Relationships between the Limit of Predictability and Initial Error in the Uncoupled and Coupled Lorenz Models

In this study, the relationship between the limit of predictability and initial error was investigated using two simple chaotic systems： the Lorenz model, which possesses a single characteristic time scale, and the coupled Lorenz model, which possesses two different characteristic time scales. The limit of predictability is defined here as the time at which the error reaches 95% of its saturation level; nonlinear behaviors of the error growth are therefore involved in the definition of the limit of predictability. Our results show that the logarithmic function performs well in describing the relationship between the limit of predictability and initial error in both models, although the coefficients in the logarithmic function were not constant across the examined range of initial errors. Compared with the Lorenz model, in the coupled Lorenz model in which the slow dynamics and the fast dynamics interact with each other--there is a more complex relationship between the limit of predictability and initial error. The limit of predictability of the Lorenz model is unbounded as the initial error becomes infinitesimally small; therefore, the limit of predictability of the Lorenz model may be extended by reducing the amplitude of the initial error. In contrast, if there exists a fixed initial error in the fast dynamics of the coupled Lorenz model, the slow dynamics has an intrinsic finite limit of predictability that cannot be extended by reducing the amplitude of the initial error in the slow dynamics, and vice versa. The findings reported here reveal the possible existence of an intrinsic finite limit of predictability in a coupled system that possesses many scales of time or motion.

Quantifying local predictability of the Lorenz system using the nonlinear local Lyapunov exponent

The nonlinear local Lyapunov exponent（NLLE） can be used as a quantification of the local predictability limit of chaotic systems. In this study, the phase-spatial structure of the local predictability limit over the Lorenz-63 system is investigated. It is found that the inner and outer rims of each regime of the attractor have a high probability of a longer than average local predictability limit, while the center part is the opposite. However, the distribution of the local predictability limit is nonuniformly organized, with adjacent points sometimes showing quite distinct error growth.The source of local predictability is linked to the local dynamics, which is related to the region in the phase space and the duration on the current regime.

A model for predicting SST in limited region-I. The dynamical equations

-Starting from physical oceanology characteristics of the China seas and for the short-term operational prediction of SST in the region, a two-dimensional (vertically integrated) primitive equation model, physically reasonable and operationally feasible,on the upper mixed layer is constructed and given here, which consists of three parts, the nondivergent residual current (the monthly mean field of the Kuroshio and its branches) equations, the dynamic forecasting equations, and the equation of model's physics consisting of surface heat flux, coolings of the upper mixed layer due to the Ekman pumping and the entrainment by gale. This model may be used primarily to forecast the sea surface temperature, and to give estimations of the mean wind-driven current and the sea level, for a period of 3-5 d. In part 1 of this series, the physical conditions for establishing model equations are discussed first, that is, 1. the existence of the upper well mixed layer in the region; 2. the distinguishability of currents of all kinds; 3. the splitting of thermodynamical equation. The equations of nondivergent residual current, and the dynamic forecasting equations with initial values and boundary conditions are also discussed.

The predictability of atmospheric and oceanic motions:Retrospect and prospects

This paper reviews the historic understanding of the predictability of atmospheric and oceanic motions, and interprets it in a general framework. On this basis, the existing challenges and unsolved problems in the study of the intrinsic predictability limit(IPL) of weather and climate events of different spatio-temporal scales are summarized. Emphasis is also placed on the structure of the initial error and model parameter errors as well as the associated targeting observation issue. Finally, the predictability of atmospheric and oceanic motion in the ensemble-probabilistic methods widely used in current operational forecasts are discussed.The necessity of considering IPLs in the framework of stochastic dynamic systems is also addressed.

