Abstract
The backward nonlinear local Lyapunov exponent method (BNLLE) is applied to quantify the predictability of warm and cold events in the Lorenz model. Results show that the maximum prediction lead times of warm and cold events present obvious layered structures in phase space. The maximum prediction lead times of each warm (cold) event on individual circles concentric with the distribution of warm (cold) regime events are roughly the same, whereas the maximum prediction lead time of events on other circles are different. Statistical results show that warm events are more predictable than cold events.
摘 要
本文基于向后非线性局部 Lyapunov 指数 (backward nonlinear local Lyapunov exponent, BNLLE) 方法, 定量研究了 Lorenz 模型中冷暖事件的可预报性. 研究结果显示, 冷暖事件的最长提前预报时间在相空间中呈现明显的层状结构. 在暖 (冷) 流型的每一圈层上, 所有暖 (冷) 事件的最长提前预报时间基本一致, 而在不同的圈层上, 暖 (冷) 事件的最长提前预报时间则不同. 基于统计结果表明, 暖事件比冷事件更加容易预报.
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Carbone, R. E., J. D. Tuttle, D. A. Ahijevych, and S. B. Trier, 2002: Inferences of predictability associated with warm season precipitation episodes. J. Atmos. Sci., 59, 2033–2056, https://doi.org/10.1175/1520-0469(2002)059<2033:IOPAWW>2.0.CO;2.
Charney, J. G, 1966: The feasibility of a global observation and analysis experiment. Bull. Amer. Meteorol. Soc., 47, 200–221, https://doi.org/10.1175/1520-0477-47.3.200.
Chen, D. K., S. E. Zebiak, A. J. Busalacchi, and M. A. Cane, 1995: An improved procedure for EI Niño forecasting: Implications for predictability. Science, 269, 1699–1702, https://doi.org/10.1126/science.269.5231.1699.
Dalcher, A., and E. Kalnay, 1987: Error growth and predictability in operational ECMWF forecasts. Tellus A, 39, 474–491, https://doi.org/10.3402/tellusa.v39i5.11774.
Dambacher, J. M., H. W. Li, and P. A. Rossignol, 2003: Qualitative predictions in model ecosystems. Ecological Modelling, 161, 79–93, https://doi.org/10.1016/S0304-3800(02)00295-8.
Ding, R. Q., and J. P. Li, 2007: Nonlinear finite-time Lyapunov exponent and predictability. Physics Letters A, 364, 396–400, https://doi.org/10.1016/j.physleta.2006.11.094.
Ding, R. Q., J. P. Li, and H. A. Kyung-Ja, 2008: Nonlinear local Lyapunov exponent and quantification of local predictability. Chinese Physics Letters, 25, 1919–1922, https://doi.org/10.1088/0256-307X/25/5/109.
Duan, W. S., and M. Mu, 2005: Applications of nonlinear optimization methods to quantifying the predictability of a numerical model for El Nino-Southern Oscillation. Progress in Natural Science, 15, 915–921, https://doi.org/10.1080/10020070512331343110.
Duan, W. S., and H. Y. Luo, 2010: A new strategy for solving a class of constrained nonlinear optimization problems related to weather and climate predictability. Adv. Atmos. Sci., 27, 741–749, https://doi.org/10.1007/s00376-009-9141-0.
Duan, W. S., and J. Y. Hu, 2016: The initial errors that induce a significant “spring predictability barrier” for El Niño events and their implications for target observation: Results from an earth system model. Climate Dyn., 46, 3599–3615, https://doi.org/10.1007/s00382-015-2789-5.
Duan, W. S., R. Q. Ding, and F. F. Zhou, 2013: Several dynamical methods used in predictability studies for numerical weather forecasts and climate prediction. Climatic and Environmental Research, 18, 524–538, https://doi.org/10.3878/j.issn.1006-9585.2012.12009. (in Chinese with English abstract)
Eckmann, J. P., and D. Ruelle, 1985: Ergodic theory of chaos and strange attractors. Reviews of Modern Physics, 57, 617–656, https://doi.org/10.1103/RevModPhys.57.617.
