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 interracial 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.

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 collabora- tors 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.

A Variant Constrained Genetic Algorithm for Solving Conditional Nonlinear Optimal Perturbations

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.

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.

Application of the Conditional Nonlinear Optimal Perturbations Method in a Theoretical Grassland Ecosystem

Using a simplified nonlinearly theoretical grassland ecosystem proposed by Zeng et al.,we study the sensitivity and nonlinear instability of the grassland ecosystem to finiteamplitude initial perturbations with the approach of conditional nonlinear optimal perturbation （CNOP）.The results show that the linearly stable grassland （desert or latent desert） states can turn to be nonlinearly unstable with finite amplitude initial perturbations.When the precipitation is between the two bifurcation points,a large enough finite amplitude initial perturbation can induce a transition between the grassland statethe desert state or the latent desert.

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.

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 procedures are formulated using an idealized model with parameterization "on-off" switches in the forcing term, the impacts of "on-off" switches on the capture of CNOP are analyzed, and three numerical experiments are conducted to check the effectiveness of GA in capturing CNOP and to analyze the impacts of different initial populations on the optimization result. The result shows that GA is competent for the capture of CNOP in the context of the idealized model with parameterization ‘on-off’ switches in this study. Finally, the advantages and disadvantages of GA in capturing CNOP are analyzed in detail.

THE APPLICATION OF CONDITIONAL NONLINEAR OPTIMAL PERTURBATION TO THE BINARY TYPHOONS INTERACTION —FENGSHEN AND FUNG-WONG

The interaction between the typhoons Fengshen and Fung-wong over the Western Pacific in 2002 is studied with the Conditional Nonlinear Optimal Perturbation(CNOP) method.The study discovered that the CNOP method reveals the process of one-way interaction between Fengshen and Fung-wong.Moreover,if the region of Fung-wong was selected for verification,the sensitivity area was mainly located in the region of Fengshen and presented a half-ring structure;if the region of Fengshen was selected for verification,most of the sensitivity areas were located in the region between the Fengshen and the subtropical high,far away from Fung-wong.This indicated that Fung-wong is mainly steered by Fengshen,but Fengshen is mainly affected by the subtropical high.The sensitivity experiment showed that the initial errors in the CNOP-identified sensitive areas have larger impacts on the verification-area prediction than those near the typhoon center and their developments take a large proportion in the whole domain.This suggests that the CNOP-identified sensitive areas do have large influence on the verification-area prediction.

A SVD-based ensemble projection algorithm for calculating the conditional nonlinear optimal perturbation

Conditional nonlinear optimal perturbation(CNOP) is an extension of the linear singular vector technique in the nonlinear regime.It represents the initial perturbation that is subjected to a given physical constraint,and results in the largest nonlinear evolution at the prediction time.CNOP-type errors play an important role in the predictability of weather and climate.Generally,when calculating CNOP in a complicated numerical model,we need the gradient of the objective function with respect to the initial perturbations to provide the descent direction for searching the phase space.The adjoint technique is widely used to calculate the gradient of the objective function.However,it is difficult and cumbersome to construct the adjoint model of a complicated numerical model,which imposes a limitation on the application of CNOP.Based on previous research,this study proposes a new ensemble projection algorithm based on singular vector decomposition(SVD).The new algorithm avoids the localization procedure of previous ensemble projection algorithms,and overcomes the uncertainty caused by choosing the localization radius empirically.The new algorithm is applied to calculate the CNOP in an intermediate forecasting model.The results show that the CNOP obtained by the new ensemble-based algorithm can effectively approximate that calculated by the adjoint algorithm,and retains the general spatial characteristics of the latter.Hence,the new SVD-based ensemble projection algorithm proposed in this study is an effective method of approximating the CNOP.

