Strategies for Model Reduction: Comparing Different Optimal Bases

D. T. Crommelin Courant Institute of Mathematical Science, Center for Atmosphere Ocean Science, New York University, New York, New York

Search for other papers by D. T. Crommelin in
Current site
Google Scholar
PubMed
Close
and
A. J. Majda Courant Institute of Mathematical Science, Center for Atmosphere Ocean Science, New York University, New York, New York

Search for other papers by A. J. Majda in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

Several different ways of constructing optimal bases for efficient dynamical modeling are compared: empirical orthogonal functions (EOFs), optimal persistence patterns (OPPs), and principal interaction patterns (PIPs). Past studies on fluid-dynamical topics have pointed out that EOF-based models can have difficulties reproducing behavior dominated by irregular transitions between different dynamical states. This issue is addressed in a geophysical context, by assessing the ability of these strategies for efficient dynamical modeling to reproduce the chaotic regime transitions in a simple atmosphere model. The atmosphere model is the well-known Charney– DeVore model, a six-dimensional truncation of the equations describing barotropic flow over topography in a β-plane channel geometry. This model is able to generate regime transitions for well-chosen parameter settings. The models based on PIPs are found to be superior to the EOF- and OPP-based models, in spite of some undesirable sensitivities inherent to the PIP method.

Corresponding author address: D. T. Crommelin, Courant Institute of Mathematical Sciences, Center for Atmosphere Ocean Science, New York University, 251 Mercer Street, New York, NY 10012. Email: crommelin@cims.nyu.edu

Abstract

Several different ways of constructing optimal bases for efficient dynamical modeling are compared: empirical orthogonal functions (EOFs), optimal persistence patterns (OPPs), and principal interaction patterns (PIPs). Past studies on fluid-dynamical topics have pointed out that EOF-based models can have difficulties reproducing behavior dominated by irregular transitions between different dynamical states. This issue is addressed in a geophysical context, by assessing the ability of these strategies for efficient dynamical modeling to reproduce the chaotic regime transitions in a simple atmosphere model. The atmosphere model is the well-known Charney– DeVore model, a six-dimensional truncation of the equations describing barotropic flow over topography in a β-plane channel geometry. This model is able to generate regime transitions for well-chosen parameter settings. The models based on PIPs are found to be superior to the EOF- and OPP-based models, in spite of some undesirable sensitivities inherent to the PIP method.

Corresponding author address: D. T. Crommelin, Courant Institute of Mathematical Sciences, Center for Atmosphere Ocean Science, New York University, 251 Mercer Street, New York, NY 10012. Email: crommelin@cims.nyu.edu

Save
  • Achatz, U., and G. Branstator, 1999: A two-layer model with empirical linear corrections and reduced order for studies of internal climate variability. J. Atmos. Sci, 56 , 31403160.

    • Search Google Scholar
    • Export Citation
  • Achatz, U., and J. D. Opsteegh, 2003a: Primitive-equation-based low-order models with seasonal cycle. Part I: Model construction. J. Atmos. Sci, 60 , 465477.

    • Search Google Scholar
    • Export Citation
  • Achatz, U., and J. D. Opsteegh, 2003b: Primitive-equation-based low-order models with seasonal cycle. Part II: Application to complexity and nonlinearity of large-scale atmosphere dynamics. J. Atmos. Sci, 60 , 478490.

    • Search Google Scholar
    • Export Citation
  • Achatz, U., G. Schmitz, and K-M. Greisinger, 1995: Principal interaction patterns in baroclinic wave life cycles. J. Atmos. Sci, 52 , 32013213.

    • Search Google Scholar
    • Export Citation
  • Armbruster, D., R. Heiland, E. J. Kostelich, and B. Nicolaenko, 1992: Phase-space analysis of bursting behavior in Kolmogorov flow. Physica D, 58 , 392401.

    • Search Google Scholar
    • Export Citation
  • Aubry, N., P. Holmes, J. L. Lumley, and E. Stone, 1988: The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech, 192 , 115173.

