Return Pre Home Next




MU Mu1,  DUAN Wansuo1  and  CHOU Jifan2

1.  LASG, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China

2.  Beijing Training Center, China Meteorological Administration, Beijing 100081, China


Recent advances in predictability studies in China (1999-2002) are summarized from the aspects as follows: new classification of predictability problems in numerical weather and climate prediction, nonlinear singular vector (NSV) and nonlinear singular value (NSVA), conditional nonlinear optimal perturbation (CNOP) and its applications to the predictability of ENSO, application of CNOP to sensitivity of thermohaline circulation to finite amplitude perturbation, monotonicity principle of predictability, dependence of model predictability on machine precision, global analysis on climate predictability, predictability of summer rainfall and monthly temperature in China, and predictability of numerical weather prediction and numerical short-term climate forecasting.

Key words: predictability, numerical prediction, conditional nonlinear optimal perturbation, monotonicity principle of predictability, computational uncertainty principle


Predictability is a fundamental issue in atmospheric sciences. However, it is very difficult to estimate quantitatively the predictability limit of the atmosphere, to explore the physical cause of predictability, and to improve the predictability of numerical model. Recently Chinese scientists have carried out some theoretical and numerical simulation studies on predictability, and have made some progresses. This paper simply introduces the advances in predictability studies in China during 1999 to 2002.



The predictability problems are usually classified into two types according to the factors which cause the uncertainties of the forecast results. The first kind of predictability is related to the initial errors, and the second kind of predictability is to the model errors. Mu et al. (2002a) presented a new classification of predictability problems in numerical weather and climate prediction on the basis of practical demands (see also Mu et al. 2002b).

Problem 1.  Assume that the initial observations and the first given values of the parameters of the model are known. At prediction time T, the prediction error in terms of a chosen measurement can be expressed by subtracting the true value of the state at time T from the prediction result. Our purpose is to gain the maximum predictable time for given maximum allowing prediction error. This problem can be reduced to a nonlinear optimization problem. Since the true value of the atmosphere cannot be obtained exactly, this nonlinear optimization problem is unsolvable. But if we know more information about the errors of initial values and parameters, the useful estimation on the maximum predictable time can be derived. Mu et al. (2002a) have established a lower bound of the maximum predictable time by investigating a corresponding solvable nonlinear optimization problem.

Problem 2. Assume that the initial observations and the first given values of the parameters of the model are known, for a given prediction time, we look for the prediction error. Similar to the above problem, since the true value of the atmosphere cannot be known exactly, it is also impossible to get the exact value of prediction error. But we can estimate it by using the information on the errors of initial observations and parameters. Mu et al. (2002a) investigated a nonlinear optimization problem, and proved that the solution of this nonlinear optimization problem yields an upper bound on the prediction errors.

Problem 3. At given prediction time and with the maximum allowing prediction error, for given initial observations and the first given parameters, we look for the allowing maximum initial error and the parameter error. Similar to the above two problems, Mu et al. (2002a) established a lower bound of the allowing maximum initial error and the parameter error by investigating a corresponding nonlinear optimization problem. Mu et al. (2002a) used the well-known Lorenz model to study the above three predictability problems by numerical approach.



It is of importance to determine the fastest growing initial perturbations in numerical weather and climate prediction and in the atmospheric research. In the linear approach for finding the fastest growing initial perturbation, it is assumed that the initial perturbation is sufficiently small such that its evolution can be governed approximately by the tangent linear model (TLM) of the nonlinear model. For a discrete TLM, the forward propagator can be expressed as a matrix, and computing the linear fastest growing perturbation is reduced to calculate the linear singular vector (LSV), which corresponds to the maximum singular value of the matrix. Lorenz (1965) introduced LSV and linear singular value (LSVA) into meteorology to investigate the predictability of the atmospheric motion. Buizza and Palmer (1995) utilized LSVs to study the patterns of the atmospheric general circulations. Recently, this method has been used to find out the initial condition for optimal growth in a coupled ocean-atmosphere model of El Niño-Southern Oscillation (ENSO), in an attempt to explore error growth and predictability of the coupled model (Xue and Cane, 1997a, b; Thompson, 1998; Samelson et al., 2001).

The motions of the atmosphere and ocean are governed by complicated nonlinear systems. This raises a few questions concerning the validity of TLM. One is how small the initial perturbation should be to guarantee the validity of TLM; another is how to determine the time interval during which the TLM is valid. For the nonlinear systems in the numerical weather and climate prediction, it is desirable and often necessary to deal with the nonlinear models themselves rather than their linear approximations.

