A Potential Vorticity and Wave Activity Diagnosis of Optimal Perturbation Evolution

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

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Abstract

A diagnosis of singular vector (SV) evolution (computed for the L2 streamfunction norm) in the Eady model using potential vorticity (PV) and Eliassen–Palm (E–P) flux diagnostics is performed. In addition, a partitioning of the vertical component of the E–P flux vector based on the results of the piecewise PV inversion is introduced to better elucidate the fundamental mechanisms for SV amplification.

The initial PV structures of the Eady model SVs on both the long- and shortwave sides of the Eady model shortwave cutoff are characterized by initially upshear tilted interior PV anomalies. The results of the PV and the E–P flux diagnostics for optimal perturbations reveal a three-stage process for the SV evolution: 1) a superposition of interior PV anomalies (diagnosed by a positive vertical component of the E–P flux), 2) a subsequent intensification (characterized by maxima in the E–P flux near the boundaries) of the SV boundary potential temperature anomalies (BTAs) by winds attributed to interior PV, and 3) finally a transient or sustained mutual interaction between the BTAs (associated with a nearly nondivergent interior E–P flux). The PV inversion demonstrates that a significant fraction of the observed SV amplification may be attributed to the initially upshear tilted PV anomalies, and that the initial BTAs, while important in describing the initial SV structure, play a minimal role in the subsequent evolution.

The results of the experiments in which the Eady model SV structures were altered by removing either the BTAs or the interior PV suggest that data assimilation schemes that more heavily weight targeted surface observations to targeted tropospheric wind and temperature observations may result in a forecast correction that is either excessive (for short optimization times) or insufficient for optimization times comparable to or larger than the time required for the PV to be rendered vertical by the shear flow.

An analysis of SVs calculated about the time-varying basic state of an NWP model reveals that the SV PV initially consists of upshear tilted structures located beneath a depressed tropopause in a region of strong low-level thermal gradient (and concomitant vertical shear). The subsequent nonlinear evolution of the SV PV in the NWP model is congruous with that observed in the Eady model: The vertical superposition of SV PV, followed by amplification of a surface thermal ridge by winds attributed to the SV PV, characterize the SV amplification. These results suggest the relevance of the SV amplification mechanisms identified in the Eady model in more realistic flows, and indicate the potential utility of applying the aforementioned diagnostic techniques to understanding SV development in observed flows.

Corresponding author address: Dr. Michael C. Morgan, Dept. of Atmospheric and Oceanic Sciences, University of Wisconsin—Madison, 1225 West Dayton St., Madison, WI 53706. Email: morgan@aurora.meteor.wisc.edu

Abstract

A diagnosis of singular vector (SV) evolution (computed for the L2 streamfunction norm) in the Eady model using potential vorticity (PV) and Eliassen–Palm (E–P) flux diagnostics is performed. In addition, a partitioning of the vertical component of the E–P flux vector based on the results of the piecewise PV inversion is introduced to better elucidate the fundamental mechanisms for SV amplification.

The initial PV structures of the Eady model SVs on both the long- and shortwave sides of the Eady model shortwave cutoff are characterized by initially upshear tilted interior PV anomalies. The results of the PV and the E–P flux diagnostics for optimal perturbations reveal a three-stage process for the SV evolution: 1) a superposition of interior PV anomalies (diagnosed by a positive vertical component of the E–P flux), 2) a subsequent intensification (characterized by maxima in the E–P flux near the boundaries) of the SV boundary potential temperature anomalies (BTAs) by winds attributed to interior PV, and 3) finally a transient or sustained mutual interaction between the BTAs (associated with a nearly nondivergent interior E–P flux). The PV inversion demonstrates that a significant fraction of the observed SV amplification may be attributed to the initially upshear tilted PV anomalies, and that the initial BTAs, while important in describing the initial SV structure, play a minimal role in the subsequent evolution.

The results of the experiments in which the Eady model SV structures were altered by removing either the BTAs or the interior PV suggest that data assimilation schemes that more heavily weight targeted surface observations to targeted tropospheric wind and temperature observations may result in a forecast correction that is either excessive (for short optimization times) or insufficient for optimization times comparable to or larger than the time required for the PV to be rendered vertical by the shear flow.

An analysis of SVs calculated about the time-varying basic state of an NWP model reveals that the SV PV initially consists of upshear tilted structures located beneath a depressed tropopause in a region of strong low-level thermal gradient (and concomitant vertical shear). The subsequent nonlinear evolution of the SV PV in the NWP model is congruous with that observed in the Eady model: The vertical superposition of SV PV, followed by amplification of a surface thermal ridge by winds attributed to the SV PV, characterize the SV amplification. These results suggest the relevance of the SV amplification mechanisms identified in the Eady model in more realistic flows, and indicate the potential utility of applying the aforementioned diagnostic techniques to understanding SV development in observed flows.

Corresponding author address: Dr. Michael C. Morgan, Dept. of Atmospheric and Oceanic Sciences, University of Wisconsin—Madison, 1225 West Dayton St., Madison, WI 53706. Email: morgan@aurora.meteor.wisc.edu

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