Editorial Type:
Article
Article Type:
Research Article

Resonance in Optimal Perturbation Evolution. Part II: Effects of a Nonzero Mean PV Gradient

H. de Vries Institute for Marine and Atmospheric Research, Utrecht University, Utrecht, Netherlands

Search for other papers by H. de Vries in
Current site
Google Scholar
PubMed Close
and
J. D. Opsteegh Institute for Marine and Atmospheric Research, Utrecht University, Utrecht, Netherlands

Search for other papers by J. D. Opsteegh in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Mar 2007
DOI:
https://doi.org/10.1175/JAS3868.1
Page(s):
695–710
Restricted access

We are aware of a technical issue preventing figures and tables from showing in some newly published articles in the full-text HTML view.
While we are resolving the problem, please use the online PDF version of these articles to view figures and tables.

Abstract

Optimal perturbations are constructed for a two-layer β-plane extension of the Eady model. The surface and interior dynamics is interpreted using the concept of potential vorticity building blocks (PVBs), which are zonally wavelike, vertically confined sheets of quasigeostrophic potential vorticity. The results are compared with the Charney model and with the two-layer Eady model without β. The authors focus particularly on the role of the different growth mechanisms in the optimal perturbation evolution.

The optimal perturbations are constructed allowing only one PVB, three PVBs, and finally a discrete equivalent of a continuum of PVBs to be present initially. On the f plane only the PVB at the surface and at the tropopause can be amplified. In the presence of β, however, PVBs influence each other’s growth and propagation at all levels. Compared to the two-layer f-plane model, the inclusion of β slightly reduces the surface growth and propagation speed of all optimal perturbations. Responsible for the reduction are the interior PVBs, which are excited by the initial PVB after initialization. Their joint effect is almost as strong as the effect from the excited tropopause PVB, which is also negative at the surface.

If the optimal perturbation is composed of more than one PVB, the Orr mechanism dominates the initial amplification in the entire troposphere. At low levels, the interaction between the surface PVB and the interior tropospheric PVBs (in particular those near the critical level) takes over after about half a day, whereas the interaction between the tropopause PVB and the interior PVBs is responsible for the main amplification in the upper troposphere. In all cases in which more than one PVB is used, the growing normal mode configuration is not reached at optimization time.

* Current affiliation: Department of Meteorology, University of Reading, Reading, United Kingdom

Corresponding author address: Dr. Hylke de Vries, Department of Meteorology, University of Reading, P.O. Box 243, Earley Gate, RG6 6BB Reading, United Kingdom. Email: h.devries@reading.ac.uk

Abstract

Optimal perturbations are constructed for a two-layer β-plane extension of the Eady model. The surface and interior dynamics is interpreted using the concept of potential vorticity building blocks (PVBs), which are zonally wavelike, vertically confined sheets of quasigeostrophic potential vorticity. The results are compared with the Charney model and with the two-layer Eady model without β. The authors focus particularly on the role of the different growth mechanisms in the optimal perturbation evolution.

The optimal perturbations are constructed allowing only one PVB, three PVBs, and finally a discrete equivalent of a continuum of PVBs to be present initially. On the f plane only the PVB at the surface and at the tropopause can be amplified. In the presence of β, however, PVBs influence each other’s growth and propagation at all levels. Compared to the two-layer f-plane model, the inclusion of β slightly reduces the surface growth and propagation speed of all optimal perturbations. Responsible for the reduction are the interior PVBs, which are excited by the initial PVB after initialization. Their joint effect is almost as strong as the effect from the excited tropopause PVB, which is also negative at the surface.

If the optimal perturbation is composed of more than one PVB, the Orr mechanism dominates the initial amplification in the entire troposphere. At low levels, the interaction between the surface PVB and the interior tropospheric PVBs (in particular those near the critical level) takes over after about half a day, whereas the interaction between the tropopause PVB and the interior PVBs is responsible for the main amplification in the upper troposphere. In all cases in which more than one PVB is used, the growing normal mode configuration is not reached at optimization time.

* Current affiliation: Department of Meteorology, University of Reading, Reading, United Kingdom

Corresponding author address: Dr. Hylke de Vries, Department of Meteorology, University of Reading, P.O. Box 243, Earley Gate, RG6 6BB Reading, United Kingdom. Email: h.devries@reading.ac.uk

Save
Cover Journal of the Atmospheric Sciences
Journal of the Atmospheric Sciences

  • Bretherton, F. P., 1966: Baroclinic instability and the short wavelength cut-off in terms of potential vorticity. Quart. J. Roy. Meteor. Soc., 92 , 335–345.

    • Search Google Scholar
    • Export Citation

  • Chang, E. K. M., 1992: Resonating neutral modes of the Eady model. J. Atmos. Sci., 49 , 2452–2463.

