Science in China Series D: Earth Sciences

, Volume 52, Issue 7, pp 883–906

Conditional nonlinear optimal perturbation: Applications to stability, sensitivity, and predictability

Article

Abstract

Conditional nonlinear optimal perturbation (CNOP) is a nonlinear generalization of linear singular vector (LSV) and features the largest nonlinear evolution at prediction time for the initial perturbations in a given constraint. It was proposed initially for predicting the limitation of predictability of weather or climate. Then CNOP has been applied to the studies of the problems related to predictability for weather and climate. In this paper, we focus on reviewing the recent advances of CNOP’s applications, which involves the ones of CNOP in problems of ENSO amplitude asymmetry, block onset, and the sensitivity analysis of ecosystem and ocean’s circulations, etc. Especially, CNOP has been primarily used to construct the initial perturbation fields of ensemble forecasting, and to determine the sensitive area of target observation for precipitations. These works extend CNOP’s applications to investigating the nonlinear dynamical behaviors of atmospheric or oceanic systems, even a coupled system, and studying the problem of the transition between the equilibrium states. These contributions not only attack the particular physical problems, but also show the superiority of CNOP to LSV in revealing the effect of nonlinear physical processes. Consequently, CNOP represents the optimal precursors for a weather or climate event; in predictability studies, CNOP stands for the initial error that has the largest negative effect on prediction; and in sensitivity analysis, CNOP is the most unstable (sensitive) mode. In multi-equilibrium state regime, CNOP is the initial perturbation that induces the transition between equilibriums most probably. Furthermore, CNOP has been used to construct ensemble perturbation fields in ensemble forecast studies and to identify sensitive area of target observation. CNOP theory has become more and more substantial. It is expected that CNOP also serves to improve the predictability of the realistic predictions for weather and climate events plays an increasingly important role in exploring the nonlinear dynamics of atmospheric, oceanic and coupled atmosphere-ocean system.

