Nonlinear Optimal Perturbation Research Articles

Verification of Long‐Term Ensemble Evapotranspiration Hindcast Using a Conditional Nonlinear Optimal Parameter Perturbation Ensemble Prediction Method on the Tibetan Plateau

AbstractIn this research, a conditional nonlinear optimal perturbation related to parameters (CNOP‐P) method is employed to propose an ensemble prediction method titled as the conditional nonlinear optimal perturbation related to parameters ensemble prediction (CNOP‐PEP) method. Within the CNOP‐PEP method, all ensemble members are generated by the CNOP‐P method. To explore the operability and validity of the CNOP‐PEP method, long‐term evapotranspiration (ET) hindcast skill is evaluated at 13 sites on the Tibetan Plateau (TP) during the period of 2001–2018 with a land surface model. Two traditional ensemble prediction methods (the stochastically perturbed parameters (SPP) method and the one‐at‐a‐time (OAT) method) are also applied to assess the ET hindcast skills. The numerical results indicate that the CNOP‐PEP method shows the best hindcast skill for estimated ET at 13 stations on the TP among the three methods (CNOP‐PEP, OAT, and SPP methods) during the period of 2001–2018. This suggests that the CNOP‐PEP method is helpful and effective for long‐term and interannual hydrological predictions, such as long‐term and interannual ET predictions. The numerical results also indicate that ensemble members with certain special properties of fast‐growing types should be selected to implement ensemble prediction compared to these ordinary and traditional samples. The CNOP‐PEP method could supply special samples to improve the ensemble prediction and hindcast skill of hydrological cycle.

Role of nonlinear transmission term in stability of hand, foot and mouth disease dynamics

Aim: This study focuses on the nonlinear characteristics of two linearly stable equilibrium states within a theoretical hand, foot and mouth disease (HFMD) dynamics model. Methods: The conditional nonlinear optimal perturbation method, the Lyapunov method and the linear singular vector method are employed. Results: By utilizing the conditional nonlinear optimal perturbation method, it is determined that the two linearly stable equilibrium states (namely the free disease and disease HFMD states) also exhibit nonlinear stability. It is found that the stabilities of the free disease and disease HFMD equilibrium states differ in terms of returning to their original states. Conclusion: It is discovered that nonlinearity plays a crucial role in reinstating the original equilibrium states.

The sampling method for optimal precursors of El Niño–Southern Oscillation events

Abstract. The El Niño–Southern Oscillation (ENSO) is a significant climate phenomenon that appears periodically in the tropical Pacific. The intermediate coupled ocean–atmosphere Zebiak–Cane (ZC) model is the first and classical one designed to numerically forecast the ENSO events. Traditionally, the conditional nonlinear optimal perturbation (CNOP) approach has been used to capture optimal precursors in practice. In this paper, based on state-of-the-art statistical machine learning techniques1, we investigate the sampling algorithm proposed in Shi and Sun (2023) to obtain optimal precursors via the CNOP approach in the ZC model. For the ZC model, or more generally, the numerical models with a large number O(104−105) of degrees of freedom, the numerical performance, regardless of the statically spatial patterns and the dynamical nonlinear time evolution behaviors as well as the corresponding quantities and indices, shows the high efficiency of the sampling method compared to the traditional adjoint method. The sampling algorithm does not only reduce the gradient (first-order information) to the objective function value (zeroth-order information) but also avoids the use of the adjoint model, which is hard to develop in the coupled ocean–atmosphere models and the parameterization models. In addition, based on the key characteristic that the samples are independently and identically distributed, we can implement the sampling algorithm by parallel computation to shorten the computation time. Meanwhile, we also show in the numerical experiments that the important features of optimal precursors can still be captured even when the number of samples is reduced sharply.

Outlet boundary condition and mean temperature gradient effects on the minimum acoustics disturbances energy in triggering nonlinear thermoacoustic instability

