• Brunet, G., and T. Warn, 1990: Rossby wave critical layers on a jet. J. Atmos. Sci.,47, 1173–1178.

  • ——, and P. H. Haynes, 1995: The nonlinear evolution of disturbances to a parabolic jet. J. Atmos. Sci.,52, 464–477.

  • Carr, L. E., and R. T. Williams, 1989: Barotropic vortex stability to perturbations from axisymmetry. J. Atmos. Sci.,46, 3177–3191.

  • Emanuel, K. E., 1997: Some aspects of hurricane inner-core dynamics and energetics. J. Atmos. Sci.,54, 1014–1026.

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

  • Gent, P. R., and J. C. McWilliams, 1986: The instability of barotropic circular vortices. Geophys. Astrophys. Fluid Dyn.,35, 209–233.

  • Guinn, T. A., and W. H. Schubert, 1993: Hurricane spiral bands. J. Atmos. Sci.,50, 3380–3403.

  • Leith, C. E., 1984: Minimum enstrophy vortices. Phys. Fluids,27, 1388–1395.

  • Llewellyn Smith, S. G., 1995: The influence of circulation on the stability of vortices to mode-one disturbances. Proc. Roy. Soc. London,451A, 747–755.

  • Michaelke, A., and A. Timme, 1967: On the inviscid instability of certain two-dimensional vortex-type flows. J. Fluid Mech.,29, 647–666.

  • Miller, J., P. B. Weichman, and M. C. Cross, 1992: Statistical mechanics, Euler’s equations, and Jupiter’s great red spot. Phys. Rev. A,45, 2328–2359.

  • Möller, J. D., and M. T. Montgomery, 1999: Vortex Rossby waves and hurricane intensification in a barotropic model. J. Atmos. Sci.,56, 1674–1687.

  • Montgomery, M. T., 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.

  • ——, and J. Enagonio, 1998: Tropical cyclogenesis via convectively forced vortex Rossby waves in a three-dimensional quasigeostrophic model. J. Atmos. Sci.,55, 3176–3207.

  • ——, J. M. Hidalgo, and P. D. Reasor, 2000: A semi-specral numerical method for modelling the vorticity dynamics of the near-core region of hurricane-like vortices. Atmospheric Science Paper 695, Department of Atmospheric Science, Colorado State University. [Available from Department of Atmospheric Science, Colorado State University, Fort Collins, CO 80523.].

  • Nolan, D. S., 1996: Axisymmetric and asymmetric vortex dynamics in convergent flows. Ph.D. thesis, Harvard University, 279 pp. [Available from University Microfilm, 305 N. Zeeb Rd., Ann Arbor, MI 48106.].

  • ——, and B. F. Farrell, 1999: Generalized stability analyses of asymmetric disturbances in one- and two-celled vortices maintained by radial inflow. J. Atmos. Sci.,56, 1282–1307.

  • Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992: Numerical Recipes. Cambridge University Press, 963 pp.

  • Reasor, P. D., M. T. Montgomery, F. D. Marks, and J. F. Gamache, 2000: Low-wavenumber structure and evolution of the hurricane inner core observed by airborne dual-Doppler radar. Mon. Wea. Rev.,128, 1653–1680.

  • Reznik, G. M., and W. K. Dewar, 1994: An analytical theory of distributed axisymmetric barotropic vortices on the beta-plane. J. Fluid Mech.,269, 301–321.

  • Schecter, D. A., 1999: On the dynamics of inviscid relaxation in 2D fluids and nonneutral plasmas. Ph.D. thesis, University of California, San Diego, 160 pp. [Available from Department of Physics, University of California, San Diego, 9500 Gilman Dr., La Jolla, CA 92093.].

  • Schubert, W. H., M. T. Montgomery, R. K. Taft, T. G. Guinn, S. R. Fulton, J. P. Kossin, and J. P. Edwards, 1999: Polygonal eyewalls, asymmetric eye contraction, and potential vorticity mixing in hurricanes. J. Atmos. Sci.,56, 1197–1223.

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

  • Smith, G. B., and M. T. Montgomery, 1995: Vortex axisymmetrization: Dependence on azimuthal wave-number or asymmetric radial structure changes. Quart. J. Roy. Meteor. Soc.,121, 1615–1650.

  • Smith, R. A., and M. N. Rosenbluth, 1990: Algebraic instability of hollow electron columns and cylindrical vortices. Phys. Rev. Lett.,64, 649–652.

