• Allen, M. R., , and L. A. Smith, 1996: Monte Carlo SSA: Detecting irregular oscillations in the presence of colored noise. J. Climate, 9 , 33733404.

    • Search Google Scholar
    • Export Citation
  • Barsugli, J. J., , and D. S. Battisti, 1998: The basic effects of atmosphere–ocean thermal coupling on midlatitude variability. J. Atmos. Sci., 55 , 477493.

    • Search Google Scholar
    • Export Citation
  • Bjerknes, J., 1969: Atmospheric teleconnections from the equatorial Pacific. Mon. Wea. Rev., 97 , 163172.

  • Blumenthal, M. B., 1991: Predictability of a coupled atmosphere–ocean model. J. Climate, 4 , 766784.

  • Buizza, R., , and T. N. Palmer, 1995: The singular-vector structure of the atmospheric general circulation. J. Atmos. Sci., 52 , 14341456.

    • Search Google Scholar
    • Export Citation
  • Chang, P., , R. Saravanan, , F. Wang, , and L. Ji, 2004: Predictability of linear coupled systems. Part II: An application to a simple model of tropical Atlantic variability. J. Climate, 17 , 14871503.

    • Search Google Scholar
    • Export Citation
  • Chen, Y-Q., , D. S. Battisti, , T. N. Palmer, , J. Barsugli, , and E. S. Sarachik, 1997: A study of the predictability of tropical Pacific SST in a coupled atmosphere–ocean model using singular vector analysis: The role of the annual cycle and the ENSO cycle. Mon. Wea. Rev., 125 , 831845.

    • Search Google Scholar
    • Export Citation
  • DelSole, T., , and P. Chang, 2003: Predictable component analysis, canonical correlation analysis, and autoaggressive models. J. Atmos. Sci., 60 , 409416.

    • Search Google Scholar
    • Export Citation
  • Farrell, B. F., , and P. J. Ioannou, 1993: Stochastic forcing of perturbation variance in unbounded shear and deformation flows. J. Atmos. Sci., 50 , 40444057.

    • Search Google Scholar
    • Export Citation
  • Farrell, B. F., , and P. J. Ioannou, 1996: Generalized stability. Part I: Autonomous operators. J. Atmos. Sci., 53 , 20252040.

  • Farrell, B. F., , and P. J. Ioannou, 1999: Perturbation growth and structure in time-dependent flows. J. Atmos. Sci., 56 , 36223639.

  • Frankignoul, C., 1995: Climate spectra and stochastic climate models. Analysis of Climate Variability: Application of Statistical Techniques, H. von Storch and A. Navarra, Eds., Springer-Verlag, 29–50.

    • Search Google Scholar
    • Export Citation
  • Gill, A. E., 1982: Atmosphere–Ocean Dynamics. Academic Press, 662 pp.

  • Hasselmann, K., 1976: Stochastic climate models. Tellus, 28 , 473485.

  • Ioannou, P. J., 1995: Nonnormality increases variance. J. Atmos. Sci., 52 , 11551158.

  • Kirtman, B. P., , and J. Shukla, 2002: Interactive coupled ensemble: A new coupling strategy for CGCMs. Geophys. Res. Lett.,29, 1367, doi:10.1029/2002GL014834.

    • Search Google Scholar
    • Export Citation
  • Latif, M., , and T. P. Barnett, 1994: Causes of decadal climate variability over the North Pacific and North America. Science, 266 , 634637.

    • Search Google Scholar
    • Export Citation
  • Latif, M., , T. P. Barnett, , M. A. Cane, , M. Flugel, , N. E. Graham, , H. van Storch, , J-S. Xu, , and S. E. Zebiak, 1994: A review of ENSO prediction studies. Climate Dyn., 9 , 167179.

    • Search Google Scholar
    • Export Citation
  • Lorenz, E. N., 1965: A study of the predictability of a 28-variable atmospheric model. Tellus, 17 , 321333.

  • Moore, A. M., , and R. Kleeman, 1996: The dynamics of error growth and predictability in a coupled model of ENSO. Quart. J. Roy. Meteor. Soc., 122 , 14051446.

    • Search Google Scholar
    • Export Citation
  • Moore, A. M., , and R. Kleeman, 1999: Stochastic forcing of ENSO by the intraseasonal oscillation. J. Climate, 12 , 11991220.

  • Neelin, J. D., , D. S. Battisti, , A. C. Hirst, , F-F. Jin, , Y. Wakata, , T. Yamagata, , and S. E. Zebiak, 1998: ENSO theory. J. Geophys. Res., 103 , 1426114290.