In this study, we theoretically investigate the impact of outlet boundary conditions and mean temperature gradients on the maximum transient growth rate of acoustical energy and the critical energy required for triggering. Our analysis encompasses open–open and open–closed thermoacoustic systems. The theoretical models developed focus on horizontal ducts with a mean temperature jump over the heat source, employing the modified King's law. By linearizing the unsteady heat release, the nonlinear thermoacoustic equations transform into linearized-delay ones. This approach enables us to predict optimal initial perturbations for linearized-delay and nonlinear systems, corresponding to maximum transient growth rates of acoustic energy over short and long periods, respectively, thus providing insights into critical energy for triggering. We find that a closed outlet leads to higher transient energy growth and a lower critical energy for triggering compared to an open outlet. The increased mean temperature gradient has a “destructive” impact on triggering in open–open systems but a “constructive” effect in open–closed systems. Raising the mean temperature ratio generally increases the critical energy for triggering in the open–open system, whereas it decreases the critical energy in the open–closed system. The critical energy for nonlinear optimal initial perturbations is notably affected by the minimum energy of critical unstable periodic solutions, while the critical energy for linearized-delay optimal initial perturbations is closely tied to the energy level of stable periodic solutions. Due to the transient energy growth rate, the critical energy for nonlinear optimal initial perturbations is significantly lower than that for linearized-delay optimal initial perturbations.

Targeted observation for the climatology temperature in the Kuroshio region based on the CNOP approach

To reduce the cost of ocean observations and improve prediction accuracy of the Kuroshio region temperature, this study investigates the related targeted observation by using the conditional nonlinear optimal perturbation (CNOP) approach. Results show that the scheme of vertical-integrated energy is more suitable for the identification of sensitive area in the related targeted observation. By conducting a set of observation system simulation experiments (OSSEs), we discovered that the sensitive areas identified by the CNOP exert substantial influence on temperature predictions within the target area. The dynamic diagnosis further indicated that the pressure gradient and Coriolis force in the momentum equations greatly contribute the development of the prediction biases. These findings implied that the implement of CNOP-based targeted observation represents a cost-effective strategy for enhancing temperature predictions in the Kuroshio region.

Coupled conditional nonlinear optimal perturbations and their application to ENSO ensemble forecasts

Coupled conditional nonlinear optimal perturbations and their application to ENSO ensemble forecasts

An Ensemble CNOP Method Based on a Pre-Screening Mechanism for Targeted Observations in the South China Sea

The uncertainty in the initial condition seriously affects the forecasting skill of numerical models. Targeted observations play an important role in reducing uncertainty in numerical prediction. The conditional nonlinear optimal perturbation (CNOP) method is a useful tool for studying adaptive observation. However, the traditional CNOP method highly relies on the adjoint model, and it is difficult to find the global optimal solution. In this paper, a pre-screening and ensemble CNOP hybrid method called PECNOP is proposed to identify optimal sensitive areas in targeted observations. PECNOP is an adjoint-free method that captures global CNOP with high probability, which can effectively solve the two major problems faced by traditional CNOP methods. We evaluated the performance of PECNOP by building an observation simulation system consisting of an ocean model and data assimilation. One of the assimilation experiments was dedicated to evaluating the stability and effectiveness of PECNOP in extreme events. The results show that, compared with traditional methods, PECNOP can stably capture the global CNOP. Extra observations and assimilation in the optimal sensitive areas identified by PECNOP can effectively improve forecasting by about 20% within 30 days. Therefore, PECNOP has potential to reduce the initial error of numerical models, which is important for improving forecasting.

Impact of Uncertainties in Land Surface Processes on Subseasonal Predictability of Heat Waves Onset Over the Yangtze River Valley

AbstractLand surface processes are strongly associated with heat waves (HWs). However, how the uncertainties in land surface processes owing to inaccurate physical parameters influence subseasonal HW predictions has rarely been explored. To examine the impact of parameter errors of land surface processes on the uncertainty of subseasonal HW predictions, five strong and long‐lasting HW events over the middle and lower reaches of the Yangtze River are investigated. Based on the Weather Research and Forecasting model, the conditional nonlinear optimal perturbation related to parameters (CNOP‐P) approach is employed to address the aforementioned issues. Numerical results demonstrate that the CNOP‐P type errors of physical parameters cause large prediction errors for five HW event onsets. Two types of CNOP‐Ps are obtained for HW events, called the type‐1 CNOP‐P and the type‐2 CNOP‐P. The type‐1 (type‐2) CNOP‐P causes an approximately 3°C (2°C) warm (cold) bias during the HW period. Surface sensible and latent heat flux errors, especially flux exchange between vegetation canopy and canopy air, provide considerable uncertainty in subseasonal HW predictions. The type‐1 (type‐2) CNOP‐P exhibits an underestimation (overestimation) of transpiration. Furthermore, it should be noted that the type‐1 CNOP‐P results in a substantial difference in soil moisture, a phenomenon that is demonstrated to be challenging to observe in the type‐2 CNOP‐P. The results indicate that understanding vegetation‐atmosphere dynamics is crucial for improving subseasonal HW predictions. Jointly lowering soil‒atmosphere and vegetation‐atmosphere uncertainty can notably improve subseasonal HW prediction skills.

