差问题,将前面确定的瞄准区范围扩大相同的幅度,讨论瞄准区范围变化对改进预报的影响。试验结果表明,增加瞄准格点数,有可能使预报效果得到改善,但是试验结果同时也暗示了单纯靠扩大瞄准...