Discrete and Continuous Spectra of the Barotropic Quasigeostrophic Vorticity Model. Part I

Minghua Zhang Institute for Terrestrial and Planetary Atmospheres, State University of New York at Stony Brook, Stony Brook, New York

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Qingcun Zeng Institute of Atmospheric Physics, Academia Sinica, Beijing, China

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Abstract

The evolution processes of small disturbances in an arbitrary basic flow can be expressed as a combination of spectral functions of the discrete spectra and continuous spectrum of a model that bear distinctly different evolutionary characteristics. Using the linearized barotropic quasigeostrophic vorticity model, this study formulates the discrete spectral solution into a form that is consistent with traditional normal modes in time and space, and the continuous spectral solution into a form with the continuum covering the range between minimum and maximum zonal angular velocities. An estimation of the bounds of the spectral points is derived to complement those derived from integral constraints. A theorem is given to describe the possible number of discrete spectral points away from the continuum.

The theoretical analysis is then used to aid the numerical identification and interpretation of discrete and continuous spectra of the model with realistic atmospheric basic zonal flows. It is shown that neutral spectral points correspond to either ultralong waves with global meridional coverage or synoptic-scale waves in low latitudes. The unstable spectral points correspond to localized waves with developing or decaying timescales longer than 2 weeks. Structures of spectral function of the continuum are also presented and discussed. They are shown to restrict on one side to the critical latitude and on the other side to the jet core under certain conditions.

Corresponding author address: Dr. Minghua Zhang, Institute for Terrestrial and Planetary Atmospheres, State University of New York at Stony Brook, Stony Brook, NY 11794-5000.

Email: mzhang@atmsci.msrc.sunysb.edu

Abstract

The evolution processes of small disturbances in an arbitrary basic flow can be expressed as a combination of spectral functions of the discrete spectra and continuous spectrum of a model that bear distinctly different evolutionary characteristics. Using the linearized barotropic quasigeostrophic vorticity model, this study formulates the discrete spectral solution into a form that is consistent with traditional normal modes in time and space, and the continuous spectral solution into a form with the continuum covering the range between minimum and maximum zonal angular velocities. An estimation of the bounds of the spectral points is derived to complement those derived from integral constraints. A theorem is given to describe the possible number of discrete spectral points away from the continuum.

The theoretical analysis is then used to aid the numerical identification and interpretation of discrete and continuous spectra of the model with realistic atmospheric basic zonal flows. It is shown that neutral spectral points correspond to either ultralong waves with global meridional coverage or synoptic-scale waves in low latitudes. The unstable spectral points correspond to localized waves with developing or decaying timescales longer than 2 weeks. Structures of spectral function of the continuum are also presented and discussed. They are shown to restrict on one side to the critical latitude and on the other side to the jet core under certain conditions.

Corresponding author address: Dr. Minghua Zhang, Institute for Terrestrial and Planetary Atmospheres, State University of New York at Stony Brook, Stony Brook, NY 11794-5000.

Email: mzhang@atmsci.msrc.sunysb.edu

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  • Burger, A. P., 1966: Instability associated with continuous spectrum in a baroclinic flow. J. Atmos. Sci.,23, 272–277.

  • Case, K. M., 1960: Stability of inviscid plane couette-flow. Phys. Fluids,3, 143–1148.

  • Charney, J. G., 1947: The dynamics of long waves in a baroclinic westerly current. J. Meteor.,4, 125–162.

  • Dickinson, R. E., and F. J. Clare, 1973: Numerical study of the unstable modes of a hyperbolic-tangent barotropic shear flow. J. Atmos. Sci.,30, 1035–1049.

  • Dikiy, L. A., and V. V. Katayev, 1971: Calculation of the planetary wave spectrum by the Galerkin method. Atmos. Oceanic Phys.,7, 1031–1038.

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

  • ——, 1985: Transient growth and damped baroclinic waves. J. Atmos. Sci.,42, 2718–2727.

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

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

  • ——, R. L. Panetta, and R. T. Pierrehumbert, 1985: Stationary external Rossby waves in vertical shear. J. Atmos. Sci.,42, 865–883.

  • Howard, L. N., 1961: Note on a paper by John W. Miles. J. Fluid Mech.,10, 509–512.

  • Kasahara, A., 1980:Effect of zonal flow on the free oscillations of a barotropic atmosphere. J. Atmos. Sci.,37, 917–929.

  • Korn, G. A., and T. M. Korn, 1968: Mathematical Handbook for Scientists and Engineers. McGraw-Hill, 1129 pp.

  • Kuo, H. L., 1949: Dynamic instability of two-dimensional non-divergent flow in a barotropic atmosphere. J. Meteor.,6, 105–122.

  • Lu, P. S., L. Lu, and Q. C. Zeng, 1986: Spectra of the barotropic quasi-geostrophic model and the evolutionary process of disturbances. Sci. Sin.11, 1225–1233.

  • Pedlosky, J., 1964a: An initial value problem in the theory of baroclinic instability. Tellus,16, 12–17.

  • ——, 1964b: The stability of currents in the atmosphere and the ocean: Part I. J. Atmos. Sci.,21, 201–219.

  • Thuburn, J., and P. H. Haynes, 1996: Bounds on the growth rate and phase velocity of instabilities in non-divergent barotropic flow on a sphere: A semicircle theorem. Quart. J. Roy. Meteor. Soc.,122, 779–787.

  • Tung, K. K., 1983: Initial-value problems for Rossby waves in a shear flow with critical level. J. Fluid Mech.,133, 443–469.

  • Watson, M., 1981: Shear instability of differential rotation in stars. Geophys. Astrophys. Fluid Dyn.,16, 285–298.

  • Yamagata, T., 1976: On trajectories of Rossby wave packets released in a lateral shear flow. J. Oceanol. Soc. Japan,32, 162–168.

  • Yanai, M., and T. Nitta, 1968: Finite difference approximation for the barotropic instability problem. J. Meteor. Soc. Japan,46, 389–403.

  • Zeng, Q. C., 1979: The Mathematical-Physical Basis of Numerical Weather Prediction. Vol. 1, Science Press, 543 pp.

  • ——, P. S. Lu, R. F. Li, and C. G. Yuan, 1986: Evolution of large scale disturbances and their interactions with mean flow in a rotating atmosphere, Part 1. Adv. Atmos. Sci.,3, 38–58.

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