Linear and Nonlinear Vortex Motion in an Asymmetric Balance Shallow Water Model

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

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J. Dominique Möller Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado

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Christopher T. Nicklas Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado

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Abstract

This work extends asymmetric balance (AB) theory to the beta plane (β-AB). The physical problem examined is the motion of a coherent vortex on a beta plane in a finite depth fluid in the absence of an environmental steering flow. A useful attribute of the β-AB formulation is that it allows one to separate the linear and nonlinear balance contributions to the vortex motion when the standard Rossby number is not small compared to unity. It is therefore well suited for testing the hurricane-motion paradigm proposed by Willoughby for equivalent barotropic dynamics.

The β-AB model is formulated first for linear shallow water dynamics on a circular vortex forced by the meridional gradient of planetary vorticity (the beta effect) in an earth-based coordinate system. The linear dynamics precludes wave–wave and wave–mean flow interactions. From incipient vortices to hurricanes, the β-AB model correctly develops the wavenumber-one asymmetries (the “beta” gyres) necessary for vortex self-advection. Cyclonic vortices move in a northwestward direction consistent with their relative strengths. In contrast to Willoughby’s predictions of a persistent acceleration in the linear problem, numerical simulations with the linear β-AB model suggest that finite drift speeds are always attained in a finite depth fluid. The present findings extend the theoretical predictions of a finite linear drift speed for stable quasigeostrophic vortices by Sutyrin and Flierl to the case of stable vortices in gradient wind balance. No evidence of a translating normal mode of zero frequency (other than the pseudo mode) is found when the beta forcing is switched off.

Nonlinear dynamics are considered next by adding the nonlinear quasigeostrophic terms to the linear balance system so that wave–wave and wave–mean flow interactions are included. Consistent with other works, asymptotic drift speeds are reduced from their linear values. Vortices develop an anticyclonic circulation in the vortex periphery and shed a Rossby wave wake in their environment. The sensitivity of vortex tracks with respect to azimuthal-wavenumber truncation is also investigated for the purposes of determining the minimal number of azimuthal modes required to make accurate long-time motion forecasts. While total relative angular momentum and eddy potential vorticity fluxes are known to be useful aids for interpreting changes in the vortex structure and intensity, the authors show in contrast to Willoughby that they give little insight into the behavior of vortex tracks at long times.

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

This work extends asymmetric balance (AB) theory to the beta plane (β-AB). The physical problem examined is the motion of a coherent vortex on a beta plane in a finite depth fluid in the absence of an environmental steering flow. A useful attribute of the β-AB formulation is that it allows one to separate the linear and nonlinear balance contributions to the vortex motion when the standard Rossby number is not small compared to unity. It is therefore well suited for testing the hurricane-motion paradigm proposed by Willoughby for equivalent barotropic dynamics.

The β-AB model is formulated first for linear shallow water dynamics on a circular vortex forced by the meridional gradient of planetary vorticity (the beta effect) in an earth-based coordinate system. The linear dynamics precludes wave–wave and wave–mean flow interactions. From incipient vortices to hurricanes, the β-AB model correctly develops the wavenumber-one asymmetries (the “beta” gyres) necessary for vortex self-advection. Cyclonic vortices move in a northwestward direction consistent with their relative strengths. In contrast to Willoughby’s predictions of a persistent acceleration in the linear problem, numerical simulations with the linear β-AB model suggest that finite drift speeds are always attained in a finite depth fluid. The present findings extend the theoretical predictions of a finite linear drift speed for stable quasigeostrophic vortices by Sutyrin and Flierl to the case of stable vortices in gradient wind balance. No evidence of a translating normal mode of zero frequency (other than the pseudo mode) is found when the beta forcing is switched off.

Nonlinear dynamics are considered next by adding the nonlinear quasigeostrophic terms to the linear balance system so that wave–wave and wave–mean flow interactions are included. Consistent with other works, asymptotic drift speeds are reduced from their linear values. Vortices develop an anticyclonic circulation in the vortex periphery and shed a Rossby wave wake in their environment. The sensitivity of vortex tracks with respect to azimuthal-wavenumber truncation is also investigated for the purposes of determining the minimal number of azimuthal modes required to make accurate long-time motion forecasts. While total relative angular momentum and eddy potential vorticity fluxes are known to be useful aids for interpreting changes in the vortex structure and intensity, the authors show in contrast to Willoughby that they give little insight into the behavior of vortex tracks at long times.

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

Email: mtm@charney.atmos.colostate.edu

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  • Aberson, S. D., and M. DeMaria, 1994: Verification of a nested barotropic hurricane track forecast model (VICBAR). Mon. Wea. Rev.,122, 2804–2815.

  • Batchelor, G. K., 1967: An Introduction to Fluid Dynamics. Cambridge University Press, 615 pp.

  • DeMaria, M., 1985: Tropical cyclone motion in a nondivergent barotropic model. Mon. Wea. Rev.,113, 1999–2110.

  • Elsberry, R. L., and R. F. Abbey, 1991: Recent advances in understanding tropical cyclone motion. Tech. Rep. NPS-MR-91-003, 92 pp. [Available from Naval Postgraduate School, Monterey, CA 93943.].

