Based on a five-variable theoretical ecosystem model, the stability of equilibrium state and the nonlinear feature of the transition between a grassland state and a desert state are investigated. The approach of the conditional nonlinear optimal perturbations （CNOPs）, which are the nonlinear generalization of the linear singular vectors （LSVs）, is adopted. The numerical results indicate that the linearly stable grassland and desert states are nonlinearly unstable to large enough initial perturbations on the condition that the moisture index # satisfies 0.3126 〈 μ 〈 0.3504. The perturbations represent some kind of anthropogenic influence and natural factors. The results obtained by CNOPs, LSVs and Lyapunov vectors （LVs） are compared to analyze the nonlinear feature of the transition between the grassland state and the desert state. Besides this, it is shown that the five-variable model is superior to the three-variable model in providing more visible signals when the transitions occur.
With the Zebiak-Cane （ZC） model, the initial error that has the largest effect on ENSO prediction is explored by conditional nonlinear optimal perturbation （CNOP）. The results demonstrate that CNOP-type errors cause the largest prediction error of ENSO in the ZC model. By analyzing the behavior of CNOPtype errors, we find that for the normal states and the relatively weak E1 Nifio events in the ZC model, the predictions tend to yield false alarms due to the uncertainties caused by CNOP. For the relatively strong E1 Nino events, the ZC model largely underestimates their intensities. Also, our results suggest that the error growth of E1 Nifio in the ZC model depends on the phases of both the annual cycle and ENSO. The condition during northern spring and summer is most favorable for the error growth. The ENSO prediction bestriding these two seasons may be the most difficult. A linear singular vector （LSV） approach is also used to estimate the error growth of ENSO, but it underestimates the prediction uncertainties of ENSO in the ZC model. This result indicates that the different initial errors cause different amplitudes of prediction errors though they have same magnitudes. CNOP yields the severest prediction uncertainty. That is to say, the prediction skill of ENSO is closely related to the types of initial error. This finding illustrates a theoretical basis of data assimilation. It is expected that a data assimilation method can filter the initial errors related to CNOP and improve the ENSO forecast skill.
Conditional nonlinear optimal perturbation (CNOP),which is a natural extension of singular vector (SV) into the nonlinear regime,is applied to ensemble prediction study by using a quasi-geostrophic model under the perfect model assumption. SVs and CNOPs have been utilized to generate the initial pertur-bations for ensemble prediction experiments. The results are compared for forecast lengths of up to 14 d. It is found that the forecast skill of samples,in which the first SV is replaced by CNOP,is com-paratively higher than that of samples composed of only SVs in the medium range (day 6―day 14). This conclusion is valid under the condition that analysis error is a kind of fast-growing ones regardless of its magnitude,whose nonlinear growth is faster than that of SV in the later part of the forecast. Fur-thermore,similarity index and empirical orthogonal function (EOF) analysis are performed to explain the above numerical results.
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.
Conditional nonlinear optimal perturbation (CNOP) is a nonlinear generalization of linear singular vector (LSV) and features the largest nonlinear evolution at prediction time for the initial perturbations in a given constraint. It was proposed initially for predicting the limitation of predictability of weather or climate. Then CNOP has been applied to the studies of the problems related to predictability for weather and climate. In this paper, we focus on reviewing the recent advances of CNOP's applications, which involves the ones of CNOP in problems of ENSO amplitude asymmetry, block onset, and the sensitivity analysis of ecosystem and ocean's circulations, etc. Especially, CNOP has been primarily used to construct the initial perturbation fields of ensemble forecasting, and to determine the sensitive area of target observation for precipitations. These works extend CNOP's applications to investigating the nonlinear dynamical behaviors of atmospheric or oceanic systems, even a coupled system, and studying the problem of the transition between the equilibrium states. These contributions not only attack the particular physical problems, but also show the superiority of CNOP to LSV in revealing the effect of nonlinear physical processes. Consequently, CNOP represents the optimal precursors for a weather or climate event; in predictability studies, CNOP stands for the initial error that has the largest negative effect on prediction; and in sensitivity analysis, CNOP is the most unstable (sensitive) mode. In multi-equilibrium state regime, CNOP is the initial perturbation that induces the transition between equilibriums most probably. Furthermore, CNOP has been used to construct ensemble perturbation fields in ensemble forecast studies and to identify sensitive area of target observation. CNOP theory has become more and more substantial. It is expected that CNOP also serves to improve the predictability of the realistic predictions for weather and climate events plays an increasingly important role in exploring th
