Editorial Type:
Article
Article Type:
Research Article

Dependence of Singular Vector Structure and Evolution on the Choice of Norm

Hyun Mee Kim Department of Atmospheric and Oceanic Sciences, University of Wisconsin—Madison, Madison, Wisconsin

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Michael C. Morgan Department of Atmospheric and Oceanic Sciences, University of Wisconsin—Madison, Madison, Wisconsin

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Print Publication:
01 Nov 2002
DOI:
https://doi.org/10.1175/1520-0469(2002)059<3099:DOSVSA>2.0.CO;2
Page(s):
3099–3116
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Abstract

A diagnosis of singular vector (SV) evolution in the Eady model for the potential enstrophy and energy norms is performed using potential vorticity (PV) inversion and Eliassen–Palm (E–P) flux diagnostics, and compared with the SV evolution for the streamfunction variance norm. The diagnostics reveal that the mechanism for SV amplification depends on the initial relative magnitudes of the interior PV and boundary temperature anomalies (BTAs). In addition, the relative magnitudes of the initial PV and BTAs are dependent on the norm chosen, the length scale of the perturbation, and the length of the optimization interval.

If the initial contribution of the PV to a given norm is larger than the contribution of the BTAs to that norm, then the SV evolution in that norm is governed by the baroclinic superposition of the interior PV followed by an amplification of the BTAs by winds attributed to the interior PV. In the other case, the mutual interaction of BTAs governs the SV evolution. The initial interior PV is most important for the energy and streamfunction variance SVs, but is less important for the potential enstrophy SVs. Excluding the longwave (i.e., wavelengths longer than the Eady instability cutoff) enstrophy norm SVs, for the shortwave SVs and for long optimization times, the importance of the initial interior PV is most apparent.

In the view of targeted observations, the sensitive regions indicated by the SV analysis can be identified with particular mechanisms for SV development. The forecast measure may be considered sensitive in some regions in the sense that the forecast measure exhibits a large response to small changes in the initial conditions in those regions. The potential enstrophy norm is identified as being dynamically sensitive at the boundaries in contrast to the energy and streamfunction variance norm in the midtroposphere. It is suggested that subjective PV diagnosis of sensitivity may be viewed as being consistent with an objective diagnosis of sensitivity using potential enstrophy norm SVs.

Corresponding author address: Hyun Mee Kim, Department of Atmospheric and Oceanic Sciences, University of Wisconsin—Madison, 1225 W. Dayton Street, Madison, WI 53706. Email: khm@mapmaker.meteor.wisc.edu

Abstract

A diagnosis of singular vector (SV) evolution in the Eady model for the potential enstrophy and energy norms is performed using potential vorticity (PV) inversion and Eliassen–Palm (E–P) flux diagnostics, and compared with the SV evolution for the streamfunction variance norm. The diagnostics reveal that the mechanism for SV amplification depends on the initial relative magnitudes of the interior PV and boundary temperature anomalies (BTAs). In addition, the relative magnitudes of the initial PV and BTAs are dependent on the norm chosen, the length scale of the perturbation, and the length of the optimization interval.

If the initial contribution of the PV to a given norm is larger than the contribution of the BTAs to that norm, then the SV evolution in that norm is governed by the baroclinic superposition of the interior PV followed by an amplification of the BTAs by winds attributed to the interior PV. In the other case, the mutual interaction of BTAs governs the SV evolution. The initial interior PV is most important for the energy and streamfunction variance SVs, but is less important for the potential enstrophy SVs. Excluding the longwave (i.e., wavelengths longer than the Eady instability cutoff) enstrophy norm SVs, for the shortwave SVs and for long optimization times, the importance of the initial interior PV is most apparent.

In the view of targeted observations, the sensitive regions indicated by the SV analysis can be identified with particular mechanisms for SV development. The forecast measure may be considered sensitive in some regions in the sense that the forecast measure exhibits a large response to small changes in the initial conditions in those regions. The potential enstrophy norm is identified as being dynamically sensitive at the boundaries in contrast to the energy and streamfunction variance norm in the midtroposphere. It is suggested that subjective PV diagnosis of sensitivity may be viewed as being consistent with an objective diagnosis of sensitivity using potential enstrophy norm SVs.

Corresponding author address: Hyun Mee Kim, Department of Atmospheric and Oceanic Sciences, University of Wisconsin—Madison, 1225 W. Dayton Street, Madison, WI 53706. Email: khm@mapmaker.meteor.wisc.edu

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