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Perturbation Growth in Baroclinic Waves

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  • 1 University of Washington, Seattle, Washington
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Abstract

Floquet theory is applied to the stability of time-periodic, nonparallel shear flows consisting of a baroclinic jet plus a neutral wave. This configuration is chosen as an idealized representation of baroclinic waves in a storm track, and the stability analysis may be helpful for understanding generic properties of the growth of forecast errors in such regions. Two useful attributes of Floquet theory relevant to this problem are that the period-average mode growth rate is norm independent, and the t→ ∞ stability limit is determined by the stability over one period. Exponentially growing Floquet modes are found for arbitrarily small departures from parallel flows.

Approximately 70% of Floquet-mode growth in energy is due to barotropic conversion, with the remainder due to zonal heat flux. Floquet-mode growth rates increase linearly with neutral wave amplitude (i.e., the “waviness” of the jet) and also increase with neutral wave wavelength. Growth rates for meridionally localized jets are approximately 40% smaller than for comparable cases with linear vertical shear (the Eady jet). Singular vectors for these flows converge to the leading Floquet mode over one basic-state period, and the leading instantaneous optimal mode closely resembles the leading Floquet mode.

Initial-value problems demonstrate that the periodic basic states are absolutely unstable, with Floquet modes spreading faster than the basic-state flow both upstream and downstream of an initially localized disturbance. This behavior dominates the convective instability of parallel-flow jets when the neutral baroclinic wave amplitude exceeds a threshold value of about 8–10 K. This result suggests that forecast errors in a storm track may spread faster, and affect upstream locations, for sufficiently wavy jets.

Corresponding author address: Gregory J. Hakim, Department of Atmospheric Sciences, University of Washington, Box 351640, Seattle, WA 98195-1640. Email: hakim@atmos.washington.edu

Abstract

Floquet theory is applied to the stability of time-periodic, nonparallel shear flows consisting of a baroclinic jet plus a neutral wave. This configuration is chosen as an idealized representation of baroclinic waves in a storm track, and the stability analysis may be helpful for understanding generic properties of the growth of forecast errors in such regions. Two useful attributes of Floquet theory relevant to this problem are that the period-average mode growth rate is norm independent, and the t→ ∞ stability limit is determined by the stability over one period. Exponentially growing Floquet modes are found for arbitrarily small departures from parallel flows.

Approximately 70% of Floquet-mode growth in energy is due to barotropic conversion, with the remainder due to zonal heat flux. Floquet-mode growth rates increase linearly with neutral wave amplitude (i.e., the “waviness” of the jet) and also increase with neutral wave wavelength. Growth rates for meridionally localized jets are approximately 40% smaller than for comparable cases with linear vertical shear (the Eady jet). Singular vectors for these flows converge to the leading Floquet mode over one basic-state period, and the leading instantaneous optimal mode closely resembles the leading Floquet mode.

Initial-value problems demonstrate that the periodic basic states are absolutely unstable, with Floquet modes spreading faster than the basic-state flow both upstream and downstream of an initially localized disturbance. This behavior dominates the convective instability of parallel-flow jets when the neutral baroclinic wave amplitude exceeds a threshold value of about 8–10 K. This result suggests that forecast errors in a storm track may spread faster, and affect upstream locations, for sufficiently wavy jets.

Corresponding author address: Gregory J. Hakim, Department of Atmospheric Sciences, University of Washington, Box 351640, Seattle, WA 98195-1640. Email: hakim@atmos.washington.edu

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