Conditional nonlinear optimal perturbation: Applications to stability, sensitivity, and predictability
 WanSuo Duan,
 Mu Mu
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
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 multiequilibrium 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 atmosphereocean system.
 Tennekes, H. (1991) Karl Popper and the accountability of numerical forecasting. ECMWF Workshop Proceedings. New Developments in Predictability. European Centre for MediumRange Weather Forecasts, London
 Thompson, P. (1957) Uncertainty of the initial state as a factor in the predictability of large scale atmospheric flow patterns. Tellus 9: pp. 275295
 Palmer, T. N., Molteni, F., Mureau, R. (1992) Ensemble prediction. ECMWF Res Department Tech Memo 188: pp. 45
 Toth, Z., Kalnay, E. (1993) Ensemble forecasting at NMC: The generation of perturbations. Bull Amer Meteor Soc 74: pp. 23172330 CrossRef
 Mu, M., Zhang, Z. Y. (2006) Conditional nonlinear optimal perturbations of a twodimensional quasigeostrophic model. J Atmos Sci 63: pp. 15871604 CrossRef
 Moore, A. M., Kleeman, R. (1996) The dynamics of error growth and predictability in a coupled model of ENSO. Q J R Meteorol Soc 122: pp. 14051446 CrossRef
 Samelson, R. G., Tziperman, E. (2001) Instability of the chaotic ENSO: The growthphase predictability barrier. J Atmos Sci 58: pp. 36133625 CrossRef
 Duan, W. S., Mu, M. (2005) Application of nonlinear optimization method to quantifying the predictability of a numerical model for El NinoSouthern Oscillation. Prog Nat Sci 1510: pp. 915921
 Mu, M., Duan, W. S., Wang, B. (2007) Seasondependent dynamics of nonlinear optimal error growth and El NinoSouthern Oscillation predictability in a theoretical model. J Geophys Res 112: pp. D10113 CrossRef
 Mu, M., Xu, H., Duan, W. S. (2007) A kind of initial errors related to “spring predictability barrier” for El Nino events in ZebiakCane model. Geophys Res Lett 34: pp. L03709 CrossRef
 Smith, L. A., Ziehmann, C., Fraedrich, K. (1999) Uncertainty dynamics and predictability in chaotic systems. Q J R Meteorol Soc 125: pp. 28552886 CrossRef
 Lorenz, E. N. (1965) A study of the predictability of a 28variable atmospheric model. Tellus 17: pp. 321333 CrossRef
 Xue, Y., Cane, M. A., Zebaik, S. E. (1997) Predictability of a coupled model of ENSO using singular vector analysis. Part I: Optimal growth in seasonal background and ENSO cycles. Mon Weather Rev 125: pp. 20432056 CrossRef
 Buizza, R., Molteni, F. (1996) The role of finitetime barotropic instability during the transition to blocking. J Atmos Sci 53: pp. 16751697 CrossRef
 Frederisen, J. S. (1997) Adjoint sensitivity and finite time normal mode disturbances during blocking. J Atmos Sci 47: pp. 24092416
 Tziperman, E., Ioannou, P. J. (2002) Transient growth and optimal excitation of thermohaline variability. J Phys Oceanogr 32: pp. 34273435 CrossRef
 Mu, M. (2000) Nonlinear singualr vectors and nonlinear singular values. Sci China Ser DEarth Sci 43: pp. 375385 CrossRef
 Mu, M., Duan, W. S., Wang, B. (2003) Conditional nonlinear optimal perturbation and its applications. Non Proc Geophys 10: pp. 493501
 Mu, M., Wang, J. C. (2001) Nonlinear fastest growing perturbation and the first kind of predictability. Sci China Ser DEarth Sci 44: pp. 11281139 CrossRef
 Mu, M., Duan, W. S. (2005) Conditional nonlinear optimal perturbation and its applications to the studies of weather and climate predictability. Chin Sci Bull 50: pp. 24012407 CrossRef
 Mu, M., Duan, W. S., Xu, H. (2006) Applications of conditional nonlinear optimal perturbation in predictability study and sensitivity analysis of weather and climate. Adv Atmos Sci 23: pp. 9921002 CrossRef