    • Search Google Scholar
    • Export Citation
  • Aubry, N., W-Y. Lian, and E. S. Titi, 1993: Preserving symmetries in the proper orthogonal decomposition. SIAM J. Sci. Comput, 14 , 483505.

    • Search Google Scholar
    • Export Citation
  • 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
  • Cazemier, W., R. W. C. P. Verstappen, and A. E. P. Veldman, 1998: Proper orthogonal decomposition and low-dimensional models for driven cavity flows. Phys. Fluids, 10 , 16851699.

    • Search Google Scholar
    • Export Citation
  • Charney, J. G., and J. G. DeVore, 1979: Multiple flow equilibria in the atmosphere and blocking. J. Atmos. Sci, 36 , 12051216.

  • Crommelin, D. T., J. D. Opsteegh, and F. Verhulst, 2004: A mechanism for atmospheric regime behavior. J. Atmos. Sci, 61 , 14061419.

  • D'Andrea, F., and R. Vautard, 2001: Extratropical low-frequency variability as a low-dimensional problem. I: A simplified model. Quart. J. Roy. Meteor. Soc, 127 , 13571374.

    • Search Google Scholar
    • Export Citation
  • DelSole, T., 2001: Optimally persistent patterns in time-varying fields. J. Atmos. Sci, 58 , 13411356.

  • De Swart, H. E., 1988: Low-order spectral models of the atmospheric circulation: A survey. Acta Appl. Math, 11 , 4996.

  • De Swart, H. E., 1989: Analysis of a six-component atmospheric spectral model: Chaos, predictability and vacillation. Physica D, 36 , 222234.

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

  • Kwasniok, F., 1997: Optimal Galerkin approximations of partial differential equations using principal interaction patterns. Phys. Rev, E55 , 53655375.

    • Search Google Scholar
    • Export Citation
  • Kwasniok, F., 2001: Low-dimensional models of the Ginzburg–Landau equation. SIAM J. Appl. Math, 61 , 20632079.

  • Kwasniok, F., 2004: Empirical low-order models of barotropic flow. J. Atmos. Sci, 61 , 235245.

  • Majda, A. J., I. Timofeyev, and E. Vanden-Eijnden, 1999: Models for stochastic climate prediction. Proc. Natl. Acad. Sci. USA, 96 , 1468714691.

    • Search Google Scholar
    • Export Citation
  • Majda, A. J., 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
  • Rinne, J., and V. Karhilla, 1975: A spectral barotropic model in horizontal empirical orthogonal functions. Quart. J. Roy. Meteor. Soc, 101 , 365382.

    • Search Google Scholar
    • Export Citation
  • Schubert, S. D., 1985: A statistical-dynamical study of empirically determined modes of atmospheric variability. J. Atmos. Sci, 42 , 317.

    • Search Google Scholar
    • Export Citation
  • Schubert, S. D., 1986: The structure, energetics and evolution of the dominant frequency-dependent three-dimensional atmospheric modes. J. Atmos. Sci, 43 , 12101237.

    • Search Google Scholar
    • Export Citation
  • Selten, F. M., 1993: Toward an optimal description of atmospheric flow. J. Atmos. Sci, 50 , 861877.

  • Selten, F. M., 1995: An efficient description of the dynamics of barotropic flow. J. Atmos. Sci, 52 , 915936.

  • Selten, F. M., 1997a: Baroclinic empirical orthogonal functions as basis functions in an atmospheric model. J. Atmos. Sci, 54 , 21002114.

    • Search Google Scholar
    • Export Citation
  • Selten, F. M., 1997b: A statistical closure of a low-order barotropic model. J. Atmos. Sci, 54 , 10861093.

  • Sirovich, L., 1989: Chaotic dynamics of coherent structures. Physica D, 37 , 126145.

  • Winkler, C. R., M. Newman, and P. D. Sardeshmukh, 2001: A linear model of wintertime low-frequency variability. Part I: Formulation and forecast skill. J. Climate, 14 , 44744494.

    • Search Google Scholar
    • Export Citation
  • Wu, C-J., 1996: Large optimal truncated low-dimensional dynamical systems. Discr. Cont. Dyn. Syst, 2 , 559583.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 682 162 10
PDF Downloads 297 85 6