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.
Initial errors and model errors are the source of prediction errors. In this study, the authors compute the conditional nonlinear optimal perturbation(CNOP)-type initial errors and nonlinear forcing singular vector(NFSV)-type tendency errors of the Zebiak-Cane model with respect to El Nio events and analyze their combined effect on the prediction errors for El Nio events. The CNOPtype initial error(NFSV-type tendency error) represents the initial errors(model errors) that have the largest effect on prediction uncertainties for El Nio events under the perfect model(perfect initial conditions) scenario. However, when the CNOP-type initial errors and the NFSVtype tendency errors are simultaneously considered in the model, the prediction errors caused by them are not amplified as the authors expected. Specifically, the prediction errors caused by the combined mode of CNOP-type initial errors and NFSV-type tendency errors are a little larger than those caused by the NFSV-type tendency errors. This fact emphasizes a need to investigate the optimal combined mode of initial errors and tendency errors that cause the largest prediction error for El Nio events.
Model errors offset by constant and time-variant optimal forcing vector approaches （termed COF and OFV, respectively） are analyzed within the framework of E1 Nifio simulations. Applying the COF and OFV approaches to the well-known Zebiak-Cane model, we re-simulate the 1997 and 2004 E1 Nifio events, both of which were poorly degraded by a certain amount of model error when the initial anomalies were generated by coupling the observed wind forcing to an ocean com- ponent. It is found that the Zebiak-Cane model with the COF approach roughly reproduced the 1997 E1 Nifio, but the 2004 E1 Nifio simulated by this approach defied an ENSO classification, i.e., it was hardly distinguishable as CP-E1 Nifio or EP-E1 Nifio. In hoth E1 Nifio simulations, substituting the COF with the OFV improved the fit between the simulations and obser- vations because the OFV better manages the time-variant errors in the model. Furthermore, the OFV approach effectively corrected the modeled E1 Nifio events even when the observational data （and hence the computational time） were reduced. Such a cost-effective offset of model errors suggests a role for the OFV approach in complicated CGCMs.
Based on the Zebiak-Cane model, the timedependent nonlinear forcing singular vector(NFSV)-type tendency errors with components of 4 and 12(denoted by NFSV-4 and NFSV-12) are calculated for predetermined El Nio events and compared with the constant NFSV(denoted by NFSV-1) from their patterns and resultant prediction errors. Specifically, NFSV-1 has a zonal dipolar sea surface temperature anomaly(SSTA) pattern with negative anomalies in the equatorial eastern Pacific and positive anomalies in the equatorial central-western Pacific. Although the first few components in NFSV-4 and NFSV-12 present patterns similar to NFSV-1, they tend to extend their dipoles farther westward; meanwhile, the positive anomalies gradually cover much smaller regions with the lag times. In addition, the authors calculate the prediction errors caused by the three kinds of NFSVs, and the results indicate that the prediction error induced by NFSV-12 is the largest, followed by the NFSV-4. However, when compared with the prediction errors caused by random tendency errors, the NFSVs generate significantly larger prediction errors. It is therefore shown that the spatial structure of tendency errors is important for producing large prediction errors. Furthermore, in exploring the tendency errors that cause the largest prediction error for El Nio events, the time-dependent NFSV should be evaluated.
Recent progress in the study of nonlinear atmospheric dynamics and related predictability of weather and climate in China （2007-2011） are briefly introduced in this article. Major achievements in the study of nonlinear atmospheric dynamics have been classified into two types： （1） progress based on the analysis of solutions of simplified control equations, such as the dynamics of NAO, the optimal precursors for blocking onset, and the behavior of nonlinear waves, and （2） progress based on data analyses, such as the nonlinear analyses of fluctuations and recording-breaking temperature events, the long-range correlation of extreme events, and new methods of detecting abrupt dynamical change. Major achievements in the study of predictability include the following： （1） the application of nonlinear local Lyapunov exponents （NLLE） to weather and climate predictability; （2） the application of condition nonlinear optimal perturbation （CNOP） to the studies of E1 Nifio-Southern Oscillation （ENSO） predictions, ensemble forecasting, targeted observation, and sensitivity analysis of the ecosystem; and （3） new strategies proposed for predictability studies. The results of these studies have provided greater understanding of the dynamics and nonlinear mecha- nisms of atmospheric motion, and they represent new ideas for developing numerical models and improving the forecast skill of weather and climate events.
