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Apparent Absolute Instability and the Continuous Spectrum

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  • 1 Department of Meteorology, The Pennsylvania State University, University Park, Pennsylvania
  • | 2 Department of Earth and Planetary Sciences, Harvard University, Cambridge, Massachusetts
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Abstract

The role of the continuous spectrum and its associated potential vorticity (PV) in absolute instability are investigated in the context of a semi-infinite version of Eady’s basic state. This flow crudely resembles the zonally averaged midlatitude atmosphere. The disturbances are composed of a zero PV part and a nonzero PV part. A closed form analytic solution is described that features a localized wave packet whose streamfunction field expands and amplifies in an absolutely unstable way. However, since an examination of the PV field associated with this disturbance reveals that the linear amplification of the localized streamfunction wave packet is induced by a nonlocalized PV field, it is clear that the seed for the expansion of the streamfunction wave packet lies not within the wave packet but upstream of the wave packet. While this precise analytic solution allows for the identification of the upstream PV anomalies, any sort of measurement error would render these upstream PV anomalies invisible. Thus, observations would be incapable of distinguishing an absolute instability seeded by PV anomalies generated within the confines of the streamfunction wave packet from the absolute instability described by the authors’ solution.

The general initial value solution is analyzed, and it is found that this apparent absolute instability is not a peculiarity of this particular solution. Absolutely unstable wave packets will be “naturally selected” over geophysically relevant timescales to dominate the flows that emerge from random disturbances to the idealized basic state. In Eady’s basic state, which is bounded aloft by a rigid lid, the natural selection mechanism only operates at wavelengths at which the normal modes of Eady’s basic state are neutral. It is suggested that an atmospheric counterpart of this natural selection process may be responsible for the medium-scale upper- and lower-tropospheric waves that have recently been identified in the observational record.

The authors prove that the group velocity of a streamfunction field attributable to eastward moving PV anomalies may, in fact, be westward.

Corresponding author address: Dr. C. H. Bishop, Dept. of Meteorology, The Pennsylvania State University, 520 Walker Building, University Park, PA 16802.

Email: cbishop@essc.psu.edu

Abstract

The role of the continuous spectrum and its associated potential vorticity (PV) in absolute instability are investigated in the context of a semi-infinite version of Eady’s basic state. This flow crudely resembles the zonally averaged midlatitude atmosphere. The disturbances are composed of a zero PV part and a nonzero PV part. A closed form analytic solution is described that features a localized wave packet whose streamfunction field expands and amplifies in an absolutely unstable way. However, since an examination of the PV field associated with this disturbance reveals that the linear amplification of the localized streamfunction wave packet is induced by a nonlocalized PV field, it is clear that the seed for the expansion of the streamfunction wave packet lies not within the wave packet but upstream of the wave packet. While this precise analytic solution allows for the identification of the upstream PV anomalies, any sort of measurement error would render these upstream PV anomalies invisible. Thus, observations would be incapable of distinguishing an absolute instability seeded by PV anomalies generated within the confines of the streamfunction wave packet from the absolute instability described by the authors’ solution.

The general initial value solution is analyzed, and it is found that this apparent absolute instability is not a peculiarity of this particular solution. Absolutely unstable wave packets will be “naturally selected” over geophysically relevant timescales to dominate the flows that emerge from random disturbances to the idealized basic state. In Eady’s basic state, which is bounded aloft by a rigid lid, the natural selection mechanism only operates at wavelengths at which the normal modes of Eady’s basic state are neutral. It is suggested that an atmospheric counterpart of this natural selection process may be responsible for the medium-scale upper- and lower-tropospheric waves that have recently been identified in the observational record.

The authors prove that the group velocity of a streamfunction field attributable to eastward moving PV anomalies may, in fact, be westward.

Corresponding author address: Dr. C. H. Bishop, Dept. of Meteorology, The Pennsylvania State University, 520 Walker Building, University Park, PA 16802.

Email: cbishop@essc.psu.edu

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