Improving numerical forecasting skill in the atmospheric and oceanic sciences by solving optimization problems is an important issue. One such method is to compute the conditional nonlinear optimal perturbation(CNOP), which has been applied widely in predictability studies. In this study, the Differential Evolution(DE) algorithm, which is a derivative-free algorithm and has been applied to obtain CNOPs for exploring the uncertainty of terrestrial ecosystem processes, was employed to obtain the CNOPs for finite-dimensional optimization problems with ball constraint conditions using Burgers' equation. The aim was first to test if the CNOP calculated by the DE algorithm is similar to that computed by traditional optimization algorithms, such as the Spectral Projected Gradient(SPG2) algorithm. The second motive was to supply a possible route through which the CNOP approach can be applied in predictability studies in the atmospheric and oceanic sciences without obtaining a model adjoint system, or for optimization problems with non-differentiable cost functions. A projection skill was first explanted to the DE algorithm to calculate the CNOPs. To validate the algorithm, the SPG2 algorithm was also applied to obtain the CNOPs for the same optimization problems. The results showed that the CNOPs obtained by the DE algorithm were nearly the same as those obtained by the SPG2 algorithm in terms of their spatial distributions and nonlinear evolutions. The implication is that the DE algorithm could be employed to calculate the optimal values of optimization problems, especially for non-differentiable and nonlinear optimization problems associated with the atmospheric and oceanic sciences.
This study examines the time and regime dependencies of sensitive areas identified by the conditional nonlinear optiflml perturbation （CNOP） method for forecasts of two typhoons. Typhoon Meari （2004） was weakly nonlinear and is herein referred to as the linear case, while Typhoon Matsa （2005） was strongly nonlinear and is herein referred to as the nonlinear case. In the linear case, the sensitive areas identified for special forecast times when the initial time was fixed resembled those identified for other forecast times. Targeted observations deployed to improve a special time forecast would thus also benefit forecasts at other times. In the nonlinear case, the similarities among the sensitive areas identified for different forecast times were more limited. The deployment of targeted observations in the nonlinear case would therefore need to be adapted to achieve large improvements for different targeted forecasts. For both cases, the closer the forecast time, the higher the similarities of the sensitive areas. When the forecast time was fixed, the sensitive areas in the linear case diverged continuously from the verification area as the forecast period lengthened, while those in the nonlinear case were always located around the initial cyclones. The deployment of targeted observations to improve a special forecast depends strongly on the time of deployment. An examination of the efficiency gained by reducing initial errors within the identified sensitive areas confirmed these results. In general, the greatest improvement in a special time forecast was obtained by identifying the sensitive areas for the corresponding forecast time period.
Here discussed is the sensitivity of simulated typhoon track and intensity over the Northwest Pacific Ocean to different cumulus schemes.The results from the 20 typhoon cases during 2003-2008 show that the simulation of typhoon track and intensity are very sensitive to cumulus schemes.The relationship between simulations of typhoon track and cumulus schemes can be case dependent.Different best tracks obtained from different case studies depend on which cumulus scheme we chose.However,simulations of typhoon intensity exhibit different features.The Kain-Fritsch scheme simulation obtains the most intensive typhoon,whereas the Betts-Miller-Janjic scheme and the Grell-Devenyi scheme obtain weaker typhoons.The sensitivity of simulated typhoon track and intensity to different cumulus schemes is due mainly to different hypotheses and precipitation calculations.The difference of simulated large scale circulations using different cumulus schemes leads to the difference of typhoon tracks.The closer the simulations are compared to observations,the less the errors of simulated typhoon tracks.The difference of simulated typhoon intensity is due mainly to the difference of simulated vertical heating of the atmosphere.These lead to different strengths of convection which causes the difference of cumulus precipitation and latent heat.The KF scheme simulation obtains the strongest vertical convection,the obvious warm core structure,more cumulus precipitation,and stronger intensity.By contrast,the BMJ scheme and the GD scheme obtain weaker convection,less cumulus precipitation,and weaker intensity.
With more and more improvements of atmosphere or ocean models,a growing number of physical processes in the form of parameterization are incorporated into the models,which,on the one hand,makes the models capable of describing the at-mospheric or oceanic movement more precisely,and on the other hand,introduces non-smoothness in the form of "on-off" switches into the models."On-off" switches enhance the nonlinearity of the models and finally result in the loss of the effec-tiveness of variational data assimilation(VDA) based on the conventional adjoint method(ADJ).This study,in virtue of the optimization ability of a genetic algorithm(GA) for non-smooth problems,presents a new GA(referred to as GA NEW) to solve the problems of the VDA with discontinuous "on-off" processes.In the GA-NEW,adaptive selection and mutation oper-ators,blend crossover operator,and elitist strategy are combined in application.In order to verify the effectiveness and feasi-bility of the GA NEW in VDA,an idealized model of partial differential equation with discontinuous "on-off" switches in the forcing term is adopted as the governing equation.By comparison with the ADJ,it is shown that the GA NEW in VDA is more effective and can yield better assimilation retrievals.In addition,VDA experiments demonstrate that the performance of a GA is greatly related to the configuration of genetic operators(selection,crossover and mutation operators) and much better results may be attained with more proper genetic operations.Furthermore,the robustness of the GA NEW to observational noise,model errors and observation density is investigated,and the results show that the GA NEW has stronger robustness than the ADJ with respect to all the three observation noises,model errors,and sparse observation.
This study investigated the impact of different verification-area designs on the sensitive areas identified using the conditional nonlinear optimal perturbation （CNOP） method for tropical cyclone targeted observations.The sensitive areas identified using the first singular vector （FSV） method,which is the linear approximation of CNOP,were also investigated for comparison.By analyzing the validity of the sensitive areas,the proper design of a verification area was developed.Tropical cyclone Rananim,which occurred in August 2004 in the northwest Pacific Ocean,was studied.Two sets of verification areas were designed;one changed position,and the other changed both size and position.The CNOP and its identified sensitive areas were found to be less sensitive to small variations of the verification areas than those of the FSV and its sensitive areas.With larger variations of the verification area,the CNOP and the FSV as well as their identified sensitive areas changed substantially.In terms of reducing forecast errors in the verification area,the CNOP-identified sensitive areas were more beneficial than those identified using FSV.The design of the verification area is important for cyclone prediction.The verification area should be designed with a proper size according to the possible locations of the cyclone obtained from the ensemble forecast results.In addition,the development trend of the cyclone analyzed from its dynamic mechanisms was another reference.When the general position of the verification area was determined,a small variation in size or position had little influence on the results of CNOP.
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.