Mu (2000) formulated a novel concept of nonlinear singular value (NSVA) and nonlinear singular vector (NSV), which is a natural generalization to the classical LSVA and LSV.

Mu and Wang (2001) used a two-dimensional quasi-geostrophic model to study the NSVA and NSV. The numerical results demonstrate that for some types of basic states, there exist local fastest growing perturbations, which correspond to the local maximum values of the functional with which NSVA and NSV are determined. But there is no such phenomenon in the case of LSVs and LSVAs due to the absence of the nonlinearity of the corresponding TLM. The local fastest growing perturbations usually possess larger norms comparing to the first NSV, which corresponds to the global maximum value of functional. Although the growth rates of the local fastest growing perturbations are smaller than the first NSVA, their nonlinear evolutions at the end of the time interval are considerably greater than that of the first NSV in terms of the chosen norm. In this case, the local fastest growing perturbations could play a more important role than the global fastest growing perturbation in the study of the predictability.

Implications of the results of Mu and Wang (2001) are clear. To investigate predictability, we should first find out all local fastest growing perturbations, then investigate their effects on the predictability. This is inconvenient in practical application. Besides, sometimes the local fastest growing perturbation with a large norm could be physically unreasonable.

These weaknesses suggest that we should investigate the nonlinear optimal perturbation with constrained conditions, which will be given below.



Mu, Duan and Wang (2003a) introduced the concept of conditional nonlinear optimal perturbation (CNOP), which is the initial perturbation corresponding to the maximum value of the functional with constraint conditions. Some applications of CNOP to the study of predictability of ENSO have been given. First, Mu and Duan (2003) used CNOP to study the precursors and the spring predictability barrier of ENSO event within the frame of a simple coupled ocean-atmosphere model for ENSO. The results suggest that for the proper constraint condition, the CNOPs of the climatological mean state evolve into ENSO events more probably than the linear singular vector (LSV). Consequently it is reasonable to regard CNOP as the optimal precursors of ENSO events. Observed anomalous monthly mean SST (°C) and depth of 20°C isotherm (m) derived from NCEP ocean reanalysis for the equatorial eastern Pacific (5°S-5°N, 150°-90°W) region verifies the existence of these optimal precursors qualitatively.

Secondly, the "spring predictability barrier" problem for ENSO event is also investigated (Mu and Duan, 2003). By computing the CNOPs of El Niño and La Niña events, it is found out that the error growth is enhanced in spring for El Nino event and suppressed in spring for La Niña event. To further investigate what causes the spring predictability barrier in the model, the CNOP of El Nino and La Niña events with strong and weak coupled ocean-atmosphere instability is also computed. It is shown that the strong coupled ocean-atmosphere instability during spring of the year is one of the causes of the spring predictability barrier. Sensitivity experiments show that the spring predictability barrier of ENSO event is more sensitive to the zonal sea surface temperature gradient than to the temperature gradient between the mixed-layer and subsurface-layer.




CNOP is also used by Mu et al. (2003b) to analyze the different sensitivities of the thermohaline circulation to finite amplitude freshwater and salt perturbations. Within the frame of a simple model for the thermohaline circulation, the impacts of nonlinearity on the evolution of the finite amplitude freshwater and salinity perturbations are investigated by CNOP approach. It is demonstrated that the thermohaline circulation is more unstable to the finite amplitude freshwater perturbation than to the salinity perturbation. From the sensitivity analysis of the thermohaline circulation to the freshwater and salinity perturbations along the bifurcation diagram, it is shown that the system becomes unstable near the bifurcation diagram regime, and a finite perturbation could lead to the shut-off of the thermohaline circulation.



The problem of predictability itself is essentially of an issue of spatial-temporal scale (Chou, 2002; Mu et al., 2002b; Li et al. 2003b). Predictability time strongly depends on the spatial-temporal scale of phenomenon studied. Moreover, it is also related to initial condition and external forcing condition. Generally, the predictability time  of the atmosphere is a function of spatial and temporal scales, initial field and external forcing, i.e.,

,                                 (1)

where D and  T are the space and time scales, respectively, X0 is the initial field, and F the external forcing. This suggests that predictability of a system is determined by the four factors of its space scale, time scale, initial condition and external forcing. By use of this function Li et al. (Li et al. 2003; see also Mu et al., 2002b) defined exactly the concepts of stable component and chaotic component in the atmosphere. They pointed out that under the same conditions of both initial field and external forcing larger, spatial-temporal scale system possesses longer predictability time. This property, which is similar to the Newton's Law of Inertia that the bigger the mass, the bigger is its inertia, is called the monotonicity principle of predictability. The principle shows that the model that is used to describe the sub-scale chaotic component (low level) of a system cannot be applied to predict the stable component (high level) of the system. To reduce blindness we should therefore focus on finding its corresponding stable component for prediction of a specified spatial-temporal scale system  (Li et al. 2003b).



