Free Waves on Barotropic Vortices. Part I: Eigenmode Structure

Michael T. Montgomery Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado

Search for other papers by Michael T. Montgomery in
Current site
Google Scholar
PubMed
Close
and
Chungu Lu Forecast Systems Laboratory, NOAA/ERL, Boulder, Colorado

Search for other papers by Chungu Lu in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

To understand the nature of coupling between a hurricane vortex and asymmetries in its near-core region, it is first necessary to have an understanding of the spectrum of free waves on barotropic vortices. As foundation for upcoming work examining the nonaxisymmetric initial-value problem in inviscid and swirling boundary layer vortex flows, the complete spectrum of free waves on barotropic vortices is examined here.

For a variety of circular vortices in gradient balance the linearized momentum and continuity equations are solved as a matrix eigenvalue problem for perturbation height and wind fields. Vortex eigensolutions are found to fall into two continuum classes. Eigenmodes with frequencies greater than the advective frequency for azimuthal wavenumber n are modified gravity–inertia waves possessing nonzero potential vorticity in the near-core region. Eigenmodes whose frequencies scale with the advective frequency comprise both gravity–inertia waves and Rossby–shear waves. Linearly superposing the Rossby–shear waves approximates the sheared disturbance solutions. For wavenumbers greater than a minimum number, Rossby–shear waves exhibit gravity wave characteristics in the near-vortex region. Although such eigenstructure changes are not anticipated by traditional scaling analyses using solely external flow parameters, a criterion extending Rossby’s characterization of “balanced” and “unbalanced” flow to that of azimuthal waves on a circular vortex is developed that correctly predicts the observed behavior from incipient vortices to hurricane-like vortices. The criterion is consistent with asymmetric balance theory. Possible applications of these results to the wave-mean-flow dynamics of geophysical vortex flows are briefly discussed.

Corresponding author address: Dr. Michael T. Montgomery, Department of Atmospheric Science, Colorado State University, Fort Collins, CO 80523.

Email: mtm@charney.atmos.colostate.edu

Abstract

To understand the nature of coupling between a hurricane vortex and asymmetries in its near-core region, it is first necessary to have an understanding of the spectrum of free waves on barotropic vortices. As foundation for upcoming work examining the nonaxisymmetric initial-value problem in inviscid and swirling boundary layer vortex flows, the complete spectrum of free waves on barotropic vortices is examined here.

For a variety of circular vortices in gradient balance the linearized momentum and continuity equations are solved as a matrix eigenvalue problem for perturbation height and wind fields. Vortex eigensolutions are found to fall into two continuum classes. Eigenmodes with frequencies greater than the advective frequency for azimuthal wavenumber n are modified gravity–inertia waves possessing nonzero potential vorticity in the near-core region. Eigenmodes whose frequencies scale with the advective frequency comprise both gravity–inertia waves and Rossby–shear waves. Linearly superposing the Rossby–shear waves approximates the sheared disturbance solutions. For wavenumbers greater than a minimum number, Rossby–shear waves exhibit gravity wave characteristics in the near-vortex region. Although such eigenstructure changes are not anticipated by traditional scaling analyses using solely external flow parameters, a criterion extending Rossby’s characterization of “balanced” and “unbalanced” flow to that of azimuthal waves on a circular vortex is developed that correctly predicts the observed behavior from incipient vortices to hurricane-like vortices. The criterion is consistent with asymmetric balance theory. Possible applications of these results to the wave-mean-flow dynamics of geophysical vortex flows are briefly discussed.

Corresponding author address: Dr. Michael T. Montgomery, Department of Atmospheric Science, Colorado State University, Fort Collins, CO 80523.

Email: mtm@charney.atmos.colostate.edu

Save
  • Broadbent, E. G., and D. W. Moore, 1979: Acoustic destabilization of vortices. Philos. Trans. Roy. Soc. London, Ser. A,290, 353–371.

