• Benettin, G., L. Galgani, and J. M. Strelcyn, 1976: Kolmogorov entropy and numerical experiments. Phys. Rev. A,14, 2338–2345.

  • ——, ——, A. Giorgilli, and J. M. Strelcyn, 1980: Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing them. Meccanica,15, 9.

  • Cvitanović, P., 1991: Periodic orbits as the skeleton of classical and quantum dynamics. Physica D,51, 138–151.

  • Eckhardt, B., and G. Ott, 1994: Periodic orbit analysis of the Lorenz attractor. Z. Phys.,B93, 259–266.

  • Farrell, B. F., 1988: Optimal excitation of neutral Rossby waves. J. Atmos. Sci.,45, 163–172.

  • ——, 1990: Small error dynamics and the predictability of atmospheric flows. J. Atmos. Sci.,47, 2409–2416.

  • Ghil, M., and J. Tavantzis, 1983: Global Hopf bifurcation in a simple climate model. SIAM J. Appl. Math.,43, 1019–1041.

  • Goldhirsch, I., P. L. Sulem, and A. Orszag, 1987: Stability and Lyapunov stability of dynamical systems: a differential approach and a numerical method. Physica D,27, 311–337.

  • Hartman, P., 1982: Ordinary Differential Equations. Birkhaü≃r, 612 pp.

  • Houtekamer, P. L., and J. Derome, 1995: Methods for ensemble prediction. Mon. Wea. Rev.,123, 2181–2196.

  • Krishnamurty, V., 1993: A predictability study of Lorenz’s 28-variable model as a dynamical system. J. Atmos. Sci.,50, 2215–2229.

  • Lacarra, J., and O. Talagrand, 1988: Short range evolution of small perturbations in a barotropic model. Tellus,40A, 81–95.

  • Legras, B., and R. Vautard, 1996: A guide to Lyapunov vectors.Seminar on Predictability, Reading, United Kingdom, ECMWF, 143–156.

  • Lichtenberg, A. J., and M. A. Lieberman, 1983: Regular and Stochastic Motion. Springer-Verlag, 259 pp.

  • Lorenz, E. N., 1963: Deterministic nonperiodic flow. J. Atmos. Sci.,20, 130–141.

  • ——, 1965: A study of the predictability of a 28-variable atmospheric model. Tellus,17, 321–333.

  • ——, 1984: The local structure of a chaotic attractor in four dimensions. Physica D,13, 90–104.

  • Molteni, F., and T. N. Palmer, 1993: Predictability and finite-time instability of the northern winter circulation. Quart. J. Roy. Meteor. Soc.,119B, 269–298.

  • ——, R. Buizza, T. N. Palmer, and T. Petroliagis, 1996: The ECMWF ensemble prediction system: Methodology and validation. Quart. J. Roy. Meteor. Soc.,122, 73–119.

  • Mukougawa, I., M. Kimoto, and S. Yoden, 1991: A relationship between local error growth and quasi-stationary states: A case study in the Lorenz system. J. Atmos. Sci.,48, 1231–1237.

  • Mureau, R., F. Molteni, and T. N. Palmer, 1993: Ensemble prediction using dynamically conditioned perturbations. Quart. J. Roy. Meteor. Soc.,119B, 299–323.

  • Nicolis, C., S. Vannitsem, and J. F. Royer, 1995: Short range predictability of the atmosphere: mechanisms for superexponential error growth. Quart. J. Roy. Meteor. Soc.,121, 705–722.

  • Oseledec, V. I., 1968: A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc.,19, 197–231.

  • Savijarvi, H., 1995: Error growth in a large numerical forecast system. Mon. Wea. Rev.,123, 212–221.

  • Shimada, I., and T. Nagashima, 1979: A numerical approach to ergodic problem of dissipative dynamical systems. Progr. Theor. Phys.,61, 1605–1616.

  • Simmons, A. J., R. Mureau, and T. Petroliagis: Error growth and estimates of predictability from the ECMWF forecasting system. Quart. J. Roy. Meteor. Soc.,121, 1739–1771.

  • Szunyogh, I., E. Kalnay, and Z. Toth, 1997: A comparison of Lyapunov vectors and optimal vectors in a low resolution GCM. Tellus,49A, 200–227.

  • Toth, Z., and E. Kalnay, 1993: Ensemble forecasting at NMC: The generation of perturbations. Bull. Amer. Meteor. Soc.,74, 2317–2330.

  • Trevisan, A., 1993: Impact of transient error growth on global average predictability measures. J. Atmos. Sci.,50, 1016–1028.

  • ——, and R. Legnani, 1995: Transient error growth and local predictability: A study in the Lorenz system. Tellus,47A, 103–117.

  • Yoden, S., and M. Nomura, 1993: Finite time Lyapunov stability analysis and its application to atmospheric predictability. J. Atmos. Sci.,50, 1531–1543.

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Periodic Orbits, Lyapunov Vectors, and Singular Vectors in the Lorenz System

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  • 1 CNR–FISBAT, Bologna, Italy
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Abstract

Some theoretical issues related to the problem of quantifying local predictability of atmospheric flow and the generation of perturbations for ensemble forecasts are investigated in the Lorenz system. A periodic orbit analysis and the study of the properties of the associated tangent linear equations are performed.

In this study a set of vectors are found that satisfy Oseledec theorem and reduce to Floquet eigenvectors in the particular case of a periodic orbit. These vectors, called Lyapunov vectors, can be considered the generalization to aperiodic orbits of the normal modes of the instability problem and are not necessarily mutually orthogonal.

The relation between singular vectors and Lyapunov vectors is clarified, and transient or asymptotic error growth properties are investigated. The mechanism responsible for super-Lyapunov growth is shown to be related to the nonorthogonality of Lyapunov vectors.

The leading Lyapunov vectors, as defined here, as well as the asymptotic final singular vectors, are tangent to the attractor, while the leading initial singular vectors, in general, point away from it. Perturbations that are on the attractor and maximize growth should be considered in meteorological applications such as ensemble forecasting and adaptive observations. These perturbations can be found in the subspace of the leading Lyapunov vectors.

Corresponding author address: Dr. Anna Trevisan, CNR–FISBAT, via Gobetti 101, 40129 Bologna, Italy.

Email: trevisan@atmos1.fisbat.bo.cnr.it

Abstract

Some theoretical issues related to the problem of quantifying local predictability of atmospheric flow and the generation of perturbations for ensemble forecasts are investigated in the Lorenz system. A periodic orbit analysis and the study of the properties of the associated tangent linear equations are performed.

In this study a set of vectors are found that satisfy Oseledec theorem and reduce to Floquet eigenvectors in the particular case of a periodic orbit. These vectors, called Lyapunov vectors, can be considered the generalization to aperiodic orbits of the normal modes of the instability problem and are not necessarily mutually orthogonal.

The relation between singular vectors and Lyapunov vectors is clarified, and transient or asymptotic error growth properties are investigated. The mechanism responsible for super-Lyapunov growth is shown to be related to the nonorthogonality of Lyapunov vectors.

The leading Lyapunov vectors, as defined here, as well as the asymptotic final singular vectors, are tangent to the attractor, while the leading initial singular vectors, in general, point away from it. Perturbations that are on the attractor and maximize growth should be considered in meteorological applications such as ensemble forecasting and adaptive observations. These perturbations can be found in the subspace of the leading Lyapunov vectors.

Corresponding author address: Dr. Anna Trevisan, CNR–FISBAT, via Gobetti 101, 40129 Bologna, Italy.

Email: trevisan@atmos1.fisbat.bo.cnr.it

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