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Charney Problem with a Generic Stratosphere

Mankin Mak 1 , Siyu Zhao 2 , and Yi Deng 3
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  • 1 Department of Atmospheric Sciences, University of Illinois at Urbana-Champaign, Champaign, Illinois
  • 2 School of Earth and Atmospheric Sciences, Georgia Institute of Technology, Atlanta, Georgia, and Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, Los Angeles, California
  • 3 School of Earth and Atmospheric Sciences, Georgia Institute of Technology, Atlanta, Georgia
Published-online:
12 Jan 2021
DOI:
https://doi.org/10.1175/JAS-D-20-0027.1
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Abstract

This paper reports a comprehensive instability analysis of a 3D Charney-like model with an observationally compatible generic stratosphere. It is found that the values of a single non-dimensional parameter Γ=λfo2βDN12 (detailed definition in text), in conjunction with representative values of four other non-dimensional parameters, would dictate the existence of multiple branches of unstable modes as a function of the zonal wavenumber. Prototype Charney mode, Green mode, and two additional structurally distinct modes (Charney+ Mode and Tropopause Mode) are identified. The latter result from the additional strong influence of the tropopause. The dynamical nature of all modes is delineated in terms of their meridional fluxes of heat and potential-vorticity. The three-dimensional structure of the ageostrophic velocity field in each mode is presented to identify its potential of inducing frontogenesis. Optimal-mode analyses are also performed to ascertain how a disturbance with a predisposed structure would explosively develop towards each type of normal mode. In view of the pivotal role of Γ in baroclinic instability and for historical reason, we name it the Charney Number. It is most instructive to think of Γ as a ratio of the meridional gradient of PV associated with the basic flow in the troposphere, λfo2DN12, to that associated with the earth’s rotation, β.

Corresponding author address: Mankin Mak, Department of Atmospheric Sciences, University of Illinois at Urbana-Champaign, Champaign, Illinois. Email: m-mak@illinois.edu

Abstract

This paper reports a comprehensive instability analysis of a 3D Charney-like model with an observationally compatible generic stratosphere. It is found that the values of a single non-dimensional parameter Γ=λfo2βDN12 (detailed definition in text), in conjunction with representative values of four other non-dimensional parameters, would dictate the existence of multiple branches of unstable modes as a function of the zonal wavenumber. Prototype Charney mode, Green mode, and two additional structurally distinct modes (Charney+ Mode and Tropopause Mode) are identified. The latter result from the additional strong influence of the tropopause. The dynamical nature of all modes is delineated in terms of their meridional fluxes of heat and potential-vorticity. The three-dimensional structure of the ageostrophic velocity field in each mode is presented to identify its potential of inducing frontogenesis. Optimal-mode analyses are also performed to ascertain how a disturbance with a predisposed structure would explosively develop towards each type of normal mode. In view of the pivotal role of Γ in baroclinic instability and for historical reason, we name it the Charney Number. It is most instructive to think of Γ as a ratio of the meridional gradient of PV associated with the basic flow in the troposphere, λfo2DN12, to that associated with the earth’s rotation, β.

Corresponding author address: Mankin Mak, Department of Atmospheric Sciences, University of Illinois at Urbana-Champaign, Champaign, Illinois. Email: m-mak@illinois.edu
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