Estimation of atmospheric predictability for multivariable system using information theory in nonlinear error growth dynamics

To estimate atmospheric predictability for multivariable system, based on information theory in nonlinear error growth dynamics, a quantitative method is introduced in this paper using multivariable joint predictability limit(MJPL) and corresponding single variable predictability limit(SVPL). The predictability limit, obtained from the evolutions of nonlinear error entropy and climatological state entropy, is not only used to measure the predictability of dynamical system with the constant climatological state entropy, but also appropriate to the case of climatological state entropy changed with time. With the help of daily NCEP-NCAR reanalysis data, by using a method of local dynamical analog, the nonlinear error entropy, climatological state entropy, and predictability limit are obtained, and the SVPLs and MJPL of the winter 500-hPa temperature field, zonal wind field and meridional wind field are also investigated. The results show that atmospheric predictability is well associated with the analytical variable. For single variable predictability, there exists a big difference for the three variables, with the higher predictability found for the temperature field and zonal wind field and the lower predictability for the meridional wind field. As seen from their spatial distributions, the SVPLs of the three variables appear to have a property of zonal distribution, especially for the meridional wind field, which has three zonal belts with low predictability and four zonal belts with high predictability. For multivariable joint predictability, the MJPL of multivariable system with the three variables is not a simple mean or linear combination of its SVPLs. It presents an obvious regional difference characteristic. Different regions have different results. In some regions, the MJPL is among its SVPLs. However, in other regions, the MJPL is less than its all SVPLs.

CFD predictions of LBO limits for aero-engine combustors using fuel iterative approximation

Lean blow-out （LBO） is critical to operational performance of combustion systems in propulsion and power generation. Current predictive tools for LBO limits are based on decadesold empirical correlations that have limited applicability for modern combustor designs. According to the Lefebvre＇s model for LBO and classical perfect stirred reactor （PSR） concept, a load parameter （LP） is proposed for LBO analysis of aero-engine combustors in this paper. The parameters contained in load parameter are all estimated from the non-reacting flow field of a combustor that is obtained by numerical simulation. Additionally, based on the load parameter, a method of fuel iterative approximation （FIA） is proposed to predict the LBO limit of the combustor. Compared with experimental data for 19 combustors, it is found that load parameter can represent the actual combustion load of the combustor near LBO and have good relativity with LBO fuel/air ratio （FAR）. The LBO FAR obtained by FIA shows good agreement with experimental data, the maximum prediction uncertainty of FIA is about ±17.5%. Because only the non-reacting flow is simulated, the time cost of the LBO limit prediction using FIA is relatively low （about 6 h for one combustor with computer equipment of CPU 2.66 GHz · 4 and 4 GB memory）, showing that FIA is reliable and efficient to be used for practical applications.

A lower flammability limit prediction model of alkane-CO2 mixtures based on flame phenomenon simulation

In the present study, a novel model is proposed to evaluate the lower flammability limit(LFL) of alkane diluted with CO2. The LFL model is based on flame phenomenon simulation(FS-LFL). The model consists of combustion, turbulence, and igniter models, which are used to characterise the combustion based on the chemical kinetics and CFD, which is not feasible with traditional methods. The flame simulation phenomenon was validated by contrast with experiment and same criterion of flammability limit in the experiment was adopted. The FS-LFL model was used to predict the LFLs of a propane-CO2 mixture and propane at various temperatures. The model performance was analysed by comparing the results with experimental data and predictions obtained from existing methods. The AARDs between the predicted and experimentally determined LFLs of the propane-CO2 mixture are 0.34%, 1.19%, and 1.35% at 30℃, 50℃, and 70℃, respectively. The model also has a good predictive power with respect to the LFLs of propane at initial temperatures ranging from 30℃–300℃, with an AARD of 2.10%. When the dilution of CO2 is 90%, the model yields a better result due to the utilisation of the chemical kinetics mechanism. This result is instructive for the use of this method in the prediction of upper flammability limits.

Mechanical Properties of Resistance Spot Welded Components of High Strength Austenitic Stainless Steel

Resistance multi-spot welding （MSW） in column, triangle and tetragonal symmetry arrangements was prepared using cold-rolled 301L high-strength sheets, and their static and fatigue properties were in- vestigated. The effects of spot welds on the fracture strengths and fatigue limits were discussed. The results show that the static strengths can be estimated using an inherent linear relationship formula of the load versus the welding area. It was based on the 28%-33% volume fraction of α′ martensite induced at the interfacial spot weld fracture because of the failure deformation. The fatigue limits of the MSW nonlin- early increase with the number of spot welds. The arrangement of spot welds in the MSW significantly affects the average fatigue limit of each spot weld; its 26% maximum reduction occurred in the triangle, and the interaction stress between spot welds led to its 18% reduction in the tetragonal symmetry. The calculated fatigue stress of all MSW loadings with their mean fatigue limits was 230-270 MPa.