Linear and Nonlinear Signatures in the Planetary Wave Dynamics of an AGCM: Phase Space Tendencies

Grant Branstator NCAR, Boulder, Colorado

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Judith Berner NCAR, Boulder, Colorado, and Meteorological Institute, University of Bonn, Bonn, Germany

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Abstract

To identify and quantify indications of linear and nonlinear planetary wave behavior, characteristics of a very long integration of an atmospheric general circulation model in a four-dimensional phase space are examined. The phase space is defined by the leading four empirical orthogonal functions of 500-hPa geopotential heights, and the primary investigated characteristic is the state dependence of mean phase space tendencies. Defining the linear component of planetary wave tendencies as that part which can be captured by a least squares fit linear operator driven by additive Gaussian white noise, the study finds that there are distinct linear and nonlinear signatures. These signatures are especially easy to see in plots of mean tendencies projected onto phase space planes. For some planes the mean tendencies are highly linear, while for others there are strong departures from linearity.

The results of the analysis are found to depend strongly on the lag time used to estimate tendencies with the linear component monotonically increasing with lag time. This is shown to result from the ergodicity of the system. Using the theory of Markov models it is possible to remove the lag-dependent component of the tendencies from the results. When this is done the projected mean dynamics in some planes is found to be almost exclusively nonlinear, while in others it is nearly linear.

In the four-dimensional space the linear component of the dynamics is largely a reflection of a westward propagating Northern Hemisphere pattern concentrated over the Pacific and North America. The nonlinear signature can be approximated by two linear functions, each operating in a different region of phase space. One region is centered around a Pacific blocking pattern while the other is centered on a state with enhanced zonal symmetry. It is concluded that reduced models of the planetary waves should strive to include these state-dependent dynamics.

Corresponding author address: Dr. Grant Branstator, NCAR, P.O. Box 3000, Boulder, CO 80307-3000. Email: branst@ucar.edu

Abstract

To identify and quantify indications of linear and nonlinear planetary wave behavior, characteristics of a very long integration of an atmospheric general circulation model in a four-dimensional phase space are examined. The phase space is defined by the leading four empirical orthogonal functions of 500-hPa geopotential heights, and the primary investigated characteristic is the state dependence of mean phase space tendencies. Defining the linear component of planetary wave tendencies as that part which can be captured by a least squares fit linear operator driven by additive Gaussian white noise, the study finds that there are distinct linear and nonlinear signatures. These signatures are especially easy to see in plots of mean tendencies projected onto phase space planes. For some planes the mean tendencies are highly linear, while for others there are strong departures from linearity.

The results of the analysis are found to depend strongly on the lag time used to estimate tendencies with the linear component monotonically increasing with lag time. This is shown to result from the ergodicity of the system. Using the theory of Markov models it is possible to remove the lag-dependent component of the tendencies from the results. When this is done the projected mean dynamics in some planes is found to be almost exclusively nonlinear, while in others it is nearly linear.

In the four-dimensional space the linear component of the dynamics is largely a reflection of a westward propagating Northern Hemisphere pattern concentrated over the Pacific and North America. The nonlinear signature can be approximated by two linear functions, each operating in a different region of phase space. One region is centered around a Pacific blocking pattern while the other is centered on a state with enhanced zonal symmetry. It is concluded that reduced models of the planetary waves should strive to include these state-dependent dynamics.

Corresponding author address: Dr. Grant Branstator, NCAR, P.O. Box 3000, Boulder, CO 80307-3000. Email: branst@ucar.edu

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  • Achatz, U., and G. Schmitz, 1997: On the closure problem in the reduction of complex atmospheric models by PIPs and EOFs: A comparison for the case of a two-layer model with zonally symmetric forcing. J. Atmos. Sci., 54 , 24522474.

    • Search Google Scholar
    • Export Citation
  • Berner, J., 1999: Weather regimes and transitions in a general circulation model. Diplomarbeit, Meteorological Institute of the University of Bonn, 138 pp.

  • Berner, J., 2003: Detection and stochastic modeling of nonlinear signatures in the geopotential height field of an atmospheric general circulation model. Bonner Meteorologische Abhandlungen, No. 58, Asgard-Verlag, 156 pp.

  • Berner, J., 2005: Linking nonlinearity and non-Gaussianity of planetary wave behavior by the Fokker–Planck equation. J. Atmos. Sci., in press.

    • Search Google Scholar
    • Export Citation
  • Branstator, G., 1987: A striking example of the atmosphere’s leading traveling pattern. J. Atmos. Sci., 44 , 23102323.

  • Branstator, G., 1990: Low-frequency patterns induced by stationary waves. J. Atmos. Sci., 47 , 629648.

  • Branstator, G., 1992: The maintenance of low-frequency atmospheric anomalies. J. Atmos. Sci., 49 , 19241945.