Effects of Wind Stress Uncertainty on Short-Term Prediction of the Kuroshio Extension State Transition Process

Abstract Based on the conditional nonlinear optimal perturbation for boundary condition method and Regional Ocean Modeling System (ROMS), this study investigates the influence of wind stress uncertainty on predicting the short-term state transitions of the Kuroshio Extension (KE). The optimal time-dependent wind stress errors that lead to maximum prediction errors are obtained for two KE stable-to-unstable and two reverse transitions, which exhibit local multieddies structures with decreasing magnitude as the end time of prediction approaches. The optimal boundary errors initially induce small oceanic errors through Ekman pumping. Subsequently, these errors grow in magnitude as oceanic internal processes take effect, which exerts significant influences on the short-term prediction of the KE state transition process. Specifically, during stable-to-unstable (unstable-to-stable) transitions, the growing error induces an overestimation (underestimation) of the meridional sea surface height gradient across the KE axis, leading to the predicted KE state being more (less) stable. Furthermore, the dynamics mechanism analysis indicates that barotropic instability is crucial for the error growth in the prediction of both the stable-to-unstable and the reverse transition processes due to the horizontal shear of flow field. But work generated by wind stress error plays a more important role in the prediction of the unstable-to-stable transitions because of the synergistic effect of strong wind stress error and strong oceanic error. Eventually, the sensitive areas have been identified based on the optimal boundary errors. Reducing wind stress errors in sensitive areas can significantly improve prediction skills, offering theoretical guidance for devising observational strategies.

Targeted Observations on Arctic Sea Ice Concentration for Improving Extended‐Range Prediction of Ural Blocking

AbstractThe predictability of certain extreme weather events can exceed the traditional 2 weeks by considering the boundary conditions. Targeted observations in sensitive areas (SA) on Arctic sea ice concentration (SIC) can improve the extended‐range (4 pentads) forecast skills of long‐lasting and strong Ural blocking (UB). The SA are determined based on the SIC optimally growing boundary errors, obtained by the conditional nonlinear optimal perturbation method. The SA are mainly located in the Barents Sea, Greenland Sea, and Okhotsk Sea. The results of observing system simulation experiments for 8 UB cases indicate that the targeted observations can remarkably improve the prediction skills of UB in the fourth pentad. Targeted observations have a positive effect on 75% of 160 experiment members, reduce 35% forecast errors of the fourth pentad mean blocking index, and perform even better when the original forecast errors are greater. Further diagnosis shows that targeted observations contribute to more accurate SIC boundary conditions in the Barents Sea, Greenland Sea, and Okhotsk Sea and reduce temperature errors in the lower and middle troposphere. It further results in well‐described westerly winds in the Ural region and its adjacent regions, corresponding to the more skillful predictions of blocking circulations. The above results supply a theoretical base for the design of Arctic SIC observations and more skillful extended‐range predictions for mid‐latitude extreme weather.

Using the Orthogonal Conditional Nonlinear Optimal Perturbations Approach to Address the Uncertainties of Tropical Cyclone Track Forecasts Generated by the WRF Model

Abstract The orthogonal conditional nonlinear optimal perturbations (O-CNOPs) approach for measuring initial uncertainties is applied to the Weather Research and Forecasting (WRF) Model to provide skillful forecasts of tropical cyclone (TC) tracks. The hindcasts for 10 TCs selected from 2005 to 2020 show that the ensembles generated by the O-CNOPs have a greater probability of capturing the true TC tracks, and the corresponding ensemble forecasts significantly outperform the forecasts made by the singular vectors, bred vectors, and random perturbations in terms of both deterministic and probabilistic skills. In particular, for two unusual TCs, Megi (2010) and Tembin (2012), the ensembles generated by the O-CNOPs successfully reproduce the sharp northward-turning track in the former and the counterclockwise loop track in the latter, while the ensembles generated by the other methods fail to do so. Moreover, additional attempts are performed on the real-time forecasts of TCs In-Fa (2021) and Hinnamnor (2022), and it is shown that O-CNOPs are very useful for improving the accuracy of real-time TC track forecasts. Therefore, O-CNOPs, together with the WRF Model, could provide a new platform for the ensemble forecasting of TC tracks with much higher skill.

