Modal and Nonmodal Symmetric Perturbations. Part II: Nonmodal Growths Measured by Total Perturbation Energy

Qin Xu NOAA/National Severe Storms Laboratory, Norman, Oklahoma

Search for other papers by Qin Xu in
Current site
Google Scholar
PubMed
Close
,
Ting Lei Institute of Atmospheric Physics, Beijing, China

Search for other papers by Ting Lei in
Current site
Google Scholar
PubMed
Close
, and
Shouting Gao Institute of Atmospheric Physics, Beijing, China

Search for other papers by Shouting Gao in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

Maximum nonmodal growths of total perturbation energy are computed for symmetric perturbations constructed from the normal modes presented in Part I. The results show that the maximum nonmodal growths are larger than the energy growth produced by any single normal mode for a give optimization time, and this is simply because the normal modes are nonorthogonal (measured by the inner product associated with the total perturbation energy norm). It is shown that the maximum nonmodal growths are produced mainly by paired modes, and this can be explained by the fact that the streamfunction component modes are partially orthogonal between different pairs and parallel within each pair in the streamfunction subspace. When the optimization time is very short (compared with the inverse Coriolis parameter), the nonmodal growth is produced mainly by the paired fastest propagating modes. When the optimization time is not short, the maximum nonmodal growth is produced almost solely by the paired slowest propagating modes and the growth can be very large for a wide range of optimization time if the parameter point is near the boundary and outside the unstable region. If the parameter point is near the boundary but inside the unstable region, the paired slowest propagating modes can contribute significantly to the energy growth before the fastest growing mode becomes the dominant component.

The maximum nonmodal growths produced by paired modes are derived analytically. The analytical solutions compare well with the numerical results obtained in the truncated normal mode space. The analytical solutions reveal the basic mechanisms for four types of maximum nonmodal energy growths: the PP1 and PP2 nonmodal growths produced by paired propagating modes and the GD1 and GD2 nonmodal growths produced by paired growing and decaying modes. The PP1 growth is characterized by the increase of the cross-band kinetic energy that excessively offsets the decrease of the along-band kinetic and buoyancy energy. The situation is opposite for the PP2 growth. The GD1 (or GD2) growth is characterized by the reduction of the initial cross-band kinetic energy (or initial along-band kinetic and buoyancy energy) due to the inclusion of the decaying mode.

Corresponding author address: Qin Xu, National Severe Storms Laboratory, 120 David L. Boren Blvd., Norman, OK 73072. Email: qin.xu@noaa.gov

Abstract

Maximum nonmodal growths of total perturbation energy are computed for symmetric perturbations constructed from the normal modes presented in Part I. The results show that the maximum nonmodal growths are larger than the energy growth produced by any single normal mode for a give optimization time, and this is simply because the normal modes are nonorthogonal (measured by the inner product associated with the total perturbation energy norm). It is shown that the maximum nonmodal growths are produced mainly by paired modes, and this can be explained by the fact that the streamfunction component modes are partially orthogonal between different pairs and parallel within each pair in the streamfunction subspace. When the optimization time is very short (compared with the inverse Coriolis parameter), the nonmodal growth is produced mainly by the paired fastest propagating modes. When the optimization time is not short, the maximum nonmodal growth is produced almost solely by the paired slowest propagating modes and the growth can be very large for a wide range of optimization time if the parameter point is near the boundary and outside the unstable region. If the parameter point is near the boundary but inside the unstable region, the paired slowest propagating modes can contribute significantly to the energy growth before the fastest growing mode becomes the dominant component.

The maximum nonmodal growths produced by paired modes are derived analytically. The analytical solutions compare well with the numerical results obtained in the truncated normal mode space. The analytical solutions reveal the basic mechanisms for four types of maximum nonmodal energy growths: the PP1 and PP2 nonmodal growths produced by paired propagating modes and the GD1 and GD2 nonmodal growths produced by paired growing and decaying modes. The PP1 growth is characterized by the increase of the cross-band kinetic energy that excessively offsets the decrease of the along-band kinetic and buoyancy energy. The situation is opposite for the PP2 growth. The GD1 (or GD2) growth is characterized by the reduction of the initial cross-band kinetic energy (or initial along-band kinetic and buoyancy energy) due to the inclusion of the decaying mode.

Corresponding author address: Qin Xu, National Severe Storms Laboratory, 120 David L. Boren Blvd., Norman, OK 73072. Email: qin.xu@noaa.gov

Save
  • Bennetts, D. A., and J. C. Sharp, 1982: The relevance of conditional symmetric instability to the prediction of mesoscale frontal rainbands. Quart. J. Roy. Meteor. Soc., 108 , 595–602.

    • Search Google Scholar
    • Export Citation
  • Buizza, R., and T. N. Palmer, 1995: The singular-vector structure of the atmospheric global circulation. J. Atmos. Sci., 52 , 1434–1456.

    • Search Google Scholar
    • Export Citation
  • Dixon, R. S., K. A. Browning, and G. J. Shutts, 2002: The relation of moist symmetric instability and upper-level potential vorticity anomalies to the observed evolution of cloud heads. Quart. J. Roy. Meteor. Soc., 128 , 839–859.

    • Search Google Scholar
    • Export Citation
  • Farrell, B. F., 1984: Modal and non-modal baroclinic waves. J. Atmos. Sci., 41 , 668–673.

  • Farrell, B. F., and P. J. Ioannou, 1996: Generalized stability theory. I. Autonomous operators. J. Atmos. Sci., 53 , 2025–2040.

  • Fovell, R., B. Rubin-Oster, and S-H. Kim, 2004: A discretely propagating nocturnal Oklahoma squall line: Observations and numerical simulations. Preprints, 22th Conf. on Severe Local Storms, Hyannis, MA, Amer. Meteor. Soc., CD-ROM, 6.1.

  • Gill, A. E., 1982: Atmosphere–Ocean Dynamics. Academic Press, 662 pp.

  • Koch, S. E., R. E. Golus, and P. B. Dorian, 1988: A mesoscale gravity wave event observed during CCOPE. Part II: Interactions between mesoscale convective systems and the antecedent waves. Mon. Wea. Rev., 116 , 2545–2569.

    • Search Google Scholar
    • Export Citation
  • Parsons, D. B., and H. P. Hobbs, 1983: The mesoscale and microscale structure and organization of clouds and precipitation in midlatitude cyclones. XI: Comparison between observational and theoretical aspects of rainbands. J. Atmos. Sci., 40 , 2377–2397.

    • Search Google Scholar
    • Export Citation
  • Uccellini, L. W., 1975: A case study of apparent gravity wave initiation of severe convective storms. Mon. Wea. Rev., 103 , 497–513.

    • Search Google Scholar
    • Export Citation
  • Xu, Q., 2007: Modal and nonmodal symmetric perturbations. Part I. Completeness of normal modes and constructions of nonmodal solutions. J. Atmos. Sci., 64 , 1745–1763.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 140 59 23
PDF Downloads 53 13 0