Keywords

optimal perturbation predictability stability sensitivity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Tennekes H. Karl Popper and the accountability of numerical forecasting. ECMWF Workshop Proceedings. New Developments in Predictability. London: European Centre for Medium-Range Weather Forecasts, 1991Google Scholar
  2. 2.
    Thompson P. Uncertainty of the initial state as a factor in the predictability of large scale atmospheric flow patterns. Tellus, 1957, 9: 275–295Google Scholar
  3. 3.
    Palmer T N, Molteni F, Mureau R, et al. Ensemble prediction. ECMWF Res Department Tech Memo, 1992, 188: 45Google Scholar
  4. 4.
    Toth Z, Kalnay E. Ensemble forecasting at NMC: The generation of perturbations. Bull Amer Meteor Soc, 1993, 74: 2317–2330CrossRefGoogle Scholar
  5. 5.
    Mu M, Zhang Z Y. Conditional nonlinear optimal perturbations of a two-dimensional quasigeostrophic model. J Atmos Sci, 2006, 63: 1587–1604CrossRefGoogle Scholar
  6. 6.
    Moore A M, Kleeman R. The dynamics of error growth and predictability in a coupled model of ENSO. Q J R Meteorol Soc, 1996, 122: 1405–1446CrossRefGoogle Scholar
  7. 7.
    Samelson R G, Tziperman E. Instability of the chaotic ENSO: The growth-phase predictability barrier. J Atmos Sci, 2001, 58: 3613–3625CrossRefGoogle Scholar
  8. 8.
    Duan W S, Mu M. Application of nonlinear optimization method to quantifying the predictability of a numerical model for El Nino-Southern Oscillation. Prog Nat Sci, 2005, 15(10): 915–921Google Scholar
  9. 9.
    Mu M, Duan W S, Wang B. Season-dependent dynamics of nonlinear optimal error growth and El Nino-Southern Oscillation predictability in a theoretical model. J Geophys Res, 2007, 112: D10113, doi: 10.1029/2005JD006981CrossRefGoogle Scholar
  10. 10.
    Mu M, Xu H, Duan W S. A kind of initial errors related to “spring predictability barrier” for El Nino events in Zebiak-Cane model. Geophys Res Lett, 2007, 34: L03709, doi: 0.1029/2006GL027412CrossRefGoogle Scholar
  11. 11.
    Smith L A, Ziehmann C, Fraedrich K. Uncertainty dynamics and predictability in chaotic systems. Q J R Meteorol Soc, 1999, 125: 2855–2886CrossRefGoogle Scholar
  12. 12.
    Lorenz E N. A study of the predictability of a 28-variable atmospheric model. Tellus, 1965, 17: 321–333CrossRefGoogle Scholar
  13. 13.
    Xue Y, Cane M A, Zebaik S E. Predictability of a coupled model of ENSO using singular vector analysis. Part I: Optimal growth in seasonal background and ENSO cycles. Mon Weather Rev, 1997, 125: 2043–2056CrossRefGoogle Scholar
  14. 14.
    Buizza R, Molteni F. The role of finite-time barotropic instability during the transition to blocking. J Atmos Sci, 1996, 53: 1675–1697CrossRefGoogle Scholar
  15. 15.
    Frederisen J S. Adjoint sensitivity and finite time normal mode disturbances during blocking. J Atmos Sci, 1997, 47: 2409–2416Google Scholar
  16. 16.
    Tziperman E, Ioannou P J. Transient growth and optimal excitation of thermohaline variability. J Phys Oceanogr, 2002, 32: 3427–3435CrossRefGoogle Scholar
  17. 17.
    Mu M. Nonlinear singualr vectors and nonlinear singular values. Sci China Ser D-Earth Sci, 2000, 43(4): 375–385CrossRefGoogle Scholar
  18. 18.
    Mu M, Duan W S, Wang B. Conditional nonlinear optimal perturbation and its applications. Non Proc Geophys, 2003, 10: 493–501Google Scholar
  19. 19.
    Mu M, Wang J C. Nonlinear fastest growing perturbation and the first kind of predictability. Sci China Ser D-Earth Sci, 2001, 44(12): 1128–1139CrossRefGoogle Scholar
  20. 20.
    Mu M, Duan W S. Conditional nonlinear optimal perturbation and its applications to the studies of weather and climate predictability. Chin Sci Bull, 2005, 50: 2401–2407CrossRefGoogle Scholar
  21. 21.
    Mu M, Duan W S, Xu H, et al. Applications of conditional nonlinear optimal perturbation in predictability study and sensitivity analysis of weather and climate. Adv Atmos Sci, 2006, 23(6): 992–1002CrossRefGoogle Scholar
  22. 22.
    Duan W S, Mu M, Wang B. Conditional nonlinear optimal perturbation as the optimal precursors for ENSO events. J Geophys Res, 2004, 109: D23105CrossRefGoogle Scholar
