Baroclinic Instability on the Sphere: Normal Modes of the Primitive and Quasi-Geostrophic Equations

A. J. Simmons U. K. Universities’ Atmospheric Modelling Group, Department of Geophysics, University of Reading, England RG6 2AU

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B. J. Hoskins U. K. Universities’ Atmospheric Modelling Group, Department of Geophysics, University of Reading, England RG6 2AU

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Abstract

The growth rate, phase speed, structure and transfer properties of normal modes of the primitive and quasi-geostrophic equations have been determined by applying an initial value technique to global nonlinear atmospheric models. Results are presented for three zonal flows that have the same vertical structure but quite different meridional variations. Use of a variety of vertical and horizontal resolutions gives important indications of truncation error.

Many properties of the unstable modes are much as found in simpler models of baroclinic instability, but spherical geometry has a significant effect on the location of the disturbances, particularly those of low zonal wavenumber, and on eddy momentum fluxes. The latter vary greatly from profile to profile, but mean meridional circulations are such as to give little net variation in the pattern of induced mean zonal surface winds. In fact, the change in vertical shear at the surface is shown to depend in the quasi-geostrophic limit only on the poleward eddy heat flux, which varies little, except in meridional position. Quasi-geostrophic solutions are generally similar to those of the primitive equations, although small differences are often of consistent sign. However, neglect of vertical eddy heat transfer, and to a lesser extent momentum transfer, is a poor approximation.

The present results are in some qualitative agreement with others obtained independently using two-level models, but such models are shown to be subject to severe quantitative error. More generally, vertical truncation error is found to give rise to spurious high-wavenumber growth.

Abstract

The growth rate, phase speed, structure and transfer properties of normal modes of the primitive and quasi-geostrophic equations have been determined by applying an initial value technique to global nonlinear atmospheric models. Results are presented for three zonal flows that have the same vertical structure but quite different meridional variations. Use of a variety of vertical and horizontal resolutions gives important indications of truncation error.

Many properties of the unstable modes are much as found in simpler models of baroclinic instability, but spherical geometry has a significant effect on the location of the disturbances, particularly those of low zonal wavenumber, and on eddy momentum fluxes. The latter vary greatly from profile to profile, but mean meridional circulations are such as to give little net variation in the pattern of induced mean zonal surface winds. In fact, the change in vertical shear at the surface is shown to depend in the quasi-geostrophic limit only on the poleward eddy heat flux, which varies little, except in meridional position. Quasi-geostrophic solutions are generally similar to those of the primitive equations, although small differences are often of consistent sign. However, neglect of vertical eddy heat transfer, and to a lesser extent momentum transfer, is a poor approximation.

The present results are in some qualitative agreement with others obtained independently using two-level models, but such models are shown to be subject to severe quantitative error. More generally, vertical truncation error is found to give rise to spurious high-wavenumber growth.

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