Application of Conditional Nonlinear Optimal Perturbation to Targeted Observation Studies of the Atmosphere and Ocean

This paper reviews progress in the application of conditional nonlinear optimal perturbation to targeted observation studies of the atmosphere and ocean in recent years, with a focus on the E1 Nifio-Southern Oscillation （ENSO）, Kuroshio path variations, and blocking events. Through studying the optimal precursor （OPR） and optimally growing initial error （OGE） of the occurrence of the above events, the similarity and localization features of OPR and OGE spatial structures have been found for each event. Ideal hindcasting experiments have shown that, if initial errors are reduced in the areas with the largest amplitude for the OPR and OGE for ENSO and Kuroshio path variations, the forecast skill of the model for these events is significantly improved. Due to the similarity between patterns of the OPR and OGE, additional observations implemented in the same sensitive region would help to not only capture the precursors, but also reduce the initial errors in the predictions, greatly increasing the forecast abilities. The similarity and localization of the spatial structures of the OPR and OGE during the onset of blocking events have also been investigated, but their application to targeted observation requires further study.

Inducing Unstable Grassland Equilibrium States Due to Nonlinear Optimal Patterns of Initial and Parameter Perturbations:Theoretical Models

Due to uncertainties in initial conditions and parameters, the stability and uncertainty of grassland ecosystem simulations using ecosystem models are issues of concern. Our objective is to determine the types and patterns of initial and parameter perturbations that yield the greatest instability and uncertainty in simulated grassland ecosystems using theoretical models. We used a nonlinear optimization approach, i.e., a conditional nonlinear optimal perturbation related to initial and parameter perturbations （CNOP） approach, in our work. Numerical results indicated that the CNOP showed a special and nonlinear optimal pattern when the initial state variables and multiple parameters were considered simultaneously. A visibly different complex optimal pattern characterizing the CNOPs was obtained by choosing different combinations of initial state variables and multiple parameters in different physical processes. We propose that the grassland modeled ecosystem caused by the CNOP-type perturbation is unstable and exhibits two aspects： abrupt change and the time needed for the abrupt change from a grassland equilibrium state to a desert equilibrium state when the initial state variables and multiple parameters are considered simultaneously. We compared these findings with results affected by the CNOPs obtained by considering only uncertainties in initial state variables and in a single parameter. The numerical results imply that the nonlinear optimal pattern of initial perturbations and parameter perturbations, especially for more parameters or when special parameters are involved, plays a key role in determining stabilities and uncertainties associated with a simulated or predicted grassland ecosystem.

A study of parameter uncertainties causing uncertainties in modeling a grassland ecosystem using the conditional nonlinear optimal perturbation method

In this paper, we apply the approach of conditional nonlinear optimal perturbation related to the parameter(CNOP-P)to study parameter uncertainties that lead to the stability(maintenance or degradation) of a grassland ecosystem. The maintenance of the grassland ecosystem refers to the unchanged or increased quantity of living biomass and wilted biomass in the ecosystem,and the degradation of the grassland ecosystem refers to the reduction in the quantity of living biomass and wilted biomass or its transformation into a desert ecosystem. Based on a theoretical five-variable grassland ecosystem model, 32 physical model parameters are selected for numerical experiments. Two types of parameter uncertainties could be obtained. The first type of parameter uncertainty is the linear combination of each parameter uncertainty that is computed using the CNOP-P method. The second type is the parameter uncertainty from multi-parameter optimization using the CNOP-P method. The results show that for the 32 model parameters, at a given optimization time and with greater parameter uncertainty, the patterns of the two types of parameter uncertainties are different. The different patterns represent physical processes of soil wetness. This implies that the variations in soil wetness(surface layer and root zone) are the primary reasons for uncertainty in the maintenance or degradation of grassland ecosystems, especially for the soil moisture of the surface layer. The above results show that the CNOP-P method is a useful tool for discussing the abovementioned problems.

Recent Progress in Applications of the Conditional Nonlinear Optimal Perturbation Approach to Atmosphere-Ocean Sciences

The conditional nonlinear optimal perturbation(CNOP for short) approach is a powerful tool for predictability and targeted observation studies in atmosphere-ocean sciences. By fully considering nonlinearity under appropriate physical constraints, the CNOP approach can reveal the optimal perturbations of initial conditions, boundary conditions, model parameters, and model tendencies that cause the largest simulation or prediction uncertainties. This paper reviews the progress of applying the CNOP approach to atmosphere-ocean sciences during the past five years. Following an introduction of the CNOP approach, the algorithm developments for solving the CNOP are discussed.Then, recent CNOP applications, including predictability studies of some high-impact ocean-atmospheric environmental events, ensemble forecast, parameter sensitivity analysis, uncertainty estimation caused by errors of model tendency or boundary condition, are reviewed. Finally, a summary and discussion on future applications and challenges of the CNOP approach are presented.

