An Analysis of Interactions between Geostrophic and Ageostrophic Modes in a Simple Model

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  • 1 National Center for Atmospheric Research, Boulder, CO 80307
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Abstract

We examine some properties of the extratropical atmosphere which act to maintain quasi-geostrophic balance. A nonlinear, f-plane, primitive equation, two-layer model is used. The momentum and temperature fields are described in terms of normal modes of the system given by the model's linear terms. These modes are classified as either geostrophic or ageostrophic depending on their associated eigenvalues. The original nonlinear equations are transformed into a system in which the modulations of the modal amplitudes by nonlinear effects are explicitly expressed in terms of the modal amplitudes themselves. This transformation facilitates a multiple-time-scale analysis.

The stability of simple finite-amplitude geostrophic solutions with respect to infinitesimal perturbations in other modes is investigated. Results are discussed for nondimensional unperturbed-state amplitudes of magnitude ε < 1. These geostrophic solutions may be unstable with respect to further geostrophic perturbations, with growth rates order ε. The ageostrophic modes in this case satisfy approximate quasi-geostrophic conditions. The geostrophic solutions may also be unstable with respect to ageostrophic perturbations, but only with growth rates of least order ε2.

The time-mean balance of ageostrophic energy is investigated using time-scale analysis and ordering arguments. Nearly resonant triad interactions between a geostrophic mode and a pair of inertial-gravity waves only weakly act to affect the geostrophic mode. This property and the separation between natural frequencies of geostrophic and ageostrophic modes suggests that energy is not readily exchanged between modes of the two types. Sufficiently strong dissipation by viscosity and diabatic effects is necessary to limit a slow accumulation of energy by inertial-gravity waves.

Abstract

We examine some properties of the extratropical atmosphere which act to maintain quasi-geostrophic balance. A nonlinear, f-plane, primitive equation, two-layer model is used. The momentum and temperature fields are described in terms of normal modes of the system given by the model's linear terms. These modes are classified as either geostrophic or ageostrophic depending on their associated eigenvalues. The original nonlinear equations are transformed into a system in which the modulations of the modal amplitudes by nonlinear effects are explicitly expressed in terms of the modal amplitudes themselves. This transformation facilitates a multiple-time-scale analysis.

The stability of simple finite-amplitude geostrophic solutions with respect to infinitesimal perturbations in other modes is investigated. Results are discussed for nondimensional unperturbed-state amplitudes of magnitude ε < 1. These geostrophic solutions may be unstable with respect to further geostrophic perturbations, with growth rates order ε. The ageostrophic modes in this case satisfy approximate quasi-geostrophic conditions. The geostrophic solutions may also be unstable with respect to ageostrophic perturbations, but only with growth rates of least order ε2.

The time-mean balance of ageostrophic energy is investigated using time-scale analysis and ordering arguments. Nearly resonant triad interactions between a geostrophic mode and a pair of inertial-gravity waves only weakly act to affect the geostrophic mode. This property and the separation between natural frequencies of geostrophic and ageostrophic modes suggests that energy is not readily exchanged between modes of the two types. Sufficiently strong dissipation by viscosity and diabatic effects is necessary to limit a slow accumulation of energy by inertial-gravity waves.

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