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Linear and nonlinear decay of cat's eyes in two-dimensional vortices, and the link to Landau poles

  • M. R. TURNER (a1) and ANDREW D. GILBERT (a1)


This paper considers the evolution of smooth, two-dimensional vortices subject to a rotating external strain field, which generates regions of recirculating, cat's eye stream line topology within a vortex. When the external strain field is smoothly switched off, the cat's eyes may persist, or they may disappear as the vortex relaxes back to axisymmetry. A numerical study obtains criteria for the persistence of cat's eyes as a function of the strength and time scale of the imposed strain field, for a Gaussian vortex profile.

In the limit of a weak external strain field and high Reynolds number, the disturbance decays exponentially, with a rate that is linked to a Landau pole of the linear inviscid problem. For stronger strain fields, but not strong enough to give persistent cat's eyes, the exponential decay of the disturbance varies: as time increases the decay slows down, because of the nonlinear feedback on the mean profile of the vortex. This is confirmed by determining the decay rate given by the Landau pole for these modified profiles. For strain fields strong enough to generate persistent cat's eyes, their location and rotation rate are determined for a range of angular velocities of the external strain field, and are again linked to Landau poles of the mean profiles, modified through nonlinear effects.



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Bajer, K., Bassom, A. P. & Gilbert, A. D. 2001 Accelerated diffusion in the centre of a vortex. J. Fluid Mech. 437, 395411.
Balmforth, N. J., Llewellyn Smith, S. G. & Young, W. R. 2001 Disturbing vortices. J. Fluid Mech. 426, 95133.
Barba, L. A. & Leonard, A. 2007 Emergence and evolution of tripole vortices from net-circulation initial conditions. Phys. Fluids 19, 017101.
Bassom, A. P. & Gilbert, A. D. 1998 The spiral wind-up of vorticity in an inviscid planar vortex. J. Fluid Mech. 371, 109140.
Bassom, A. P. & Gilbert, A. D. 1999 The spiral wind-up and dissipation of vorticity and a passive scalar in a strained planar vortex. J. Fluid Mech. 398, 245270.
Benney, D. J. & Bergeron, R. F. 1969 A new class of nonlinear waves in parallel flows. Stud. Appl. Maths 48, 181204.
Benzi, R., Paladin, G., Patarnello, S., Santangelo, P. & Vulpiani, A. 1986 Intermittency and coherent structures in two-dimensional turbulence. J. Phys. A: Math Gen. 19, 37713784.
Bernoff, A. J. & Lingevitch, J. F. 1994 Rapid relaxation of an axisymmetric vortex. Phys. Fluids 6, 37173723.
Brachet, M. E., Meneguzzi, M., Politano, H. & Sulem, P. L. 1988 The dynamics of freely decaying two-dimensional turbulence. J. Fluid Mech. 194, 333349.
Briggs, R. J., Daugherty, J. D. & Levy, R. H. 1970 Role of Landau damping in crossed-field electron beams and inviscid shear flow. Phys. Fluids 13, 421432.
Brown, S. N. & Stewartson, K. 1978 The evolution of the critical layer of a Rossby wave. Part II. Geophys. Astrophys. Fluid Dyn. 10, 124.
Corngold, N. R. 1995 Linear response of the two-dimensional pure electron plasma: Quasimodes for some model profiles. Phys. Plasmas 2, 620628.
Dritschel, D. G. 1989 Contour dynamics and contour surgery: numerical algorithms for extended high-resolution modelling of vortex dynamics in two-dimensional, inviscid, incompressible flows. Comput. Phys. Rep. 10, 77146.
Dritschel, D. G. 1998 On the persistence of non-axisymmetric vortices in inviscid two-dimensional flows. J. Fluid Mech. 371, 141155.
Fornberg, B. 1977 A numerical study of 2-D turbulence. J. Comput. Phys. 25, 131.
Guinn, T. A. & Schubert, W. H. 1993 Hurricane spiral bands. J. Atmos. Sci. 50, 33803403.
Haberman, R. 1972 Critical layers in parallel flows. Stud. Appl. Maths 51, 139161.
Hall, I. M., Bassom, A. P. & Gilbert, A. D. 2003 The effect of fine structure on the stability of planar vortices. Eur. J. Mech. B Fluids 22, 179198.
Koumoutsakos, P. 1997 Inviscid axisymmetrization of an elliptical vortex. J. Comput. Phys. 138, 821857.
Le Dizès, S. 2000 Non-axisymmetric vortices in two-dimensional flows. J. Fluid Mech. 406, 175198.
Legras, B. & Dritschel, D. 1993 Vortex stripping and the generation of high vorticity gradients in two-dimensional flows. Appl. Sci. Res. 51, 445455.
Lingevitch, J. F. & Bernoff, A. J. 1995 Distortion and evolution of a localized vortex in an irrotational flow. Phys. Fluids 7, 10151026.
Macaskill, C., Bassom, A. P. & Gilbert, A. D. 2002 Nonlinear wind-up in a strained planar vortex. Eur. J. Mech. B Fluids 21, 293306.
McWilliams, J. C. 1984 The emergence of isolated coherent vortices in turbulent flow. J. Fluid Mech. 146, 2143.
Melander, M. V., McWilliams, J. C. & Zabusky, N. J. 1987 Axisymmetrization and vorticity-gradient intensification of an isolated two-dimensional vortex through filamentation. J. Fluid Mech. 178, 137159.
Moffatt, H. K. & Kamkar, H. 1983 On the time-scale associated with flux expulsion. In: Stellar and Planetary Magnetism (ed. Soward, A. M.), pp. 9197. Gordon & Breach.
Montgomery, M. T. & Kallenbach, R. J. 1997 A theory for vortex Rossby waves and its application to spiral bands and intensity changes in hurricanes. Q. J. R. Met. Soc. 123, 435465.
Rhines, P. B. & Young, W. R. 1983 How rapidly is a passive scalar mixed within closed streamlines? J. Fluid Mech. 133, 133145.
Rossi, L. F., Lingevitch, J. F. & Bernoff, A. J. 1997 Quasi-steady monopole and tripole attractors for relaxing vortices. Phys. Fluids 9, 23292338.
Schecter, D. A., Dubin, D. H. E., Cass, A. C., Driscoll, C. F., Lansky, I. M. & O'Neil, T. M. 2000 Inviscid damping of asymmetries on a two-dimensional vortex. Phys. Fluids 12, 23972412.
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows, pp. 487489. Springer.
Smith, F. T. & Bodonyi, R. J. 1982 Nonlinear critical layers and their development in streaming-flow stability. J. Fluid Mech. 118, 165185.
Smith, G. B. & Montgomery, M. T. 1995 Vortex axisymmetrization: Dependence on azimuthal wave-number or asymetric radial structure changes. Q. J. R. Met. Soc. 121, 16151650.
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Linear and nonlinear decay of cat's eyes in two-dimensional vortices, and the link to Landau poles

  • M. R. TURNER (a1) and ANDREW D. GILBERT (a1)


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