Adjoint-free calculation method for conditional nonlinear optimal perturbations

Adjoint-free calculation method is proposed to compute conditional nonlinear optimal perturbations(CNOP) combined with initial perturbations and model parameter perturbations. The new approach avoids the use of adjoint technique in the optimization process. CNOPs respectively generated by ensemble-based and adjoint-based methods are compared based on a simple theoretical model.

The Influence of Arctic Sea Ice Concentration Perturbations on Subseasonal Predictions of North Atlantic Oscillation Events

The influence of Arctic sea ice concentration (SIC) on the subseasonal prediction of the North Atlantic Oscillation (NAO) event is investigated by utilizing the Community Atmospheric Model version 4. The optimal Arctic SIC perturbations which exert the greatest influence on the onset of an NAO event from a lead of three pentads (15 days) are obtained with a conditional nonlinear optimal perturbation approach. Numerical results show that there are two types of optimal Arctic SIC perturbations for each NAO event, with one weakening event (marked as type-1) and another strengthening event (marked as type-2). For positive NAO events, type-1 optimal SIC perturbations mainly show positive SIC anomalies in the Greenland, Barents, and Okhotsk Seas, while type-2 perturbations mainly feature negative SIC anomalies in these regions. For negative NAO events, the optimal SIC perturbations have almost opposite patterns to those in positive events, although there are some differences among these SIC perturbations due to different atmospheric initial conditions. Further diagnosis reveals that the optimal Arctic SIC perturbations first modify the surface turbulent heat flux and the temperature in the lower troposphere via diabatic processes. Afterward, the temperature in the low troposphere is mainly affected by dynamic advection. Finally, potential vorticity advection plays a crucial role in the 500-hPa geopotential height prediction in the northern North Atlantic sector during pentad 4, which influences NAO event prediction. These results highlight the importance of Arctic SIC on NAO event prediction and the spatial characteristics of the SIC perturbations may provide scientific support for target observations of SIC in improving NAO subseasonal predictions.

Optimal nonlinear excitation of decadal variability of the North Atlantic thermohaline circulation

Nonlinear development of salinity perturbations in the Atlantic thermohaline circulation(THC) is investigated with a three-dimensional ocean circulation model,using the conditional nonlinear optimal perturbation method.The results show two types of optimal initial perturbations of sea surface salinity,one associated with freshwater and the other with salinity.Both types of perturbations excite decadal variability of the THC.Under the same amplitude of initial perturbation,the decadal variation induced by the freshwater perturbation is much stronger than that by the salinity perturbation,suggesting that the THC is more sensitive to freshwater than salinity perturbation.As the amplitude of initial perturbation increases,the decadal variations become stronger for both perturbations.For salinity perturbations,recovery time of the THC to return to steady state gradually saturates with increasing amplitude,whereas this recovery time increases remarkably for freshwater perturbations.A nonlinear(advective) feedback between density and velocity anomalies is proposed to explain these characteristics of decadal variability excitation.The results are consistent with previous ones from simple box models,and highlight the importance of nonlinear feedback in decadal THC variability.

Three-Dimensional Structure of Optimal Nonlinear Excitation for Decadal Variability of the Thermohaline Circulation

The decadal variability of the North Atlantic thermohaline circulation(THC) is investigated within a three-dimensional ocean circulation model using the conditional nonlinear optimal perturbation method. The results show that the optimal initial perturbations of temperature and salinity exciting the strongest decadal THC variations have similar structures: the perturbations are mainly in the northwestern basin at a depth ranging from 1500 to 3000 m. These temperature and salinity perturbations act as the optimal precursors for future modifications of the THC, highlighting the importance of observations in the northwestern basin to monitor the variations of temperature and salinity at depth. The decadal THC variation in the nonlinear model initialized by the optimal salinity perturbations is much stronger than that caused by the optimal temperature perturbations, indicating that salinity variations might play a relatively important role in exciting the decadal THC variability. Moreover, the decadal THC variations in the tangent linear and nonlinear models show remarkably different characteristics, suggesting the importance of nonlinear processes in the decadal variability of the THC.