  • Whitaker, N., and B. Turkington, 1994: Maximum entropy states for rotating vortex patches. Phys. Fluids,6, 3963–3973.

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The Algebraic Growth of Wavenumber One Disturbances in Hurricane-like Vortices

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  • 1 Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado
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Abstract

An exact solution for the evolution of linearized perturbations of azimuthal wavenumber one on inviscid vortices was previously discovered for nondivergent vorticity dynamics on an f plane. The longtime asymptotics for this exact solution have been shown to allow an algebraic instability with unbounded growth even in the absence of exponentially growing modes. The necessary requirement for this instability is that there exist a local maximum in the basic-state angular velocity other than at the center of circulation. Hurricanes are naturally occurring examples of such vortices, due to the relatively calm eye and intense vorticity in the eyewall region. In this paper, the dynamics of this algebraic instability are studied in the context of the near-core dynamics of hurricanes.

The longtime asymptotic solution can be written as a sum of three parts: a discrete mode whose amplitude grows in time as t1/2, an excitation of the neutral pseudomode (vortex displacement) that is constant in time, and residual terms that decay in time as t−1/2. A remarkable feature of the solution is that the discrete mode requires the decaying residuals to support its growth; without them it remains constant in amplitude. These residuals are shown to be a collection of sheared vortex-Rossby waves that are trapped in the core of the vortex. The explicit mechanism by which these waves sustain the longtime growth of the instability is investigated. The decaying vortex-Rossby waves are found to continuously amplify the growing discrete mode through the production of perturbation vorticity via interaction with the basic-state vorticity gradient. This is fundamentally different from the classic instability mechanism in barotropic shear flows, often interpreted in terms of discrete, counterpropagating vortex-Rossby waves.

The instability manifests itself as a growing wobble of the low-vorticity core of the vortex, resulting in a net transport of high vorticity to the vortex center. A fully nonlinear simulation is performed to study the behavior of the instability as its amplitude becomes large. The steady growth of the instability leads to secondary instabilities and vorticity mixing in the vortex core. The implications for hurricanes of the algebraic instability and its large-amplitude nonlinear dynamics are discussed.

Corresponding author address: Dr. David S. Nolan, Department of Atmospheric Science, Colorado State University, Fort Collins, CO 80523.

Email: nolan@chandra.atmos.colostate.edu

Abstract

An exact solution for the evolution of linearized perturbations of azimuthal wavenumber one on inviscid vortices was previously discovered for nondivergent vorticity dynamics on an f plane. The longtime asymptotics for this exact solution have been shown to allow an algebraic instability with unbounded growth even in the absence of exponentially growing modes. The necessary requirement for this instability is that there exist a local maximum in the basic-state angular velocity other than at the center of circulation. Hurricanes are naturally occurring examples of such vortices, due to the relatively calm eye and intense vorticity in the eyewall region. In this paper, the dynamics of this algebraic instability are studied in the context of the near-core dynamics of hurricanes.

The longtime asymptotic solution can be written as a sum of three parts: a discrete mode whose amplitude grows in time as t1/2, an excitation of the neutral pseudomode (vortex displacement) that is constant in time, and residual terms that decay in time as t−1/2. A remarkable feature of the solution is that the discrete mode requires the decaying residuals to support its growth; without them it remains constant in amplitude. These residuals are shown to be a collection of sheared vortex-Rossby waves that are trapped in the core of the vortex. The explicit mechanism by which these waves sustain the longtime growth of the instability is investigated. The decaying vortex-Rossby waves are found to continuously amplify the growing discrete mode through the production of perturbation vorticity via interaction with the basic-state vorticity gradient. This is fundamentally different from the classic instability mechanism in barotropic shear flows, often interpreted in terms of discrete, counterpropagating vortex-Rossby waves.

The instability manifests itself as a growing wobble of the low-vorticity core of the vortex, resulting in a net transport of high vorticity to the vortex center. A fully nonlinear simulation is performed to study the behavior of the instability as its amplitude becomes large. The steady growth of the instability leads to secondary instabilities and vorticity mixing in the vortex core. The implications for hurricanes of the algebraic instability and its large-amplitude nonlinear dynamics are discussed.

Corresponding author address: Dr. David S. Nolan, Department of Atmospheric Science, Colorado State University, Fort Collins, CO 80523.

Email: nolan@chandra.atmos.colostate.edu

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