Evans, E., N. Bhatti, J. Kinney, L. Pann, P. Malaquias, S. C. Yang, E. Kalnay, and J. Hansen, 2004: RISE: Undergraduates find that regime changes in Lorenz’s model are predictable. Bull. Amer. Meteorol. Soc., 85, 520–524, https://doi.org/10.1175/BAMS-85-4-520.
Feng, J., R. Q. Ding, D. Q. Liu, and J. P. Li, 2014: The application of nonlinear local Lyapunov vectors to ensemble predictions in Lorenz systems. J. Atmos. Sci., 71, 3554–3567, https://doi.org/10.1175/JAS-D-13-0270.1.
Fraedrich, K., 1986: Estimating the dimensions of weather and climate attractors. J. Atmos. Sci., 43, 419–432, https://doi.org/10.1175/1520-0469(1986)043<0419:ETDOWA>2.0.CO;2.
Fraedrich, K., 1987: Estimating weather and climate predictability on attractors. J. Atmos. Sci., 44, 722–728, https://doi.org/10.1175/1520-0469(1987)044<0722:EWACPO>2.0.CO;2.
Islam, S., R. L. Bras, and K. A. Emanuel, 1993: Predictability of mesoscale rainfall in the tropics. J. Appl. Meteorol., 32, 297–310, https://doi.org/10.1175/1520-0450(1993)032<0297:POMRIT>2.0.CO;2.
Lavaysse, C., G. Naumann, L. Alfieri, P. Salamon, and J. Vogt, 2019: Predictability of the European heat and cold waves. Clim. Dyn., 52, 2481–2495, https://doi.org/10.1007/s00382-018-4273-5.
Leith, C., 1965: Numerical simulation of the earth’s atmosphere. Methods in Computational Physics, Academic Press, 1–28.
Leith, C. E., 1978: Predictability of climate. Nature, 276, 352–355, https://doi.org/10.1038/276352a0.
Li, J. P., and R. Q. Ding, 2011: Temporal-spatial distribution of atmospheric predictability limit by local dynamical analogs. Mon. Wea. Rev., 139, 3265–3283, https://doi.org/10.1175/MWR-D-10-05020.1.
Li, J. P., and R. Q. Ding, 2013: Temporal-spatial distribution of the predictability limit of monthly sea surface temperature in the global oceans. International Journal of Climatology, 33, 1936–1947, https://doi.org/10.1002/joc.3562.
Li, X., R. Q. Ding, and J. P. Li, 2019: Determination of the backward predictability limit and its relationship with the forward predictability limit. Adv. Atmos. Sci., 36, 669–677, https://doi.org/10.1007/s00376-019-8205-z.
Lorenz, E. N., 1963: Deterministic nonperiodic flow. J. Atmos. Sci., 20, 130–141, https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.
Lorenz, E. N., 1982: Atmospheric predictability experiments with a large numerical model. Tellus, 34, 505–513, https://doi.org/10.1111/j.2153-3490.1982.tb01839.x.
Lorenz, E. N., 1996: Predictability: A problem partly solved. Proc. ECMWF Seminar on Predictability, Vol. I, Reading, United Kingdom, ECMWF, 1–18.
Mintz, Y., 1968: Very long-term global integration of the primitive equations of atmospheric motion: An experiment in climate simulation. Causes of Climatic Change, D. E. Billings et al., Eds., Springer, 20–36, https://doi.org/10.1007/978-1-935704-38-6_3.
Mu, M., W. S. Duan, and J. C. Wang, 2002: The predictability problems in numerical weather and climate prediction. Adv. Atmos. Sci., 19, 191–204, https://doi.org/10.1007/s00376-002-0016-x.
Mu, M., W. S. Duan, and B. Wang, 2003: Conditional nonlinear optimal perturbation and its applications. Nonlinear Processes in Geophysics, 10, 493–501, https://doi.org/10.5194/npg-10-493-2003.