Optimal precursor of the transition from Kuroshio large meander to straight path

We used the conditional nonlinear optimal perturbation(CNOP) method to explore the optimal precursor of the transition from Kuroshio large meander(LM) to straight path within a barotropic inflowoutflow model,and found that large amplitudes of the optimal precursor are mainly located in the east of Kyushu,which implies that perturbations in the region are important for the transition from LM to straight path.Furthermore,we investigated the transition processes caused by the optimal precursor,and found that these processes could be divided into three stages.In the first stage,a cyclonic eddy is advected to the formation region of the Kuroshio large meander,which enhances the LM path and causes a cyclonic eddy to shed from the Kuroshio mainstream.This process causes the LM path to change into a small meander path.Subsequently,the small meander is maintained for a period because the vorticity advection is balanced by the beta effect in the second stage.In the third stage,the small meander weakens and the straight path ultimately forms.The positive vorticity advecting downstream is responsible for this process.The exploration of the optimal precursor will conduce to improve the prediction of the transition processes from LM path to straight path,and its spatial structure can be used to guide Kuroshio targeted observation studies.

Is Model Parameter Error Related to a Significant Spring Predictability Barrier for El Nio events? Results from a Theoretical Model

Within a theoretical ENSO model, the authors investigated whether or not the errors superimposed on model parameters could cause a significant ＂spring predictability barrier＂ （SPB） for El Nio events. First, sensitivity experiments were respectively performed to the air-sea coupling parameter, α and the thermocline effect coefficient μ. The results showed that the uncertainties superimposed on each of the two parameters did not exhibit an obvious season-dependent evolution; furthermore, the uncertainties caused a very small prediction error and consequently failed to yield a significant SPB. Subsequently, the conditional nonlinear optimal perturbation （CNOP） approach was used to study the effect of the optimal mode （CNOP-P） of the uncertainties of the two parameters on the SPB and to demonstrate that the CNOP-P errors neither presented a unified season-dependent evolution for different El Nio events nor caused a large prediction error, and therefore did not cause a significant SPB. The parameter errors played only a trivial role in yielding a significant SPB. To further validate this conclusion, the authors investigated the effect of the optimal combined mode （i.e. CNOP error） of initial and model errors on SPB. The results illustrated that the CNOP errors tended to have a significant season-dependent evolution, with the largest error growth rate in the spring, and yielded a large prediction error, inducing a significant SPB. The inference, therefore, is that initial errors, rather than model parameter errors, may be the dominant source of uncertainties that cause a significant SPB for El Nio events. These results indicate that the ability to forecast ENSO could be greatly increased by improving the initialization of the forecast model.

The Roles of Spatial Locations and Patterns of Initial Errors in the Uncertainties of Tropical Cyclone Forecasts

In this study, a series of sensitivity experiments were performed for two tropical cyclones （TCs）, TC Longwang （2005） and TC Sinlaku （2008）, to explore the roles of locations and patterns of initial errors in uncertainties of TC forecasts. Specifically, three types of initial errors were generated and three types of sensitive areas were determined using conditional nonlinear optimal perturbation （CNOP）, first singular vector （FSV）, and composite singular vector （CSV） methods. Additionally, random initial errors in randomly selected areas were considered. Based on these four types of initial errors and areas, we designed and performed 16 experiments to investigate the impacts of locations and patterns of initial errors on the nonlinear developments of the errors, and to determine which type of initial errors and areas has the greatest impact on TC forecasts. Overall, results from the experiments indicate the following： （1） The impact of random errors introduced into the sensitive areas was greater than that of errors themselves fixed in the randomly selected areas. From the perspective of statisticul analysis, and by comparison, the impact of random errors introduced into the CNOP target area was greatest. （2） The initial errors with CNOP, CSV, or FSV patterns were likely to grow faster than random errors. （3） The initial errors with CNOP patterns in the CNOP target areas had the greatest impacts on the final verification forecasts.