Skip to main content

Nonlinear local Lyapunov exponent and atmospheric predictability research

Abstract

Because atmosphere itself is a nonlinear system and there exist some problems using the linearized equations to study the initial error growth, in this paper we try to use the error nonlinear growth theory to discuss its evolution, based on which we first put forward a new concept: nonlinear local Lyapunov exponent. It is quite different from the classic Lyapunov exponent because it may characterize the finite time error local average growth and its value depends on the initial condition, initial error, variables, evolution time, temporal and spatial scales. Based on its definition and the atmospheric features, we provide a reasonable algorithm to the exponent for the experimental data, obtain the atmospheric initial error growth in finite time and gain the maximal prediction time. Lastly, taking 500 hPa height field as example, we discuss the application of the nonlinear local Lyapunov exponent in the study of atmospheric predictability and get some reliable results: atmospheric predictability has a distinct spatial structure. Overall, predictability shows a zonal distribution. Prediction time achieves the maximum over tropics, the second near the regions of Antarctic, it is also longer next to the Arctic and in subtropics and the mid-latitude the predictability is lowest. Particularly speaking, the average prediction time near the equation is 12 days and the maximum is located in the tropical Indian, Indonesia and the neighborhood, tropical eastern Pacific Ocean, on these regions the prediction time is about two weeks. Antarctic has a higher predictability than the neighboring latitudes and the prediction time is about 9 days. This feature is more obvious on Southern Hemispheric summer. In Arctic, the predictability is also higher than the one over mid-high latitudes but it is not pronounced as in Antarctic. Mid-high latitude of both Hemispheres (30°S–60°S, 30°–60°N) have the lowest predictability and the mean prediction time is just 3–4 d. In addition, predictability varies with the seasons. Most regions in the Northern Hemisphere, the predictability in winter is higher than that in summer, especially in the mid-high latitude: North Atlantic, North Pacific and Greenland Island. However in the Southern Hemisphere, near the Antarctic regions (60°S–90°S), the corresponding summer has higher predictability than its winter, while in other areas especially in the latitudes of 30°S–60°S, the prediction does not change obviously with the seasons and the average time is 3–5 d. Both the theoretical and data computation results show that nonlinear local Lyapunov exponent and the nonlinear local error growth really may measure the predictability of the atmospheric variables in different temporal and spatial scales.

This is a preview of subscription content, access via your institution.

References

  1. 1

    Chou J F. Predictability of the atmosphere, Adv Atmos Sci, 1989, 6: 335–346

    Google Scholar 

  2. 2

    Thompson P D. Uncertainty of initial state as a factor in the predictability of large-scale atmospheric flow patterns. Tellus, 1957, 9: 275–295

    Google Scholar 

  3. 3

    Lorenz E N. Three approaches to atmospheric predictability. Bull Ame Meteor Soc, 1969, 50: 345–349

    Google Scholar 

  4. 4

    Charney J G, Fleagle R G, Riehl H, et al. The feasibility of a global observation and analysis experiment. Bull Amer Meteor Soc, 1966, 47: 200–220.

    Google Scholar 

  5. 5

    Smagorinsky J. Problems and promises of deterministic extended range forecasting. Bull Amer Meteor Soc, 1969, 50: 286–311

    Google Scholar 

  6. 6

    Lorenz E N. A study of the predictability of a 28 variable atmospheric model. Tellus, 1965, 17: 321–333

    Google Scholar 

  7. 7

    Lorenz E N. Atmospheric predictability experiments with a large numerical model. Tellus, 1982, 34: 505–513

    Google Scholar 

  8. 8

    Li J P, Zeng Q C, Chou J F. Computational uncertainty principle in nonlinear ordinary differential equationas I. numerical results. Sci China Ser E-Eng Mater Sci, 2000,43(5): 449–460

    Google Scholar 

  9. 9

    Li J P, Zeng Q C, Chou J F. Computational uncertainty principle in nonlinear ordinary differential equations II. theoretical analysis. Sci China Ser E-Eng Mater Sci, 2001, 44(1): 55–74

    Article  Google Scholar 

  10. 10

    Mu M, Li J P, Chou J F, et al. Theoretical research on the predictability of climate system. Clim Environ Res (in Chinese), 2002, 7(2): 227–235

    Google Scholar 

  11. 11

    Feng G L, Dai X G, Wang A H, et al. The study of the predictability in chaotic systems. Chin Phys (in Chinese), 2001, 50: 606–611

    Google Scholar 

  12. 12

    Gao X Q, Feng G L, Chou J F, et al. On the predictability of chaotic systems with respect to maximally effective computation time. Acta Meteorol Sin, 2003, 19: 134–139

    Google Scholar 

  13. 13

    Dalcher A, Kalnay E. Error growth and predictability in operational ECMWF forecasts. Tellus A, 1987, 39: 474–491

