Editorial Type:
Article
Article Type:
Research Article

Linear Amplification and Error Growth in the 2D Eady Problem with Uniform Potential Vorticity

Claude Fischer Météo-France, Centre National de Recherches Météorologiques/Groupe de Météorologie à Moyenne Echelle, Toulouse, France

Search for other papers by Claude Fischer in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Nov 1998
DOI:
https://doi.org/10.1175/1520-0469(1998)055<3363:LAAEGI>2.0.CO;2
Page(s):
3363–3380
Restricted access

Abstract

The concept of a singular mode underlies optimal linear amplification theories. This concept is studied in the frame of the two-dimensional, quasigeostrophic Eady problem with uniform potential vorticity. Analytical solutions are produced for the relevant physical norms. Exact relations are also derived for the amplifications, which give the lower and upper bounds to any linear development. Results show significant differences in the structure of the singular modes, as well as in the associated amplifications, when the horizontal wavenumber is varied or the inner product is changed. It is found that the singular modes can depart significantly from the normal modes, though the dynamics of the problem are very simple. Comparisons with previous works are also performed. Finally, the derived equations are used to present the linear evolution of error growth within the Eady problem, as predicted by a Kalman filter. Considerations on the spectral space error covariance matrix are made, and a particular case of error dynamics in the 2D physical space is shown. The derivation of the general algebraic solutions is included in the appendix.

Corresponding author address: Dr. Claude Fischer, Météo-France, CNRM/GMME, 42, Avenue Gustave Coriolis, 31057 Toulouse, Cedex, France.

Abstract

The concept of a singular mode underlies optimal linear amplification theories. This concept is studied in the frame of the two-dimensional, quasigeostrophic Eady problem with uniform potential vorticity. Analytical solutions are produced for the relevant physical norms. Exact relations are also derived for the amplifications, which give the lower and upper bounds to any linear development. Results show significant differences in the structure of the singular modes, as well as in the associated amplifications, when the horizontal wavenumber is varied or the inner product is changed. It is found that the singular modes can depart significantly from the normal modes, though the dynamics of the problem are very simple. Comparisons with previous works are also performed. Finally, the derived equations are used to present the linear evolution of error growth within the Eady problem, as predicted by a Kalman filter. Considerations on the spectral space error covariance matrix are made, and a particular case of error dynamics in the 2D physical space is shown. The derivation of the general algebraic solutions is included in the appendix.

Corresponding author address: Dr. Claude Fischer, Météo-France, CNRM/GMME, 42, Avenue Gustave Coriolis, 31057 Toulouse, Cedex, France.

Save
Cover Journal of the Atmospheric Sciences
Journal of the Atmospheric Sciences

  • Bishop, C. H., 1993: On the behaviour of baroclinic waves undergoing horizontal deformation. II: Error-bound amplification and Rossby wave diagnostics. Quart. J. Roy. Meteor. Soc.,119, 241–267.

  • Bouttier, F., 1994: Sur la prévision de la qualité des prévisions météorologiques. Ph.D. thesis, Université P. Sabatier-Toulouse III, 240 pp.

  • Bretherton, F. P., 1966: Baroclinic instability and the short wave cutoff in terms of potential vorticity. Quart. J. Roy. Meteor. Soc.,92, 335–345.

  • Davies, H., and C. Bishop, 1994: Eady edge waves and rapid development. J. Atmos. Sci.,51, 1930–1946.

  • Eady, E., 1949: Long waves and cyclone waves. Tellus,1, 33–52.

  • Farrell, B. F., 1982: The initial growth of disturbances in a baroclinic flow. J. Atmos. Sci.,39, 1663–1686.

  • ——, 1984: Modal and non-modal baroclinic waves. J. Atmos. Sci.,41, 668–673.

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

  • Ghil, M., 1989: Meteorological data assimilation for oceanographers. Part I: Description and theoretical framework. Dyn. Atmos. Oceans,13, 171–218.

  • Gill, A., 1982: Atmosphere–Ocean Dynamics. International Geophysics Series, Vol. 30, Academic Press, 662 pp.

  • Held, I. M., 1985: Pseudomomentum and the orthogonality of modes in shear flows. J. Atmos. Sci.,42, 2280–2288.

  • Holton, J. R., 1979: An Introduction to Dynamic Meteorology. International Geophysics Series, Vol. 23, Academic Press, 391 pp.

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

  • Joly, A., 1995: The stability of steady fronts and the adjoint method:Nonmodal frontal waves. J. Atmos. Sci.,52, 3082–3108.

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

  • Kalman, R., 1960: A new approach to linear filtering and prediction problems. Trans. ASME,82D, 35–45.

  • ——, and R. Bucy, 1961: New results in linear filtering and prediction theory. Trans. ASME,83D, 95–108.

  • Mukougawa, H., and, T. Ikeda, 1994: Optimal excitation of baroclinic waves in the Eady model. J. Meteor. Soc. Japan,72, 499–513.

  • O’Brien, E., 1992: Optimal growth rates in the quasigeostrophic initial value problem. J. Atmos. Sci.,49, 1557–1570.

  • Rotunno, R., and M. Fantini, 1989: Petterssen’s type B cyclogenesis in terms of discrete, neutral Eady modes. J. Atmos. Sci.,46, 3599–3604.

  • Snyder, C., 1996: Summary of an informal workshop on adaptive observations and FASTEX. Bull. Amer. Meteor. Soc.,77, 953–961.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 364 167 6
PDF Downloads 72 23 1