Linear Stability of the Three-Dimensional Semigeostrophic Model in Geometric Coordinates

Shuzhan Ren Department of Physics, University of Toronto, Toronto, Ontario, Canada

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Abstract

Motivated by the recent work on the stability properties of balanced dynamics, the author investigates in this paper the stability of parallel basic flow in the baroclinic semigeostrophic (SG) model in geometric coordinates. The linearized baroclinic SG equation with a nonconstant Coriolis parameter is presented. Conservation equations for two wave-activity invariants, analogous to the pseudomomentum and pseudoenergy in baroclinic quasigeostrophic dynamics but with the contributions from lateral boundaries, are derived and then used to examine the stability properties of the SG model. It is found that the lateral boundary contributions to the invariants, which were often ignored, are important to the stability mechanism of the SG model.

The general properties of normal mode disturbances in the SG model are investigated. Results obtained in this work include an orthogonality relation, the semicircle theorem to bound the phase speed of normal mode disturbances, and an estimate of upper bound on unstable growth rate.

Corresponding author address: Dr. Shuzhan Ren, Data Assimilation and Satellite Meteorology Division, Environment Canada-ARMA, 4905 Dufferin Street, Downsview, ON M3H ST4, Canada.

Email: shuzhan@smarts8.tor.ec.gc.ca

Abstract

Motivated by the recent work on the stability properties of balanced dynamics, the author investigates in this paper the stability of parallel basic flow in the baroclinic semigeostrophic (SG) model in geometric coordinates. The linearized baroclinic SG equation with a nonconstant Coriolis parameter is presented. Conservation equations for two wave-activity invariants, analogous to the pseudomomentum and pseudoenergy in baroclinic quasigeostrophic dynamics but with the contributions from lateral boundaries, are derived and then used to examine the stability properties of the SG model. It is found that the lateral boundary contributions to the invariants, which were often ignored, are important to the stability mechanism of the SG model.

The general properties of normal mode disturbances in the SG model are investigated. Results obtained in this work include an orthogonality relation, the semicircle theorem to bound the phase speed of normal mode disturbances, and an estimate of upper bound on unstable growth rate.

Corresponding author address: Dr. Shuzhan Ren, Data Assimilation and Satellite Meteorology Division, Environment Canada-ARMA, 4905 Dufferin Street, Downsview, ON M3H ST4, Canada.

Email: shuzhan@smarts8.tor.ec.gc.ca

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  • Allen, J. S., J. A. Barth., and P. A. Newberger, 1990: On intermediate models for barotropic continental shelf and slop flow fields. Part I: Formulation and comparison of exact solutions. J. Phys. Oceanogr.,20, 1017–1042.

  • Barth, J. A., 1989: Stability of a coastal upwelling front. Part I: Model development and a stability theorem. J. Geophys. Res.,94, 10 844–10 856.

  • Charney, J., and M. E. Stern, 1962: On the stability of internal baroclinic jets in a rotating atmosphere. J. Atmos. Sci.,19, 113–126.

  • Duffy, D. G., 1976: The application of the semigeostrophic equations to the frontal instability problem. J. Atmos. Sci.,33, 2322–2337.

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

  • Eliassen, A., 1948: The quasi-static equations of motion. Geofys. Publ.,17, 5–44.

  • ——, 1983: The Charney-Stern theorem on barotropic-baroclinic instability. Pure Appl. Geophys.,121, 563–572.

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

  • Hoskins, B. J., 1975: The geostrophic momentum approximation and semi-geostrophic equations. J. Atmos. Sci.,32, 233–242.

  • ——, 1976: Baroclinic waves and frontogenesis. Part I: Introduction and Eady waves. Quart. J. Roy. Meteor. Soc.,102, 103–122.

  • ——, and F. P. Bretherton, 1972: Atmospheric frontogenesis models:Mathematical formulation and solution. J. Atmos. Sci.,29, 11–37.

  • ——, and N. V. West, 1979: Baroclinic waves and frontogenesis. Part II: Uniform potential vorticity jet flows—Cold and warm fronts. J. Atmos. Sci.,36, 1663–1680.

  • Kushner, P. J., and T. G. Shepherd, 1995a: Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics. Part 1: Pseudomomentum-based theory. J. Fluid. Mech.,290, 67–104.

  • ——, and ——, 1995b: Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics. Part 2: Pseudoenergy-based theory. J. Fluid. Mech.,290, 105–129.

  • ——, M. E McIntyre, and T. G. Shepherd, 1998: Coupled Kelvin-waves and mirage-wave instability in semigeostrophic dynamics. J. Phys. Oceanogr.,28, 513–518.

  • Magnusdottir, G., and W. H. Shubert, 1990: The generalization of semigeostrophic theory to the β plane. J. Atmos. Sci.,47, 1714–1720.

  • ——, and ——, 1991: Semigeostrophic theory on the hemisphere. J. Atmos. Sci.,48, 1449–1456.

  • Mu, M., and X. Wang, 1992: Nonlinear stability criteria for the motion of three-dimensional quasi-geostrophic flow on a β-plane. Nonlinearity,5, 353–371.

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

  • Ren, S. Z., 1999: Further results on the stability of rapidly rotating vortices in the asymmetric balance formulation. J. Atmos. Sci., in press.

  • Schär, C., and H. C. Davies, 1990: An instability of mature cold front. J. Atmos. Sci.,47, 929–950.

  • Shapiro, L. J., and M. T. Montgomery, 1993: A three-dimensional balance theory for rapidly rotating vortices. J. Atmos. Sci.,50, 3322–3335.

  • Shutts, G. J., 1989: Planetary semi-geostrophic equations derived from Hamilton’s principle. J. Fluid. Mech.,208, 545–573.

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