    Google Scholar 

  14. 14

    Arpe K, Klinker E. Systematic errors of the ECMWF operational forecasting model in mid-latitudes. Quart J Roy Meteor Soc, 1986, 112: 181–202

    Article  Google Scholar 

  15. 15

    Yang P C, Chen L T. The predictability of El Niño/Southern Oscillation. J Atmos Sci (in Chinese), 1990, 14: 64–71

    Google Scholar 

  16. 16

    Yang P C, Liu J L, Yang S W. The strange attractor of low atmospheric movement. J Atmos Sci (in Chinese), 1990, 14: 335–341

    Google Scholar 

  17. 17

    Zheng Z G, Liu S D. Computation of Lyapunov exponent and fractal dimension by using atmospheric turbulent data. Acta Meteorol Sin (in Chinese), 1988, 41–48

  18. 18

    Farmer J D, Ott E, Yorke J A. The dimension of chaotic attractors. Physica D, 1983, 7: 153–180

    Article  Google Scholar 

  19. 19

    Grassberger P, Procaccia I. Dimensions and entropies of strange attractors from a fluctuating dynamics approach. Physica D, 1984, 13: 34–54

    Article  Google Scholar 

  20. 20

    Legras B, Ghil M. Persistent anomalies, blocking and variations in atmospheric predictability. J Atmos Sci, 1985, 42: 433–471

    Article  Google Scholar 

  21. 21

    Nese J M. PhD dissertation, Pennsylvania State University, 1989

  22. 22

    Nese J M. Quantifying local predictability in phase space. Physica D, 1989, 35: 237–250

    Article  Google Scholar 

  23. 23

    Farrell B F. Small error dynamics and the predictability of atmospheric flows. J Atmos Sci, 1990, 47: 2409–2416

    Article  Google Scholar 

  24. 24

    Houtekamer P L. Variation of the predictability in a low-order spectral model of the atmospheric circulation. Tellus A, 1991, 43(3): 177190

    Google Scholar 

  25. 25

    Yoden S, Nomura M. Finite-time Lyapunov stability analysis and its application to Atmospheric Predictability. J Atmos Sci, 1993, 50: 1531–1543

    Article  Google Scholar 

  26. 26

    Kazantsev E. Local Lyapunov exponents of the quasi-geostrophic ocean dynamics. Appl Math Comp, 1999, 104: 217–257

    Article  Google Scholar 

  27. 27

    Ziehmann C, Smith L A, Jürgen K. Localized Lyapunov exponents and the prediction of predictability. Phys Lett A, 2000, 271: 237–251

    Article  Google Scholar 

  28. 28

    Oseledec V I. A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans Moscow Math Soc, 1968, 19: 197–231

    Google Scholar 

  29. 29

    Lacarra J F, Talagrand O. Short-range evolution of small perturbations in a barotropics model. Tellus A, 1988, 40: 81–95

    Article  Google Scholar 

  30. 30

    Tanguay M, Bartello P, Gauthier P. Four-dimension data assimilation with a wide range of scales. Tellus A, 1995, 47: 974–997

    Article  Google Scholar 

  31. 31

    Mu M, Guo H, Wang J F, et al. The impact of nonlinear stability and instability on the validity of the tangent linear model. Adv Atmos Sci, 2000, 17(3): 375–390

    Google Scholar 

  32. 32

    Wolf A, Swift J B, Swinney H L, et al. Determining Lyapunov Exponents from a time series. Physica D, 1985, 16: 285–317

    Article  Google Scholar 

  33. 33

    Li J P, Chou J F. Some problems existed in estimating fractal dimension of attractor with one-dimensional time series. Acta Meteorol Sin (in Chinese), 1996, 54(3): 312–323

    Google Scholar 

  34. 34

    Reichler T, Roads J O. Time-space distribution of Long-Range Atmospheric Predictability. J Atmos Sci, 2004, 61(3): 249–263

    Article  Google Scholar 

  35. 35

    Kumar A, Schubert S D, Suarez M S. Variability and predictability of 200-mb seasonal mean heights during summer and winter. J Geophys Res, 2003, 108(D5), 4169, doi: 10.102/2002JD002728

    Article  Google Scholar 

  36. 36

    Bacmeister J T, Pegion P J, Schubert S D, et al. Atlas of seasonal means simulated by the NSIPP1 atmospheric GCM, NASA Tech. Memo. 104606, Goddard Space Flight Center, 2000, 194

  37. 37

    Trenberth K E. Potential predictability of geopotential heights over the Southern Hemisphere. Mon Wea Rev, 1985, 113: 54–64

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Li Jianping.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Chen, B., Li, J. & Ding, R. Nonlinear local Lyapunov exponent and atmospheric predictability research. SCI CHINA SER D 49, 1111–1120 (2006). https://doi.org/10.1007/s11430-006-1111-0

Download citation

Keywords

  • nonlinear
  • local
  • Lyapunov exponent
  • atmospheric predictability
  • maximal prediction time