The traditional researches on model predictability do not take machine precision into account. Based on the latest analysis, Li et al. (2000a,b; 2001) pointed out that this traditional manner is improper. On one hand, numerical methods themselves are not entire accuracy; on the other hand any computer used to run models possesses finite precision. This implies that model predictability time depends not only on numerical scheme, but also on machine precision as well as on model itself. Therefore, the model predictability time from the traditional researches is neither the predictability time of the real atmosphere or climate nor definite measure of optimal predictability time of model.

The results of Li et al. (2000a,b, 2001) indicated that round-off error due to finite machine precision has very significant influence in long-term numerical solutions of integration of model, and this finite precision in practice causes the computational uncertainty principle (Li et al., 2001). The principle suggests that there is a limit to the ability of effective simulation of computer. This limit is inherent and is independent of the objects simulated (more precisely except a zero measure set). The extent of capacity of effective simulation, however, usually depends on the objects simulated. In the light of the computational uncertainty principle a new approach can be presented to study model predictability time; that is, through carrying out optimal calculation of numerical model its maximum effective computational time (MECT) is obtained and then its predictability time in practice may be estimated. By use of the theory of the computational uncertainty principle Li et al. (2000b, 2003) presented an optimal numerical integration method of step-by-step adjustment, which may be used to arrive at the best predictability of numerical model.



The cell-to-cell mapping method (Hsu 1980, 1987; Chou 1986) is a powerful tool for globally analyzing nonlinear system. This method may be applied to quantitatively investigate the problem of predictability and to obtain global predictability limit of system for infinite initial conditions (Chou 1989). By introducing the cell-to-cell mapping method, Fan et al. (1999) studied the predictability of climate in a most simplified air-sea coupled model. Their results indicated that there exists a maximum predictability limit in climate prediction, and for the prediction beyond the daily predictability limit, mean value is predictable. They also obtained and discussed some related quantitative results. Moreover, their study implied that coupling mechanism could prolong predictability time, and improvement of observational error also might extend maximum predictability time.




By use of 500 hPa geopotential height data and China's 160-satation rainfall data, Zhu (1999) studied predictability of summer rainfall in China based on the relation between rainfall pattern and circulation in summer and preceding winter. Her results showed that the characteristics of simultaneous and preceding circulations are significantly different among the different rainfall patterns, and the predictability of summer rainfall is different in different areas of China, and there is more predictability in the lower and middle reaches of the Yangtze River than other regions over the East China.

Yue et al. (1999) investigated climatic noise and potential predictability of monthly mean temperature in China based on the analysis of China's 74-station temperature data. Their results implied that generally the climatic noise of monthly mean temperature in China increases with latitude and altitude, and varies with seasons, and the potential predictability possesses strongly seasonal and regional dependencies, and that the monthly mean temperature over China is potentially predictable at statistical significance level of 0.10. They further pointed out that for different seasons regional model could be a hopeful approach to predict the monthly mean temperature over China.



For the need of daily weather prediction and a call for the definition of ambient field as the initial input of regional forecasting model, Li et al. (2001) preliminarily diagnosed the predictability of T106 objective analysis/forecasting field and also discussed the possibilities to make the forecast of the prediction error in the T106 itself. They found that the major error source in T106 model is actually due to the fixed error existing in the T106 objective analysis field, and presented an assembly objective analysis field for weather forecaster to make correct analysis, and the field may be the better initial-boundary condition as input being used in the regional forecasting model.

By using an ensemble of nine 17-year (1980-1996) hindcast experiments conducted with the first generation of General Circulation Model (IAP L2 AGCM 1.1) of the Institute of Atmospheric Physics, Zhao et al. (2000) applied the Analysis of Variance (ANOVA) technique to study the predictability of numerical short-term climate prediction. The results showed that the predictability of atmospheric seasonal variations induced by SST anomalies is higher in tropics and moderate in extratropics except some areas. In extratropics, the predictability is higher in spring (MAM) than in summer (JJA). The predictability over ocean is generally higher than over land. In China, especially, the predictability decreases from the South China Sea to the northwest for the fields of precipitation, sea level pressure and air temperature.

