Editorial Type:
Article
Article Type:
Research Article

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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Print Publication:
01 Nov 1998
DOI:
https://doi.org/10.1175/1520-0469(1998)055<3392:LSOTTD>2.0.CO;2
Page(s):
3392–3402
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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.

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.

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