• Artuso, R., E. Aurell, and P. Cvitanović, 1990a: Recycling of strange sets I: Cycle expansions. Nonlinearity,3, 325–359.

  • ——, ——, and ——, 1990b: Recycling of strange sets II: Applications. Nonlinearity,3, 361–386.

  • Buizza, R., 1995: The impact of orographic forcing on barotropic unstable singular vectors. J. Atmos. Sci.,52, 1457–1472.

  • ——, and T. Palmer, 1995: The singular vector structure of the atmospheric general circulation. J. Atmos. Sci.,52, 1434–1456.

  • ——, J. Tribbia, F. Molteni, and T. Palmer, 1993: Computation of unstable structures for a numerical weather prediction model. Tellus,45A, 388–407.

  • Christiansen, F., P. Cvitanović, and V. Putkaradze, 1997: Spatio-temporal chaos in terms of unstable recurrent patterns. Nonlinearity,10, 55–70.

  • Coddington, E., and N. Levinson, 1955: Theory of Ordinary Differential Equations. McGraw-Hill, 429 pp.

  • Cvitanović, P., R. Artuso, R. Mainieri, and G. Vatay, cited 2000: Classical and quantum chaos. Niels Bohr Institute, Copenhagen, Denmark, 682 pp. [Available online at http://www.nbi.dk/ChaosBook/.].

  • Ehrendorfer, M., and J. Tribbia, 1997: Optimal prediction of forecast error covariances through singular vectors. J. Atmos. Sci.,54, 286–313.

  • Epstein, E., 1969: Stochastic dynamic prediction. Tellus,21, 739–759.

  • Farrell, B., 1989: Optimal excitation of baroclinic waves. J. Atmos. Sci.,46, 1193–1206.

  • ——, and P. Ioannou, 1996: Generalized stability theory. Part I: Autonomous operators. J. Atmos. Sci.,53, 2025–2040.

  • Joly, A., and A. Thorpe, 1991: The stability of time-dependent flows:An application to fronts in developing baroclinic waves. J. Atmos. Sci.,48, 163–182.

  • Klein, P., and J. Pedlosky, 1986: A numerical study of baroclinic instability at large supercriticality. J. Atmos. Sci.,43, 1243–1262.

  • Legras, B., and R. Vautard, 1996: A guide to Lyapunov vectors. Proc. 1995 ECMWF Seminar on Predictability, Vol. I, Reading, United Kingdom, European Centre for Medium-Range Weather Forecasts, 143–156.

  • Leith, C., 1974: Theoretical skill of Monte Carlo forecasts. Mon. Wea. Rev.,102, 409–418.

  • Lorenz, E., 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.

  • Pedlosky, J., 1971: Finite-amplitude baroclinic waves with small dissipation. J. Atmos. Sci.,28, 587–597.

  • ——, 1987: Geophysical Fluid Dynamics. Springer-Verlag, 710 pp.

  • ——, and C. Frenzen, 1980: Chaotic and periodic behavior of finite-amplitude baroclinic waves. J. Atmos. Sci.,37, 1177–1196.

  • Phillips, N., 1954: Energy transformations and meridional circulations associated with simple baroclinic waves in a two-level, quasi-geostrophic model. Tellus,6, 273–286.

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

  • Toth, Z., and E. Kalnay, 1997: Ensemble forecasting at NCEP and the breeding method. Mon. Wea. Rev.,125, 3297–3319.

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

  • ——, and F. Pancotti, 1998: Periodic orbits, Lyapunov vectors, and singular vectors in the Lorenz system. J. Atmos. Sci.,55, 390–398.

  • Vannitsem, S., and C. Nicolis, 1997: Lyapunov vectors and error growth patterns in a T21L3 quasigeostrophic model. J. Atmos. Sci.,54, 347–361.

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Periodic Orbits and Disturbance Growth for Baroclinic Waves

R. M. SamelsonCollege of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, Oregon

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Abstract

The growth of linear disturbances to stable and unstable time-periodic basic states is analyzed in an asymptotic model of weakly nonlinear, baroclinic wave–mean interaction. In this model, an ordinary differential equation for the wave amplitude is coupled to a partial differential equation for the zonal-flow correction. Floquet vectors, the eigenmodes for linear disturbances to the oscillatory basic states, split into wave-dynamical and decaying zonal-flow modes. Singular vectors reflect the structure of the Floquet vectors: the most rapid amplification and decay are associated with the wave-dynamical Floquet vectors, while the intermediate singular vectors closely follow the decaying zonal-flow Floquet vectors. Singular values depend strongly on initial and optimization times. For initial times near wave amplitude maxima, the Floquet decomposition of the leading singular vector depends relatively weakly on optimization time. For the unstable oscillatory basic state in the chaotic regime, the leading Floquet vector is tangent to the large-scale structure of the attractor, while the leading singular vector is not. However, corresponding inferences about the accessibility of disturbed states rely on the simple attractor geometry, and may not easily generalize. The primary mechanism of disturbance growth on the wave timescale in this model involves a time-dependent phase shift along the basic wave cycle. The Floquet vectors illustrate that modal disturbances to time-dependent basic states can have time-dependent spatial structure, and that the latter need not indicate nonmodal dynamics. The dynamical splitting reduces the “butterfly effect,” the ability of small-scale disturbances to influence the evolution of an unstable large-scale flow.

Corresponding author address: Dr. R. M. Samelson, College of Oceanic and Atmospheric Sciences, 104 Ocean Admin. Bldg., Oregon State University, Corvallis, OR 97331-5503.

Email: rsamelson@oce.orst.edu

Abstract

The growth of linear disturbances to stable and unstable time-periodic basic states is analyzed in an asymptotic model of weakly nonlinear, baroclinic wave–mean interaction. In this model, an ordinary differential equation for the wave amplitude is coupled to a partial differential equation for the zonal-flow correction. Floquet vectors, the eigenmodes for linear disturbances to the oscillatory basic states, split into wave-dynamical and decaying zonal-flow modes. Singular vectors reflect the structure of the Floquet vectors: the most rapid amplification and decay are associated with the wave-dynamical Floquet vectors, while the intermediate singular vectors closely follow the decaying zonal-flow Floquet vectors. Singular values depend strongly on initial and optimization times. For initial times near wave amplitude maxima, the Floquet decomposition of the leading singular vector depends relatively weakly on optimization time. For the unstable oscillatory basic state in the chaotic regime, the leading Floquet vector is tangent to the large-scale structure of the attractor, while the leading singular vector is not. However, corresponding inferences about the accessibility of disturbed states rely on the simple attractor geometry, and may not easily generalize. The primary mechanism of disturbance growth on the wave timescale in this model involves a time-dependent phase shift along the basic wave cycle. The Floquet vectors illustrate that modal disturbances to time-dependent basic states can have time-dependent spatial structure, and that the latter need not indicate nonmodal dynamics. The dynamical splitting reduces the “butterfly effect,” the ability of small-scale disturbances to influence the evolution of an unstable large-scale flow.

Corresponding author address: Dr. R. M. Samelson, College of Oceanic and Atmospheric Sciences, 104 Ocean Admin. Bldg., Oregon State University, Corvallis, OR 97331-5503.

Email: rsamelson@oce.orst.edu

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