  • Branstator, G., and S. E. Haupt, 1998: An empirical model of barotropic atmospheric dynamics and its response to tropical forcing. J. Climate, 11 , 26452667.

    • Search Google Scholar
    • Export Citation
  • Branstator, G., A. Mai, and D. Baumhefner, 1993: Identification of highly predictable flow elements for spatial filtering of medium- and extended-range numerical forecasts. Mon. Wea. Rev., 121 , 17861802.

    • Search Google Scholar
    • Export Citation
  • Brockwell, P., and R. Davis, 1987: Time Series: Theory and Methods. Springer Verlag, 519 pp.

  • Chandrasekhar, S., 1943: Stochastic problems in physics and astronomy. Rev. Mod. Phys., 15 , 189.

  • Charney, J., and J. D. DeVore, 1979: Multiple flow equilibria in the atmosphere and blocking. J. Atmos. Sci., 36 , 12051216.

  • Cheng, X., and J. Wallace, 1993: Cluster analysis of the Northern Hemisphere wintertime 500-hPa height field: Spatial patterns. J. Atmos. Sci., 50 , 26742696.

    • Search Google Scholar
    • Export Citation
  • Corti, S., F. Molteni, and T. N. Palmer, 1999: Signature of recent climate change in frequencies of natural atmospheric circulation regimes. Nature, 398 , 799802.

    • Search Google Scholar
    • Export Citation
  • Crommelin, D. T., 2003: Regime transitions and heteroclinic connections in a barotropic atmosphere. J. Atmos. Sci., 60 , 229246.

  • Crommelin, D. T., and A. Majda, 2004: Strategies for model reduction: Comparing different optimal bases. J. Atmos. Sci., 61 , 22062217.

    • Search Google Scholar
    • Export Citation
  • DelSole, T., 2000: A fundamental limitation of Markov models. J. Atmos. Sci., 57 , 21582168.

  • DelSole, T., and B. Farrell, 1995: A stochastically excited linear system as a model for quasigeostrophic turbulence: Analytic results for one- and two-layer fluids. J. Atmos. Sci., 52 , 25312547.

    • Search Google Scholar
    • Export Citation
  • Hannachi, A., 1997a: Low-frequency variability in a GCM: Three-dimensional flow regimes and their dynamics. J. Climate, 10 , 13571379.

    • Search Google Scholar
    • Export Citation
  • Hannachi, A., 1997b: Weather regimes in the Pacific from a GCM. Part II: Dynamics and stability. J. Atmos. Sci., 54 , 13341348.

  • Hannachi, A., and A. O’Neill, 2001: Atmospheric multiple equilibria and non-Gaussian behaviour in model simulations. Quart. J. Roy. Meteor. Soc., 127 , 939958.

    • Search Google Scholar
    • Export Citation
  • Hasselmann, K., 1988: PIPs and POPs: The reduction of complex dynamical systems using principal interaction and oscillation patterns. J. Geophys. Res., 93 , 1101511021.

    • Search Google Scholar
    • Export Citation
  • Hoskins, B., and D. Karoly, 1981: The steady linear response of a spherical atmosphere to thermal and orographic forcing. J. Atmos. Sci., 38 , 11791196.

    • Search Google Scholar
    • Export Citation
  • Hsu, C., and F. Zwiers, 2001: Climate change in recurrent regimes and modes of atmospheric variability. J. Geophys. Res., 106 , 2014520159.

    • Search Google Scholar
    • Export Citation
  • Itoh, H., and M. Kimoto, 1999: Weather regimes, low-frequency oscillations, and principal patterns of variability: A perspective of extratropical low-frequency variability. J. Atmos. Sci., 56 , 26842705.

    • Search Google Scholar
    • Export Citation
  • Kalnay, E., and Coauthors, 1996: The NCEP/NCAR 40-Year Reanalysis Project. Bull. Amer. Meteor. Soc., 77 , 437471.

  • Kimoto, M., and M. Ghil, 1993a: Multiple flow regimes in the Northern Hemisphere winter. Part I: Methodology and hemispheric regimes. J. Atmos. Sci., 50 , 26252643.

    • Search Google Scholar
    • Export Citation
  • Kimoto, M., and M. Ghil, 1993b: Multiple flow regimes in the Northern Hemisphere winter. Part II: Sectorial regimes and preferred transitions. J. Atmos. Sci., 50 , 26452673.

    • Search Google Scholar
    • Export Citation
  • Kushnir, Y., 1987: Retrograding wintertime low-frequency disturbances over the north Pacific ocean. J. Atmos. Sci., 44 , 27272742.

  • Kwasniok, F., 1996: The reduction of complex dynamical systems using principal interaction patterns. Physica D, 92 , 2860.

  • Legras, B., and M. Ghil, 1985: Persistent anomalies, blocking and variations in atmospheric predictability. J. Atmos. Sci., 42 , 433471.