 Duan, W. S., Mu, M., Wang, B. (2004) Conditional nonlinear optimal perturbation as the optimal precursors for ENSO events. J Geophys Res 109: pp. D23105 CrossRef
 Mu, M., Sun, L., Henk, D. A. (2004) The sensitivity and stability of the ocean’s thermocline circulation to finite amplitude freshwater perturbations. J Phys Oceanogr 34: pp. 23052315 CrossRef
 Sun, L., Mu, M., Sun, D. J. (2005) Passive mechanism decadal variation of thermohaline circulation. J Geophys Res 110: pp. C07025 CrossRef
 Liu, Y. M. (2008) Maximum principle of conditional nonlinear optimal perturbation (in Chinese). J East Chin Norm Univ (Nat Sci) 2: pp. 131134
 Riviere, O., Lapeyre, G., Talagrand, O. (2008) Nonlinear generalization of singular vectors: Behavior in a baroclinic unstable flow. J Atmos Sci 65: pp. 18961911 CrossRef
 Terwisscha van Scheltinga, A. D. (2007) Data assimilation with implicit ocean models. Institute for Marine and Atmospheric Research, Utrecht University, Utrecht
 Powell, M. J. D. (1983) VMCWD: A Fortran subroutine for constrained optimization. Acm Sigmap Bull 32: pp. 416 CrossRef
 Birgin, E. G., Martinez, J. M., Raydan, M. (2000) Nonmonotone spectral projected gradient methods for convex sets. Siam J Opt 10: pp. 11961211 CrossRef
 Jiang, Z. N., Mu, M., Wang, D. H. (2008) Conditional nonlinear optimal perturbation of a T21L3 quasigeostrophic model. Q J R Meteorol Soc 134: pp. 10271038 CrossRef
 Rex, D. F. (1950) Blocking action in the middle troposphere and its effects upon regional climate. I: An aerological study of blocking action. Tellus 2: pp. 196211 CrossRef
 Molteni, F., Palmer, T. N. (1993) Predictability and finitetime instability of the northern winter circulation. Quart J Roy Meteor Soc 119: pp. 269298 CrossRef
 Buizza, R., Molteni, F. (1996) The role of finitetime barotropic instability during the transition to blocking. J Atmos Sci 53: pp. 16751697 CrossRef
 Frederiksen, J. S. (2000) Singular vector, finitetime normal modes, and error growth during blocking. J Atmos Sci 57: pp. 312333 CrossRef
 Mu, M., Jiang, Z. N. (2008) A method to find out the perturbations triggering the blocking onset: Conditional nonlinear optimal perturbations. J Atmos Sci 65: pp. 39353946 CrossRef
 Jin, F. F., An, S. I., Timmermann, A. (2003) Strong El Nino events and nonlinear dynamical heating. Geophys Res Lett 30: pp. 1120 CrossRef
 An, S. I., Jin, F. F. (2004) Nonlinearity and asymmetry of ENSO. J Clim 17: pp. 23992412 CrossRef
 Wang, B., An, S. I. (2001) Why the properties of El Nino changed during the late 1970s. Geophys Res Lett 28: pp. 37093712 CrossRef
 Duan, W. S., Mu, M. (2006) Investigating decadal variability of El NinoSouthern Oscillation asymmetry by conditional nonlinear optimal perturbation. J Geophys Res 111: pp. C07015 CrossRef
 Philander, S. G. H. (1983) El Nino Southern Oscillation phenomena. Nature 302: pp. 295 CrossRef
 Jin, F. F. (1997) An equatorial ocean recharge paradigm for ENSO. Part I: Conceptual model. J Atmos Sci 54: pp. 811829 CrossRef
 Rodgers, K. B., Friederichs, P., Latif, M. (2004) Tropical Pacific decadal variability and its relation to decadal modulations of ENSO. J Clim 17: pp. 37613774 CrossRef
 Duan, W. S., Xu, H., Mu, M. (2008) Decisive role of nonlinear temperature advection in El Nino and La Nina amplitude asymmetry. J Geophys Res 113: pp. C01014 CrossRef
 Wang, B., Fang, Z. (1996) Chaotic oscillation of tropical climate: A dynamic system theory for ENSO. J Atmos Sci 53: pp. 27862802 CrossRef
 Zebiak, S. E., Cane, A. (1987) A model El NinoSouthern Oscillation. Mon Weather Rev 115: pp. 22622278 CrossRef
 Webster, P. J., Yang, S. (1992) Monsoon and ENSO: Selectively interactive systems. Q J R Meteorol Soc 118: pp. 877926 CrossRef
 Wang C, Picaut J. Understanding ENSO physicsA review. In: Wang C Z, Xie S P, Carton J A, eds. Earth’s Climate: The OceanAtmosphere Interaction. Geophys Monogr, 2004, 147: 21–48