  23. 23.
    Mu M, Sun L, Henk D A. The sensitivity and stability of the ocean’s thermocline circulation to finite amplitude freshwater perturbations. J Phys Oceanogr, 2004, 34: 2305–2315CrossRefGoogle Scholar
  24. 24.
    Sun L, Mu M, Sun D J, et al. Passive mechanism decadal variation of thermohaline circulation. J Geophys Res, 2005, 110: C07025, doi: 10.1029/2005JC002897CrossRefGoogle Scholar
  25. 25.
    Liu Y M. Maximum principle of conditional nonlinear optimal perturbation (in Chinese). J East Chin Norm Univ (Nat Sci), 2008, (2): 131–134Google Scholar
  26. 26.
    Riviere O, Lapeyre G, Talagrand O. Nonlinear generalization of singular vectors: Behavior in a baroclinic unstable flow. J Atmos Sci, 2008, 65: 1896–1911CrossRefGoogle Scholar
  27. 27.
    Terwisscha van Scheltinga A D. Data assimilation with implicit ocean models. PhD dissertation. Utrecht: Institute for Marine and Atmospheric Research, Utrecht University, 2007. 119Google Scholar
  28. 28.
    Powell M J D. VMCWD: A Fortran subroutine for constrained optimization. Acm Sigmap Bull, 1983, 32: 4–16CrossRefGoogle Scholar
  29. 29.
    Birgin E G, Martinez J M, Raydan M. Nonmonotone spectral projected gradient methods for convex sets. Siam J Opt, 2000, 10(4): 1196–1211CrossRefGoogle Scholar
  30. 30.
    Jiang Z N, Mu M, Wang D H. Conditional nonlinear optimal perturbation of a T21L3 quasi-geostrophic model. Q J R Meteorol Soc, 2008, 134: 1027–1038CrossRefGoogle Scholar
  31. 31.
    Rex D F. Blocking action in the middle troposphere and its effects upon regional climate. I: An aerological study of blocking action. Tellus, 1950, 2: 196–211CrossRefGoogle Scholar
  32. 32.
    Molteni F, Palmer T N. Predictability and finite-time instability of the northern winter circulation. Quart J Roy Meteor Soc, 1993, 119: 269–298CrossRefGoogle Scholar
  33. 33.
    Buizza R, Molteni F. The role of finite-time barotropic instability during the transition to blocking. J Atmos Sci, 1996, 53: 1675–1697CrossRefGoogle Scholar
  34. 34.
    Frederiksen J S. Singular vector, finite-time normal modes, and error growth during blocking. J Atmos Sci, 2000, 57: 312–333CrossRefGoogle Scholar
  35. 35.
    Mu M, Jiang Z N. A method to find out the perturbations triggering the blocking onset: Conditional nonlinear optimal perturbations. J Atmos Sci, 2008, 65: 3935–3946CrossRefGoogle Scholar
  36. 36.
    Jin F F, An S I, Timmermann A, et al. Strong El Nino events and nonlinear dynamical heating. Geophys Res Lett, 2003, 30: 1120, doi: 10.1029/2002GL016356CrossRefGoogle Scholar
  37. 37.
    An S I, Jin F F. Nonlinearity and asymmetry of ENSO. J Clim, 2004, 17: 2399–2412CrossRefGoogle Scholar
  38. 38.
    Wang B, An S I. Why the properties of El Nino changed during the late 1970s. Geophys Res Lett, 2001, 28: 3709–3712CrossRefGoogle Scholar
  39. 39.
    Duan W S, Mu M. Investigating decadal variability of El Nino-Southern Oscillation asymmetry by conditional nonlinear optimal perturbation. J Geophys Res, 2006, 111: C07015, doi: 10.1029/2005JC003458CrossRefGoogle Scholar
  40. 40.
    Philander S G H. El Nino Southern Oscillation phenomena. Nature, 1983, 302: 295CrossRefGoogle Scholar
  41. 41.
    Jin F F. An equatorial ocean recharge paradigm for ENSO. Part I: Conceptual model. J Atmos Sci, 1997, 54: 811–829CrossRefGoogle Scholar
  42. 42.
    Rodgers K B, Friederichs P, Latif M. Tropical Pacific decadal variability and its relation to decadal modulations of ENSO. J Clim, 2004, 17: 3761–3774CrossRefGoogle Scholar
  43. 43.
    Duan W S, Xu H, Mu M. Decisive role of nonlinear temperature advection in El Nino and La Nina amplitude asymmetry. J Geophys Res, 2008, 113: C01014, doi: 10.1029/2006JC003974CrossRefGoogle Scholar
  44. 44.
    Wang B, Fang Z. Chaotic oscillation of tropical climate: A dynamic system theory for ENSO. J Atmos Sci, 1996, 53: 2786–2802CrossRefGoogle Scholar
  45. 45.
    Zebiak S E, Cane A. A model El Nino-Southern Oscillation. Mon Weather Rev, 1987, 115: 2262–2278CrossRefGoogle Scholar
  46. 46.
    Webster P J, Yang S. Monsoon and ENSO: Selectively interactive systems, Q J R Meteorol Soc, 1992, 118: 877–926CrossRefGoogle Scholar