  • Fiorino, M., and R. L. Elsberry, 1989: Some aspects of vortex structure related to tropical cyclone motion. J. Atmos. Sci.,46, 975–990.

  • Flatau, M., 1992: The role of baroclinic processes in a tropical cyclone motion. Colorado State University Atmospheric Science Paper 488, 142 pp. [Available from Dept. of Atmospheric Science, Colorado State University, Fort Collins, CO 80523.].

  • Flierl, G. R., M. E. Stern, and J. A. Whitehead, 1983: The physical significance of modons: Laboratory experiments and general integral constraints. Dyn. Atmos. Oceans,7, 233–263.

  • Franklin, J. L., S. E. Feuer, J. Kaplan, and S. D. Aberson, 1996: Tropical cyclone motion and surrounding flow relationships: Searching for the Beta gyres in omega dropwindsonde datasets. Mon. Wea. Rev.,124, 64–84.

  • Kallenbach, R. J., and M. T. Montgomery, 1995: Symmetrization, vortex Rossby waves, and hurricane motion in an asymmetric balance model. Colorado State University Atmospheric Science Paper 588, 78 pp. [Available from R. Kallenbach, Dept. of Atmospheric Science, Colorado State University, Fort Collins, CO 80523.].

  • Llewellyn Smith, S. G., 1997: The motion of a non-isolated vortex on the beta plane. J. Fluid Mech.,346, 149–179.

  • McWilliams, J. C., and G. R. Flierl, 1979: On the evolution of isolated, nonlinear vortices. J. Phys. Oceanogr.,9, 1155–1182.

  • ——, and P. R. Gent, 1986: The evolution of sub-mesoscale, coherent vortices on the beta-plane. Geophys. Astrophys. Fluid Dyn.,35, 235–255.

  • ——, ——, and N. J. Norton, 1986: The evolution of balanced, low-mode vortices on the beta-plane. J. Phys. Oceanogr.,16, 838–855.

  • Möller, J. D., and S. C. Jones, 1998: Potential vorticity inversion for tropical cyclones using the asymmetric balance theory. J. Atmos. Sci.,55, 259–282.

  • ——, and M. T. Montgomery, 1999: Vortex Rossby-waves and their influence on hurricane intensification in a barotropic model. J. Atmos. Sci., in press.

  • 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.

  • Nicklas, C. T., and M. T. Montgomery, 1996: Hurricane motion on a beta plane in an asymmetric balance model. Colorado State University Atmospheric Science Paper 610, 80 pp. [Available from C. T. Nicklas, Dept. of Atmospheric Science, Colorado State University, Fort Collins, CO 80523.].

  • 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.

  • Ross, R. J., and Y. Kurihara, 1992: A simplified scheme to simulate asymmetries due to the beta effect in barotropic vortices. J. Atmos. Sci.,49, 1620–1628.

  • Shapiro, L. J., 1998: Convective asymmetries and tropical cyclone evolution. Preprints, Symposium on Tropical Cyclone Intensity Change, Phoenix, AZ, Amer. Meteor. Soc., 80–81.

  • ——, and K. V. Ooyama, 1990: Barotropic vortex evolution on a beta plane. J. Atmos. Sci.,47, 170–187.

  • ——, 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 and its dependence on azimuthal wavenumber or asymmetric radial structure changes. Quart. J. Roy. Meteor. Soc.,121, 1615–1650.

  • Smith, R. B., 1993: A hurricane beta-drift law. J. Atmos. Sci.,50, 3213–3215.

  • ——, X. Li, and B. Wang, 1997: Scaling laws for barotropic vortex beta-drift. Tellus,49A, 474–485.

  • Smith, R. K., and W. Ulrich, 1993: Vortex motion in relation to the absolute vorticity gradient of the vortex environment. Quart. J. Roy. Meteor. Soc.,119, 207–215.

  • ——, ——, and G. Dietachmayer, 1990: A numerical study of tropical cyclone motion using a barotropic model. I: The role of vortex asymmetries. Quart. J. Roy. Meteor. Soc.,116, 337–362.

  • ——, H. C. Weber, and A. Kraus, 1995: On the symmetric circulation of a moving hurricane. Quart. J. Roy. Meteor. Soc.,121, 945–952.

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

  • Sutyrin, G. G., and G. R. Flierl, 1994: Intense vortex motion on the beta plane: Development of the beta gyres. J. Atmos. Sci.,51, 773–790.

  • Willoughby, H. E., 1988: Linear motion of a shallow-water-barotropic vortex. J. Atmos. Sci.,45, 1906–1928.

  • ——, 1990: Linear normal modes of a shallow-water barotropic vortex. J. Atmos. Sci.,47, 2141–2148.

  • ——, 1992: Linear motion of a shallow-water barotropic vortex as an initial-value problem. J. Atmos. Sci.,49, 2015–2031.

  • ——, 1994: Nonlinear motion of a shallow water barotropic vortex. J. Atmos. Sci.,51, 3722–3744.

  • ——, 1995: Normal-mode initialization of barotropic vortex motion models. J. Atmos. Sci.,52, 4501–4514.

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