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Nonlinear and free-surface effects on the spin-down of barotropic axisymmetric vortices

Published online by Cambridge University Press:  26 April 2006

Leo R. M. Maas
Leo R. M. Maas
Affiliation:
Netherlands Institute for Sea Research, P.O. Box 59, 1790 AB Den Burg, The Netherlands
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Abstract

The spin-down of a barotropic axisymmetric vortex, such as observed in laboratory models, is examined analytically. In addition to the classical, self-similar Ekman decay due to viscous effects (Greenspan & Howard 1963), which is characterized by an azimuthal velocity profile with the position of its maximum velocity fixed and a decay time equal to the Ekman timescale. the effects of nonlinearity and a free surface are considered separately.

The Ekman circulation in the radial and vertical planes whose strength is determined by the vorticity of the overlying fluid, leads to radial advection of the azimuthal velocity. This nonlinearity results in a nonlinear kinematic wave equation for the circulation and leads to the outward/inward propagation of the position of maximum azimuthal velocity for cyclonic/anticyclonic vortices. The associated steepening of the azimuthal velocity profile may lead to a shock formation when the absolute vorticity of the initial profile is negative at a certain radius. For anticyclonic vortices having a monotonically increasing angular velocity profile this shock formation occurs at the core. For such vortices (or arbitrary cyclonic vortices) this dynamical ‘breaking’ criterion is, despite significant differences in the physics concerned, identical to Rayleigh's kinematical criterion for the onset of centrifugal instability.

For a dynamically active free-surface fluid the spin-down of a decaying vortex is prolonged by a radially dependent factor proportional to the Froude number. This conclusion holds both in a cylinder with a parabolic bottom (mimicking the shape of the free surface of a fluid in solid-body rotation) and in a flat-bottomed cylinder. In view of the constancy of background vorticity the former geometry is relevant for a comparison to geophysical f-plane vortices. The latter geometry, however, is more easily established in a laboratory experiment, but the evolution of the azimuthal velocity profile is much more complicated and depends on the initial azimuthal velocity profile in a highly convoluted way.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 246 , January 1993 , pp. 117 - 141
Copyright
© 1993 Cambridge University Press

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References

Benton, E. R. 1979 Vorticity dynamics in spin-up from rest. Phys. Fluids 22, 1250– 1251.Google Scholar
Benton, E. R. & Clark, A. 1974 Spin-up. Ann. Rev. Fluid Mech. 6, 257– 280.Google Scholar
Berman, A. S., Bradford, J. & Lundgren, T. S. 1983 Two-fluid spin-up in a centrifuge. J. Fluid Mech. 84, 411– 431.Google Scholar
Carslaw, H. S. & Jaeger, J. C. 1959 Conduction of Heat in Solids. Clarendon.
Carton, X. J. & McWilliams, J. C. 1989 Barotropic and baroclinic instabilities of axisymmetric vortices in a quasi-geostrophic model. In Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence (ed. J. C. J. Nihoul & B. M. Jamart), pp. 225– 244. Elsevier.
Cederlöf, U. 1988 Free-surface effects on spin-up. J. Fluid Mech. 187, 395– 407.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.
Goller, H. & Ranov, T. 1968 Unsteady rotating flow in a cylinder with a free surface. Trans. ASME D: J. Basic Engng 90, 445– 454.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Greenspan, H. P. & Howard, L. N. 1963 On a time-dependent motion of a rotating fluid. J. Fluid Mech. 17, 385– 404.Google Scholar
Heijst, G. J. F. Van & Kloosterziel, R. C. 1989 Tripolar vortices in a rotating fluid. Nature 338. 569– 571.Google Scholar
Heijst, G. J. F. Van, Kloosterziel, R. C. & Williams, C. W. M. 1991 Laboratory experiments on the tripolar vortex in a rotating fluid. J. Fluid Mech. 225, 301– 331.Google Scholar
Kármán, T. Von 1921 Über laminäre und turbulente Reibung. Z. Angew. Math. Mech. 1, 233– 252.Google Scholar
Kloosterziel, R. C. 1990 Barotropic vortices in a rotating fluid. Ph.D. thesis. University of Utrecht.
Kloosterziel, R. C. & Heijst, G. J. F. Van 1989 On tripolar vortices. In Mesoscale /Synoptic Coherent Structures in Geophysical Turbulence (ed. J. C. J. Nihoul & B. M. Jamart). pp. 609– 625. Elsevier.
Kloosterztel, R. C. & Heijst, G. J. F. Van 1991 An experimental study of unstable barotropic vortices in a rotating fluid. J. Fluid Mech. 223, 1– 24.Google Scholar
Kloosterziel, R. C. & Heijst, G. J. F. Van 1992 The evolution of stable barotropic vortices in a rotating free-surface fluid. J. Fluid Mech. 239, 607– 629.Google Scholar
Melander, M. V., Me Williams, J. C. & Zabusky, N. J. 1987 Axisymmetrization and vorticity-gradient intensification of an isolated two-dimensional vortex through filamentation. J. Fluid Mech. 178, 137– 159.Google Scholar
Michalke, A. & Timme, A. 1967 On the inviscid instability of certain two-dimensional vortex-type flows. J. Fluid Mech. 29, 647– 666.Google Scholar
Mied, R. P. 1989 The decay of mesoscale vortices. In Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence (ed. J. C. J. Nihoul & B. M. Jamart), pp. 135– 147. Elsevier.
Myint-U, Y. & Debnath, L. 1987 Partial Differential-Equations for Scientists and Engineers. North-Holland.
Nikiforov, A. F. & Uvarov, V. B. 1988 Special Functions of Mathematical Physics. Birkhäuser.
O'Donnell, J. & Linden, P. F. 1991 Free-surface effects on the spin-up of fluid in a rotating cylinder. J. Fluid Mech. 232, 439– 453.Google Scholar
Ou, H. W. & Gordon, A. L. 1986 Spin-down of baroclinic eddies under sea-ice. J. Geophys. Res. 91, 7623– 7630.Google Scholar
Prudnikov, A. P., Brychkov, Yu. A. & Marichev. 0.1. 1986 Integrals and Series, Vol. 2. Gordon and Breach.
Rayleigh, Lord 1916 On the dynamics of revolving fluids. Proc. R. Soc. Lond. A 93, 148– 154.Google Scholar
Rogers, M. H. & Lance, G. N. 1960 The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating disk. J. Fluid Mech. 7. 617– 631.Google Scholar
Saffman, P. G. & Baker, G. R. 1979 Vortex interactions. Ann. Rev. Fluid Mech. 11, 95– 122.Google Scholar
Venezian, G. 1970 Nonlinear spin-up. Topics in Ocean Engineering, Vol. 2, pp. 87– 96. Gulf Publishing Co.
Watkins, W. B. & Hussey, R. G. 1977 Spin-up from rest in a cylinder. Phys. Fluids 20, 1596– 1604.Google Scholar
Wedemeyer, E. H. 1964 The unsteady flow within a spinning cylinder. J. Fluid Mech. 20. 383– 399.Google Scholar
Weidman, P. D. 1976 On the spin-up and spin-down of a rotating fluid. Part 1. Extending the Wedemeyer model. J. Fluid Mech. 77, 658– 708.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley-Interscience.
Zandbergen, P. J. & Dijkstra, D. 1987 Von Kármán swirling flows. Ann. Rev. Fluid Mech. 19, 465– 491.Google Scholar