• Asselin, R. A., 1972: Frequency filter for time integrations. Mon. Wea. Rev.,100, 487–490.

  • Barkmeijer, J., P. Houtekamer, and X. Wang, 1993: Validation of a skill prediction method. Tellus,45A, 424–434.

  • Buizza, R., 1994: Sensitivity of optimal unstable structures. Quart. J. Roy. Meteor. Soc.,120, 429–451.

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

  • Ehrendorfer, M., and R. M. Errico, 1995: Mesoscale predictability and the spectrum of optimal perturbations. J. Atmos. Sci.,52, 3475–3500.

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

  • ——, R. M. Errico, and K. D. Raeder, 1999: Singular-vector perturbation growth in a primitive equation model with moist physics. J. Atmos. Sci.,56, 1627–1648.

  • Errico, R. M., 1991: Theory and application of nonlinear normal mode initialization. NCAR Tech. Note NCAR/TN-344+IA, 137 pp. [Available from National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307.].

  • ——, 2000: Interpretations of the total energy and rotational energy norms applied to determination of singular vectors. Quart. J. Roy. Meteor. Soc.,126, 1581–1599.

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

  • Hoskins, B. J., and A. J. Simmons, 1975: A multi-layer spectral model and the semi-implicit method. Quart. J. Roy. Meteor. Soc.,101, 637–655.

  • Houtekamer, P. L., 1995: The construction of optimal perturbations. Mon. Wea. Rev.,123, 2888–2898.

  • Lorenz, E. N., 1955: Available potential energy and the maintenance of the general circulation. Tellus,7, 157–167.

  • ——, 1960: Energy and numerical weather prediction. Tellus,12, 364–373.

  • Machenhauer, B., 1979: The spectral method. Numerical Methods Used in Atmospheric Models Vol. II, GARP Publications Series, Vol. 17, A. Kasahara, Ed., World Meteorological Organization, 121–275.

  • Mak, M., 1991: Influences of the earth’s sphericity in the quasi-geostrophic theory. J. Meteor. Soc. Japan,69, 497–510.

  • Marshall, J., and F. Molteni, 1993: Toward a dynamical understanding of planetary-scale flow regimes. J. Atmos. Sci.,50, 1792–1818.

  • Molteni, F., and S. Corti, 1998: Long-term fluctuations in the statistical properties of low-frequency variability: Dynamical origin and predictability. Quart. J. Roy. Meteor. Soc.,124, 495–526.

  • Oortwijn, J., and J. Barkmeijer, 1995: Perturbations that optimally trigger weather regimes. J. Atmos. Sci.,52, 3932–3944.

  • Palmer, T. N., R. Gelaro, J. Barkmeijer, and R. Buizza, 1998: Singular vectors, metrics, and adaptive observations. J. Atmos. Sci.,55, 633–653.

  • Pedlosky, J., 1987: Geophysical Fluid Dynamics. Springer, 710 pp.

  • Reynolds, C. A., 1999: A comparison of analysis and forecast correction techniques: Impact of negative dissipation. Mon. Wea. Rev.,127, 2576–2596.

  • ——, and T. N. Palmer, 1998: Decaying singular vectors and their impact on analysis and forecast correction. J. Atmos. Sci.,55, 3005–3023.

  • ——, and R. M. Errico, 1999: Convergence of singular vectors toward Lyapunov vectors. Mon. Wea. Rev.,127, 2309–2323.

  • Salmon, R., 1998: Lectures on Geophysical Fluid Dynamics. Oxford University Press, 378 pp.

  • Snyder, C., and A. Joly, 1998: Development of perturbations within growing baroclinic waves. Quart. J. Roy. Meteor. Soc.,124, 1961–1983.

  • Swanson, K., R. Vautard, and C. Pires, 1998: Four-dimensional variational assimilation and predictability in a quasi-geostrophic model. Tellus,50A, 369–390.

  • ——, T. N. Palmer, and R. Vautard, 2000: Observational error structures and the value of advanced assimilation techniques. J. Atmos. Sci.,57, 1327–1340.

  • Talagrand, O., 1981: A study of the dynamics of four-dimensional data assimilation. Tellus,33, 43–60.

  • Thorpe, A. J., and H. Volkert, 1997: Potential vorticity: A short history of its definitions and uses. Meteor. Z., N.F.,6, 275–280.

  • 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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The Total Energy Norm in a Quasigeostrophic Model

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  • 1 Institute for Meteorology and Geophysics, University of Vienna, Vienna, Austria
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Abstract

Total energy E as the sum of kinetic and available potential energies is considered here for quasigeostrophic (QG) dynamics. The discrete expression for E is derived for the QG model formulation of Marshall and Molteni. While E is conserved by the nonlinear unforced model equations, an analogous expression in terms of perturbed fields is, in general, not conserved for tangent-linearized versions of the model, thereby allowing for growth (or decay) in this total energy norm. Examples for structures linearly growing optimally (i.e., the so-called singular vectors) in terms of either the total energy or just the kinetic energy norm are briefly illustrated and contrasted. It is argued that E might preferably be used (rather than kinetic energy) in predictability and data assimilation studies that are based on the QG model considered here.

Corresponding author address: Dr. Martin Ehrendorfer, Institute for Meteorology and Geophysics, University of Vienna, Hohe Warte 38, A-1190 Vienna, Austria.

Email: martin.ehrendorfer@univie.ac.at

Abstract

Total energy E as the sum of kinetic and available potential energies is considered here for quasigeostrophic (QG) dynamics. The discrete expression for E is derived for the QG model formulation of Marshall and Molteni. While E is conserved by the nonlinear unforced model equations, an analogous expression in terms of perturbed fields is, in general, not conserved for tangent-linearized versions of the model, thereby allowing for growth (or decay) in this total energy norm. Examples for structures linearly growing optimally (i.e., the so-called singular vectors) in terms of either the total energy or just the kinetic energy norm are briefly illustrated and contrasted. It is argued that E might preferably be used (rather than kinetic energy) in predictability and data assimilation studies that are based on the QG model considered here.

Corresponding author address: Dr. Martin Ehrendorfer, Institute for Meteorology and Geophysics, University of Vienna, Hohe Warte 38, A-1190 Vienna, Austria.

Email: martin.ehrendorfer@univie.ac.at

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