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Symmetric Stability of Compressible Zonal Flows on a Generalized Equatorial β Plane

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  • 1 Department of Physics, University of Toronto, Toronto, Ontario, Canada
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Abstract

Sufficient conditions are derived for the linear stability with respect to zonally symmetric perturbations of a steady zonal solution to the nonhydrostatic compressible Euler equations on an equatorial β plane, including a leading order representation of the Coriolis force terms due to the poleward component of the planetary rotation vector.

A version of the energy–Casimir method of stability proof is applied: an invariant functional of the Euler equations linearized about the equilibrium zonal flow is found, and positive definiteness of the functional is shown to imply linear stability of the equilibrium. It is shown that an equilibrium is stable if the potential vorticity has the same sign as latitude and the Rayleigh centrifugal stability condition that absolute angular momentum increase toward the equator on surfaces of constant pressure is satisfied. The result generalizes earlier results for hydrostatic and incompressible systems and for systems that do not account for the nontraditional Coriolis force terms.

The stability of particular equilibrium zonal velocity, entropy, and density fields is assessed. A notable case in which the effect of the nontraditional Coriolis force is decisive is the instability of an angular momentum profile that decreases away from the equator but is flatter than quadratic in latitude, despite its satisfying both the centrifugal and convective stability conditions.

* Current affiliation: Laboratoire de Physique des Oceans, IFREMER, Brest, France

Corresponding author address: T. G. Shepherd, Department of Physics, University of Toronto, 60 St. George Street, Toronto, ON M5S 1A7, Canada. Email: tgs@atmosp.physics.utoronto.ca

Abstract

Sufficient conditions are derived for the linear stability with respect to zonally symmetric perturbations of a steady zonal solution to the nonhydrostatic compressible Euler equations on an equatorial β plane, including a leading order representation of the Coriolis force terms due to the poleward component of the planetary rotation vector.

A version of the energy–Casimir method of stability proof is applied: an invariant functional of the Euler equations linearized about the equilibrium zonal flow is found, and positive definiteness of the functional is shown to imply linear stability of the equilibrium. It is shown that an equilibrium is stable if the potential vorticity has the same sign as latitude and the Rayleigh centrifugal stability condition that absolute angular momentum increase toward the equator on surfaces of constant pressure is satisfied. The result generalizes earlier results for hydrostatic and incompressible systems and for systems that do not account for the nontraditional Coriolis force terms.

The stability of particular equilibrium zonal velocity, entropy, and density fields is assessed. A notable case in which the effect of the nontraditional Coriolis force is decisive is the instability of an angular momentum profile that decreases away from the equator but is flatter than quadratic in latitude, despite its satisfying both the centrifugal and convective stability conditions.

* Current affiliation: Laboratoire de Physique des Oceans, IFREMER, Brest, France

Corresponding author address: T. G. Shepherd, Department of Physics, University of Toronto, 60 St. George Street, Toronto, ON M5S 1A7, Canada. Email: tgs@atmosp.physics.utoronto.ca

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