Nonlinear optimal perturbation of turbulent channel flow as a precursor of extreme events

This work aims at studying the mechanisms behind the occurrence of extreme dissipation events in a channel flow, identifying nonlinear optimal perturbations as potential precursors of these events. Nonlinear optimal perturbations with respect to a generic turbulent instantaneous snapshot are computed for the first time using a direct-adjoint algorithm in the channel flow at $Re_{\tau }\approx 180$ . The resulting initial perturbation displays the upstream tilting characteristic of Orr's mechanism and is positioned along the interfaces between two opposite-sign velocity streaks of the pre-existing turbulent field. Such a perturbation induces a sudden breakdown of the pre-existing structures and a heavier tail in the dissipation probability density function distribution. Different mechanisms are at play during this process: the high shear present at the interface between coherent low- and high-momentum regions is exploited to break down the larger structures and drive energy to small scales. This energy cascade is fed by an enhanced lift-up effect that produces intense streaks near the wall. It is found that the optimal perturbation grows exponentially during the first phase of its evolution reflecting the existence of a secondary modal instability of the streaks. To corroborate the results, the conditional spatiotemporal proper orthogonal decomposition (POD) analysis of Hack & Schimdt (J. Fluid Mech., vol. 907, 2021, A9) is performed both in the perturbed and in the unperturbed flow, showing a clear agreement between the two cases and with the reference study. Thus, the optimal perturbation at initial time can be considered as a precursor of extreme events.

Optimal initial errors related to the prediction of the vertical thermal structure and their application to targeted observation: A 3-day hindcast case study in the northern South China Sea

This 3-day hindcast case study focuses on the effects of the initial errors, a non-negligible factor in oceanic prediction, on VTS prediction in the NSCS and provides scientific guidance on observation design through targeted observation. Based on the Conditional Nonlinear Optimal Perturbation (CNOP) approach and the Regional Ocean Modeling System, the impact of optimal initial errors (OIEs) on the 3-day VTS prediction is estimated and their nonlinear evolution mechanism is clarified. The OIEs, horizontally distributed along the 1000 m isobath and vertically confined within the upper 50 m, experience non-localized evolution and cause a rapid increase in the root mean square errors by about 100 times during the prediction period. Eddy energetics analysis indicates that barotropic instabilities constitute a key factor in the evolution of the OIEs. In addition, sensitive areas are identified for targeted observation based on the vertically integrated total energy of the OIEs. Sensitive experiments demonstrate that the prediction errors caused by random initial errors in the sensitive areas are larger than those in other nearby areas. Observing system simulation experiments (OSSEs) further suggest that eliminating the initial errors in sensitive areas can improve VTS prediction by over 50%. This study reveals the patterns and growth dynamic mechanism of CNOP-type OIEs in 3-day VTS prediction in the NSCS and provides scientific guidance for operational observation design.

Application of the Observation‐Oriented CNOP‐P Sensitivity Analysis Method in Evapotranspiration Simulation and Prediction Over the Tibetan Plateau

AbstractTo improve the ability to accurately simulate and predict evapotranspiration (ET) over the Tibetan Plateau (TP), an observation‐oriented sensitivity analysis (SA) method based on a conditional nonlinear optimal parameter perturbation (CNOP‐P) method is employed. With the observation‐oriented CNOP‐P SA method, a sensitive parameter subset among all parameters is first determined through comparison with observations of ET across the TP. Then, the sensitive parameter combination is optimized to evaluate the ET simulation ability and prediction skill over the TP. A traditional one‐at‐a‐time (OAT) method is also employed, and the results are compared with the numerical results obtained with the observation‐oriented CNOP‐P SA method. The results show that the identified parameter combination using the observation‐oriented CNOP‐P SA method is more sensitive than that using the OAT method for the estimation of ET at 13 sites on the TP. The above results demonstrate that it is important to determine the sensitive parameter combination using the observation‐oriented SA method to improve the ET simulation ability and prediction skill over the TP.