Mu, M., F. F. Zhou, and H. L. Wang, 2009: A method for identifying the sensitive areas in targeted observations for tropical cyclone prediction: Conditional nonlinear optimal perturbation. Mon. Wea. Rev., 137, 1623–1639, https://doi.org/10.1175/2008MWR2640.1.
Mukougawa, H., M. Kimoto, and S. Yoden, 1991: A relationship between local error growth and quasi-stationary states: Case study in the Lorenz system. J. Atmos. Sci., 48, 1231–1237, https://doi.org/10.1175/1520-0469(1991)048<1231:ARBLEG>2.0.CO;2.
Nese, J. M., 1989: Quantifying local predictability in phase space. Physica D: Nonlinear Phenomena, 35, 237–250, https://doi.org/10.1016/0167-2789(89)90105-X.
Palmer, T. N., 1993: Extended-range atmospheric prediction and the Lorenz model. Bull. Amer. Meteorol. Soc., 74, 49–66, https://doi.org/10.1175/1520-0477(1993)074<0049:ERAPAT>2.0.CO;2.
Reynolds, C. A., P. J. Webster, and E. Kalnay, 1994: Random error growth in NMC’s global forecasts. Mon. Wea. Rev., 122, 1281–1305, https://doi.org/10.1175/1520-0493(1994)122<1281:REGING>2.0.CO;2.
Smagorinsky, J., 1969: Problems and promises of deterministic extended range forecasting. Bull. Amer. Meteorol. Soc., 50, 286–312, https://doi.org/10.1175/1520-0477-50.5.286.
Snyder, C., and F. Q. Zhang, 2003: Assimilation of simulated Doppler radar observations with an ensemble Kalman filter. Mon. Wea. Rev., 131, 1663–1677, https://doi.org/10.1175//2555.1.
Tang, Y. M., H. Lin, and A. M. Moore, 2008: Measuring the potential predictability of ensemble climate predictions. J. Geo-phys. Res. Atmos., 113, D04108, https://doi.org/10.1029/2007jd008804.
Thompson, P. D., 1957: Uncertainty of initial state as a factor in the predictability of large scale atmospheric flow patterns. Tellus, 9, 275–295, https://doi.org/10.1111/j.2153-3490.1957.tb01885.x.
Weisheimer, A., F. J. Doblas-Reyes, T. Jung, and T. N. Palmer, 2011: On the predictability of the extreme summer 2003 over Europe. Geophys. Res. Lett., 38, L05704, https://doi.org/10.1029/2010GL046455.
Wolf, A., J. B. Swift, H. L. Swinney, and J. A. Vastano, 1985: Determining Lyapunov exponents from a time series. Physica D: Nonlinear Phenomena, 16, 285–317, https://doi.org/10.1016/0167-2789(85)90011-9.
Yoden, S., and M. Nomura, 1993: Finite-time Lyapunov stability analysis and its application to atmospheric predictability. J. Atmos. Sci., 50, 1531–1543, https://doi.org/10.1175/1520-0469(1993)050<1531:FTLSAA>2.0.CO;2.
Ziehmann, C., L. A. Smith, and J. Kurths, 2000: Localized Lyapunov exponents and the prediction of predictability. Physics Letters A, 271, 237–251, https://doi.org/10.1016/S0375-9601(00)00336-4.
Acknowledgements
This work was jointly supported by the National Natural Science Foundation of China (Grant No. 41790474) and the National Program on Global Change and Air-Sea Interaction (GASI-IPOVAI-03 GASI-IPOVAI-06).
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Article Highlights
• A new method is introduced to quantify predictabilities of warm and cold events, and gives their algorithms.
• Statistical results indicate that warm events are more predictable than cold events.
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Li, X., Ding, R. & Li, J. Quantitative Comparison of Predictabilities of Warm and Cold Events Using the Backward Nonlinear Local Lyapunov Exponent Method. Adv. Atmos. Sci. 37, 951–958 (2020). https://doi.org/10.1007/s00376-020-2100-5
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DOI: https://doi.org/10.1007/s00376-020-2100-5
Key words
- backward nonlinear local Lyapunov exponent
- maximum prediction lead time
- layered structure
- statistical result