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Barotropic Instability of Planetary-Scale Flows

R. J. HaarsmaRoyal Netherlands Meteorological Institute, De Bilt, The Netherlands

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J. D. OpsteeghRoyal Netherlands Meteorological Institute, De Bilt, The Netherlands

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Abstract

The relevance of barotropic instability for the observed low-frequency variability in the atmosphere is investigated. The stability properties of the shallow-water equations on a sphere are computed for small values of Lamb's parameter (F = α2Ω2/gHe) where a is the earth's radius, Ω its angular velocity, g gravity and He the equivalent depth. For small values of F these equations describe the horizontal structure of external and deep internal modes that are basically barotropic in the troposphere.

The stability of simple zonal flows, as well as free and forced planetary Rossby waves, has been computed as a function of F. This is done numerically using a hemispheric spectral model with a T13 truncation. For F = 0 we have tried to interpret the numerical results by analytically computing the stability properties of the flow when only one triad is considered. The results show that for increasing F the critical amplitudes for instability decrease slightly, but in the area of instability both growth rate and frequency of the perturbations decrease with increasing F. The horizontal structure of the perturbations changes only slightly. In most cases the instability process occurs within one triad which is the triad closest to resonance. An analysis in terms of unstable triads stems equally relevant for zonal and for nonzonal flows. The stability properties of the observed 400 mb Northern Hemisphere winter climatological flow show the same dependence on F as found for simple flow patterns: both growth rate and frequency of the perturbations decrease for increasing F.

Abstract

The relevance of barotropic instability for the observed low-frequency variability in the atmosphere is investigated. The stability properties of the shallow-water equations on a sphere are computed for small values of Lamb's parameter (F = α2Ω2/gHe) where a is the earth's radius, Ω its angular velocity, g gravity and He the equivalent depth. For small values of F these equations describe the horizontal structure of external and deep internal modes that are basically barotropic in the troposphere.

The stability of simple zonal flows, as well as free and forced planetary Rossby waves, has been computed as a function of F. This is done numerically using a hemispheric spectral model with a T13 truncation. For F = 0 we have tried to interpret the numerical results by analytically computing the stability properties of the flow when only one triad is considered. The results show that for increasing F the critical amplitudes for instability decrease slightly, but in the area of instability both growth rate and frequency of the perturbations decrease with increasing F. The horizontal structure of the perturbations changes only slightly. In most cases the instability process occurs within one triad which is the triad closest to resonance. An analysis in terms of unstable triads stems equally relevant for zonal and for nonzonal flows. The stability properties of the observed 400 mb Northern Hemisphere winter climatological flow show the same dependence on F as found for simple flow patterns: both growth rate and frequency of the perturbations decrease for increasing F.

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