  • Chan, W. M., K. Shariff, and T. H. Pulliam, 1993: Instabilities of two-dimensional inviscid compressible vortices. J. Fluid Mech.,253, 173–209.

  • Dewar, W. K., and P. D. Killworth, 1995: On the stability of oceanic rings. J. Phys. Oceanogr.,25, 1467–1487.

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

  • ——, 1984: Modal and non-modal baroclinic waves. J. Atmos. Sci.,41, 668–673.

  • Flatau, M., and D. E. Stevens, 1989: Barotropic and inertial instabilities in the hurricane outflow layer. Geophys. Astrophys. Fluid Dyn.,47, 1–18.

  • Ford, R., 1994: The instability of an axisymmetric vortex with monotonic potential vorticity in rotating shallow water. J. Fluid Mech.,280, 303–334.

  • Fung, I. Y., 1977: The organization of spiral rainbands in a hurricane. Ph.D. dissertation, Massachusetts Institute of Technology, 140 pp.

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

  • Hunter, C., 1983: Galactic dynamics. Lect. Appl. Math.,20, 199–203.

  • Kurihara, Y., 1976: On the development of spiral bands in a tropical cyclone. J. Atmos. Sci.,33, 940–958.

  • Lindzen, R. D., 1967: Planetary waves on beta-planes. Mon. Wea. Rev.,95, 441–451.

  • Longuet-Higgins, M. S., 1968: The eigenfunctions of Laplace’s tidal equations over a sphere. Philos. Trans. Roy. Soc. London, Ser. A,262, 511–607.

  • Matsuno, T., 1966: Quasi-geostrophic motions in the equatorial area. J. Meteor. Soc. Japan,44, 25–43.

  • McWilliams, J. C., 1985: A uniformly valid model spanning the regimes of geostrophic and isotropic, stratified turbulence: Balanced turbulence. J. Atmos. Sci.,42, 1773–1774.

  • Montgomery, M. T., and L. J. Shapiro, 1995: Generalized Charney–Stern and Fjortoft theorems for rapidly rotating vortices. J. Atmos. Sci.,52, 1829–1833.

  • ——, and R. J. Kallenbach, 1997: A theory for vortex Rossby waves and its application to spiral bands and intensity changes in hurricanes. Quart. J. Roy. Meteor. Soc.,123, 435–465.

  • Pearce, R. P., 1993: A critical review of progress in tropical cyclone physics including experimentation with numerical models. Tropical Cyclone Disasters. Proceedings of the ICSU/WMO International Symposium, Peking University Press, 45–60.

  • Phillips, N. A., 1965: Elementary Rossby waves. Tellus,XVII, 295–301.

  • Riehl, H., 1963: Some relations between wind and thermal structure of steady state hurricanes. J. Atmos. Sci.,20, 276–287.

  • Shapiro, L. J., and M. T. Montgomery, 1993: A three-dimensional balance theory for rapidly rotating vortices. J. Atmos. Sci.,50, 3322–3335.

  • Spall, M. A., and J. C. McWilliams, 1992: Rotational and gravitational influences on the degree of balance in the shallow-water equation. Geophys. Astrophys. Fluid Dyn.,64, 1–29.

  • Stepaniak, D., W. Schubert, and M. T. Montgomery, 1996: An initial-value problem study of free waves on a barotropic vortex. Atmospheric Science Bluebook #600, 74 pp.

  • Thomson, W., 1879: On gravitational oscillations of rotating water. Proc. Roy. Soc. Edinburgh, Session 1878–79, 92–99.

  • ——, 1887: Stability of fluid motion: Rectilinear motion of viscous fluid between two parallel planes. Philos. Mag.,24, 188–196.

  • Willoughby, H. E., 1977: Inertia-buoyancy waves in hurricanes. J. Atmos. Sci.,34, 1028–1039.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 379 74 17
PDF Downloads 104 36 6