Wang and Zhu (2000a, b) reviewed studies on seasonal climate prediction, and evaluated the level and skill of short-term climate prediction, and discussed the predictability of short-term climate forecast. They pointed out that the scale score of seasonal prediction at present is 0.2-0.3 for temperature and 0.1-0.2 for precipitation, which correspond to accuracy 60%-65% and 55%-60%, and that the theoretical limit of monthly- and seasonal-scale climate prediction is about 6-12 months.

Employing the IAP 2-level AGCM and LASG 9-level spectral AGCM, Long and Li (2001) simulated the influence of positive SSTAs with different duration over the eastern equatorial Pacific on the subtropical high over the western Pacific. Their results showed that the predictability of summer subtropical high over the western Pacific is determined by both the SSTAs over the equatorial eastern Pacific and the atmospheric internal dynamical process.

Acknowledgments: This work was jointly supported by the National Key Basic Research Project Research on the Formation mechanism and Prediction Theory of Severe Weather  Disasters in China (No. G1998040910), the National Natural Science Foundation of China (Nos. 40023001, 40075015, 40221503), and  Chinese Academy of Sciences (No. KZCX2-208).


Buizza, R., and T. N. Palmer, 1995: The singular-vector structure of the atmospheric global circulation. J. Atmos. Sci., 52, 1434-1456.
Chou J. F., 1986: Some general properties of the atmospheric model in H space, R space, point mapping, cell mapping, in: Proceedings of International Summer Colloquium on Nonlinear Dynamics of the Atmosphere (Edited by Zeng Qingcun), 10-20 Aug. Beijing: Science Press, 187-189.
Chou J. F., 1989: Predictability of the atmosphere. Adv. Atmos. Sci., 6, 335-346.
Chou J. F., 2002: Nonlinearity and Complexity in Atmospheric Sciences. Beijing: China Meteorological Press, 131-166.
Fan X. G., Zhang H. L., and Chou J. F., 1999: Global study on climate predictability. Acta Meteor. Sin., 57, 190-197 (in Chinese) .
Hsu, C. S., 1980: A generalized theory of cell to cell mapping dynamical systems. ASME J. Appl. Mech., 47, 931-939.
Hsu, C. S., 1987: Cell to Cell MappingA Method of Global Analysis for Nonlinear System. New York: Springer Verlag, pp 358.
Li C., Wang Y., Pan Y. N., Zhang L. J., and Yu B., 2001:  The preliminary analyses on the reliability and predictability of T106 objective analysis/forecasting field, Sci. Meteor. Sin., 21, 379-391 (in Chinese) .
Li J. P., Zeng Q. C., and Chou J. F., 2000a: Computational uncertainty principle in nonlinear ordinary differential equations. I Numerical results. Science in China (Series E), 43, 449-460.
Li J. P., 2000b: Computational uncertainty principle: meaning and implication. Bull. Chinese Academy Sci., 15, 428-430. (in Chinese)
Li J. P., Zeng Q. C., and Chou J. F., 2001: Computational uncertainty principle in nonlinear ordinary differential equations: II Theoretical analysis. Science in China (Series E), 44, 55-74.
Li J. P., and Chou J. F., 2003a: The global analysis theory of climate system and its applications. Chinese Sci. Bull., 48, in press.
Li J. P., and Chou J. F., 2003b: Advances in nonlinear atmospheric dynamics. Chinese J. Atmos. Sci., 27, in press.
Lorenz, E. N., 1965: A study of the predictability of a 28-variable atmospheric model. Tellus, 17, 321-333.
Long Z. X., and Li C. Y., 2001: Simulating influence of possible sea surface temperature anomalies over the eastern equatorial on subtropical high over the western Pacific, Chinese J. Atmos. Sci., 25, 145-159.
Mu M., 2000: Nonlinear singular vectors and nonlinear singular values. Science in China (D), 43, 375-385.
Mu M., and Wang J. C., 2001: Nonlinear fastest growing perturbation and the first kind of predictability. Science in China (D), 44, 1128-1139.
Mu M., Duan W. S., and Wang J. C., 2002a: The predictability problems in numerical weather and climate prediction. Adv. Atmos. Sci., 19, 191-204.
Mu M., Li J. P., Chou J. F., Duan W. S., and Wang J. C., 2002b: Theoretical research on the predictability of climate system. Clim. Envir. Res., 7, 227-235. (in Chinese)
Mu M., and Duan W. S., 2003: A new approach to study ENSO predictability: conditional nonlinear optimal perturbation. Chinese Sci. Bull., to appear.
Mu M., Duan W. S., and B. Wang, 2003a: Conditional nonlinear optimal perturbation and its application. Nonlinear Processes in Geophysics., submitted.