Influence of Physically Constrained Initial Perturbations on the Predictability of Mei-Yu Heavy Precipitation

Abstract Based on the conditional nonlinear optimal perturbation (CNOP) approach, the predictability of mei-yu heavy precipitation and its underlying physical processes is investigated. As an extension of our previous work, the practical predictability of heavy precipitation events is studied using more realistic initial perturbations than previously considered. The initial perturbation reflects certain physical connections among multiple variables, including zonal and meridional winds, potential temperature (T), and water vapor mixing ratio (Q). Two types of initial perturbations for the CNOP are identified, with similar spatial distributions but opposite signs and resulting effects. The accumulated precipitation is strengthened with mostly positive perturbations in the T and Q components for the CNOP and weakened by negative perturbations. Comparing downscaling (DOWN) perturbations and random perturbations (RPs) with the CNOP, it is found that the CNOP and DOWN perturbations exhibit particularly large- and mesoscale spatial structures, respectively, while the RPs yield a spatial distribution with mostly convective-scale features. Also, the CNOP results in the largest error growth and forecast uncertainty, especially for Q, followed by the DOWN perturbations; those in the RPs are the smallest. These results provide important implications for optimizing the initial perturbations of convection-permitting ensemble prediction systems, especially precipitation forecasts. Moreover, it is suggested that small-scale related variables, that is, those associated with vertical motion and microphysical processes, are much less predictable than thermodynamic variables, and the errors grow through distinct physical processes for the three types of initial perturbations, that is, those with flow-dependent features.

An approach to refining the ground meteorological observation stations for improving PM2.5 forecasts in the Beijing–Tianjin–Hebei region

Abstract. This paper investigates how to refine the ground meteorological observation network to greatly improve the PM2.5 concentration forecasts by identifying sensitive areas for targeted observations that are associated with a total of 48 forecasts in eight heavy haze events during the years of 2016–2018 over the Beijing–Tianjin–Hebei (BTH) region. The conditional nonlinear optimal perturbation (CNOP) method is adopted to determine the sensitive area of the surface meteorological fields for each forecast, and a total of 48 CNOP-type errors are obtained including wind, temperature, and water vapor mixing ratio components. It is found that, although all the sensitive areas tend to locate within and/or around the BTH region, their specific distributions are dependent on the events and the start times of the forecasts. Based on these sensitive areas, the current ground meteorological stations within and around the BTH region are refined to form a cost-effective observation network, which makes the relevant PM2.5 forecasts starting from different initial times for varying events assimilate fewer observations, but overall, it achieve the forecasting skill comparable to and even higher than that obtained by assimilating all ground station observations. This network sheds light on the idea that some of the current ground stations within and around the BTH region are very useless for improving the PM2.5 forecasts in the BTH region and can be greatly scattered to avoid unnecessary work.

An adjoint-free algorithm for conditional nonlinear optimal perturbations (CNOPs) via sampling

Abstract. In this paper, we propose a sampling algorithm based on state-of-the-art statistical machine learning techniques to obtain conditional nonlinear optimal perturbations (CNOPs), which is different from traditional (deterministic) optimization methods.1 Specifically, the traditional approach is unavailable in practice, which requires numerically computing the gradient (first-order information) such that the computation cost is expensive, since it needs a large number of times to run numerical models. However, the sampling approach directly reduces the gradient to the objective function value (zeroth-order information), which also avoids using the adjoint technique that is unusable for many atmosphere and ocean models and requires large amounts of storage. We show an intuitive analysis for the sampling algorithm from the law of large numbers and further present a Chernoff-type concentration inequality to rigorously characterize the degree to which the sample average probabilistically approximates the exact gradient. The experiments are implemented to obtain the CNOPs for two numerical models, the Burgers equation with small viscosity and the Lorenz-96 model. We demonstrate the CNOPs obtained with their spatial patterns, objective values, computation times, and nonlinear error growth. Compared with the performance of the three approaches, all the characters for quantifying the CNOPs are nearly consistent, while the computation time using the sampling approach with fewer samples is much shorter. In other words, the new sampling algorithm shortens the computation time to the utmost at the cost of losing little accuracy.