  47. 47.
    Wang C, Picaut J. Understanding ENSO physics-A review. In: Wang C Z, Xie S P, Carton J A, eds. Earth’s Climate: The Ocean-Atmosphere Interaction. Geophys Monogr, 2004, 147: 21–48Google Scholar
  48. 48.
    Chen D, Cane M A, Kaplan A, et al. Predictability of El Nino over the past 148 years. Nature, 2004, 428: 733–736CrossRefGoogle Scholar
  49. 49.
    Charney J G. The dynamics of long waves in a baroclinic westerly current. J Meteor, 1947, 4: 135–162Google Scholar
  50. 50.
    Eady E T. Long waves and cyclone waves. Tellus, 1949, 1: 33–52Google Scholar
  51. 51.
    Farrell B F. The initial growth of disturbances in baroclinic flows. J Atmos Sci, 1982, 39: 1663–1686CrossRefGoogle Scholar
  52. 52.
    Lacarra J F, Talagrand O. Short-range evolution of small perturbations in a barotropic model. Tellus, 1988, 40: 81–95Google Scholar
  53. 53.
    Badger J, Hoskins B J. Simple initial value problems and mechanisms for baroclinic growth. J Atmos Sci, 2001, 58: 38–49CrossRefGoogle Scholar
  54. 54.
    Jiang S, Jin F F, Ghil M. Multiple equilibria and aperiodic solutions in a wind-driven doublegyre, shallow-water model. J Phys Oceanogr, 1995, 25: 764–786CrossRefGoogle Scholar
  55. 55.
    Dijkstra H A. Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Nino. 2nd ed. Dordrecht: Springer, 2005Google Scholar
  56. 56.
    Dijkstra H A, De Ruijter W P M. Finite amplitude stability of the wind-driven ocean circulation. Geophys Astrophys Fluid Dyn, 1996, 83: 1–31CrossRefGoogle Scholar
  57. 57.
    Wu X G, Mu M. Impact of horizontal diffusion on the nonlinear stability of thermohaline circulation in a modified box model. J Phys Ocenogr, 2009, 39: 798–805CrossRefGoogle Scholar
  58. 58.
    Mu M, Wang B. Nonlinear instability and sensitivity of a theoretical grassland ecosystem to finite-amplitude perturbations. Nonlinear Process Geophys, 2007, 14: 409–423Google Scholar
  59. 59.
    Zeng Q C, Lu P S, Zeng X D. Maximum simplified dynamic model of grass field ecosystem with two variables. Sci China Ser B, 1994, 37: 94–103Google Scholar
  60. 60.
    Zeng X D, Shen S H, Zeng X B, et al. Multiple equilibrium states and the abrupt transitions in a dynamical system of soil water interacting with vegetation. Geophys Res Lett, 2004, 31: 5501, doi:10.1029/2003GL018910CrossRefGoogle Scholar
  61. 61.
    Zeng Q C, Zeng X D. An analytical dynamic model of grass field ecosystem with two variables. Ecol Model, 1996, 85: 187–196CrossRefGoogle Scholar
  62. 62.
    Chao J P, Zhang G K, Yuan X M. A preliminary investigation for the formation of pressure jump produced by the mountain in a two model (in Chinese). Acta Meteorol Sin, 1964, 34: 233–241Google Scholar
  63. 63.
    Chao J P. A preliminary analysis of the interaction between convection development and ambient environment (in Chinese). Acta Meteorol Sin, 1962, 32: 11–18Google Scholar
  64. 64.
    Houtekamer P L, Derome J. Methods for ensemble prediction. Mon Weather Rev, 1995, 123: 2181–2196CrossRefGoogle Scholar
  65. 65.
    Hamill T M, Snyder C, Morss R E. A comparison of probabilistic forecasts from bred, singular-vector, and perturbed observation ensembles. Mon Weather Rev, 2000, 128: 1835–1851CrossRefGoogle Scholar
  66. 66.
    Palmer T N, Gelaro R, Barkmeuer J, et al. Singular vectors, metrics, and adaptive observations. J Atmos Sci, 1998, 55: 633–653CrossRefGoogle Scholar
  67. 67.
    Mu M, Jiang Z N. A new approach to the generation of initial perturbations for ensemble prediction: Conditional nonlinear optimal perturbation. Chin Sci Bull, 2008, 53(13): 2062–2068CrossRefGoogle Scholar
  68. 68.
    Mu M, Wang H, Zhou F F. A preliminary application of conditional nonlinear optimal perturbation to adaptive observation (in Chinese). Chin J Atmos Sci, 2007, 31(6): 1102–1112Google Scholar

Copyright information

© Science in China Press and Springer-Verlag GmbH 2009

Authors and Affiliations

  1. 1.State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric PhysicsChinese Academy of SciencesBeijingChina

Personalised recommendations