    • Search Google Scholar
    • Export Citation
  • Noble, B., , and J. W. Daniel, 1988: Applied Linear Algebra, 3d ed. Prentice Hall, 500 pp.

  • Penland, M. C., , and P. Sardeshmukh, 1995: The optimal growth of tropical sea surface temperature anomalies. J. Climate, 8 , 20002024.

  • Schneider, T., , and S. M. Griffies, 1999: A conceptual framework for predictability studies. J. Climate, 12 , 31333155.

  • Thompson, C. J., , and D. S. Battisti, 2000: A linear stochastic dynamical model of ENSO. Part I: Model development. J. Climate, 13 , 28182832.

    • Search Google Scholar
    • Export Citation
  • Tippett, M. K., , and P. Chang, 2003: Some theoretical considerations on predictability of linear stochastic dynamics. Tellus, 55A , 148157.

    • Search Google Scholar
    • Export Citation
  • Trefethen, L. N., 1997: Pseudospectra of linear operators. SIAM Rev., 39 , 383406.

  • Weiss, J. B., 2003: Coordinate invariance in stochastic dynamical systems. Tellus, 55A , 208219.

  • Wunsch, C., 1999: The interpretation of short climate records, with comments on the North Atlantic and Southern Oscillations. Bull. Amer. Meteor. Soc., 80 , 245256.

    • Search Google Scholar
    • Export Citation
  • Xue, Y., , M. A. Cane, , and S. E. Zebiak, 1997: Predictability of a coupled model of ENSO using singular vector analysis. Part I: Optimal growth in seasonal background and ENSO cycles. Mon. Wea. Rev., 125 , 20432056.

    • Search Google Scholar
    • Export Citation
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Predictability of Linear Coupled Systems. Part I: Theoretical Analyses

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  • 1 Department of Oceanography, Texas A&M University, College Station, Texas
  • | 2 National Center for Atmospheric Research, Boulder, Colorado
  • | 3 Center for Ocean–Land–Atmosphere Studies, Calverton, Maryland
  • | 4 Department of Oceanography, Texas A&M University, College Station, Texas
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Abstract

The predictability of stochastically forced linear systems is investigated under the condition that an ensemble of forecasts are each initialized at the true state but driven by different realizations of white noise. Some important issues of predictability are brought out by analytically investigating a stochastically driven, damped inertial oscillator. These issues are then studied in a generic context without reference to any specific linear stochastic system. The predictability is measured primarily by the mean-square forecast error normalized by the mean climatological variance. If the dynamical operator is normal, then this metric depends only on the real eigenvalues (i.e., the normal mode damping rates) and is independent of the existence of spectral peaks at nonzero frequency caused by oscillatory eigenmodes. It is shown that a nonnormal system is more predictable by this measure than a normal system with identical eigenvalues for short lead times. The noise structure that optimizes this metric can be calculated analytically and shown to approach the stochastic optimals in the limit of short lead times; this calculation gives bounds on predictability that depend only on the dynamical operator. The most predictable components, or spatial structures, can also be determined analytically and are shown to approach the empirical orthogonal functions (EOFs) of the variance in the limit of short lead times. In a companion paper, these concepts are applied to the predictability analysis of a simple coupled climate model of tropical Atlantic variability.

Corresponding author address: Dr. Ping Chang, Department of Oceanography, Texas A&M University, College Station, TX 77843-3146. Email: ping@ocean.tamu.edu

Abstract

The predictability of stochastically forced linear systems is investigated under the condition that an ensemble of forecasts are each initialized at the true state but driven by different realizations of white noise. Some important issues of predictability are brought out by analytically investigating a stochastically driven, damped inertial oscillator. These issues are then studied in a generic context without reference to any specific linear stochastic system. The predictability is measured primarily by the mean-square forecast error normalized by the mean climatological variance. If the dynamical operator is normal, then this metric depends only on the real eigenvalues (i.e., the normal mode damping rates) and is independent of the existence of spectral peaks at nonzero frequency caused by oscillatory eigenmodes. It is shown that a nonnormal system is more predictable by this measure than a normal system with identical eigenvalues for short lead times. The noise structure that optimizes this metric can be calculated analytically and shown to approach the stochastic optimals in the limit of short lead times; this calculation gives bounds on predictability that depend only on the dynamical operator. The most predictable components, or spatial structures, can also be determined analytically and are shown to approach the empirical orthogonal functions (EOFs) of the variance in the limit of short lead times. In a companion paper, these concepts are applied to the predictability analysis of a simple coupled climate model of tropical Atlantic variability.

Corresponding author address: Dr. Ping Chang, Department of Oceanography, Texas A&M University, College Station, TX 77843-3146. Email: ping@ocean.tamu.edu

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