 Chen, D., Cane, M. A., Kaplan, A. (2004) Predictability of El Nino over the past 148 years. Nature 428: pp. 733736 CrossRef
 Charney, J. G. (1947) The dynamics of long waves in a baroclinic westerly current. J Meteor 4: pp. 135162
 Eady, E. T. (1949) Long waves and cyclone waves. Tellus 1: pp. 3352
 Farrell, B. F. (1982) The initial growth of disturbances in baroclinic flows. J Atmos Sci 39: pp. 16631686 CrossRef
 Lacarra, J. F., Talagrand, O. (1988) Shortrange evolution of small perturbations in a barotropic model. Tellus 40: pp. 8195
 Badger, J., Hoskins, B. J. (2001) Simple initial value problems and mechanisms for baroclinic growth. J Atmos Sci 58: pp. 3849 CrossRef
 Jiang, S., Jin, F. F., Ghil, M. (1995) Multiple equilibria and aperiodic solutions in a winddriven doublegyre, shallowwater model. J Phys Oceanogr 25: pp. 764786 CrossRef
 Dijkstra, H. A. (2005) Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Nino. Springer, Dordrecht
 Dijkstra, H. A., Ruijter, W. P. M. (1996) Finite amplitude stability of the winddriven ocean circulation. Geophys Astrophys Fluid Dyn 83: pp. 131 CrossRef
 Wu, X. G., Mu, M. (2009) Impact of horizontal diffusion on the nonlinear stability of thermohaline circulation in a modified box model. J Phys Ocenogr 39: pp. 798805 CrossRef
 Mu, M., Wang, B. (2007) Nonlinear instability and sensitivity of a theoretical grassland ecosystem to finiteamplitude perturbations. Nonlinear Process Geophys 14: pp. 409423
 Zeng, Q. C., Lu, P. S., Zeng, X. D. (1994) Maximum simplified dynamic model of grass field ecosystem with two variables. Sci China Ser B 37: pp. 94103
 Zeng, X. D., Shen, S. H., Zeng, X. B. (2004) Multiple equilibrium states and the abrupt transitions in a dynamical system of soil water interacting with vegetation. Geophys Res Lett 31: pp. 5501 CrossRef
 Zeng, Q. C., Zeng, X. D. (1996) An analytical dynamic model of grass field ecosystem with two variables. Ecol Model 85: pp. 187196 CrossRef
 Chao, J. P., Zhang, G. K., Yuan, X. M. (1964) A preliminary investigation for the formation of pressure jump produced by the mountain in a two model (in Chinese). Acta Meteorol Sin 34: pp. 233241
 Chao, J. P. (1962) A preliminary analysis of the interaction between convection development and ambient environment (in Chinese). Acta Meteorol Sin 32: pp. 1118
 Houtekamer, P. L., Derome, J. (1995) Methods for ensemble prediction. Mon Weather Rev 123: pp. 21812196 CrossRef
 Hamill, T. M., Snyder, C., Morss, R. E. (2000) A comparison of probabilistic forecasts from bred, singularvector, and perturbed observation ensembles. Mon Weather Rev 128: pp. 18351851 CrossRef
 Palmer, T. N., Gelaro, R., Barkmeuer, J. (1998) Singular vectors, metrics, and adaptive observations. J Atmos Sci 55: pp. 633653 CrossRef
 Mu, M., Jiang, Z. N. (2008) A new approach to the generation of initial perturbations for ensemble prediction: Conditional nonlinear optimal perturbation. Chin Sci Bull 53: pp. 20622068 CrossRef
 Mu, M., Wang, H., Zhou, F. F. (2007) A preliminary application of conditional nonlinear optimal perturbation to adaptive observation (in Chinese). Chin J Atmos Sci 31: pp. 11021112
 Title
 Conditional nonlinear optimal perturbation: Applications to stability, sensitivity, and predictability
 Journal

Science in China Series D: Earth Sciences
Volume 52, Issue 7 , pp 883906
 Cover Date
 20090701
 DOI
 10.1007/s1143000900903
 Print ISSN
 10069313
 Online ISSN
 18622801
 Publisher
 SP Science in China Press
 Additional Links
 Topics
 Keywords

 optimal perturbation
 predictability
 stability
 sensitivity
 Industry Sectors
 Authors

 WanSuo Duan ^{(1)}
 Mu Mu ^{(1)}
 Author Affiliations

 1. State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, 100029, China