    • Search Google Scholar
    • Export Citation
  • Leith, C., 1971: Atmospheric predictability and two-dimensional turbulence. J. Atmos. Sci., 28 , 145161.

  • Madden, R. A., 1979: Observations of large-scale traveling Rossby waves. Rev. Geophys. Space Phys., 17 , 19351949.

  • Majda, A., I. Timofeyev, and E. Vanden-Eijnden, 2001: A mathematical framework for stochastic climate models. Comm. Pure Appl. Math., 54 , 891974.

    • Search Google Scholar
    • Export Citation
  • Majda, A., I. Timofeyev, and E. Vanden-Eijnden, 2003: Systematic strategies for stochastic mode reduction in climate. J. Atmos. Sci., 60 , 17051722.

    • Search Google Scholar
    • Export Citation
  • Mo, K., and M. Ghil, 1987: Statistics and dynamics of persistent anomalies. J. Atmos. Sci., 44 , 877901.

  • Monahan, A., L. Pandolfo, and J. Fyfe, 2001: The preferred structure of variability of the northern hemisphere atmospheric circulation. Geophys. Res. Lett., 28 , 10191022.

    • Search Google Scholar
    • Export Citation
  • Palmer, T. N., 1999: A nonlinear dynamical perspective on climate prediction. J. Climate, 12 , 575591.

  • Penland, C., 1989: Random forcing and forecasting using principal oscillation pattern analysis. Mon. Wea. Rev., 117 , 21652185.

  • Penland, C., and T. Magorian, 1993: Prediction of Niño-3 sea surface temperatures using linear inverse modeling. J. Climate, 6 , 10671076.

    • Search Google Scholar
    • Export Citation
  • Penland, C., and P. Sardeshmukh, 1995: The optimal growth of tropical sea surface temperature anomalies. J. Climate, 8 , 19992024.

  • Rossby, C-G., 1939: Relation between variations in the intensity of the zonal circulation of the atmosphere and displacements of the semi-permanent centers of action. J. Mar. Res., 2 , 3855.

    • Search Google Scholar
    • Export Citation
  • Salby, M., 1984: A survey of planetary-scale traveling waves: The state of theory and observations. Rev. Geophys. Space Phys., 22 , 209236.

    • Search Google Scholar
    • Export Citation
  • Sardeshmukh, P., C. Penland, and M. Newman, 2001: Rossby waves in a fluctuating medium. Stochastic Climate Models, P. Imkeller and J.-S. von Storch, Eds., Progress In Probability, Vol. 49, Birkäuser Verlag, 369–384.

    • Search Google Scholar
    • Export Citation
  • Selten, F., and G. Branstator, 2004: Preferred regime transition routes and evidence of an unstable periodic orbit in a baroclinic model. J. Atmos. Sci., 61 , 22672282.

    • Search Google Scholar
    • Export Citation
  • Siegert, S., R. Friedrich, and J. Peinke, 1998: Analysis of data sets of stochastic systems. Phys. Lett. A, 243 , 275280.

  • Simmons, A. J., J. M. Wallace, and G. W. Branstator, 1983: Barotropic wave propagation and instability, and atmospheric teleconnection patterns. J. Atmos. Sci., 40 , 13631392.

    • Search Google Scholar
    • Export Citation
  • Sura, P., 2002: Noise-induced transitions in a barotropic β-plane channel. J. Atmos. Sci., 59 , 97110.

  • Sura, P., and J. Barsugli, 2002: A note on estimating drift and diffusion parameters from time series. Phys. Lett. A., 305 , 304311.

  • von Storch, H., G. Buerger, R. Schnur, and J-S. von Storch, 1995: Principal oscillation patterns: A review. J. Climate, 8 , 377400.

  • Wallace, J., and D. Gutzler, 1981: Teleconnections in the geopotential height field during the Northern Hemisphere winter. Mon. Wea. Rev., 109 , 784812.

    • Search Google Scholar
    • Export Citation
  • Wallace, J. M., and D. W. J. Thompson, 2002: The Pacific center of action of the Northern Hemisphere annular mode: Real or artifact? J. Climate, 15 , 19871991.

    • Search Google Scholar
    • Export Citation
  • Wiin-Nielsen, A., 1979: Steady states and stability properties of a low-order barotropic system with forcing and dissipation. Tellus, 31 , 375386.

    • Search Google Scholar
    • Export Citation
  • Williamson, D. L., 1983: Description of the NCAR Community Climate Model (CCM0B). NCAR Tech. Note NCAR/TN-210+STR, 88 pp.

  • Williamson, D. L., and G. Williamson, 1983: Circulation statistics from January and July simulations with the NCAR Community Climate Model (CCM0B). NCAR Tech. Note NCAR/TN-244+STR, 112 pp.

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