Impacts of Initial Zonal Current Errors on the Predictions of Two Types of El Niño Events

AbstractPerturbations in the thermocline and surface zonal current (ZC) play crucial roles in the evolutions of the eastern Pacific (EP) and central Pacific (CP) El Niño events, respectively. Whereas numerous studies have examined the influence of initial uncertainties in ocean temperature on the predictability of El Niño, only a few studies investigated the impact of the initial ZC. Using an air‐sea coupling model, the conditional nonlinear optimal perturbation (CNOP) approach was employed to investigate the maximum impact of initial ZC errors on the El Niño prediction. The optimal initial ZC errors (denoted as CNOP‐Us) that have the severest impact on the El Niño prediction are found to mainly concentrate in the western and central tropical Pacific. The CNOP‐Us cause larger errors in the CP El Niño prediction than in the EP El Niño prediction. Additionally, CNOP‐Us cause rapid sea surface temperature error growth in spring in the EP El Niño prediction but in summer in the CP El Niño prediction. Dynamically, the former is related to the large uncertainties in the meridional current in spring caused by CNOP‐Us, while the latter is related to the strong ZC errors in summer. According to the distributions of CNOP‐Us, reducing the initial ZC errors in the western and central tropical Pacific may be vital in weakening the predictability barrier phenomena and improving the predictions of El Niño diversity.

A paralleled embedding high-dimensional Bayesian optimization with additive Gaussian kernels for solving CNOP

Conditional Nonlinear Optimal Perturbation (CNOP) is widely used in atmospheric and oceanic predictability studies. Solving CNOP is essentially a nonlinear optimization problem with certain constraints. One method to solve CNOP is the intelligent optimization algorithm. But it may not always obtain the satisfactory solution and efficiency because of the local exploitation of individual particles. The Bayesian optimization algorithm can avoid this local stagnation by building a global probability grasp for the optimization space of CNOP. Nonetheless, the limit of approximately 20-dimensional optimization space hampers its application in solving CNOP of high-dimensional numerical models. To overcome this bottleneck, we propose a paralleled embedding high-dimensional Bayesian optimization with additive Gaussian kernels (PEBOA) algorithm, which mainly consists of the feature extraction process and low-dimensional optimization process. In PEBOA, a feature extraction method (deepFE) using the convolutional Autoencoder with residual connections and customized constraint-preserved loss function is designed, which compresses 10 million dimensional data to relatively low-dimensional search spaces, from tens to hundreds of dimensions. Then, we propose the strategy of additive Gaussian kernels, with which the Bayesian optimization algorithm can optimize in relatively low-dimensional search spaces. Concretely, a kernel handles a subspace, and a subspace can handle dimensions approximately not more than 20. Additionally, the modified acquisition function can sample multiple candidates simultaneously with the aim of accelerating the optimization process. We apply PEBOA to solve CNOP of Regional Ocean Modeling System (ROMS) for identifying optimal initial errors of upstream Kuroshio transport variation, which is a 10 million dimensional and time-consuming problem. The computational performance of PEBOA and physical mechanisms of the obtained CNOP are analyzed. Experimental results indicate that deepFE excels Principal Component Analysis (PCA) in terms of relative error ratio, the magnitude of objective function values, and the obtained CNOP pattern. Besides, compared to deepFE-based Particle Swarm Optimization, PEBOA has better solving efficiency with 3.4% larger objective function values and 2.2 times greater likelihood of obtaining the valid CNOP. Furthermore, the modified acquisition function reduces computation time by about 71.0% with four cores. The physical mechanism analysis shows that CNOPs obtained by PEBOA are almost identical to the adjoint method, which can cause an anomalous increase (decrease) in upstream Kuroshio transport and maintain physics consistency.

Optimal Precursors for Central Pacific El Niño Events in GFDL CM2p1

Abstract Compared to the canonical eastern Pacific El Niño, the understanding and ability to predict the central Pacific (CP)-type event still need further improvement. In this study, the principal component analysis–based particle swarm optimization algorithm (PPSO) is applied in Geophysical Fluid Dynamics Laboratory Climate Model version 2p1 (GFDL CM2p1) to obtain the optimal precursors (OPRs) for CP El Niño events, based on the conditional nonlinear optimal perturbation (CNOP) method. For this, three normal years with neither El Niño nor La Niña events, i.e., three cases, are chosen as the reference states. The obtained OPRs for these cases exhibit a consistent positive sea surface temperature (SST) perturbation distribution in the subtropical North Pacific (20°–40°N, 175°E–140°W), which is further proven to be crucial for the evolution of CP El Niño based on the Northern and Southern Hemisphere significance test results. Mechanically, these positive SST perturbations are enhanced and reach the equatorial Pacific via wind–evaporation–SST (WES) feedback to evolve into a CP El Niño at the end of the year. The nonlinear approach is adopted to investigate the predictability of CP El Niño events and can shed some light on future studies.