  • Charney, J. G., 1947: The dynamics of long waves in a baroclinic westerly current. J. Atmos. Sci., 4 , 135–162.

  • Charney, J. G., and M. E. Stern, 1962: On the stability of internal baroclinic jets in a rotating atmosphere. J. Atmos. Sci., 19 , 159–172.

    • Search Google Scholar
    • Export Citation

  • Davies, H. C., and C. H. Bishop, 1994: Eady edge waves and rapid development. J. Atmos. Sci., 51 , 1930–1946.

  • De Vries, H., and J. D. Opsteegh, 2005: Optimal perturbations in the Eady model: Resonance versus PV unshielding. J. Atmos. Sci., 62 , 492–505.

    • Search Google Scholar
    • Export Citation

  • De Vries, H., and J. D. Opsteegh, 2006: Dynamics of singular vectors in the semi-infinite Eady model: Nonzero β but zero mean PV gradient. J. Atmos. Sci., 63 , 547–564.

    • Search Google Scholar
    • Export Citation

  • De Vries, H., and J. D. Opsteegh, 2007: Resonance in optimal perturbation evolution. Part I: Two-layer Eady model. J. Atmos. Sci., 64 , 673–694.

    • Search Google Scholar
    • Export Citation

  • Dirren, S., and H. C. Davies, 2004: Combined dynamics of boundary and interior perturbations in the Eady setting. J. Atmos. Sci., 61 , 1549–1565.

    • Search Google Scholar
    • Export Citation

  • Eady, E. T., 1949: Long waves and cyclone waves. Tellus, 1 , 33–52.

  • Farrell, B. F., 1982: The initial growth of disturbances in a baroclinic flow. J. Atmos. Sci., 39 , 1663–1686.

  • 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. Part I: Autonomous operators. J. Atmos. Sci., 53 , 2025–2040.

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

  • Green, J. S. A., 1960: A problem in baroclinic stability. Quart. J. Roy. Meteor. Soc., 86 , 237–251.

  • Harnik, N., and R. S. Lindzen, 1998: The effect of basic-state potential vorticity gradients on the growth of baroclinic waves and the height of the tropopause. J. Atmos. Sci., 55 , 344–360.

    • Search Google Scholar
    • Export Citation

  • Heifetz, E., and J. Methven, 2005: Relating optimal growth to counterpropagating Rossby waves in shear instability. Phys. Fluids, 17 .064107, doi:10.1063/1.1937064.

    • Search Google Scholar
    • Export Citation

  • Heifetz, E., C. H. Bishop, B. J. Hoskins, and J. Methven, 2004a: The counter-propagating Rossby-wave perspective on baroclinic instability. I: Mathematical basis. Quart. J. Roy. Meteor. Soc., 130 , 211–231.

    • Search Google Scholar
    • Export Citation

  • Heifetz, E., J. Methven, B. J. Hoskins, and C. H. Bishop, 2004b: The counter-propagating Rossby-wave perspective on baroclinic instability. II: Application to the Charney model. Quart. J. Roy. Meteor. Soc., 130 , 233–258.

    • Search Google Scholar
    • Export Citation

  • Hoskins, B. J., M. E. McIntyre, and A. W. Robertson, 1985: On the use and significance of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc., 111 , 877–946.

    • Search Google Scholar
    • Export Citation

  • Juckes, M. N., 1994: Quasigeostrophic dynamics of the tropopause. J. Atmos. Sci., 51 , 2756–2768.

  • Morgan, M. C., 2001: A potential vorticity and wave activity diagnosis of optimal perturbation evolution. J. Atmos. Sci., 58 , 2518–2544.

    • Search Google Scholar
    • Export Citation

  • Morgan, M. C., and C. C. Chen, 2002: Diagnosis of optimal perturbation evolution in the Eady model. J. Atmos. Sci., 59 , 169–185.

  • Mukougawa, H., and T. Ikeda, 1994: Optimal excitation of baroclinic waves in the Eady model. J. Meteor. Soc. Japan, 72 , 499–513.

  • Müller, J. C., 1991: Baroclinic instability in a two-layer, vertically semi-infinite domain. Tellus, 43A , 275–284.

  • Pedlosky, J., 1987: Geophysical Fluid Dynamics. Prentice Hall, 710 pp.

  • Rivest, C., C. A. Davis, and B. F. Farrell, 1992: Upper-tropospheric synoptic-scale waves. Part I: Maintenance as Eady normal modes. J. Atmos. Sci., 49 , 2108–2119.

    • Search Google Scholar
    • Export Citation

  • Robinson, W. A., 1989: On the structure of potential vorticity in baroclinic instability. Tellus, 41 , 275–284.

  • Thorncroft, C. D., and B. J. Hoskins, 1990: Frontal cyclogenesis. J. Atmos. Sci., 47 , 2317–2336.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 180 90 36
PDF Downloads 53 10 0